We provide three dimensional geometry practice exercises, instructions, and a learning material that allows learners to study outside of the classroom. We focus on three dimensional geometry skills mastery so, below you will get all questions that are also asking in the competition exam beside that classroom.
List of three dimensional geometry Questions
| Question No | Questions | Class |
|---|---|---|
| 1 | Perimeter of triangle whose vertices are (0,4,0),(3,4,0) and ( (0,4,4), ) is A . 10 B. 12 c. 25 D. 15 |
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| 2 | 87. (a) 315 (b) 256 (c) 84 The length of the projection of the line segment joining the points (5,-1, 4) and (4,-1,3) on the plane, x+y+z= JEE M 2018] WIN |
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| 3 | Prove that the points ( boldsymbol{A}= ) ( (1,2,3), B(3,4,7), C(-3,-2,-5) ) are collinear ( & ) find the ratio in which ( B ) divides ( boldsymbol{A C} ) A .2: 5 B . 2: 3 c. 2: 8 D. 2: 7 |
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| 4 | In geometry, we take a point, a line and a plane as undefined terms. A. True B. False c. Ambiguous D. Data Insufficient |
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| 5 | Find the shortest distance between the skew lines: ( l_{1}: frac{x-1}{2}=frac{y+1}{1}=frac{z-2}{4} ) ( l_{2}: frac{x+2}{4}=frac{y-0}{-3}=frac{z+1}{1} ) |
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| 6 | The direction ratios of the line joining the points (4,3,-5) and (-2,1,-8) are A ( cdot frac{6}{7}, frac{2}{7}, frac{3}{7} ) в. 6,2,3 c. 5,8,0 D. 3,7,9 |
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| 7 | If a point ( boldsymbol{P} ) from where line drawn cuts coordinates axes at ( A ) and ( B ) (with ( A ) on ( x-text { axis and } B text { on } y-text { axis }) ) satisfies ( alpha frac{x^{2}}{P B^{2}}+beta frac{y^{2}}{P A^{2}}=1, ) then ( alpha+beta ) is ( A cdot 1 ) B. 2 ( c .3 ) D. 4 |
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| 8 | The planes ( 2 x-y+4 z=5 ) and ( 5 x- ) ( 2.5 y+10 z=6 ) are A. Parallel B. Perpendicular c. Intersect D. intersect ( x ) axis |
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| 9 | The number of straight line that are equally inclined to the three dimensional co- ordinate axes, is |
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| 10 | ( boldsymbol{A}=(mathbf{1}, mathbf{2}, mathbf{3}), boldsymbol{B}=(mathbf{4}, mathbf{5}, mathbf{7}), boldsymbol{C}= ) ( (-4,3,-6), D=(2, k, 2) ) are four points. If the lines ( A B ) and ( C D ) are parallel, then ( k= ) ( A cdot 0 ) в. -9 ( c .9 ) D. 2 |
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| 11 | Algebraic sum of intercepts made by the plane ( x+3 y-4 z+6=0 ) on the axes is A. 7 B. 0 ( c cdot frac{13}{2} ) ( D cdot-frac{13}{2} ) |
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| 12 | The following lines are ( hat{boldsymbol{r}}=(hat{boldsymbol{i}}+hat{boldsymbol{j}})+ ) ( lambda(hat{i}+2 hat{j}-hat{k})+mu(-hat{i}+hat{j}-2 hat{k}) ) A. collinear B. skew-lines c. co-planar lines D. parallel lines |
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| 13 | Find the point on Z-axis which are at a distance ( sqrt{21} ) unit from the point (1,2,3) |
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| 14 | If ( boldsymbol{A}=(mathbf{2},-mathbf{3}, mathbf{1}), boldsymbol{B}=(mathbf{3},-mathbf{4}, mathbf{6}) ) and ( boldsymbol{C} ) is a point of trisection of ( A B, ) then ( C_{y}= ) A ( cdot frac{11}{3} ) B. -11 c. ( frac{10}{3} ) D. ( frac{-11}{3} ) |
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| 15 | The distance between the circumcentre and the ortho centre of the triangle formed by the points (2,1,5),(3,2,3) and (4,0,4) is A ( cdot sqrt{6} ) B. ( frac{sqrt{6}}{2} ) c. ( 2 sqrt{6} ) D. |
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| 16 | Find the coordinates of a point equidistant from four points ( boldsymbol{O}(mathbf{0}, mathbf{0}, mathbf{0}), boldsymbol{A}(ell, mathbf{0}, mathbf{0}), boldsymbol{B}(mathbf{0}, boldsymbol{m}, boldsymbol{0}) ) and ( boldsymbol{C}(mathbf{0}, mathbf{0}, boldsymbol{n}) ) |
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| 17 | Find the direction cosines of a line which makes equal angles with the coordinate axes. |
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| 18 | The projections of a directed line segment on the coordinate axes are 12,4,3 respectively. What are the direction cosines of the |
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| 19 | If ( boldsymbol{A}=(mathbf{4}, mathbf{1}, mathbf{5}) ) and ( boldsymbol{B}=(mathbf{3}, mathbf{4}, mathbf{5}) ) The direction ratios of ( overline{A B} ) are |
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| 20 | x-2 y. 2+2 56. Let the line – lie in the plane 3 -5 2 . x+3y-az+B=0. Then (a, b) equals [2009] (a) (-6,7) (b) (5,-15) (c) (-5,5) (d) (6,-17) |
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| 21 | If ( boldsymbol{A}=(mathbf{5},-mathbf{1}, mathbf{1}), boldsymbol{B}=(mathbf{7},-mathbf{4}, mathbf{7}), boldsymbol{C}= ) ( (1,-6,10), D=(-1,-3,4) . ) Then ( A B C D ) is a A . square B. rectangle c. rhombus D. none of these |
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| 22 | Find the ratio in which ( 2 x+3 y+5 z= ) 1 divides the line joining the points (1,0,-3) and (1,-5,7) A .1: 2 B . 2: 1 c. 3: 2 D. 2: 3 |
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| 23 | A plane mirror is placed at the origin so that the direction ratios of its normal ( operatorname{are}(1,-1,1) . ) A ray of light, coming along the positive direction of the ( x ) axis, strikes the mirror. The direction ( operatorname{cosines} ) of the reflected ray are A ( cdot frac{1}{3}, frac{2}{3}, frac{2}{3} ) B. ( -frac{1}{3}, frac{2}{3}, frac{2}{3} ) c. ( -frac{1}{3},-frac{2}{3},-frac{2}{3} ) D. ( -frac{1}{3},-frac{2}{3}, frac{2}{3} ) |
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| 24 | Find the square of the distance between the points whose cartesian coordinates are: (-1,1,3),(0,5,6) |
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| 25 | Consider three vectors ( vec{P}=hat{i}+widehat{j}+ ) ( widehat{k} ; overrightarrow{boldsymbol{q}}=2 widehat{hat{boldsymbol{i}}}+4 widehat{boldsymbol{j}}-widehat{boldsymbol{k}} ) and ( overrightarrow{boldsymbol{r}}=boldsymbol{2} hat{boldsymbol{i}}+boldsymbol{4} widehat{boldsymbol{j}}+ ) 3 ( widehat{k} ). If ( vec{p}, vec{q} ) and ( vec{r} ) denotes the position vector of three non-collinear points, then the equation of the plane containing these points is A. ( 2 x-3 y+1=0 ) B. ( x-3 y+2 z=0 ) c. ( 3 x-y+z-3=0 ) D. ( 3 x-y-2=0 ) |
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| 26 | The d.r’s of the line of intersection of the planes ( boldsymbol{x}+boldsymbol{y}+boldsymbol{z}-mathbf{1}=mathbf{0} ) and ( mathbf{2} boldsymbol{x}+ ) ( 3 y+4 z-7=0 ) are A .1,2,-3 в. 2,1,-3 c. 4,2,-6 D. 1,-2,1 |
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| 27 | Find the distance of point (1,-2,3) from the plane ( boldsymbol{x}-boldsymbol{y}+boldsymbol{z}=mathbf{5} ) measured parallel to the line whose direction cosines are proportional to 2,3,-6 |
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| 28 | Let two planes ( p_{1}: 2 x-y+z=2, ) and ( boldsymbol{p}_{2}: boldsymbol{x}+boldsymbol{2} boldsymbol{y}-boldsymbol{z}=boldsymbol{3} ) are given. The image of plane ( P_{1} ) in the plane mirror ( P_{2} ) is A. ( x+7 y-4 z+5=0 ) B. ( 3 x+4 y-5 z+9=0 ) c. ( 7 x-y+2 z-9=0 ) 0 D. ( 7 x+y+9 z+9=0 ) |
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| 29 | If the points ( (h, 3,-4),(0,-7,10) ) and ( (1, k, 3) ) are collinear, then ( h+k ) is ( mathbf{A} cdot mathbf{4} ) B. c. -4 D. 14 |
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| 30 | Find the equation of the plane containing the line 2x -y+z -3 = 0, 3x +y+z= 5 and at a distance of Ta from the point (2,1,-1). (2005 – 2 Marks) |
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| 31 | If ( P(x, y, z) ) moves such that ( x=0, z= ) ( 0, ) then the locus of ( P ) is the line whose d.cs are A . ( y ) -axis B. 1,0,0 c. 0,1,0 D. 0,0,0 |
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| 32 | In three dimensions, the coordinate axes of a rectangular cartesian coordinate system are A. three mutually parallel lines B. three mutually perpendicular lines c. two mutually perpendicular lines and any two parallel D. None of these |
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| 33 | If line ( frac{boldsymbol{x}-mathbf{2}}{mathbf{3}}=frac{boldsymbol{y}-mathbf{4}}{mathbf{4}}=frac{boldsymbol{z}+mathbf{2}}{mathbf{1}} ) is parallel to planes ( mu x+3 y-2 z+d= ) 0 and ( x-2 lambda y+z=0, ) then value of ( lambda ) and ( mu ) are A ( cdotleft(mu=4, lambda=-frac{2}{3}right. ) B. ( mu=-6, lambda=-2 ) c. ( _{mu}=frac{-10}{3}, lambda=frac{-1}{2} ) D. ( mu=frac{-10}{3}, lambda=frac{1}{2} ) |
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| 34 | Vectors ( vec{A}, vec{B} ) and ( vec{C} ) are such that ( vec{A} ) ( vec{B}=0 . ) Then the vector parallel to ( vec{A} ) is A. ( vec{B} ) and ( vec{C} ) в. ( vec{A} times vec{B} ) c. ( vec{B}+vec{C} ) D. ( vec{B} times vec{C} ) |
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| 35 | Find the equation of the plane passing through (2,0,1) and (3,-3,4) and perpendicular to ( boldsymbol{x}-mathbf{2} boldsymbol{y}+boldsymbol{z}=mathbf{6} ) | 12 |
| 36 | Number of points having positive integral co-ordinate lying on the plane ( x+2 y+3 z=15 ) is ( n, ) then ( frac{n}{2} ) is equal to ( mathbf{A} cdot mathbf{6} ) B. 8 c. 9 D. |
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| 37 | The vector equation of the plane passes through the points ( A & B ) with position vector ( 2 hat{i}+hat{j}-hat{k} &-hat{i}+3 hat{j}+4 hat{k} ) respectively ( & ) Ler to the plane ( bar{r} cdot(hat{i}-2 hat{j}+4 hat{k})=10 ) is A ( cdot bar{r} cdot(18 hat{i}+17 hat{j}-3 hat{k})=49 ) B . ( bar{r} .(18 hat{i}-17 hat{j}-3 hat{k})+22=0 ) c. ( bar{r} .(18 hat{i}+17 hat{j}+4 hat{k})=25 ) D・ ( bar{r} .(18 hat{i}+17 hat{j}+4 hat{k})=24 ) |
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| 38 | The vector equation of the line ( frac{x-2}{2}= ) ( frac{2 y-5}{-3}, z=-1 ) is ( vec{r}= ) ( left(2 hat{i}+frac{5}{2} hat{j}-hat{k}right)+lambdaleft(2 hat{i}-frac{3}{2} hat{j}+x hat{k}right) ) where ( x ) is equal to ( mathbf{A} cdot mathbf{0} ) B. c. 2 D. 3 |
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| 39 | If ( P(x, y, z) ) is a point on the line segment joining ( Q(2,2,4) ) and ( R(3,5,6) ) such that the projection of ( overrightarrow{O P} ) on the axes are ( frac{13}{5}, frac{19}{5}, frac{26}{5} ) respectively, then ( P ) divides ( Q R ) in ratio A .1: 3 B. 2: 3 c. 3: 2 D. 3: 1 |
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| 40 | f ( (p, q, r) ) is equidistant from (1,2,-3),(2,-3,1) and ( (-3,1,2), ) then ( boldsymbol{p}+boldsymbol{q}+boldsymbol{r}= ) A . -1 B. c. 0 D. |
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| 41 | The acute angle between two lines such that the direction cosines ( l, boldsymbol{m}, boldsymbol{n} ) of each of them satisfy the equation ( l+ ) ( boldsymbol{m}+boldsymbol{n}=mathbf{0} ) and ( l^{2}+boldsymbol{m}^{2}-boldsymbol{n}^{2}=mathbf{0} ) is ( A cdot 30 ) B . 45 ( c cdot 60 ) D. ( 15^{circ} ) |
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| 42 | Plane ( a x+b y+c z=1 ) intersect axes ( operatorname{in} A, B, C ) respectively. If ( Gleft(frac{1}{6},-frac{1}{3}, 1right) ) is a centroid of ( triangle A B C ) then ( a+b+ ) ( 3 c=-1 ) A ( cdot frac{4}{3} ) B. 4 ( c cdot 2 ) D. |
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| 43 | The equation of the plane passing through the straight line ( frac{x-1}{2}= ) ( frac{boldsymbol{y}+mathbf{1}}{mathbf{- 1}}=frac{boldsymbol{z}-mathbf{3}}{mathbf{4}} ) and perpendicular to plane ( boldsymbol{x}+mathbf{2} boldsymbol{y}+boldsymbol{z}=mathbf{1 2} ) is: A. ( 9 x+2 y-5 z+8=0 ) в. ( 9 x+2 y-5 z+10=0 ) c. ( 9 x-2 y+5 z+6=0 ) D. ( 9 x-2 y-5 z+4=0 ) |
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| 44 | Derive the equation of the locus of a point equivalent from the points (1,-2,3) and (-3,4,2) | 12 |
| 45 | If vector ( vec{a}=4 hat{i}+5 hat{j}-3 hat{k} ) and ( vec{b}=5 hat{i}+ ) ( 3 hat{j}+8 hat{k} ) then value of ( frac{text { projection of vector b on a }}{text { projection of vector a on b }} ) is : A ( cdot frac{7}{5} ) B. ( frac{2}{7} ) ( c cdot frac{5}{8} ) D. None of these. |
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| 46 | Find ( a, b, c ) if ( a(1,3,2)+b(1,-5,6)+ ) ( c(2,1,-2)=(4,10,-8) ) |
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| 47 | A variable plane at a distance of 1 unit from the origin cuts the co-ordinate axes at ( A, B ) and ( C . ) If the centroid ( D(x, y, z) ) of triangle ( A B C ) satisfies the relation ( frac{1}{x^{2}}+ ) ( frac{1}{y^{2}}+frac{1}{z^{2}}=k, ) then the value of ( k ) is A . 3 B. 1 c. ( 1 / 3 ) D. |
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| 48 | The points (-5,12),(-2,-3),(9,-10),(6,5) taken in order, form A. Parallelogram B. rectangle c. rhombus D. square |
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| 49 | The distance from the origin to the centroid of the tetrahedron formed by the points ( (0,0,0),(a, 0,0),(0, b, 0),(0,0, c) ) is: A ( cdot frac{sqrt{a+b+c}}{4} ) B. ( frac{sqrt{a+b+c}}{3} ) c. ( frac{sqrt{a^{2}+b^{2}+c^{2}}}{16} ) D. ( frac{sqrt{a^{2}+b^{2}+c^{2}}}{4} ) |
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| 50 | A point ( P ) lies on a line whose ends are ( A(1,2,3) ) and ( B(2,10,1) . ) If ( z ) component of ( boldsymbol{P} ) is ( boldsymbol{7}, ) then the coordinates of ( boldsymbol{P} ) are A ( cdot(-1,-14,7) ) B. (1,-14,7) c. (-1,14,7) D. (1,14,7) |
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| 51 | A parallelopiped ‘S’ has base points A, B, C and D and upper face points A’, B’, C and D’. This parallelopiped is compressed by upper face A’B’C’D’ to form a new parallelopiped ‘T” having upper face points A”, B”, C” and D”. Volume of parallelopiped Tis 90 percent of the volume of parallelopiped S. Prove that the locus of ‘A”?, is a plane |
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| 52 | If 5,7,6 are the sums of the ( x, y ) intercepts; ( boldsymbol{y}, boldsymbol{z} ) intercepts, ( boldsymbol{z}, boldsymbol{x} ) intercepts respectively of a plane then the perpendicular distance from the origin to that plane is A ( cdot frac{144}{61} ) в. ( frac{12}{sqrt{61}} ) c. ( frac{sqrt{61}}{12} ) D. ( frac{61}{144} ) |
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| 53 | Find the angle between the following pairs of lines: ( frac{x-1}{2}=frac{y-2}{3}=frac{z-3}{-3} ) and ( frac{x+3}{-1}= ) ( frac{boldsymbol{y}-mathbf{5}}{mathbf{8}}=frac{boldsymbol{z}-mathbf{1}}{mathbf{4}} ) |
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| 54 | The vector equation of the plane which is at a distance of ( frac{3}{sqrt{14}} ) from the origin and the normal from the origin is ( 2 hat{i}- ) ( mathbf{3} hat{boldsymbol{j}}+hat{boldsymbol{k}} ) is A ( . vec{r} .(2 hat{i}-3 hat{j}+hat{k})=3 ) B . ( vec{r} .(hat{i}+hat{j}+hat{k})=9 ) c. ( vec{r} .(hat{i}+2 hat{j})=3 ) D. ( vec{r} .(2 hat{i}+hat{k})=3 ) |
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| 55 | If a line makes angles ( alpha, beta, gamma ) with axes of co-ordinates, then ( cos 2 alpha+cos 2 beta+ ) ( cos 2 gamma ) is equla to A . -2 B. – c. 1 D. 2 |
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| 56 | Name the octants in which the following points lie: ( (1,2,3),(4,-2,3)(4,-2,-5),(4,2,-5) ) |
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| 57 | The perpendicular distance of ( vec{A}(1,4,-2) ) from the segment BC where ( vec{B} ) (2,1,-2) and ( vec{C}(0,-5,1) ) is ( A cdot frac{3}{7} sqrt{26} ) B ( cdot frac{6}{7} sqrt{26} ) ( mathbf{c} cdot frac{4}{7} sqrt{26} ) ( D cdot frac{2}{7} sqrt{26} ) |
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| 58 | Find the direction cosines of perpendicular from the origin to the plane ( bar{r}(2 hat{i}+3 hat{j}+6 hat{k})+7=0 ) |
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| 59 | Perpendiculars ( A P, A Q ) and ( A R ) are drawn to the ( x-, y- ) and ( z- ) axes, respectively from the point ( boldsymbol{A}(mathbf{1},-mathbf{1}, mathbf{2}) . ) The A.M. of ( A P^{2}, A Q^{2} ) and ( A R^{2} ) is A .4 B. 5 ( c .3 ) D. |
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| 60 | If ( boldsymbol{A}(boldsymbol{6},-mathbf{7}, mathbf{0}), boldsymbol{B}(mathbf{1 6},-mathbf{1 9},-mathbf{4}) ) ( C(0,3,-6) ) and ( D(2,-5,10) ) are four points in space, then the point of intersection of the lines ( A B ) and ( C D ) is A ( cdot(2,1,-1) ) в. (1,1,2) D. does not exist as the lines are skew |
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| 61 | A plane meet the co-ordinate axes in ( A, B, C ) such that the centroid of triangle ( A B C ) is ( (a, b, c) . ) If equation of plane ( frac{x}{a}+frac{y}{b}+frac{z}{c}=k(k neq 0), ) then the value of ( k ) equals A . 2 B. 3 ( c cdot 4 ) D. 5 |
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| 62 | Two system of rectangular axes have the same origin. If a plane cuts them at distances, ( a, b, c ) and ( a_{1}, b_{1}, c_{1} ) from the origin, then A ( cdot frac{1}{a^{2}}+frac{1}{b^{2}}+frac{1}{c^{2}}=frac{1}{a_{1}^{2}}+frac{1}{b_{1}^{2}}+frac{1}{c_{1}^{2}} ) B. ( frac{1}{a^{2}}-frac{1}{b^{2}}+frac{1}{c^{2}}=frac{1}{a_{1}^{2}}-frac{1}{b_{1}^{2}}+frac{1}{c_{1}^{2}} ) C ( cdot a^{2}+b^{2}+c^{2}=a_{1}^{2}+b_{1}^{2}+c_{1}^{2} ) D cdot ( a^{2}-b^{2}+c^{2}=a_{1}^{2}-b_{1}^{2}+c_{1}^{2} ) |
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| 63 | ox, oy are positive x-axis, positive ( y ) axis respectively where ( boldsymbol{O}=(mathbf{0}, mathbf{0}, mathbf{0}) ) The ( d . c . s ) of the llne which bisects ( angle x o y ) are ( mathbf{A} cdot 1,1,0 ) B. ( frac{1}{sqrt{2}}, frac{1}{sqrt{2}}, 0 ) c. ( frac{1}{sqrt{2}}, 0, frac{1}{sqrt{2}} ) D. 0,0,1 |
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| 64 | Find the equation of the set of points which are equidistant from the points (1,2,3) and (3,2,-1) | 12 |
| 65 | The ratio in which the plane ( bar{r} .(bar{i}-2 bar{j}+ ) ( mathbf{3} bar{k})=17 ) divides the line joining the points ( -2 bar{i}+4 bar{j}+7 bar{k} ) and ( 3 bar{i}-5 bar{j}+8 bar{k} ) is A. 1: 10 B. 3: 10 ( c .3: 5 ) D. 1: 5 |
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| 66 | Consider three planes ( 2 x+p y+6 z= ) ( mathbf{8}, boldsymbol{x}+mathbf{2} boldsymbol{y}+boldsymbol{q} boldsymbol{z}=mathbf{5} ) and ( boldsymbol{x}+boldsymbol{y}+mathbf{3} boldsymbol{z}=mathbf{4} ) These planes do not have any common point of intersection if- A. ( p=2, q neq 3 ) B . ( p neq 2, q neq 3 ) c. ( p neq 2, q=3 ) D. ( p=2, q=3 ) |
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| 67 | The ratio in which the surface ( x^{2}+ ) ( y^{2}+z^{2}=25 ) divides the line joining (0,1,2) and (3,4,5) is ( frac{a pm sqrt{b}}{c} ) then ( boldsymbol{a}+boldsymbol{b}+boldsymbol{c}= ) |
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| 68 | The direction cosine of a line which is perpendicular to both the lines whose direction ratios are 1,2,2 and 0,2,1 are A ( cdot frac{-2}{3}, frac{1}{3}, frac{2}{3} ) B. ( frac{2}{3}, frac{-1}{3}, frac{2}{3} ) c. ( frac{2}{3}, frac{1}{3}, frac{-2}{3} ) D. ( frac{2}{3}, frac{-1}{3}, frac{-2}{3} ) |
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| 69 | Three lines are given by ( vec{r}=lambda hat{i}, lambda epsilon R ) ( overrightarrow{boldsymbol{r}}=boldsymbol{mu}(hat{boldsymbol{i}}+hat{boldsymbol{j}}), boldsymbol{n} boldsymbol{epsilon} boldsymbol{R} ) and ( overrightarrow{boldsymbol{r}}=boldsymbol{v}(hat{boldsymbol{i}}+hat{boldsymbol{j}}+ ) ( hat{boldsymbol{k}}), boldsymbol{v} boldsymbol{epsilon} boldsymbol{R} ) Let the lines cut the plane ( boldsymbol{x}+boldsymbol{y}+boldsymbol{z}= ) 1 at he points ( A, B ) and ( C ) respectively. If the area of the triangle ( A B C ) is ( triangle ) then the value of ( (6 triangle)^{2} ) equals A . 0.75 в. ( 0 . ) ( c .0 .85 ) D. 0.65 |
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| 70 | Find the distance of a point (3,-5) from the line ( 3 x-4 y-5=0 ) | 12 |
| 71 | The direction cosines of a vector ( hat{boldsymbol{i}}+ ) ( hat{boldsymbol{j}}+sqrt{mathbf{2}} hat{boldsymbol{k}} ) are A ( cdot frac{1}{2}, frac{1}{2}, 1 ) B. ( frac{1}{sqrt{2}}, frac{1}{sqrt{2}}, frac{1}{2} ) c. ( frac{1}{2}, frac{1}{2}, frac{1}{sqrt{2}} ) D. ( frac{1}{sqrt{2}}, frac{1}{sqrt{2}}, frac{1}{sqrt{2}} ) |
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| 72 | Find the equation of plane with intercepts 2,3 and 4 on the ( x, y ) and ( z ) axis respectively. | 12 |
| 73 | If the extremities of a diagonal of a square are (1,-2,3) and (4,2,3) then the area of the square is A . 25 B. 50 c. ( frac{25}{2} ) D. ( sqrt{50} ) |
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| 74 | The points ( (10,7,0),(6,6-1) ) and (6,9,-4) form a A. Right -angled triangle B. Isosceles triangle ( c cdot ) Both (1)( &(2) ) D. Equilateral triangle |
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| 75 | If ( bar{a}, bar{b} ) and ( bar{c} ) are non-zero non collinear vectors and ( theta(neq 0, pi) ) is the angle between ( bar{b} ) and ( bar{c} ) if ( (bar{a} times bar{b}) times bar{c}=frac{1}{2}|bar{b}| bar{c} mid bar{a} ) then ( sin theta= ) A ( cdot sqrt{frac{2}{3}} ) B. ( frac{sqrt{3}}{2} ) ( c cdot frac{4 sqrt{2}}{3} ) D. ( frac{2 sqrt{2}}{3} ) |
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| 76 | The point which is equidistant from the points (-1,1,3),(2,1,2),(0,5,6) and (3,2,2) is A ( cdot(-1,3,4) ) в. (3,1,4) c. (1,3,4) D. (4,1,3) |
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| 77 | Find the coordinates of the point where the line ( frac{boldsymbol{x}+mathbf{1}}{mathbf{2}}=frac{boldsymbol{y}+boldsymbol{2}}{mathbf{3}}=frac{boldsymbol{z}+mathbf{3}}{mathbf{4}} ) meets the plane ( boldsymbol{x}+boldsymbol{y}+boldsymbol{4} boldsymbol{z}=boldsymbol{6} ) |
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| 78 | Number of lines is space which are equally inclined to three co-ordinate axes are? A .2 B. 4 ( c .6 ) D. 8 |
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| 79 | A cube of side 5 has one vertex at the point ( (1,0,-1), ) and the three edges from this vertex are, respectively, parallel to the negative ( x ) and ( y ) axes and positive z-axis. Find the coordinates of the other vertices of the cube. A. (1,0,1) B. (0,-1,0) c. (0,0,-1) D. (1,0,0) |
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| 80 | 45. The two lines x = ay+b, z= cy+d; and x = a’y+b’, [2006|| z=c’y+d’ are perpendicular to each other if (a) aa’+cc’=-1 (b) aa’t.cc’ = 1 (c) 9+ =-1 (d) 9+6=1 |
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| 81 | Distance between two parrallel lines, ( overline{boldsymbol{r}}=overline{boldsymbol{a}}_{1}+boldsymbol{lambda} overline{boldsymbol{b}} ) and ( overline{boldsymbol{r}}=overline{boldsymbol{a}}_{2}+boldsymbol{mu} overline{boldsymbol{b}}, ) is given by A ( cdot d=mid frac{left(bar{a}_{2}-bar{a}_{1}right)}{hat{b}} ) B . ( d=midleft(bar{a}_{2}-bar{a}_{1}right) times hat{b} ) c. ( d=midleft(bar{a}_{2}+bar{a}_{1}right) times hat{b} ) D . ( d=midleft(bar{a}_{2}-bar{a}_{1}right) ) |
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| 82 | If a line ( O P ) of length ( r ) (Where ‘ ( O ) ‘ is the origin) makes an angle ( alpha ) with ( x ) -axis and lies on the xz-plane, then what are the coordinates of ( P ? ) A ( cdot(r cos alpha, 0, r sin alpha) ) B . ( (0,0, r sin alpha) ) ( mathbf{c} cdot(r cos alpha, 0,0) ) D ( cdot(0,0, r cos alpha) ) |
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| 83 | Find the equation of the plane passing through the points (2,3,-4) and (1,-1,3) and parallel to the ( x- ) axis. | 12 |
| 84 | Find the angle between the line whose direction cosines are given by ( l+m+ ) ( boldsymbol{n}=mathbf{0} ) and ( l^{2}+boldsymbol{m}^{2}=boldsymbol{n}^{2} ) |
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| 85 | 63. If the angle between the line x=> lin _y-1 Z-3 – and the plane 2 x +2y + 33 =4 is cos” (193), then aequals x + 2y + 3z=4 is cos-1 , then a equals 12011 [2011] |
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| 86 | The intercepts of the plane ( 2 x-3 y+ ) ( mathbf{5} z-mathbf{3 0}=mathbf{0} ) are A. 15,-10,6 в. 5,10,6 c. ( 1 / 8,-1 / 6,1 / 4 ) D. 3,-4,6 |
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| 87 | If ( vec{P}(1,5,4) ) and ( vec{Q}(4,-1,-2), ) find the direction ratio of ( overrightarrow{P Q} ) | 12 |
| 88 | 39. Equation of the plane containing the straight line and perpendicular to the plane containing the (2010) x y z straight lines – === 3 4 2 (a) x+2y – 2z=0 (c) x-2y+z=0 x y z is 2 (b) 3x + 2y – 2z=0 (d) 5x + 2y – 4z=0 |
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| 89 | ( A ) point ( C ) with position vector ( frac{3 a+4 b-5 c}{3} ) (where ( a, b ) and ( c ) are non co-planar vectors) divides the line joining ( A ) and ( B ) in the ratio ( 2: 1 . ) If the position vector of ( A ) is ( a-2 b+3 c, ) then the position vector of ( boldsymbol{B} ) is A ( .2 a+3 b-4 c ) B . ( 2 a-3 b+4 c ) c. ( 2 a+3 b+4 c ) D. ( a+3 b-4 c ) |
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| 90 | The coordinates of a point which is equidistant from the point ( (0,0,0),(a, 0,0),(0, b, 0) ) and ( (0,0, c) ) are given by ( ^{mathbf{A}} cdotleft(frac{a}{2}, frac{b}{2}, frac{c}{2}right) ) в. ( left(frac{-a}{2}, frac{-b}{2}, frac{c}{2}right) ) ( ^{mathrm{c}}left(frac{a}{2}, frac{-b}{2}, frac{-c}{2}right) ) D ( cdotleft(frac{-a}{2}, frac{b}{2}, frac{-c}{2}right) ) |
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| 91 | Find the magnitude of the shortage distance between the lines ( , frac{x-8}{3}= ) ( frac{y+9}{-16}=frac{z-10}{7} ; frac{x-15}{3}=frac{y-29}{8}=frac{z-5}{-5} ) |
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| 92 | A line making angles ( 45^{circ} ) and ( 60^{circ} ) with the positive direction of ( x- ) axis and ( y- ) axis respectively. Then the angle made by the line with positive direction of ( z- ) axis is A ( .60^{circ} ) B. ( 120^{circ} ) ( mathbf{c} cdot 60^{circ} ) or ( 120^{circ} ) D. None of these |
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| 93 | The equation to the altitude of the altitude triangle formed by ( (1,1,1) cdot(1,2,3),(2,-1,1) ) through (1,1,1) is A ( cdot bar{r}=(bar{i}+bar{j}+bar{k})+t(bar{i}-bar{j}-2 bar{k}) ) B ( cdot bar{r}=(bar{i}-bar{j}+bar{k})+t(bar{i}+bar{j}-2 bar{k}) ) ( mathbf{c} cdot bar{r}=(bar{i}+bar{j}+bar{k})+t(bar{i}-bar{j}+2 bar{k}) ) D ( cdot bar{r}=(bar{i}-bar{j}-bar{k})+t(bar{i}+bar{j}-2 bar{k}) ) |
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| 94 | Cartesian equation of the plane ( bar{r}= ) ( (1+lambda-mu) bar{i}+(2-lambda) bar{j}+(3-2 lambda+ ) ( 2 mu) bar{k} ) is : A ( .2 x+y=5 ) в. ( 2 x-y=5 ) c. ( 2 x+z=5 ) D. ( 2 x-mathrm{z}=5 ) |
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| 95 | Unit vector perpendicular to the plane passing through the points ( hat{mathbf{i}}-hat{mathbf{j}}+ ) ( 2 hat{k}, 2 hat{i}-hat{k} ) and ( 2 hat{j}+hat{k} ) is A ( cdot frac{2 hat{i}-hat{j}+hat{k}}{sqrt{6}} ) ( ^{text {В } cdot frac{2 hat{i}+hat{j}+hat{k}}{sqrt{6}}} ) c. ( frac{2 hat{i}+hat{j}-hat{k}}{sqrt{6}} ) D. None of these |
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| 96 | Find the direction cosines of two lines which are connected by the relations ( l+m+n=0 ) and ( m n-2 n l-2 l m= ) ( mathbf{0} ) |
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| 97 | Line ( overrightarrow{boldsymbol{r}}=(hat{boldsymbol{i}}-hat{boldsymbol{j}}+hat{boldsymbol{k}})+boldsymbol{t}(boldsymbol{2} hat{boldsymbol{i}}-hat{boldsymbol{j}}+hat{boldsymbol{k}}) ) contained in a plane to which vector ( vec{n}=3 hat{i}-2 hat{j}+lambda hat{k} ) is normal. Find the value of ( lambda ). Also find the vector equation of the plane. |
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| 98 | Three vertices of a tetrahedron are (0,0,0),(6,-5,-1) and ( (-4,1,3) . ) If the centroid of the tetrahedron be (1,-2,5) then the fourth vertex is A. (2,-4,18) в. (2,-4,-18) ( ^{c} cdotleft(frac{3}{4}, frac{-3}{2}, frac{7}{4}right) ) D. none of these |
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| 99 | Assertion If a line makes angles ( boldsymbol{alpha}, boldsymbol{beta}, boldsymbol{gamma} ) with ( O X, O Y, O Z ) respectively, then ( sin ^{2} alpha+sin ^{2} beta+sin ^{2} gamma=2 ) Reason If ( l=cos alpha, m=cos beta, n=cos gamma, ) are direction cosines of a line, then ( l^{2}+ ) ( boldsymbol{m}^{2}+boldsymbol{n}^{2}=mathbf{1} ) A. Both Assertion & Reason are individually true & Reason is correct explanation of Assertion B. Both Assertion & Reason are individually true but Reason is not the ,correct (proper) explanation of Assertion C. Assertion is true but Reason is false D. Assertion is false but Reason is true |
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| 100 | The plane ( x=0 ) divides the joinning of (-2,3,4) and (1,-2,3) in the ratio A .2: 1 B. 1: 2 c. 3: 2 D. -4: 3 |
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| 101 | The image of the line ( frac{boldsymbol{x}-mathbf{1}}{mathbf{3}}=frac{boldsymbol{y}-mathbf{3}}{mathbf{1}}= ) ( frac{z-4}{-5} ) in the plane ( 2 x-y+z+3=0 ) is the line? A. ( frac{x-3}{3}=frac{y+5}{1}=frac{z-2}{-5} ) в. ( frac{x-3}{-3}=frac{y+5}{-1}=frac{z-2}{5} ) c. ( frac{x+3}{3}=frac{y-5}{1}=frac{z-2}{-5} ) D. ( frac{x+3}{-3}=frac{y-5}{-1}=frac{z+2}{5} ) |
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| 102 | If a line makes angles ( alpha, beta, gamma ) with the coordinate axes, then the value of ( cos 2 alpha+cos 2 beta+cos 2 gamma ) is ( A cdot 3 ) B. – – ( c cdot 2 ) D. – |
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| 103 | ( begin{array}{ll}text { List I } & text { List II } \ text { 1) d.c’s of } x-text { axis } & text { a) }(1,1,1) \ text { 2) d.c’s of } y-text { axis } & text { b) } \ text { 3) d.c’s of } z-text { axis } & left(frac{1}{sqrt{3}} frac{1}{sqrt{3}}, frac{1}{sqrt{3}}right) \ begin{array}{l}text { 4) d.c’s of a line makes equal } \ text { angles with axes }end{array} & text { c) }(1,0,0) \ begin{array}{l}text { d) }(0,1,0) \ text { e) }(0,0,1)end{array}end{array} ) The correct order for 1,2,3,4 is ( mathbf{A} cdot c, d, e, b ) B. ( a, b, c, e ) ( mathbf{C} cdot c, d, a, b ) D. ( b, c, a, e ) |
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| 104 | If the centroid of the tetrahedron ( O A B C, ) where ( A, B, C ) are given by ( (alpha, 5,6),(1, beta, 4),(3,2, gamma) ) respectively be ( 1,-1,2, ) then value of ( alpha^{2}+beta^{2}+gamma^{2} ) equals A ( cdot alpha^{2}+beta^{2} ) B. ( gamma^{2}+beta^{2} ) c. ( alpha^{2}+gamma^{2} ) D. None of these |
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| 105 | The shortest distance between the lines ( frac{boldsymbol{x}-mathbf{5}}{mathbf{4}}=frac{boldsymbol{y}-mathbf{7}}{-mathbf{5}}=frac{boldsymbol{z}+mathbf{3}}{-mathbf{5}} ) and ( frac{boldsymbol{x}-mathbf{8}}{mathbf{4}}= ) ( frac{y-7}{-5}=frac{z-5}{-5} ) is A . 45 B . 46 c. 47 D. 48 |
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| 106 | If ( overrightarrow{mathbf{A}} times overrightarrow{mathbf{B}}=overrightarrow{mathbf{B}} times overrightarrow{mathbf{A}}, ) then the angle between ( A ) and ( B ) is A . ( pi ) в. ( pi / 3 ) c. ( pi / 2 ) D . ( pi / 4 ) |
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| 107 | ( boldsymbol{A}=(-1,2,-mathbf{3}), boldsymbol{B}=(mathbf{5}, mathbf{0},-mathbf{6}), boldsymbol{C}= ) (0,4,-1) are the vertices of a triangle. The d.c’s of the internal bisector of ( angle mathrm{BAC} ) are? ( ^{mathbf{A}} cdotleft(frac{25}{sqrt{714}}, frac{-8}{sqrt{714}}, frac{-5}{sqrt{714}}right) ) в. ( left(frac{5}{sqrt{74}}, frac{6}{sqrt{74}}, frac{8}{sqrt{74}}right) ) ( ^{mathbf{C}} cdotleft(frac{25}{sqrt{714}}, frac{8}{sqrt{714}}, frac{5}{sqrt{714}}right) ) D. ( left(frac{-5}{sqrt{74}}, frac{6}{sqrt{74}}, frac{-8}{sqrt{74}}right) ) |
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| 108 | Equation of plane parallel to ( 3 x+4 y+ ) ( mathbf{5} boldsymbol{z}-mathbf{6}=mathbf{0}, mathbf{6} boldsymbol{x}+mathbf{8} boldsymbol{y}+mathbf{1 0} boldsymbol{z}-mathbf{1 6}=mathbf{0} ) and equidistant from them is A. ( 3 x+4 y+5 z=7 ) B. ( 3 x+4 y+5 z=10 ) c. ( 6 x+8 y+10 z=0 ) D. ( 6 x+8 y+10 z=3 ) |
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| 109 | Assertion (A): The points ( boldsymbol{A}(mathbf{2}, mathbf{9}, mathbf{1 2}), boldsymbol{B}(mathbf{1}, mathbf{8}, mathbf{8}), boldsymbol{C}(mathbf{2}, mathbf{1 1}, mathbf{8}) boldsymbol{D}(mathbf{1}, mathbf{1 2} ) are the vertices of a rhombus Reason ( (mathrm{R}): A B=B C=C D=D A ) and ( boldsymbol{A C}=boldsymbol{B D} ) A. Both A and R are individually true and R is the correct explanation of B. Both A and R individually true but R is not the correct explanation of A c. ( A ) is true but ( R ) is false D. Both A and R false |
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| 110 | Two equat ions ( vec{r} . vec{n}_{1}=q_{1} ) and ( vec{r} . vec{n}_{2}=q_{2} ) represent two perpendicular planes, where ( vec{n}_{1} ) and ( vec{n}_{2} ) are two unit vectors. One of these plane is rotated through an angle 45 about line of intersection of two given planes then equation of plane in new position can be This question has multiple correct options A ( cdot vec{r} cdotleft(vec{n}_{1}+vec{n}_{2}right)=q_{1}-q_{2} ) В ( cdot vec{r} cdot(overrightarrow{n_{1}}+overrightarrow{n_{2}})=q_{1}+q_{2} ) c. ( vec{r} .left(vec{n}_{1}-vec{n}_{2}right)=q_{1}+q_{2} ) D ( cdot vec{r} cdot(overrightarrow{n_{1}}-overrightarrow{n_{2}})=q_{1}-q_{2} ) |
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| 111 | 13. The shortest distance from the plane 12x+4y+3z =327 to the sphere x2 + y2 + z2 + 4x – 2y – 6z =155 is (2) 39 (b) 26 (c) 11 (d) 13 |
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| 112 | 25. -3 y-k _Z intersect, then If the lines x-1 y+1 Z-1 2 3 4 ” the value of k is (a) 3/2 (b) 9/2 and 2 1 (c) – 2/9 (2004S) (d) – 3/2 |
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| 113 | ( boldsymbol{P}(mathbf{0}, mathbf{5}, mathbf{6}), boldsymbol{Q}(mathbf{1}, mathbf{4}, mathbf{7}), boldsymbol{R}(mathbf{2}, mathbf{3}, mathbf{7}) ) and ( S(3,5,16) ) are four points in the space. The point nearest to the origin ( boldsymbol{O}(mathbf{0}, mathbf{0}, mathbf{0}) ) is A. ( P ) B. ( Q ) ( c . R ) D. ( S ) |
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| 114 | Show that the lines whose d.c’s are given by ( 2 l+2 m-n=0, m n+n l+ ) ( l m=0 ) are perpendicular to each other |
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| 115 | The coordinates of any point, which lies on ( x ) axis are A ( .(0, x, 0) ) в. ( (x, 0,0) ) c. ( (x, x, 0) ) D. ( (x, x, x) ) |
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| 116 | If ( boldsymbol{A}=(mathbf{1}, mathbf{2}, mathbf{3}), boldsymbol{B}=(mathbf{2}, mathbf{3}, mathbf{4}) ) and ( boldsymbol{A} boldsymbol{B} ) is produced upto ( C ) such that ( 2 A B=B C ) then ( C= ) A. (5,4,6) в. (6,2,4) c. (4,5,6) D. (6,4,5) |
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| 117 | If ( P(x, y, z) ) is a point on the line segment joining ( Q(2,2,4) ) and ( boldsymbol{R}(boldsymbol{3}, boldsymbol{5}, boldsymbol{6}) ) such that the projection of ( O P ) on the axes are ( frac{13}{5}, frac{19}{5}, frac{26}{5} ) respectively, then ( P ) divides ( Q R ) in the ratio A .1: 2 B. 3: 2 ( c cdot 2: 3 ) D. 1: 3 |
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| 118 | Find the intersection of the line ( x- ) ( mathbf{2} boldsymbol{y}+mathbf{4} boldsymbol{z}+mathbf{4}=mathbf{0}, boldsymbol{x}+boldsymbol{y}+boldsymbol{z}-mathbf{8}=mathbf{0} ) with the plane ( boldsymbol{x}-boldsymbol{y}+mathbf{2} boldsymbol{z}+mathbf{1}=mathbf{0} ) |
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| 119 | The coordinates of a point ( mathrm{P} ) are (3,12,4) w.r.t origin ( 0, ) then the direction cosines of ( O P ) are ( mathbf{A} cdot 3,12,4 ) B. ( frac{1}{4}, frac{1}{3}, frac{1}{2} ) c. ( frac{3}{sqrt{13}}, frac{1}{sqrt{13}}, frac{2}{sqrt{13}} ) D. ( frac{3}{13}, frac{12}{13}, frac{4}{13} ) |
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| 120 | The angle between vectors ( (bar{M} times bar{N}) ) and ( (overline{boldsymbol{N}} times overline{boldsymbol{M}}) ) is then ( mathbf{A} cdot 0^{circ} ) B . ( 60^{circ} ) ( c .90^{circ} ) D. ( 180^{circ} ) |
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| 121 | If the plane ( 3 x+2 y+6 z=6 ) intersects the coordinate axes at ( A, B, C ) then the area of the ( Delta A B C ) is ( mathbf{A} cdot 49 ) B. 7 ( c cdot frac{7}{2} ) D. ( frac{11}{2} ) |
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| 122 | Arrange the points: ( mathbf{A}(1,2- ) ( mathbf{3}), mathbf{B}(-mathbf{1}, mathbf{2},-mathbf{3}), mathbf{C}(-mathbf{1},-mathbf{2}-mathbf{3}) ) and ( mathbf{D}(mathbf{1},-mathbf{2},-mathbf{3}) ) in the increasing order of their octant numbers: A. ( A, B, C, D ) в. ( B, C, D, A ) c. ( C, D, A, B ) D. ( D, C, B, A ) |
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| 123 | Distance of the point ( boldsymbol{P}(overrightarrow{boldsymbol{p}}) ) from the line ( vec{r}=vec{a}+lambda vec{b} ) is – A ( cdot(vec{a}-vec{p})+frac{((vec{p}-vec{a}) cdot vec{b}) vec{b}}{|vec{b}|^{2}} mid ) B. ( (vec{b}-vec{p})+frac{((vec{p}-vec{a}) cdot vec{b}) vec{b}}{|vec{b}|^{2}} mid ) c. ( quadleft|(vec{a}-vec{p})+frac{((vec{p}-vec{b}) cdot vec{b}) vec{b}}{|vec{b}|^{2}}right| ) D. None of these. |
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| 124 | A line makes angle ( theta_{1}, theta_{2}, theta_{3}, theta_{4} ) with the diagonals of the cube. Show that ( cos ^{2} theta_{1}+cos ^{2} theta_{2}+cos ^{2} theta_{3}+cos ^{2} theta_{4}= ) ( frac{4}{3} ? ) |
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| 125 | if a line makes angles ( alpha, beta, gamma, delta ) with four diagonals a cube then value of ( sin ^{2} alpha+sin ^{2} beta+sin ^{2} gamma+sin ^{2} delta ) equals ( A cdot 2 ) B. ( frac{4}{3} ) ( c cdot frac{8}{3} ) D. |
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| 126 | Derive the coordinates of the points ( R(x, y, z) ) dividing the line joining the points ( boldsymbol{P}left(boldsymbol{x}_{1}, boldsymbol{y}_{1}, boldsymbol{z}_{1}right) ) and ( boldsymbol{Q}left(boldsymbol{x}_{2}, boldsymbol{y}_{2}, boldsymbol{z}_{2}right) ) internally in the ratio ( m: n ) |
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| 127 | Find the value of ( lambda ) for which the four points ( A, B, C, D ) with position vectors ( -widehat{boldsymbol{j}}-widehat{boldsymbol{k}} ; boldsymbol{4} hat{boldsymbol{i}}+boldsymbol{5} hat{boldsymbol{j}}+boldsymbol{lambda} hat{boldsymbol{k}} ; boldsymbol{3} hat{boldsymbol{i}}+boldsymbol{9} hat{boldsymbol{j}}+boldsymbol{4} widehat{boldsymbol{k}} ) and ( -4 hat{i}+4 widehat{j}+4 widehat{k} ) are coplanar. |
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| 128 | If direction cosines of two lines are proportional to (2,3,-6) and (3,-4,5) then the acute angle between them is ( ^{mathbf{A}} cdot cos ^{-1}left(frac{49}{36}right) ) B. ( cos ^{-1}left(frac{18 sqrt{2}}{35}right) ) ( c cdot 96^{circ} ) D. ( cos ^{-1}left(frac{18}{35}right) ) |
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| 129 | A non-zero vector ( vec{a} ) is parallel to the line of intersection of the plane determined by the vectors ( hat{i}, hat{i}+hat{j} ) and the plane determined by the vectors ( hat{i}-hat{j}, hat{i}+hat{k} ) The angle between ( vec{a} ) and ( hat{i}-2 hat{j}+2 hat{k} ) is A. в. c. D. ( frac{pi}{2} ) |
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| 130 | The intercept made by the plane ( vec{r} cdot vec{n}= ) ( q ) on the ( x ) -axis is A ( cdot frac{q}{hat{i} cdot vec{n}} ) в. ( frac{hat{i} cdot vec{n}}{q} ) ( c cdot frac{hat{i} cdot q}{n} ) D. ( frac{q}{|vec{n}|} ) |
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| 131 | The point ( P ) is the intersection of the straight line joining the points ( Q(2,3,5) ) and ( R(1,-1,4) ) with the plane ( 5 x-4 y-z=1 . ) If ( S ) is the foot of the perpendicular drawn from the point ( T(2,1,4) ) to ( Q R, ) then the length of the line segment ( P S ) is A ( cdot frac{1}{sqrt{2}} ) B. ( sqrt{2} ) ( c cdot 2 ) D. ( 2 sqrt{2} ) |
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| 132 | Equation of the plane containing the ( operatorname{lines} overline{boldsymbol{r}}=(overline{boldsymbol{i}}-boldsymbol{2} overline{boldsymbol{j}}+overline{boldsymbol{k}})+boldsymbol{t}(overline{boldsymbol{i}}+mathbf{2} overline{boldsymbol{j}}-overline{boldsymbol{k}}) ) ( boldsymbol{boldsymbol { r }}=(overline{boldsymbol{i}}+mathbf{2} overline{boldsymbol{j}}-overline{boldsymbol{k}})+boldsymbol{s}(overline{boldsymbol{i}}+overline{boldsymbol{j}}+mathbf{3} overline{boldsymbol{k}}) ) is A. ( bar{r}(7 bar{i}-4 bar{j}-bar{k})=14 ) В. ( bar{r}(bar{i}+2 bar{j}-bar{k})=10 ) c. ( bar{r}(bar{i}+bar{j}+3 bar{k})=20 ) D. ( bar{r}(bar{i}-2 bar{j}+bar{k})=27 ) |
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| 133 | 18. A tetrahedron has vertices at O(0, 0, 0), A(1,2,1) B(2,1,3) and C(-1,1,2). Then the angle between the faces OAB and ABC will be [2003] (a) 90° (b) cos- (c) cos-1( 37 (d) 30° 31 |
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| 134 | Find the length and foot of the perpendicular from the point ( (mathbf{7}, mathbf{1 4}, mathbf{5}) ) to the plane ( 2 x+4 y-z=2 ) | 12 |
| 135 | The circum centre of the triangle formed by the points (2,5,1),(1,4,-3) and (-2,7,-3) is A. (6,0,1) () В. (0,6,-1) c. (-1,6,2) D. (6,1,-2) |
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| 136 | Graph ( x^{2}+y^{2}=4 ) in ( 3 D ) looks like A . Circle B. Cylinder c. Hemisphere D. sphere |
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| 137 | Write the equations for the ( x ) -and ( y ) axes. |
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| 138 | 16. The image of the point (-1,3,4) in the plane x-2y = 0 is ( 17 19 41 (b) (15,11,4) (2006) 10 U (d) None of these |
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| 139 | Find the image of : (-2,3,4) in the ( y z ) -plane |
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| 140 | The distance of the point (1,-2,3) from the plane ( boldsymbol{x}-boldsymbol{y}+boldsymbol{z}=mathbf{5} ) measured parallel to the line ( frac{x}{2}=frac{y}{3}=frac{z-1}{-6} ) is A . B . 2 ( c cdot 4 ) D. None of these |
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| 141 | ff ( left(x_{1}, y_{1}, z_{1}right) ) and ( Bleft(x_{2}, y_{2}, z_{2}right) ) are two points such that the direction ( operatorname{cosines} ) of ( A B ) are ( l, m, n ) then ( l=frac{x_{2}-x_{1}}{|A B|}, m=frac{y_{2}-y_{1}}{|A B|}, n=frac{z_{2}-z_{1}}{|A B|} ) |
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| 142 | Four vertices of a tetrahedron are (0,0,0),(4,0,0),(0,-8,0) and ( (0,0,12) . ) Its centroid has the coordinates A ( cdotleft(frac{4}{3},-frac{8}{3}, 4right) ) В. (2,-4,6) c. (1,-2,3) a 5 D. none of these |
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| 143 | If ( boldsymbol{A}=(-2,3,4), B=(1,2,3) ) are two points and ( P ) is the point of intersection of ( A B ) and ( z x ) -plane, then ( P_{x}+P_{y}+ ) ( boldsymbol{P}_{z}= ) ( mathbf{A} cdot mathbf{6} ) B. -8 c. 8 D. |
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| 144 | If the points ( (1,1, p) ) and (-3,0,1) be equidistant from the plane ( vec{r} .(3 hat{i}+ ) ( 4 hat{j}-12 hat{k})+13=0, ) then find the value of ( p ) |
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| 145 | Find the ratio in which YZ-plane divides the line joining ( A(2,4,5) ) and ( B(3,5,-4) . ) Also find the point of intersection. |
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| 146 | ( operatorname{Points} boldsymbol{A}(boldsymbol{3}, boldsymbol{2}, boldsymbol{4}), boldsymbol{B}left(frac{boldsymbol{3} boldsymbol{3}}{boldsymbol{5}}, frac{boldsymbol{2} boldsymbol{8}}{boldsymbol{5}}, frac{boldsymbol{3} boldsymbol{8}}{boldsymbol{5}}right), ) and ( C(9,8,10) ) are given. The ratio in which ( B ) divides ( overline{A C} ) is A . 5: 3 B . 2: 1 c. 1: 3 D. 3: 2 |
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| 147 | The image of the point (-1,3,4) in the plane ( boldsymbol{x}-mathbf{2} boldsymbol{y}=mathbf{0} ) is A ( cdot(15,11,4) ) в. ( left(frac{9}{5},-frac{13}{5}, 4right) ) ( c cdot(8,4,4) ) D. None of these |
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| 148 | 28. P, and P, are planes passing through origin. L, and L, are two line on P, and P2 respectively such that their intersection is origin. Show that there exists points A, B, C, whose permutation A’, B’, C can be chosen such that (i) Ais on L,, B on P, but not on L, and C not on P, (ii) A’ is on L,, B’ on P, but not on L and C not on P2 |
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| 149 | 26. Find the equation of plane passing through (1,1,1) & parallel to the lines L,, L, having direction ratios (1,0,-1), (1,-1,0) Find the volume of tetrahedron formed by origin and the points where these planes intersect the coordinate axes. (2001.2 Mau |
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| 150 | 6. A line makes the same angle , with each of the x and z axis. If the angle ß, which it makes with y-axis, is such that sin B = 3 sin?e, then cos2e equals [2004] т. |
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| 151 | Find the coordinates of the point ( boldsymbol{P} ) which divides the join of ( boldsymbol{A}(-2,5) ) and ( B(3,-5) ) in the ratio 2: 3 |
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| 152 | If a line makes an angle of ( frac{pi}{4} ) with the positive direction of each of ( x ) -axis and ( boldsymbol{y} ) -axis, then the angle that the line makes with the positive direction of ( z ) axis is- A ( cdot frac{pi}{3} ) в. c. ( frac{pi}{2} ) D. |
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| 153 | The eartesian equations of the line are ( mathbf{3} boldsymbol{x}+mathbf{1}=mathbf{6} boldsymbol{y}-mathbf{2}=mathbf{1}-boldsymbol{z} . ) Find its equation in vector form and find direction ratios of the line. |
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| 154 | The projections of a line segment on ( x, y ) and ( z ) axes are respectively ( sqrt{2}, 3,5 ) The length of the line segment is ( mathbf{A} cdot mathbf{6} ) B. 11 c. 8 D. 5 |
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| 155 | ( operatorname{Given} boldsymbol{A}(mathbf{1},-mathbf{1}, mathbf{0}) ; boldsymbol{B}(mathbf{3}, mathbf{1}, mathbf{2}) ) ( C(2,-2,4) ) and ( D(-1,1,-1) ) which of the following points neither lie on ( boldsymbol{A B} ) nor on ( C D ? ) A ( .(2,2,4) ) В. (2,-2,4) c. (2,0,1) D. (0,-2,-1) |
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| 156 | Directions ratio of two lines are ( 3,-2, k ) and ( -2, k, 4 . ) Find ( k ) if the lines are perpendicular to each other. |
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| 157 | Find the distance of the point ( P(3,4,4) ) from the point, where the line joining the point ( A(3,-4,-5) ) and ( B(2,-3,1) ) intersects the plane ( 2 x+y+z=7 ) | 12 |
| 158 | Two vectors ( vec{A} ) and ( vec{B} ) inclined at an angle ( theta ) have a resultant ( vec{R} ) which makes an angle ( alpha ) with ( vec{A} ) and angle ( beta ) with ( vec{B} ). Let the magnitudes of the vectors ( vec{A}, vec{B} ) and ( vec{R} ) be represented by ( A ) B and R respectively. Which of the following relations is not correct? A . Asinalpha ( =B sin beta ) B. ( R sin alpha=B sin (alpha+beta) ) c. ( R sin beta=operatorname{Asin}(alpha+beta) ) D. None of these |
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| 159 | ( A=(2,3,0) ) and ( B=(2,1,2) ) are two points. If the points ( P, Q ) are on the line ( A B ) such that ( A P=P Q=Q B, ) then ( boldsymbol{P Q}= ) ( A cdot 2 sqrt{2} ) B. ( 6 sqrt{2} ) ( c cdot sqrt{frac{8}{9}} ) D. ( sqrt{2} ) |
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| 160 | Point, Plane: ( (0,0,0), 3 x-4 y+12 z= ) 3 |
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| 161 | The angle between two diagonals of a cube is. A ( .30^{circ} ) B . ( 45^{circ} ) c. ( cos ^{-1}left(frac{1}{3}right) ) D. ( cos ^{-1}left(frac{1}{sqrt{3}}right) ) |
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| 162 | If direction numbers of two lines are ( a, b, c ) and ( b-c, c-a, a-b ) prove that they are perpendicular to each other. |
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| 163 | The direction cosines of a line equally inclined to three mutually perpendicular lines having D.C.’s as ( ell_{1} m_{1} n_{1}: ell_{2} m_{2} n_{2}: ell_{3} m_{3} n_{3} ) are A. ( l_{1}+l_{2}+l_{3}, m_{1}+m_{2}+m_{3}, n_{1}+n_{2}+n_{3} ) B ( cdotleft(pm frac{1}{sqrt{3}}, pm frac{1}{sqrt{3}}, pm frac{1}{sqrt{3}}right) ) ( ^{mathbf{c}} cdotleft(pm frac{1}{sqrt{2}}, pm frac{1}{sqrt{3}}, pm frac{1}{sqrt{4}}right) ) D. none of these |
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| 164 | The direction cosines of the line which is perpendicular to the lines with direction cosines proportional to (1,-2,-2)( &(0,2,1) ) are ( ^{mathrm{A}} cdotleft(frac{2}{3},-frac{1}{3}, frac{2}{3}right) ) в. ( left(frac{2}{3}, frac{1}{3}, frac{2}{3}right) ) c. ( left(frac{2}{3}, frac{1}{3}, frac{-2}{3}right) ) D. ( left(frac{-2}{3}, frac{1}{3}, frac{2}{3}right) ) |
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| 165 | A plane ( pi ) makes intercept 3 and 4 respectively on z-axis and x-axis. If ( pi ) is parallel to y-axis, then its equation is A. ( 3 x+4 z=12 ) B. ( 3 z+4 x=12 ) c. ( 3 y+4 z=12 ) D. ( 3 z+4 y=12 ) |
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| 166 | Distance between ( boldsymbol{A}(mathbf{4}, mathbf{5}, mathbf{6}) ) from origin ( boldsymbol{O} ) is A ( cdot 25 sqrt{3} ) B. ( sqrt{77} ) c. ( 3 sqrt{5} ) D. Data Insufficient |
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| 167 | If ( vec{A}=-4 hat{i}+3 hat{j} ) and ( vec{B}=2 hat{i}+5 hat{j} ) and ( vec{C}=vec{A} times vec{B} ) then ( vec{C} ) makes an angle of : A ( cdot 45^{0} ) with ( mathrm{x} ) -axis B. ( 180^{circ} ) with ( Y ) -axis ( c cdot 0^{0} ) with ( mathrm{z} ) -axis D. ( 180^{circ} ) with ( z ) -axis |
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| 168 | Show that ( A(3,-2) ) is a point trisection of the line segment joining the points (2,1) and (5,-8) Also find the co-ordinates of the other points of trisections. |
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| 169 | A point on the line ( frac{boldsymbol{x}-mathbf{1}}{mathbf{1}}=frac{boldsymbol{y}-mathbf{2}}{mathbf{2}}= ) ( frac{z+1}{3} ) at a distance ( sqrt{6} ) from the origin is This question has multiple correct options ( mathbf{A} cdotleft(frac{-5}{7}, frac{-10}{7}, frac{13}{7}right) ) в. (1,2,-1) ( ^{mathbf{C}} cdotleft(frac{5}{7}, frac{10}{7}, frac{-13}{7}right) ) D. (-1,-2,1) |
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| 170 | Let the co – ordinates of the point where the line joining the points (2,-3,1),(3,-4,-5) cuts the plane ( mathbf{2} boldsymbol{x}+boldsymbol{y}+boldsymbol{z}=mathbf{7} ) be ( (boldsymbol{x}, boldsymbol{y}, boldsymbol{z}) . ) Find ( boldsymbol{x}+ ) ( boldsymbol{y}+boldsymbol{z} ? ) |
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| 171 | The plane ( X O Z ) divides the join of (1,-1,5) and (2,3,4) in the ratio ( lambda: 1 ) then ( lambda ) is A . -3 B. ( -1 / 3 ) ( c cdot 3 ) D. ( 1 / 3 ) |
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| 172 | The distance of the point ( ,(-1,-5,-10) ) from the point intersection of the line, ( frac{x-2}{3}=frac{y+1}{4}=frac{z-2}{12} ) and the plane ( x- ) ( boldsymbol{y}+boldsymbol{z}=mathbf{5}, ) is ( A cdot 13 ) B. 1 c. 12 D. none of these |
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| 173 | If ( overrightarrow{P O}+overrightarrow{O Q}=overrightarrow{Q O}+overrightarrow{O R}, ) prove that the points ( P, Q, R ) are collinear. |
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| 174 | 17. The radius of the circle in which the sphere x2 + y2 + 2? + 2x – 2y – 42 -19 = 0 is cut by the plane x +2y + 2z+ 7 = 0 is [2003] (a) 4 (6) 1 (c) 2 (d) 3 |
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| 175 | Let the vector ( vec{a}, vec{b}, vec{c} ) and ( vec{d} ) be such ( operatorname{that}(overrightarrow{boldsymbol{a}} times overrightarrow{boldsymbol{b}}) times(overrightarrow{boldsymbol{c}} times overrightarrow{boldsymbol{d}})=mathbf{0} cdot operatorname{Let} boldsymbol{P}_{1} ) and ( P_{2} ) be planes determined by the pairs of vectors ( vec{a}, vec{b} ) and ( vec{c}, vec{d} ) respectively then the angle between ( P_{1} ) and ( P_{2} ) is A. 0 в. ( frac{pi}{4} ) c. ( frac{pi}{3} ) D. ( frac{pi}{2} ) |
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| 176 | Find the shortest distance between the skew lines ( r=(6 i+2 j+2 k)+t(i- ) ( 2 j+2 k) ) and ( F=(-4 i-k)+s(3 i- ) ( 2 j-2 k) ) where s,t are scalars. |
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| 177 | Find the direction cosines ( l, m, n ) of a line which are connected by the relation ( l+m-n=0 ) and ( 2 m l-2 m n+n l= ) 0 This question has multiple correct options A ( cdot frac{-2}{sqrt{6}}, frac{1}{sqrt{6}}, frac{-1}{sqrt{6}} ) в. ( frac{2}{sqrt{6}}, frac{-1}{sqrt{6}}, frac{1}{sqrt{6}} ) c. ( frac{-2}{sqrt{6}}, frac{-1}{sqrt{6}}, frac{-1}{sqrt{6}} ) D. ( frac{2}{sqrt{6}}, frac{1}{sqrt{6}}, frac{1}{sqrt{6}} ) |
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| 178 | Verify the following (i) (0,7,-10),(1,6,-6) and (4,9,-6) are the vertices of an isosceles triangle (ii) (0,7,10),(-1,6,6) and (-4,9,6) are the vertices of a right angled triangle (iii) (-1,2,1),(1,-2,5),(4,-7,8) and (2,-3,4) are the vertices of a parallelogram |
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| 179 | Show that the points ( boldsymbol{A}(mathbf{0}, mathbf{1}, mathbf{2}), boldsymbol{B}(mathbf{2},-mathbf{1}, mathbf{3}) ) and ( boldsymbol{C}(mathbf{1},-mathbf{3}, mathbf{1}) ) are vertices of an isosceles right-angled triangle. |
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| 180 | 47. The equation of the plane passing through the point (1,1,1) and perpendicular to the planes 2x+y-2z=5 and 3x – 6y- 2z=7, is (JEE Adv. 2017) (a) 14x +2y-15z=1 (b) 14x – 2y + 15z=27 (c) 14x +2y+ 15z=31 (d) -14x + 2y + 15z=3 |
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| 181 | The distance of point ( A(-2,3,1) ) from the PQ through ( P(-3,5,2), ) which makes equal angles with the axes is- A ( cdot frac{2}{sqrt{3}} ) в. ( sqrt{frac{14}{3}} ) c. ( frac{16}{sqrt{3}} ) D. ( frac{5}{sqrt{3}} ) |
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| 182 | The cartesian from of equation a line passing through the point position vector ( 2 hat{i}-hat{j}+2 hat{k} ) and is in the direction of ( -2 hat{i}+hat{j}+hat{k}, ) is A ( frac{x-2}{-2}=frac{y+1}{1}=frac{z-2}{1} ) B. ( frac{x+4}{-2}=frac{y-1}{1}=frac{z+2}{1} ) c. ( frac{x+2}{4}=frac{y-1}{-1}=frac{z-1}{2} ) D. None of these |
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| 183 | Find the vector equation of the plane whose cartesian form of equation is ( 3 x- ) ( 4 y+2 z=5 ) | 12 |
| 184 | Column I shows some vector equations. Match Column I with the value of angle between ( vec{A} ) and ( vec{B} ) given in Column II Column I Column I |
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| 185 | The point ( P(x, y, z) ) lies in the first octant and its distance from the origin is 12 units. If the position vector of ( P ) make ( 45^{circ} ) and ( 60^{circ} ) with the ( x ) -axis and ( y ) axis respectively, then the coordinates of ( boldsymbol{P} ) are A ( cdot(3 sqrt{3}, 6,3 sqrt{2}) ) B. ( (4 sqrt{3}, 8,4 sqrt{2}) ) c. ( (6 sqrt{2}, 6,6,) ) D. ( (6,6,6 sqrt{2}) ) E ( .(4 sqrt{2}, 8,4 sqrt{3}) ) |
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| 186 | The projection of a line segment joining the points ( boldsymbol{P}left(boldsymbol{x}_{1}, boldsymbol{y}_{1}, boldsymbol{z}_{1},right) ) and ( Qleft(x_{1}, y_{1}, z_{1},right) ) on another line whose DC’s are ( l, m, n ) is given by This question has multiple correct options A ( cdot lleft(x_{1}+x_{2}right)+mleft(y_{2}+y_{2}right)+nleft(z_{1}+z_{2}right) ) B. ( 2left[frac{left(l x_{2}+m y_{2}+n z_{2}right)}{2}-frac{left(l x_{1}+m y_{1}+n z_{1}right)}{2}right. ) c. ( lleft(x_{2}-x_{1}right)+mleft(y_{2}-y_{1}right)+nleft(z_{2}-z_{1}right) ) D. ( frac{x_{2}-x_{1}}{l}+frac{y_{2}-y_{1}}{m}+frac{z_{2}-z_{1}}{n} ) |
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| 187 | The point ( P ) is on the ( y ) -axis. If ( P ) is equidistant from (1,2,3) and (2,3,4) then ( boldsymbol{P}_{boldsymbol{y}}= ) A ( cdot frac{15}{2} ) B. 15 c. 30 D. ( frac{3}{2} ) |
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| 188 | If the planes ( boldsymbol{x}-boldsymbol{b} boldsymbol{z}=mathbf{0}, boldsymbol{c} boldsymbol{x}-boldsymbol{y}+=mathbf{0} ) and ( b x+a y-z=0, ) pass through a line, then find the value of ( a^{2}+b^{2}+ ) ( c^{2}+2 a b c ) A. B. c. -1 D. ( frac{1}{2} ) |
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| 189 | The vector ( vec{P} ) makes ( 120^{circ} ) with the ( x- ) axis and the vector ( vec{Q} ) makes ( 30^{circ} ) with ( boldsymbol{y}- ) axis. What is the resultant vector? A. ( P+Q ) в. ( P-Q ) c. ( sqrt{P^{2}+Q^{2}} ) D. ( sqrt{P^{2}-Q^{2}} ) |
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| 190 | 90. The equation of the line passing through (-4, 2, to the plane x + 2y – Z-5 = 0 and intersecting the passing through (-4, 3, 1), parallel -2. x+l -3 1 | y-3 2.. (JEEM 2019-9 Jan (M) -1 ) 12 1 4 () 4 +3 2+1 (Jet 2 () | 13 1 |
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| 191 | The variable plane ( (2 lambda+1) x+ ) ( (3-lambda) y+z=4 ) always passes through the line A ( cdot frac{x}{0}=frac{y}{0}=frac{x+4}{1} ) в. ( frac{x}{1}=frac{y}{2}=frac{z}{-3} ) c. ( frac{x}{1}=frac{y}{2}=frac{z-4}{-7} ) D. none of these |
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| 192 | Find the point where the line of intersection of the planes ( x-2 y+z= ) 1 and ( x+2 y-2 z=5 ) intersects the plane ( 3 x+2 y+z+6=0 ) A ( cdot P(1,-2,-4) ) B . ( P(1,2,-4) ) c. ( P(1,-2,4) ) D. None of these |
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| 193 | 26. The intersection of the spheres x2 + y2 + z2 + 7x-2y-z = 13 and x2 + y2 + 22 – 3x + 3y + 4z = 8 is the same as the intersection of one of the sphere and the plane [2004] (a) 2x – y – z=1 (b) x-2y-z=1 (©) x-y–2z=1 (d) x-y-z = 1 |
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| 194 | The d.rs of the lines ( boldsymbol{x}=boldsymbol{a} boldsymbol{y}+boldsymbol{b}, boldsymbol{z}= ) ( boldsymbol{c} boldsymbol{y}+boldsymbol{d} ) are: A. ( 1, a, c ) B. ( a, 1, c ) c. ( b, 1, c ) D. ( c, a, 1 ) |
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| 195 | The circum radius of the triangle formed by the points (1,2,-3),(2,-3,1) and (-3,1,2) is: A ( cdot sqrt{14} ) B. 14 c. ( sqrt{13} ) D. |
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| 196 | If ( hat{i}, hat{j} ) and ( hat{k} ) represents unit vectors along the ( x, y ) and ( z- ) axes respectively then find the value of angle ( theta ) between the vectors ( hat{i}+hat{j}+hat{k} ) and ( hat{i}+hat{j} ) |
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| 197 | Find the shortest distance between the line ( boldsymbol{x}=mathbf{1}+boldsymbol{t}, boldsymbol{y}=mathbf{1}+boldsymbol{6} boldsymbol{t}, boldsymbol{z}=boldsymbol{2} boldsymbol{t}, boldsymbol{t} in boldsymbol{R} ) and ( boldsymbol{x}=mathbf{1}+mathbf{2 k}, boldsymbol{y}=mathbf{5}+mathbf{1 5 k}, boldsymbol{z}= ) ( -2+6 k, k in R ) |
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| 198 | 29 (0) Tu, , A plane which is perpendicular to two planes 2x – 2y+z=0 and x-y + 2z= 4, passes through (1, -2, 1). The distance of the plane from the point (1,2,2) is (2006 – 3M, -1) (a) o (b) 1 (c) √ (d) 252 |
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| 199 | A line makes an angle ( theta ) with each of the ( x- ) and ( z^{-} ) axes. If the angle ( beta, ) which it makes with the ( y ) -axis, is such that ( sin ^{2} beta=3 sin ^{2} theta, ) then ( cos ^{2} theta ) equals- A ( cdot frac{2}{3} ) B. ( frac{1}{5} ) ( c cdot frac{3}{5} ) D. ( frac{2}{5} ) |
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| 200 | Find the vector equation of line joining the points (2,1,3) and (-4,3,-1) ( mathbf{A} cdot bar{r}=2(1-3 lambda) bar{i}-(1+2 lambda) bar{j}-(3-4 lambda) bar{k} ) B . ( bar{r}=2(1-3 lambda) bar{i}-(1+2 lambda) bar{j}+(3-4 lambda) bar{k} ) C ( . bar{r}=2(1-3 lambda) bar{i}+(1+2 lambda) bar{j}+(3-4 lambda) bar{k} ) D. ( bar{r}=2(1+3 lambda) bar{i}+(1+2 lambda) bar{j}+(3+4 lambda) bar{k} ) |
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| 201 | Show that LHS=RHS i.e. ( frac{2}{9} times 3=frac{2}{9} times ) ( frac{3}{1}=frac{2 times 3}{9 times 1}=frac{6}{9}=frac{2}{3} ) | 12 |
| 202 | A plane meets the axes in ( A, B ) and ( C ) such that centroid of the ( triangle A B C ) is ( (1,2,3) . ) The equation of the plane is A ( cdot x+frac{y}{2}+frac{z}{3}=1 ) В ( cdot frac{x}{3}+frac{y}{6}+frac{z}{9}=1 ) c. ( x+2 y+3 z=1 ) D. None of these |
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| 203 | If the projections of the line segment ( A B ) on the coordinate axes are 2,3,6 then the square of the sine of the angle made by ( boldsymbol{A B} ) with ( boldsymbol{x}=mathbf{0}, ) is A ( cdot frac{3}{7} ) B. ( frac{3}{49} ) ( c cdot frac{4}{7} ) D. ( frac{40}{49} ) |
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| 204 | Find the equation of the plane in scalar product form ( hat{r}=(2 hat{i}+hat{k})+lambda hat{i}+ ) ( mu(hat{i}+2 hat{j}-3 hat{k}) ) |
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| 205 | If ( boldsymbol{A}=(mathbf{1}, mathbf{2}, mathbf{3}), boldsymbol{B}=(mathbf{2}, mathbf{1 0}, mathbf{1}), boldsymbol{Q} ) are collinear points and ( Q_{x}=-1, ) then ( boldsymbol{Q}_{z}= ) A . -3 B. 7 ( c cdot-14 ) D. – |
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| 206 | The extremities of a diagonal of a rectangular parallelopiped whose faces are parallel to the reference planes are (-2,4,6) and ( (3,16,6) . ) The length of the base diagonal is A . 13 B. ( sqrt{13} ) c. ( 2 sqrt{13} ) D. 169 |
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| 207 | The coordinates of the points in which the line joining the points (2,5,-7) and (-3,-1,8) are intersected by the ( y-z ) plane are A ( cdotleft(0, frac{13}{5},-1right) ) B ( cdotleft(0, frac{-13}{5},-2right) ) D. ( left(0, frac{13}{5}, frac{2}{5}right) ) |
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| 208 | Find the vector and cartesian equations of the plane passing through the points ( A(1,1,-2), B(1,2,1) ) and ( C(2,-1,1) ) | 12 |
| 209 | The projections of a directed line segment on the coordinate axes 12,4,3 The direction cosines of the line are A ( cdot frac{12}{13},-frac{4}{13}, frac{3}{13} ) B. ( -frac{12}{13},-frac{4}{13}, frac{3}{13} ) c. ( frac{12}{13}, frac{4}{13}, frac{3}{13} ) D. none of these |
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| 210 | The direction cosines to two lines at right angles are (1,2,3) and ( left(-2, frac{1}{2}, frac{1}{3}right), ) then the direction cosine perpendicular to both given lines are: A ( cdot sqrt{frac{25}{219}} cdot sqrt{frac{19}{2198}} cdot sqrt{frac{729}{21988}} ) B. ( sqrt{frac{24}{2198} cdot sqrt{frac{38}{2198}}} sqrt{frac{730}{21988}} ) c. ( frac{1}{3},-2, frac{-7}{2} ) D. None of the above |
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| 211 | If the points whose position vectors are ( mathbf{2} overline{mathbf{i}}+overline{boldsymbol{j}}+overline{boldsymbol{k}}, mathbf{6} overline{mathbf{i}}-overline{boldsymbol{j}}+mathbf{2} overline{mathbf{k}} ) and ( mathbf{1 4} overline{mathbf{i}}-mathbf{5} overline{mathbf{j}}+ ) ( p bar{k} ) are collinear then the value of ( mathbf{p} ) is ( A cdot 2 ) B. 4 ( c cdot 6 ) D. 8 |
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| 212 | A point ( P ) lies on the line whose end points are ( boldsymbol{A}(mathbf{1}, mathbf{2}, mathbf{3}) ) and ( boldsymbol{B}(mathbf{2}, mathbf{1 0}, mathbf{1}) ) If ( z ) -co-ordinate of ( P ) is ( 7, ) find sum of its other co – ordinates. |
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| 213 | ( A=(2,4,5) ) and ( B=(3,5,-4) ) are two points. If the ( x y ) -plane, ( y z ) -plane divide ( A B ) in the ratios ( a: b, p: q ) respectively then ( frac{a}{b}+frac{p}{q}= ) A. ( frac{7}{15} ) в. ( frac{-7}{12} ) c. ( frac{7}{12} ) D. ( frac{22}{25} ) |
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| 214 | The ratio in which the line joining points (2,4,5) and (3,5,-4) divide YZ -plane is ( mathbf{A} cdot-2: 3 ) B. 2: 3 c. -3: 2 D. 3: 2 |
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| 215 | On a plane are two points ( A ) and ( B ) at a distance of 5 units apart. The number of straight lines in this plane which are at distance of 2 units from ( A ) and 3 units from ( mathrm{B} ) are: A . 1 B. 2 ( c .3 ) D. 4 |
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| 216 | Find the equation of the plane passing through the point (1,-2,1) and perpendicular to the line joining the points ( boldsymbol{A}(mathbf{3}, mathbf{2}, mathbf{1}) ) and ( boldsymbol{B}(mathbf{1}, mathbf{4}, mathbf{2}) ) | 12 |
| 217 | Find the shortest distance between the line ( frac{x-3}{3}=frac{y-8}{-1}=frac{z-3}{1} ) and the line of intersection of the planes ( 2 x+ ) ( 5 y-z+47=0 ) and ( 2 x+y+z+7= ) ( mathbf{0} ) |
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| 218 | The projection of a directed line segment on the co-ordinate axes are ( 12,4,3, ) the DC’s of the line are A ( cdot frac{-12}{13}, frac{-4}{13}, frac{-3}{13} ) B. ( frac{12}{13}, frac{4}{13}, frac{3}{13} ) c. ( frac{12}{13}, frac{-4}{13}, frac{3}{13} ) D. ( frac{12}{13}, frac{4}{13}, frac{-3}{13} ) |
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| 219 | The plane ( 2 x-(1+lambda) y+3 z=0 ) passes through the intersection of the planes A. ( 2 x-y=0 ) and ( y+3 z=0 ) в. ( 2 x-y=0 ) and ( y-3 z=0 ) c. ( 2 x+3 z=0 ) and ( y=0 ) D. None of the above |
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| 220 | If ( x y ) -plane and ( y z ) -plane divides the line segment joining ( A(2,4,5) ) and ( B(3,5,-4) ) in the ratio a:b and p:q respectively then value of ( left(frac{a}{b}, frac{p}{q}right) ) may be A ( cdot frac{23}{12} ) B. ( frac{7}{5} ) ( c cdot frac{7}{12} ) D. ( frac{21}{10} ) |
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| 221 | The vector equation of the plane through the point (1,-2,-3) and parallel to the vectors (2,-1,3) and (2,3,-6) is ( bar{r}= ) A ( cdot(1+2 t+2 s) bar{i}-(2+t-3 s) bar{j}-(3-3 t+6 s) bar{k} ) B. ( (1+2 t+2 s) bar{i}+(2+t+3 s) bar{j}-(3+3 t+6 s) bar{k} ) c. ( (1+2 t+2 s) bar{i}+(2+t+3 s) bar{j}+(3+3 t+6 s) bar{k} ) D. ( (1+2 t+2 s) bar{i}+(2+t-3 s) bar{j}+(3+3 t+6 s) bar{k} ) |
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| 222 | If ( O ) is origin ( O P=3 ) with direction ratios proportional to -1,2,-2 then what are the coordinates of ( P ? ) |
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| 223 | Direction ratios of the line which is perpendicular to the lines with direction ratios -1,2,2 and 0,2,1 are ( mathbf{A} cdot 1,1,2 ) B. 2,-1,2 c. -2,1,2 D. 2,1,-2 |
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| 224 | ( boldsymbol{A}=(mathbf{1},-mathbf{2}, mathbf{3}), boldsymbol{B}=(2,1,3), boldsymbol{C}=(4,2, ) 1) and ( G=(-1,3,5) ) is the centroid of the tetrahedron ( A B C D . ) Then the fourth coordinate is A. (11,11,13) В. (-11,11,45) c. (-11,11,13) D. (11,13,11) |
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| 225 | A line d.c’s proportional to (2,1,2) meets each of the lines ( boldsymbol{x}=boldsymbol{y}+boldsymbol{a}=boldsymbol{z} ) and ( x+a=2 y=2 z . ) Then the coordinates of each of the points of intersection are given by A . ( (3 a, 2 a, 3 a) ;(a, a, 2 a) ) в. ( (3 a, 2 a, 3 a) ;(a, a, a) ) C. ( (3 a, 3 a, 3 a) ;(a, a, a) ) D. ( (2 a, 3 a, 3 a) ;(2 a, a, a) ) |
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| 226 | If the plane a ( 2 x-3 y+5 z-2=0 ) divides the line segment joining (1,2,3) and ( (2,1, k) ) in the ratio 9: 11 then ( k ) is A . в. -2 c. -10 D. ( -frac{1}{2} ) |
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| 227 | 37. A line with positive direction cosines passes through point P(2,-1,2) and makes equal angles with the coordinata axes. The line meets the plane 2x+y+z=9 at point Q. The length of the line segment PQ equals (2009) (a) 1 (b) √ (c) √3 (d) 2 |
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| 228 | The distance of origin from the image of (1,2,3) in plane ( x-y+z=5 ) is A ( cdot sqrt{17} ) B. ( sqrt{29} ) c. ( sqrt{34} ) D. ( sqrt{41} ) |
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| 229 | If ( |vec{A} times vec{B}|=sqrt{3} vec{A} cdot vec{B} ) then the value of ( |vec{A}+vec{B}| ) is: ( ^{A} cdotleft(A^{2}+B^{2}+frac{A B}{sqrt{3}}right)^{1 / 2} ) в. ( A+B ) c. ( left(A^{2}+B^{2}+sqrt{3} A Bright)^{1 / 2} ) D. ( left(A^{2}+B^{2}+A Bright)^{1 / 2} ) |
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| 230 | Find the vector equation of the line through ( A(3,4,-7) ) and ( B(6,-1,1) ) | 12 |
| 231 | If the orthocentre, circumcentre of a triangle are (-3,5,2),(6,2,5) respectively then the centroid of the triangle is ( mathbf{A} cdot(3,3,4) ) В. ( left(frac{3}{2}, frac{7}{2}, frac{9}{2}right) ) c. (9,9,12) D. ( left(frac{9}{2} frac{-3}{2}, frac{3}{2}right) ) |
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| 232 | The Cartesian equation of line ( 6 x- ) ( mathbf{2}=mathbf{3} boldsymbol{y}+mathbf{1}=mathbf{2} z-mathbf{2} ) is given by ( ^{text {A } cdot frac{3 x-1}{3}}=frac{3 y+1}{6}=frac{z-1}{3} ) B. ( frac{3 x+1}{3}=frac{3 y-1}{6}=frac{z-1}{3} ) c. ( frac{3 x-1}{3}=frac{3 y-1}{6}=frac{z-1}{3} ) D. ( frac{3 x-1}{6}=frac{3 y-1}{3}=frac{z-1}{3} ) |
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| 233 | The direction ratios of the line ( boldsymbol{x}-boldsymbol{y}+ ) ( z-5=0=x-3 y-6 ) are A. 3,1,-2 в. 2,-4,1 c. ( frac{3}{sqrt{14}}, frac{1}{sqrt{14}}, frac{-2}{sqrt{14}} ) D. ( frac{2}{sqrt{14}}, frac{-4}{sqrt{14}}, frac{1}{sqrt{14}} ) |
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| 234 | A plane is at a distance of 5 units from the origin and perpendicular to the vector ( 2 hat{i}+hat{j}+2 hat{k} . ) The equation of the plane is A ( . vec{r} .(2 hat{imath}+hat{j}-2 hat{k})=15 ) в. ( vec{r} .(2 hat{i}+hat{j}-hat{k})=15 ) c. ( vec{r} .(2 hat{i}+hat{j}+2 hat{k})=15 ) D. ( vec{r} .(hat{i}+hat{j}+2 hat{k})=15 ) E ( . vec{r} .(2 hat{i}-hat{j}+2 hat{k})=15 ) |
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| 235 | Find vector equation for the line passing through the points ( 3 bar{i}+4 bar{j}- ) ( mathbf{7} bar{k}, overline{boldsymbol{i}}-overline{boldsymbol{j}}+mathbf{6} overline{boldsymbol{k}} ) A ( . bar{r}=(3-2 lambda) bar{i}+(4-5 lambda) bar{j}+(-7+13 lambda) bar{k} ) B. ( bar{r}=(2 lambda) bar{i}+(4+5 lambda) bar{j}+(-7-13 lambda) bar{k} ) c. ( bar{r}=(3-2 lambda) bar{i}-(4-5 lambda) bar{j}+(-7+13 lambda) bar{k} ) D. ( bar{r}=(3-2 lambda) bar{i}+(4-5 lambda) bar{j}-(-7+13 lambda) bar{k} ) |
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| 236 | (-1,-5,-7) lies in Octant ( A ) B. VII ( c cdot v ) D. II |
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| 237 | If ( boldsymbol{A}(cos boldsymbol{alpha}, sin boldsymbol{alpha}, boldsymbol{0}), boldsymbol{B}(cos boldsymbol{beta}, sin beta, boldsymbol{0}) ) ( C(cos gamma, sin gamma, 0) ) are vertices of ( Delta A B C ) and let [ begin{array}{l} cos alpha+cos beta+cos gamma=3 a, sin alpha+ \ sin beta+sin gamma=3 b, text { then correct } end{array} ] matching of the following is: List:1 begin{tabular}{ll} A. Circumcentre & ( 1 .(3 a, 3 b, 0) ) \ cline { 0 } end{tabular} B. Centroid [ 2 .(0,0,0) ] c. Ortho centre [ 3 .(a, b, 0) ] A . 432 в. 231 c. 123 D. 234 |
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| 238 | If ( bar{a}, bar{b} ) are the position vectors of ( A ) and ( B ) then one of the following points lie on ( A B ) A ( cdot frac{2(bar{a}+bar{b})}{3} ) в. ( frac{(bar{a}-bar{b})}{3} ) c. ( frac{(bar{a}+bar{b})}{3} ) ( ^{mathrm{D}} cdot frac{2 bar{a}+2 bar{b}}{3} ) E. None of these |
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| 239 | The perimeter of triangle with vertices at (1,0,0),(0,1,0) and (0,0,1) is : ( mathbf{A} cdot mathbf{3} ) B . 2 ( c cdot 2 sqrt{2} ) D. ( 3 sqrt{2} ) |
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| 240 | Show that the points are collinear (1) ( boldsymbol{A}(mathbf{3}, mathbf{2},-mathbf{4}), boldsymbol{B}(mathbf{9}, mathbf{8},-mathbf{1 0}), boldsymbol{C}(-mathbf{2},-mathbf{3}, mathbf{1}) ) |
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| 241 | The points ( A ) and ( B ) have co-ordinate (1,2,4) and (-1,3,5) respectively. Find ( A B ) and its magnitude. | 12 |
| 242 | Prove the ( boldsymbol{A}(-mathbf{5}, mathbf{4}), boldsymbol{B}(-mathbf{1},-mathbf{2}), boldsymbol{C}(mathbf{5}, mathbf{2}) ) are the vertices of ( n ) isosceles right angled triangle? |
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| 243 | If ( P(x, y, z) ) is a point on the line segment joining ( A(2,2,4) ) and ( B(3,5,6) ) such that projection of ( overline{O P} ) on axes are ( frac{13}{5}, frac{19}{5}, frac{26}{5} ) respectively, then ( P ) divide ( A B ) in the ratio A .3: 2 B. 2: 3 c. 1: 2 D. 1: 3 |
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| 244 | The direction cosine of a line equally inclined to the axes are A ( cdot frac{1}{3}, frac{1}{3}, frac{1}{3} ) B. ( -frac{1}{3},-frac{1}{3},-frac{1}{3} ) c. ( frac{1}{sqrt{3}}, frac{1}{sqrt{3}}, frac{1}{sqrt{3}} ) D. none of these |
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| 245 | If ( l, m, n ) are d.c’s of vector ( overline{O P} ) then maximum value of ( l ) mn is A ( cdot frac{1}{sqrt{3}} ) B. ( frac{1}{2 sqrt{3}} ) c. ( frac{1}{3 sqrt{3}} ) D. ( frac{2}{sqrt{3}} ) |
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| 246 | The lines ( frac{x-2}{1}=frac{y-3}{1}=frac{z-4}{-k} & ) ( frac{boldsymbol{x}-mathbf{1}}{boldsymbol{k}}=frac{boldsymbol{y}-boldsymbol{4}}{boldsymbol{2}}=frac{boldsymbol{z}-mathbf{5}}{mathbf{1}} ) are coplanar if A ( . k=0 ) or -1 B. ( k=1 ) or -1 c. ( k=0 ) or -3 D. ( k=3 ) or -3 |
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| 247 | If ( A B perp B C, ) then the value of ( lambda ) equal where ( boldsymbol{A}(mathbf{2 k}, mathbf{2}, mathbf{3}), boldsymbol{B}(boldsymbol{k}, mathbf{1}, mathbf{5}), boldsymbol{C}(boldsymbol{3}+ ) ( k, 2,1) ) ( mathbf{A} cdot mathbf{3} ) B. ( c .-3 ) D. ( -frac{1}{3} ) |
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| 248 | The equation of altitude through ( B ) to side ( A C ) is A. ( r=k+t(7 i-10+2 k) ) в. ( r=k+t(-9 i+6 j-2 k) ) c. ( r=k+t(7 i-10 j-2 k) ) D. ( r=k+t(7 i+10 j+2 k) ) |
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| 249 | Equation of the plane passing through the point of intersection of ( x+2 y+=0 ) and ( 2 x+y=0 ) and which is perpendicular to ( 5 x+6 z=0 ) This question has multiple correct options A. ( x=0 ) B. ( y=0 ) ( c cdot z=0 ) D. x+y=0 E. ( x+z=0 ) |
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| 250 | Solve: ( sqrt{mathbf{2}} boldsymbol{x}+sqrt{mathbf{3}} boldsymbol{y}=mathbf{0} ) ( sqrt{mathbf{3}} boldsymbol{x}-sqrt{mathbf{8}} boldsymbol{y}=mathbf{0} ) |
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| 251 | Find the vector equation of the plane through the points (2,1,-1) and (-1,3,4) and perpendicular to the plane ( boldsymbol{x}-mathbf{2} boldsymbol{y}+mathbf{4} boldsymbol{z}=mathbf{1 0} ) |
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| 252 | Write the direction cosines of the line whose cartesian equations are ( 2 x= ) ( mathbf{3} boldsymbol{y}=-boldsymbol{z} ) |
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| 253 | If the angles made by a straight line with the coordinate axes are ( alpha, frac{pi}{2}- ) ( boldsymbol{alpha}, boldsymbol{beta} ) then ( boldsymbol{beta}= ) A . 0 в. ( frac{pi}{6} ) c. ( frac{pi}{2} ) ( D ) |
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| 254 | Find the coordinates of the point which divides the line segment joining the points (-2,3,5) and (1,-4,6) in the ratio (i) 2: 3 internally (ii) 2: 3 externally |
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| 255 | If the distance between a point ( boldsymbol{P} ) and the point (1,1,1) on the line ( frac{x-1}{3}= ) ( frac{boldsymbol{y}-mathbf{1}}{mathbf{4}}=frac{boldsymbol{z}-mathbf{1}}{mathbf{1 2}} ) is ( mathbf{1 3}, ) then the coordinates of ( boldsymbol{P} ) are A. (3,4,12) в. ( left(frac{3}{13}, frac{4}{13}, frac{12}{13}right) ) c. (4,5,13) D. (40,53,157) |
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| 256 | If a line makes angles ( boldsymbol{alpha}, boldsymbol{beta}, boldsymbol{gamma} ) with positive directions of ( mathrm{X}, mathrm{Y}, mathrm{Z} ) -axes, what is the value of ( sin ^{2} alpha+sin ^{2} beta+sin ^{2} gamma ) ( ? ) |
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| 257 | ( l=m=n=1 ) represents the direction cosines of A. ( x ) -axis B. ( y ) -axis c. ( z ) -axis D. none of these |
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| 258 | The coordinates of the foot of the perpendicular drawn from of the origin to a plane are ( (12,-4,3) . ) Find the equation of the plane. |
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| 259 | What is the angle between ( vec{P} times vec{Q} ) and ( vec{P}+vec{Q} ? ) ( mathbf{A} cdot mathbf{0} ) в. ( frac{pi}{2} ) ( c . pi ) D. ( frac{3 pi}{2} ) |
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| 260 | If 0 is the origin and the coordinates of ( P ) is ( (1,2,-3), ) then find the equation of the plane passing through P and perpendicular to OP. A. ( x-2 y-3 z=-15 ) в. ( x+2 y-3 z=14 ) c. ( x-2 y+3 z=15 ) D. ( x-2 y-3 z=15 ) |
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| 261 | If ( 4 x+4 y-k z=0 ) is the equation of the plane through the origin that contains the line ( frac{boldsymbol{x}-mathbf{1}}{mathbf{2}}=frac{boldsymbol{y}+mathbf{1}}{mathbf{3}}=frac{boldsymbol{z}}{mathbf{4}} ) then ( boldsymbol{k}= ) A . 1 B. 3 ( c .5 ) D. |
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| 262 | The sum of the intercepts on the coordinate axes of the plane passing through the point (-2,-2,2) and containing the line joining the points (1,-1,2) and (1,1,1) is? A . 12 B. -8 ( c .-4 ) D. 4 |
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| 263 | If a line makes angles ( boldsymbol{alpha}, boldsymbol{beta}, boldsymbol{gamma} ) with positive axes, then the range of ( sin alpha sin beta+sin beta sin gamma+sin gamma sin alpha ) is ( ^{mathrm{A}} cdotleft(frac{-1}{2}, 1right) ) в. ( left(frac{1}{2}, 2right) ) c. (-1,2) D. (-1,2] |
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| 264 | Let the line ( frac{boldsymbol{x}-mathbf{2}}{mathbf{3}}=frac{boldsymbol{y}-mathbf{1}}{-mathbf{5}}=frac{boldsymbol{z}+mathbf{2}}{mathbf{2}} ) lie in the plane ( x+3 y-alpha z+beta=0 . ) Then ( (alpha, beta) ) equals: A ( cdot(-6,7) ) B ( cdot(5,-15) ) c. (-5,5) D. (6,-17) |
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| 265 | Find the ratio in which the ( X Y ) – plane divides ( A B ) if is (1,2,3) and ( B ) is (-3,4,-5) Also find the positive vector of the point of division. |
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| 266 | Find the measure of the angle between two lines if their direction cosines ( ell, boldsymbol{m}, boldsymbol{n} ) satisfy ( ell+boldsymbol{m}-boldsymbol{n}=mathbf{0}, ell^{2}+ ) ( boldsymbol{m}^{2}-boldsymbol{n}^{2}=mathbf{0} ) |
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| 267 | What are the DR’s of vector parallel to (2,-1,1) and (3,4,-1)( ? ) A ( cdot(1,5,-2) ) B ( cdot(-2,-5,2) ) ( mathbf{c} cdot(-1,5,2) ) D ( cdot(-1,-5,-2) ) |
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| 268 | The equation of motion of a rocket are: ( boldsymbol{x}=mathbf{2} boldsymbol{t}, boldsymbol{y}=-boldsymbol{4} boldsymbol{t}, boldsymbol{z}=boldsymbol{4} boldsymbol{t}, ) where the time ( t ) is given in seconds and the coordinate of a moving point in kilometers. At what distance will the rocket be from the starting point ( O(0,0,0) ) in 10 seconds? ( mathbf{A} cdot 60 mathrm{km} ) B. ( 30 mathrm{km} ) c. ( 45 mathrm{km} ) D. None of these |
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| 269 | Point ( D ) has coordinates as (3,4,5) Referring to the given figure, find the coordinates of point ( boldsymbol{E} ) ( mathbf{B} cdot(0,4,5) ) C. (0,5,4) D. (0,3,4) |
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| 270 | The point (0,-2,5) lies on the ( A cdot z ) axis B. x axis c. xy plane D. yz plane E . xz plane |
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| 271 | ( left(cos ^{-1} l+cos ^{-1} m+cos ^{-1} nright) ) is equal to A . ( 90^{circ} ) B. ( 50^{circ} ) ( c cdot 180^{circ} ) D. None of these |
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| 272 | Find the distance of the point (2,3,5) from the ( x y- ) plane | 12 |
| 273 | Find the equation of the plane passing through the point (2,-1,1) and through the line of intersection of the planes ( vec{r} ). ( (2 hat{i}-3 hat{j}+hat{k})=3 ) and ( vec{r} cdot(hat{i}+5 hat{j}- ) ( hat{boldsymbol{k}})=mathbf{4 . 4} ) |
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| 274 | Find the direction cosines of the line: ( frac{x-1}{2}=-y=frac{z+1}{2} ) |
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| 275 | Find the equation to the plane through the point (-1,3,2) and perpendicular to the planes ( x+2 y+2 z=11 ) and ( 3 x+3 y+2 z=15 ) |
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| 276 | figure bounded by non coplanar triangular planes. So, a tetrahedron has four non-coplanar points as its vertices. Suppose a tetrehedron has points A,B,C,D as its vertices which have coordinates ( left(boldsymbol{x} mathbf{1}, boldsymbol{y} mathbf{1}, boldsymbol{z}_{1}right)left(boldsymbol{x}_{2}, boldsymbol{y}_{2}, boldsymbol{z}_{2}right) ) ( left(x_{3}, y_{3}, z_{3}right) ) and ( left(x_{4}, y_{4}, z_{4}right), ) respectively in a rectangular three dimensional space. Then, the coordinates of its centroid are ( left[frac{boldsymbol{x}_{1}+boldsymbol{x}_{2}+boldsymbol{x}_{3}+boldsymbol{x}_{4}}{boldsymbol{4}}, frac{boldsymbol{y}_{1}+boldsymbol{y}_{2}+boldsymbol{y}_{3}+boldsymbol{y}_{4}}{boldsymbol{4}}right. ) Let a tetrahedron have three of its vertices represented by the points (0,0,0),(6,5,1) and (4,1,3) and its centroid lies at the point (1,2,5) Now, answer the following question. The coordinate of the fourth vertex of the tetrahedron is: A ( cdot(-6,2,16) ) B. (1,-2,13) c. (-2,4,-2) D. (1,-1,1) |
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| 277 | The distance of the point ( 3 hat{i}+5 hat{k} ) from the line parallel to the vector ( 6 hat{i}+hat{j}- ) ( 2 hat{k} ) and passing through the point ( 8 hat{i}+ ) ( mathbf{3} hat{boldsymbol{j}}+hat{boldsymbol{k}} ) is A . 1 B . 2 ( c .3 ) D. |
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| 278 | If ( z=cos frac{pi}{6}+i sin frac{pi}{6}, ) then A ( cdot|z|=1, arg z=frac{pi}{4} ) в ( cdot|z|=1, arg z=frac{pi}{6} ) с. ( quad|z|=frac{sqrt{3}}{2}, arg z=frac{5 pi}{24} ) D. ( quad|z|=frac{sqrt{3}}{2}, arg z=tan ^{-1} frac{1}{sqrt{2}} ) |
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| 279 | The vertices of a triangle are ( (2,3,5),(-1,3,2),(3,5,-2), ) then the angles are ( ^{mathrm{B}} cos ^{-1}left(frac{1}{sqrt{5}}right), 90^{circ}, cos ^{-1}left(frac{sqrt{5}}{sqrt{3}}right) ) c. ( 30^{circ}, 60^{circ}, 90^{circ} ) D ( cdot cos ^{-1}left(frac{1}{sqrt{3}}right), 90^{circ}, cos ^{-1}(sqrt{frac{2}{3}}) ) |
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| 280 | An equation of a plane containing the lines ( r=a_{1}+t b_{1} ) and ( r=a_{2}+t b_{2} ) where ( left[boldsymbol{a}_{2}-boldsymbol{a}_{1}, boldsymbol{b}_{1}, boldsymbol{b}_{2}right]=boldsymbol{0} ) is A ( cdotleft[r-a_{1}, b_{1}, b_{2}right]=0 ) B . ( left[r-a_{2}, b_{1}, b_{2}right]=0 ) c. ( left[r-a_{2}, a_{1}, b_{2}right]=0 ) D . ( left[r-a, a_{2}, b_{2}right]=0 ) |
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| 281 | f ( A(2,1,3), B(2,-3,4), C(-1,2,7) ) are the three points ; find a point D such that ( |overrightarrow{A B}|=frac{1}{2}|overrightarrow{C D}| ) and ( A B|| C D ) |
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| 282 | The distance between the line ( r=2 hat{i}- ) ( 2 hat{j}+3 hat{k}+lambda(hat{i}-hat{j}+4 hat{k}) ) and the plane ( r cdot(hat{i}+5 hat{j}+hat{k})=5 ) is A ( cdot frac{10}{9} ) в. ( frac{10}{3 sqrt{3}} ) c. ( frac{10}{3} ) D. None of these |
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| 283 | The name of the figure formed by the points (-1,-3,4),(5,-1,1),(7,-4,7) and (1,-6,10) is a A. square B. rhombus c. parallelogram D. rectangle |
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| 284 | The ratio in which the plane ( 2 x+3 y- ) ( mathbf{2} z+mathbf{7}=mathbf{0} ) divides the line segment joining the points (-1,1,3),(2,3,5) is A .3: 5 B. 7: 5 ( mathbf{c} cdot 9: 11 ) D. 1: 5 externally |
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| 285 | State the following statement is True or False If two distinct lines are intersecting each other in a plane then they cannot have more than one point in common. A. True B. False |
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| 286 | The direction cosines of the ray ( boldsymbol{P}(1,-2,4) ) and ( Q(-1,1,-2) ) are A ( cdot(-2,-3,-6) ) B ( cdot(2,-3,-6) ) ( ^{C} cdotleft(frac{2}{7}, frac{3}{7}, frac{6}{7}right) ) D ( cdotleft(-frac{2}{7}, frac{3}{7},-frac{6}{7}right) ) |
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| 287 | Find the equation of the plane through the line ( boldsymbol{P}=boldsymbol{a} boldsymbol{x}+boldsymbol{b} boldsymbol{y}+boldsymbol{c} boldsymbol{z}+boldsymbol{d}=mathbf{0} ) ( boldsymbol{Q}=boldsymbol{a}^{prime} boldsymbol{x}+boldsymbol{b}^{prime} boldsymbol{y}+boldsymbol{c}^{prime} boldsymbol{z}+boldsymbol{d}^{prime}=mathbf{0} ) and parallel to the line ( frac{x}{l}=frac{y}{m}=frac{z}{n} ) A ( cdot Pleft(a^{prime} l+b^{prime} m+c^{prime} nright)+Q(a l+b m+c n)=0 ) B – ( Pleft(a^{prime} l+b^{prime} m+c^{prime} nright)-Q(a l+b m+c n)=0 ) c. ( Qleft(a^{prime} l+b^{prime} m+c^{prime} nright)-P(a l+b m+c n)=0 ) D. ( Qleft(a^{prime} l+b^{prime} m+c^{prime} nright)+P(a l+b m+c n)=0 ) |
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| 288 | 69. t-3 y = and If the line X-1 y +1 Z-1 1 2 3 4 intersect, then k is equal to: [2012] (a) -1 |
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| 289 | find the equation of a line passing through the point (1,2,-4) and perpendicular to two lines. ( vec{r}= ) ( (8 hat{i}-19 hat{j}+10 hat{k})+ ) ( boldsymbol{lambda}(boldsymbol{3} hat{boldsymbol{i}}-boldsymbol{1 6} hat{boldsymbol{j}}+boldsymbol{7} hat{boldsymbol{k}}) ) and ( overrightarrow{boldsymbol{r}}= ) ( (15 hat{i}+29 hat{j}+5 hat{k})+ ) ( boldsymbol{mu}(mathbf{3} hat{boldsymbol{i}}+quad boldsymbol{8} hat{boldsymbol{j}}-boldsymbol{5} hat{boldsymbol{k}}) ) |
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| 290 | The line passing through the points ( (5,1, a) ) and ( (3, b, 1) ) crosses the ( y z ) plane at the point ( left(0, frac{17}{2}, frac{-13}{2}right) . ) Then A ( cdot a=2, b=8 ) В. ( a=4, b=6 ) c. ( a=6, b=4 ) D. ( a=8, b=2 ) 2 |
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| 291 | ff ( y ) varies directly as ( x ) and ( y=12 ) when ( x=4, ) then find the linear equation. |
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| 292 | Cartesian equation of a line is ( frac{x-5}{3}= ) ( frac{boldsymbol{y}+boldsymbol{4}}{mathbf{7}}=frac{boldsymbol{z}-boldsymbol{6}}{boldsymbol{2}} . ) Write it in vector form | 12 |
| 293 | Name three undefined terms. A. Point B. Line c. Plane D. All of the above |
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| 294 | Find a unit vector normal to the plane through the points (1,1,1),(-1,2,3) and (2,-1,3) |
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| 295 | If the line joining the points (-1,2,3),(2,-1,4) is perpendicular to the line joining the points ( (x,-2,4),(1,2,3) ) then ( x= ) ( mathbf{A} cdot mathbf{3} ) B. 10 ( c cdot frac{-3}{10} ) D. ( frac{-10}{3} ) |
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| 296 | The equation of the plane passing through the straight line ( frac{x-1}{2}= ) ( frac{boldsymbol{y}+mathbf{1}}{mathbf{- 1}}=frac{boldsymbol{z}-mathbf{3}}{mathbf{4}} ) and perpendicular to the plane ( boldsymbol{x}+mathbf{2} boldsymbol{y}+boldsymbol{z}=mathbf{1 2} ) is A. ( 9 x+2 y-5 z+4=0 ) В. ( 9 x-2 y-5 z+4=0 ) c. ( 9 x+2 y+5 z+4=0 ) D. None of these |
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| 297 | 92. x-1 y+1 z-2 If the line, meets the plane, x+2y+ 2 3 4 3z= 15 at a point P, then the distance of P from the origin in [JEEM 2019-9 April (M) (a) V5/2 (b) 215 (c) 9/2 (d) 7/2 |
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| 298 | Let ( boldsymbol{O} ) be the origin and ( boldsymbol{A} ) be the point ( (64,0) . ) If ( P ) and ( Q ) divide ( O A ) in the ratio ( mathbf{1}: mathbf{2}: mathbf{3}, ) then the point ( boldsymbol{P} ) is ( ^{mathbf{A}} cdotleft(frac{32}{3}, 0right) ) в. (32,0) ( ^{mathbf{c}} cdotleft(frac{64}{3}, 0right) ) D. (16,0) E ( cdotleft(frac{16}{3}, 0right) ) |
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| 299 | Calculate the distance between the points (-3,6,7) and (2,-1,4) in ( 3 D ) space. A . 4.36 в. 5.92 c. 7.91 D. 9.11 E . 22.25 |
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| 300 | Find the equation of the plane which passes through the point (3,2,0) and contains the line ( frac{boldsymbol{x}-mathbf{3}}{mathbf{1}}=frac{boldsymbol{y}-mathbf{6}}{mathbf{5}}= ) ( frac{z-4}{4} ? ) |
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| 301 | The direction ratios of the line, given by the planes ( x-y+z-5=0, x-3 y-6=0 ) are A ( .(3,1,-2) ) в. (2,-4,1) c. (1,-1,1) D. (0,2,1) |
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| 302 | Using section formula prove that the point ( (0.7 .-7),(1,4,-5) ) and (-1,10,-9) are collinear. |
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| 303 | Find the vector equation of the line joining points ( 2 hat{i}+hat{j}+3 hat{k} ) and ( -4 hat{i}+ ) ( mathbf{3} hat{boldsymbol{j}}-hat{boldsymbol{k}} ) | 12 |
| 304 | Assertion The point ( boldsymbol{A}(boldsymbol{3}, boldsymbol{1}, boldsymbol{6}) ) is the mirror image of the point ( B(1,3,4) ) in the plane ( x- ) ( boldsymbol{y}+boldsymbol{z}=mathbf{5} ) Reason The plane ( x-y+z=5 ) bisects the line segment joining ( boldsymbol{A}(mathbf{3}, mathbf{1}, boldsymbol{6}) ) and ( boldsymbol{B}(mathbf{1}, boldsymbol{3}, boldsymbol{4}) ) A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion c. Assertion is correct but Reason is incorrect D. Both Assertion and Reason are incorrect |
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| 305 | Show that the plane whose vector equation is ( vec{r} .(hat{i}+2 hat{j}-hat{k})=3 ) contains the line ( vec{r}=hat{i}+hat{j}+lambda(2 hat{i}+hat{j}+4 hat{k}) ) | 12 |
| 306 | A unit vector parallel to the intersection of the planes ( vec{r} cdot(hat{i}-hat{j}+hat{k})=5 ) and ( vec{r} ) ( (2 hat{i}+hat{j}-3 hat{k})=4 ) can be This question has multiple correct options ( frac{2 hat{i}+5 hat{j}+3 hat{k}}{sqrt{38}} ) B. ( frac{2 hat{i}-5 hat{j}+3 hat{k}}{sqrt{38}} ) c. ( frac{-2 hat{i}-5 hat{j}-3 hat{k}}{sqrt{38}} ) D. ( frac{-2 hat{i}+5 hat{j}-3 hat{k}}{sqrt{38}} ) |
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| 307 | The length of the perpendicular from the vertex ( D ) on the opposite face is A ( cdot frac{14}{sqrt{6}} ) в. ( frac{2}{sqrt{6}} ) c. ( frac{3}{sqrt{6}} ) D. none of these |
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| 308 | The distance of the point (1,-2,3) from the plane ( boldsymbol{x}-boldsymbol{y}+boldsymbol{z}=mathbf{5} ) measured parallel to the line ( frac{x}{2}=frac{y}{3}=frac{z-1}{-6} ) is A . B . 2 ( c cdot 4 ) D. none of these |
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| 309 | The circum radius of the triangle formed by the points (0,0,0),(0,0,12) and (3,4,0) is A. ( sqrt{156} ) B. 13 c. ( frac{13}{2} ) D. |
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| 310 | What is the equation of the plane containing the parallel lines ( boldsymbol{r}=boldsymbol{a}+boldsymbol{t c}, boldsymbol{r}=boldsymbol{b}+boldsymbol{p} boldsymbol{c} ) ( mathbf{A} cdot r cdot(b-a) times c=[a b c] ) B cdot ( r cdot(a-b) times c=[a b c] ) C ( cdot r cdot(b-a) times c=-[a b c] ) D ( cdot r c times(b-a) times c=[a b c] ) |
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| 311 | The direction cosines of the line joining the points (2,3,-1) and (3,-2,1) are в. ( frac{1}{sqrt{30}},-sqrt{frac{5}{6}} cdot sqrt{frac{2}{15}} ) c. ( frac{-1}{30}, frac{1}{6},-frac{1}{15} ) D. none of these |
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| 312 | The ratio in which the line joining (3,4,-7) and (4,2,1) is dividing the ( x y ) plane A .3: 4 B . 2: 1 ( c cdot 7: 1 ) D. 4: 3 |
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| 313 | ( G(1,1,-2) ) is the centroid of the triangle ( A B C ) and ( D ) is the mid point of ( boldsymbol{B C} cdot ) If ( boldsymbol{A}=(-1,1,-4), ) then ( boldsymbol{D}= ) ( ^{mathbf{A}} cdotleft(frac{1}{2}, 1, frac{-5}{2}right) ) в. (5,1,2) c. (-5,-1,-2) D. (2,1,-1) |
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| 314 | Let ( S ) be the set of all real values of ( lambda ) such that a plane passing through the points ( left(-lambda^{2}, 1,1right),left(1,-lambda^{2}, 1right) ) and ( left(1,1,-lambda^{2}right) ) also passes through the point ( (-1,-1,1) . ) Then ( S ) is equal to: A ( cdot(sqrt{3}) ) B. ( {sqrt{3}-sqrt{3}} ) begin{tabular}{l} c. {1,-1} \ hline end{tabular} D. {3,-3} |
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| 315 | Find the Cartesian equation of ( vec{r} ) ( (2 hat{i}+3 hat{j}-4 hat{k})=1 ) | 12 |
| 316 | A point on XOZ-plane divides the join of (5,-3,-2) and (1,2,-2) at ( ^{mathrm{A}} cdotleft(frac{13}{5}, 0,-2right) ) в. ( left(frac{13}{5}, 0,2right) ) c. (5,0,2) D. (5,0,-2) |
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| 317 | A swimmer can swim ( 2 mathrm{km} ) in 15 minutes in a lake and in a river he can swim a distance of ( 4 mathrm{km} ) in 20 minutes along the stream. If a paper boat is put in the river, then the distance covered by it in ( 2 frac{1}{2} 2 ) hours will be ( mathbf{A} cdot 18 mathrm{km} ) B. ( 12 mathrm{km} ) c. ( 8 mathrm{km} ) D. ( 10 mathrm{km} ) |
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| 318 | The points ( A(-1,3,0), B(2,2,1) ) and ( C(1,1,3) ) determine a plane. The distance of the plane ( A, B, C ) from the point ( D(5,7,8) ) is A . ( sqrt{66} ) B. ( sqrt{71} ) c. ( sqrt{73} ) D. ( sqrt{76} ) |
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| 319 | If ( l_{1}, m_{1}, n_{1} ) and ( l_{2}, m_{2}, n_{2} ) are DCs of the two lines inclined to each other at an angle ( theta ), then the DCs of the internal bisector of the angle between these lines are A ( cdot frac{l_{1}+l_{2}}{2 sin frac{theta}{2}}, frac{m_{1}+m_{2}}{2 sin frac{theta}{2}}, frac{n_{1}+n_{2}}{2 sin frac{theta}{2}} ) в. ( frac{l_{1}+l_{2}}{2 cos frac{theta}{2}}, frac{m_{1}+m_{2}}{2 cos frac{theta}{2}}, frac{n_{1}+n_{2}}{2 cos frac{theta}{2}} ) c. ( frac{l_{1}-l_{2}}{2 sin frac{theta}{2}}, frac{m_{1}-m_{2}}{2 sin frac{theta}{2}}, frac{n_{1}-n_{2}}{2 sin frac{theta}{2}} ) D. ( frac{l_{1}-l_{2}}{2 cos frac{theta}{2}}, frac{m_{1}-m_{2}}{2 cos frac{theta}{2}}, frac{n_{1}-n_{2}}{2 cos frac{theta}{2}} ) |
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| 320 | Find the equation of the line in vector and in cartesion form that passes through the point with position vector ( mathbf{2} hat{mathbf{i}}-hat{mathbf{j}}+mathbf{4} hat{boldsymbol{k}} ) and is in the direction ( hat{mathbf{i}}+ ) ( mathbf{2} hat{boldsymbol{j}}-hat{boldsymbol{k}} ) |
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| 321 | The number of octants in which ( Z ) coordinate is positive is A . 2 B. 3 ( c cdot 4 ) D. 1 |
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| 322 | The length of the normal from origin to the plane ( x+2 y-2 z=9 ) is equal to A . 2 units B. 3 units c. 4 units D. 5 units |
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| 323 | 16. Consider three points P=(-sin(-a), -cos ), Q = (cos(-a), sin B) and R=(cos(B-a +0), sin(6-0)), where 0<a,ß,e<I (2008 Then, (a) Plies on the line segment RQ (b) Q lies on the line segment PR (C) R lies on the line segment OP (d) P, Q, R are non-collinear |
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| 324 | A plane passes through (1,-2,1) and is perpendicular to the planes ( 2 x-2 y+ ) ( z=0 ) and ( x-y+2 z=4 . ) Then the distance of the plane from the point (1,2,2) is ( mathbf{A} cdot mathbf{0} ) B. ( c cdot sqrt{2} ) ( D cdot 2 sqrt{2} ) |
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| 325 | If the distance between the plane, ( 23 x-10 y-2 z+48=0 ) and the plane containing the lines ( frac{boldsymbol{x}+mathbf{1}}{mathbf{2}}=frac{boldsymbol{y}-boldsymbol{3}}{boldsymbol{4}}= ) ( frac{z+1}{3} ) and ( frac{x+3}{2}=frac{y+2}{6}= ) ( frac{z-1}{lambda}(lambda epsilon R) ) is equal to ( frac{k}{sqrt{633}} ) then ( k ) is equal to |
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| 326 | Prove that ( (vec{A} times vec{B})^{2}=A^{2} B^{2}-(vec{A} times vec{B})^{2} ) | 12 |
| 327 | A line makes an angle ( alpha, beta, gamma ) with the ( X, Y, Z ) axes. Then ( sin ^{2} alpha+sin ^{2} beta+ ) ( sin ^{2} gamma= ) A . B. 2 ( c cdot frac{3}{2} ) D. |
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| 328 | Show that the points (1,2,3),(7,0,1) and (-2,3,4) are collinear. |
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| 329 | Find the direction cosines of the sides of the triangles whose vertices are (3,5,-4),(-1,1,2) and (-5,-5,-2) |
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| 330 | The image of the point ( boldsymbol{P}(boldsymbol{alpha}, boldsymbol{beta}, gamma) ) by the plane ( l x+m y+n z=0 ) is the point ( boldsymbol{Q}left(boldsymbol{alpha}^{prime}, boldsymbol{beta}^{prime}, boldsymbol{gamma}^{prime}right) . ) Then A ( cdot alpha^{2}+beta^{2}+gamma^{2}=l^{2}+m^{2}+n^{2} ) B . ( alpha^{2}+beta^{2}+gamma^{2}=alpha^{2}+beta^{2}+gamma^{2} ) ( mathbf{c} cdot alpha alpha^{prime}+beta beta^{prime}+gamma gamma^{prime}=0 ) D ( cdot lleft(alpha-alpha^{prime}right)+mleft(beta-beta^{prime}right)+nleft(gamma-gamma^{prime}right)=0 ) |
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| 331 | Find the vector equation of the plane passing through the points (2,5,-3),(-2,-3,5) and (5,3,-3) | 12 |
| 332 | The distance of the point (1,-2,3) from the plane ( boldsymbol{x}-boldsymbol{y}+boldsymbol{z}=mathbf{5} ) measured parallel to the line ( frac{x}{2}=frac{y}{3}=frac{z-1}{-6} ) is ( mathbf{A} cdot mathbf{1} ) B . 2 ( c cdot 4 ) D. None of these |
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| 333 | Find the vector equation of the plane passing through (1,2,3) and perpendicular to the plane ( vec{r} cdot(hat{i}+2 hat{j}-5 hat{k})+9=0 ) |
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| 334 | The distance of the point (1,1,1) from the plane passing through the points (2,1,1),(1,2,1) and (1,1,2) is A ( cdot frac{1}{sqrt{3}} ) B. ( c cdot sqrt{3} ) D. None of these |
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| 335 | If the d.c’s of a line are ( left(frac{1}{c}, frac{1}{c}, frac{1}{c}right), ) find ( c . ) | 12 |
| 336 | The position vectors ( vec{a}, vec{b}, vec{c} ) of three points satisfy the relation ( 2 vec{a}+7 vec{b}+ ) ( mathbf{5} vec{c}=overrightarrow{0} . ) Are these points collinear? |
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| 337 | If a line makes angles ( 90^{circ} ) and ( 60^{circ} ) respectively with the positive directions of ( x ) and ( y ) axes, find the angle which it makes with the positive direction of ( z- ) axis. |
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| 338 | ( operatorname{Given} boldsymbol{A}(mathbf{1},-mathbf{1}, mathbf{0}) ; boldsymbol{B}(mathbf{3}, mathbf{1}, mathbf{2}) ; boldsymbol{C}(mathbf{2},-mathbf{2}, mathbf{4}) ) and ( D(-1,1,-1) ) which of the following points neither lie on ( A B ) nor on ( C D ) A ( .(2,2,4) ) В. (2,-2,4) c. (2,0,1) D. (0,-2,-1) |
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| 339 | Find the direction cosines of the line which is perpendicular to the lines which direction cosines proportional to 1,-2,-2 and 0,2,1 |
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| 340 | If ( A, B ) are the feet of the perpendiculars from (2,4,5) to the ( x ) -axis, ( y ) -axis respectively, then the distance ( A B ) is A ( .2 sqrt{5} ) B. ( sqrt{29} ) c. ( sqrt{41} ) D. ( 3 sqrt{5} ) |
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| 341 | Find the distance of the point (2,12,5) from the point of intersection of the line ( vec{r}=2 hat{i}-4 hat{j}+2 hat{k}+lambda(3 hat{i}+4 hat{j}+2 hat{k}) ) and the plane ( vec{r}(hat{i}-2 hat{j}+hat{k})=0 ) |
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| 342 | The distance of point (-1,-5,-10) from the point of intersection of ( frac{x-2}{3}=frac{y+1}{4}=frac{-2}{12} ) and plane ( x ) ( boldsymbol{y}+boldsymbol{z}=mathbf{5} ) is : A . 10 B. 8 ( c cdot 2 ) D. 13 |
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| 343 | The line ( frac{x-2}{3}=frac{y+1}{2}=frac{z-1}{-1} ) intersects the curve ( boldsymbol{x} boldsymbol{y}=boldsymbol{c}^{2}, boldsymbol{z}=mathbf{0}, ) if ( boldsymbol{c}= ) A. ( 5 sqrt{5} ) B. ( 4 sqrt{5} ) c. ( sqrt{5} ) D. ( 2 sqrt{5} ) |
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| 344 | Find the ratio in which the YZ-plane divides the line segment formed by joining the points (-2,4,7) and (3,-5,8) | 12 |
| 345 | Find the direction cosines of the vector ( overrightarrow{boldsymbol{r}}=(boldsymbol{6} hat{boldsymbol{i}}+boldsymbol{2} hat{boldsymbol{j}}-boldsymbol{3} hat{boldsymbol{k}}) ) |
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| 346 | Four vertices of a tetrahedron are (0,0,0),(4,0,0),(0,-8,0) and ( (0,0,12), ) Its centroid has the coordinates A ( cdotleft(frac{4}{3},-frac{8}{3}, 4right) ) В. (2,-4,6) c. (1,-2,3) a 5 D. none of these |
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| 347 | Find the equation of a plane which is parallel to the plane ( x-2 y+2 z=5 ) and whose distance from the point (1,2,3) is 1 |
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| 348 | Let the direction – cosines of the line which is equally inclined to the axis be ( pm frac{1}{sqrt{k}} . ) Find ( k ? ) A .2 B. 3 c. 5 D. 6 |
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| 349 | If the extremities of a diagonal of a square are (1,-2,3) and (2,-3,5) then the length of its side is: A ( cdot sqrt{6} ) B. ( sqrt{3} ) c. ( sqrt{5} ) D. ( sqrt{7} ) |
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| 350 | ( operatorname{can} frac{1}{sqrt{3}}, frac{2}{sqrt{3}}, frac{-2}{sqrt{3}} ) be the direction cosines of any directed line? A. Yes B. No c. cannot say D. None of these |
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| 351 | The centroid of Tetraheadron (3,4,5),(2,5,9),(5,2,8),(2,5,2) |
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| 352 | The distance between (0,1,-1) and the point of intersection of the line ( frac{x}{2}= ) ( frac{boldsymbol{y}-mathbf{1}}{mathbf{3}}=frac{boldsymbol{z}+mathbf{1}}{mathbf{4}} ) and the plane ( boldsymbol{x}+boldsymbol{y}+ ) ( z=9 ) is A. ( sqrt{29} ) в. ( frac{1}{2} sqrt{29} ) ( c cdot frac{4}{9} ) D. ( frac{2}{9} sqrt{29} ) |
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| 353 | If the d.rs of two lines are 1,-2,3 and ( 2,0,1, ) then the d.rs of the line perpendicular to both the given lines is A. -2,5,4 в. 2,-5,4 c. 2,5,-4 D. 1,5,-4 |
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| 354 | The direction cosines of ( A B ) are -2,2,1 If ( boldsymbol{A} equiv(mathbf{4}, mathbf{1}, mathbf{5}) ) and ( l(boldsymbol{A} boldsymbol{B})=mathbf{6 u n i t s} ) find the coordinates of ( boldsymbol{B} ) |
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| 355 | Match the statements/expressions in List 1 with the values given in List 2 |
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| 356 | If ( O A B C ) is a tetrahedron such that the ( boldsymbol{O} boldsymbol{A}^{2}+boldsymbol{B} boldsymbol{C}^{2}=boldsymbol{O} boldsymbol{B}^{2}+boldsymbol{C} boldsymbol{A}^{2}=boldsymbol{O} boldsymbol{C}^{2}+ ) ( A B^{2}, ) then which of the following is/are correct A. ( A B perp O C ) в. ( O B neq C A ) c. ( O C=A B ) D. ( A B perp B C ) |
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| 357 | Line through origin and parallel to ( Y ) – axis is ( dots dots ) A ( cdot frac{x}{1}=frac{y}{0}=frac{2}{0} ) B. ( frac{x}{0}=frac{y}{1}=frac{z}{0} ) ( mathbf{c} cdot frac{x}{1}=frac{y}{0}=frac{z}{1} ) D ( cdot frac{x}{1}=frac{y}{1}=frac{z}{0} frac{z}{0} ) |
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| 358 | A lines makes angles ( frac{boldsymbol{alpha}}{2}, frac{boldsymbol{beta}}{2}, frac{gamma}{2} ) with positive direction of coordinate axes, then ( cos alpha+cos beta+cos gamma ) is equal to A . -1 B. 1 ( c cdot 2 ) D. 3 |
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| 359 | The cartesian equation of a line is ( frac{boldsymbol{x}+mathbf{3}}{mathbf{2}}=frac{boldsymbol{y}-mathbf{5}}{mathbf{4}}=frac{boldsymbol{z}+mathbf{6}}{mathbf{2}} ) find the vector equation of the line? |
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| 360 | Let ( A(-1,0) ) and ( B(2,0) ) be two points. A point ( M ) moves in the plane in such a way that ( angle M B A=2 angle M A B ). Then the point ( M ) moves along A. A straight line B. A parabola c. An ellipse D. A hyperbola |
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| 361 | The equation of the plane passing through the intersection of the planes ( mathbf{3} boldsymbol{x}-boldsymbol{y}+mathbf{2} boldsymbol{z}-mathbf{4}=mathbf{0} ) and ( boldsymbol{x}+boldsymbol{y}+boldsymbol{z}- ) ( mathbf{2}=mathbf{0} ) and passing through the point ( A(2,2,1) ) is given by? A. ( 7 x+5 y-4 z-8=0 ) в. ( 7 x-5 y+4 z-8=0 ) c. ( 5 x-7 y+4 z-8=0 ) D. ( 5 x+7 y-4 z+8=0 ) |
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| 362 | Show that ( boldsymbol{A}(mathbf{1}, mathbf{1}, mathbf{2}), boldsymbol{B}(mathbf{2}, mathbf{1}, mathbf{3}) ) and ( C(1,3,5) ) are not collinear. |
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| 363 | Find the coordinates of the point, which divides the line segment joining the points ( boldsymbol{A}(mathbf{2},-mathbf{6}, mathbf{8}) ) and ( boldsymbol{B}(-mathbf{1}, mathbf{3},-mathbf{4}) ) externally in the ratio 1: 3 |
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| 364 | Name the octants in which the following points lie: ( (mathbf{5}, mathbf{2}, mathbf{3}) ) |
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| 365 | The ratio in which the plane ( vec{r} cdot(vec{i}- ) ( mathbf{2} overrightarrow{boldsymbol{j}}+boldsymbol{3} overrightarrow{boldsymbol{k}})=17 ) divides the line joining the points ( -2 vec{i}+4 vec{j}+7 vec{k} ) and ( 3 vec{i}- ) ( mathbf{5 j}+mathbf{8 k} ) is A .1: 5 B. 1: 10 ( c .3: 5 ) D. 3: 10 |
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| 366 | Find the distance between the point (7,2,4) and the plane determined by the points ( boldsymbol{A}(mathbf{2}, mathbf{5},-mathbf{3}), boldsymbol{B}(-mathbf{2},-mathbf{3}, mathbf{5}) ) and (5,3,-3) |
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| 367 | The distance from the origin to the centroid of the tetrahedron formed by the points (0,0,0),(3,0,0),(0,4,0),(0,0,5) is A. ( frac{sqrt{3+4+5}}{4} ) B. ( frac{sqrt{3+4+5}}{3} ) c. ( frac{sqrt{3^{2}+4^{2}+5^{2}}}{16} ) D. ( frac{sqrt{3^{2}+4^{2}+5^{2}}}{4} ) |
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| 368 | The value(s) of ( lambda ), for which the triangle with vertices ( (mathbf{6}, mathbf{1 0}, mathbf{1 0}),(mathbf{1}, mathbf{0},-mathbf{5}) ) and ( (6,-10, lambda) ) will be a right angled triangle is/ are A . 1 в. ( frac{70}{3}, 0 ) c. 35 D. ( 0,-frac{70}{3} ) |
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| 369 | Find the ratio in which the yz-plane divides the join of the points (-2,4,7) and (3,-5,8) and also find the ( c o ) ordinates of the point of intersection of this line with the ( y z ) – plane. A ( cdot lambda=frac{2}{3} ) and ( left(0, frac{2}{5}, frac{37}{5}right) ) B. ( lambda=frac{1}{3} ) and ( left(frac{-3}{4}, frac{7}{4}, frac{29}{4}right) ) c. ( lambda=frac{2}{3} ) and ( left(frac{-3}{4}, frac{7}{4}, frac{29}{4}right) ) D ( lambda=frac{1}{3} ) and ( left(0, frac{2}{5}, frac{37}{5}right) ) |
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| 370 | The direction cosines of the normal to the plane ( 5 y+4=0 ) are? A ( cdot 0, frac{-4}{5}, 0 ) в. 0,1,0 c. 0,-1,0 D. None of these |
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| 371 | The equation of the plane that passes through the points (1,0,2),(-1,1,2),(5,0,3) is A. ( x+2 y-4 z+7=0 ) B. ( x+2 y-3 z+7=0 ) c. ( x-2 y+4 z+7=0 ) D. ( 2 y-4 z-7+x=0 ) |
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| 372 | The extremities of a diagonal of a rectangular parallelopiped whose faces are parallel to the reference planes are (-2,4,6) and ( (3,16,6) . ) The length of the base diagonal is ( A cdot 7 ) B. 10 c. 11 D. 13 |
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| 373 | The equation of a line is ( 5 x-3= ) ( mathbf{1 5 y}+mathbf{7}=mathbf{3}-mathbf{1 0 z} . ) Write the direction ( operatorname{cosines} ) of the line. |
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| 374 | Find the direction cosines of a line that pass through the point ( boldsymbol{P}(mathbf{1}, mathbf{4}, boldsymbol{6}) ) and ( Q(5,1,11) ) and is so directed that it make an acute angle with the positive direction of ( boldsymbol{y}-boldsymbol{a} boldsymbol{x} boldsymbol{i} boldsymbol{s} ) |
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| 375 | The distance of point ( boldsymbol{P}left(boldsymbol{x}_{1}, boldsymbol{y}_{1}, boldsymbol{z}_{1}right) ) to the plane ( a x+b y+c z+D=0 ) is given by A ( cdot frac{left|a x_{1}+b y_{1}+c z_{1}-dright|}{sqrt{a+b+c}} ) B. ( frac{left|a x_{1}+b y_{1}+c z_{1}right|}{sqrt{a^{2}+b^{2}+c^{2}}} ) c. ( frac{left|a x_{1}+b y_{1}+c z_{1}-dright|}{sqrt{a^{2}+b^{2}+c^{2}}} ) D. None of these |
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| 376 | A plane passes through (1,-2,1) and is perpendicular to two planes ( 2 x-2 y+ ) ( z=0 ) and ( x-y+2 z=4 . ) The distance of the plane from the point is (1,2,2) ( A cdot 0 ) B. ( c cdot sqrt{2} ) D. ( 2 sqrt{2} ) |
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| 377 | intb, z = c.ytd 2003 14. The two lines x=ay+b,z=cy+d and x=a’y+ will be perpendicular, if and only if (a) aa’+cc’+1 = 0 (b) aa’ + bb’ + cc’ + 1 = 0 (c) aa’+bb’ +cc’ = 0 (d) (a+a’)(b+b) +c+c’)=0. |
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| 378 | A line passes through the points (6,-7,-1) and ( (2,-3,1) . ) The direction cosines of the line so directed that the angle made by it with the positive direction of x-axis is acute, is? A ( cdot frac{2}{3},-frac{2}{3},-frac{1}{3} ) B. ( -frac{2}{3}, frac{2}{3}, frac{1}{3} ) c. ( frac{2}{3},-frac{2}{3}, frac{1}{3} ) D. ( frac{2}{3}, frac{2}{3}, frac{1}{3} ) |
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| 379 | x – 4 The va The value of k such that y-2 z-k 22. lies in the plane 2x – 4y +z=7, is (2003) (a) 7 (b) -7 (d) 4 (c) no real value . |
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| 380 | A plane makes intercept 1,2,3 on the co-ordinate axes. If the distance from origin is ( p ) then find the value of ( p ) |
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| 381 | The equation of plane containing the lines ( vec{gamma}=vec{alpha}+overrightarrow{lambda beta} ) and ( vec{gamma}=vec{beta}+mu vec{alpha} ) ( mathbf{A} cdotleft[begin{array}{lll}vec{gamma} & vec{alpha} & vec{beta}end{array}right]=vec{alpha} cdot vec{beta} ) B ( cdotleft[begin{array}{lll}vec{gamma} & vec{alpha} & vec{beta}end{array}right]=0 ) ( mathbf{c} cdotleft[begin{array}{lll}vec{alpha} & vec{beta} & vec{gamma}end{array}right]=vec{alpha} cdot vec{beta} ) D. None of these |
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| 382 | is a sa mu on 19. Consider a pyramid OPQRS located in the first octant (x20. y=0,220) with O as origin, and OP and OR along the x-axis and the y-axis, respectively. The base OPQR of the pyramid is a square with OP=3. The point S is directly above the mid-point, T of diagonal OQ such that TS=3. Then (JEE Adv. 2016) (a) the acute angle between OQ and OS is (b) the equation of the plane containing the triangle OOS is x-y=0 the length of the perpendicular from P to the plane (c) containing the triangle OQS is 5 the perpendicular distance from 0 to the straight lin (d) containing RS is |
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| 383 | Show that the points ( boldsymbol{A}(-mathbf{3} hat{boldsymbol{i}}+boldsymbol{3} hat{boldsymbol{j}}+mathbf{5} hat{boldsymbol{k}}), boldsymbol{B}(hat{boldsymbol{i}}+mathbf{2} hat{boldsymbol{j}}+mathbf{3} hat{boldsymbol{k}}), boldsymbol{C} ) are collinear. |
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| 384 | The points ( (boldsymbol{k}-mathbf{1}, boldsymbol{k}+mathbf{2}),(boldsymbol{k}, boldsymbol{k}+ ) 1) ( ,(k+1, k) ) are collinear for A. any value of ( k ) B. ( k=-frac{1}{2} ) only c. no value of ( k ) D. integral values of ( k ) only |
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| 385 | If ( boldsymbol{A} ) is ( (2,4,5), ) and ( B ) is (-7,-2,8) then which of the following is collinear with ( A ) and ( B ) is A ( .(1,2,6) ) В. (2,-1,6) c. (-1,2,6) D. (2,6,-1) |
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| 386 | A given unit vector is orthogonal to ( mathbf{5} hat{mathbf{i}}+mathbf{2} hat{mathbf{j}}+mathbf{6} hat{boldsymbol{k}} ) and coplanar with ( hat{mathbf{i}}-hat{boldsymbol{j}}+ ) ( hat{k} ) and ( 2 hat{i}+hat{j}+hat{k} ) then the vector is? A ( cdot frac{3 hat{j}-hat{k}}{sqrt{10}} ) ( ^{text {В } cdot frac{6 hat{i}-5 hat{k}}{sqrt{61}}} ) c. ( frac{2 hat{i}-5 hat{k}}{sqrt{29}} ) ( frac{2 hat{i}+hat{j}-2 hat{k}}{3} ) |
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| 387 | to the plane 68. A equation of a plane parallel to the x-2y + 2z-5 = 0 and at a unit distance from the origin 2012 (a) x-2y+ 2z – 3 =0 (c) x2y + 2z – 1 =0 (b) x-2y + 2z+1=0 (d) x-2y + 2z+5=0 |
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| 388 | The direction ratios of the joining ( A(1,2,1) ) and (2,1,2) are A .3,3,3 B. -1,1,-1 c. 3,1,3 D. ( frac{1}{sqrt{3}}, frac{1}{sqrt{3}}, frac{1}{sqrt{3}} ) |
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| 389 | ( boldsymbol{A}=(mathbf{2}, mathbf{4}, mathbf{5}) ) and ( boldsymbol{B}=(mathbf{3}, mathbf{5},-mathbf{4}) ) are two points. If the ( X Y ) -plane, ( Y Z ) -plane divide ( A B ) in the ratio ( a: b ) and ( p: q ) respectively, then ( frac{a}{b}+frac{p}{q}= ) A ( cdot frac{23}{12} ) B. ( frac{-7}{12} ) c. ( frac{7}{12} ) D. ( frac{-22}{15} ) |
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| 390 | The pairs ( bar{a}, bar{b} ) and ( bar{c}, bar{d} ) each determine a plane. Then the planes are parallel if ( (overline{boldsymbol{a}} times overline{boldsymbol{c}}) times(overline{boldsymbol{b}} times overline{boldsymbol{d}})= ) ( mathbf{0} 2)(overline{boldsymbol{a}} times overline{boldsymbol{c}}) cdot(overline{boldsymbol{b}} times overline{boldsymbol{d}})=mathbf{0} ) ( (bar{a} times bar{b}) times(bar{c} times bar{d})= ) ( mathbf{0} 4)(bar{a} times bar{b}) cdot(bar{c} times bar{d})=0 ) |
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| 391 | If a ray makes angles ( alpha, beta, gamma ) and ( delta ) with the four diagonals of a cube and ( mathbf{A}: cos ^{2} boldsymbol{alpha}+cos ^{2} boldsymbol{beta}+cos ^{2} boldsymbol{gamma}+cos ^{2} boldsymbol{delta} ) ( mathbf{B}: sin ^{2} boldsymbol{alpha}+sin ^{2} boldsymbol{beta}+sin ^{2} boldsymbol{gamma}+sin ^{2} boldsymbol{delta} ) ( mathbf{C}: cos 2 boldsymbol{alpha}+cos 2 boldsymbol{beta}+cos 2 gamma+cos 2 boldsymbol{delta} ) Arrange ( A, B, C ) in descending order A. ( B, A, C ) в. ( A, B, C ) c. ( C, A, B ) D. ( B, C, A ) |
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| 392 | If ( R ) divides the line segment joining ( P(2, ) 3, 4) and ( Q(4,5,6) ) in the ratio -3: 2 then the value of the parameter which represents R is в. ( =(10,9,8) ) ( mathbf{c} .=(10,8,9) ) ( mathbf{D} .=(9,10,8) ) |
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| 393 | Find the coordinates of a point on ( y ) -axis which are at a distance of ( 5 sqrt{2} ) from the point ( boldsymbol{P}(boldsymbol{3},-boldsymbol{2}, boldsymbol{5}) ) |
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| 394 | Find the equation of the set of points ( P ) the sum of whose distances from ( boldsymbol{A}(mathbf{4}, mathbf{0}, mathbf{0}) ) and ( boldsymbol{B}(-mathbf{4}, mathbf{0}, mathbf{0}) ) is equal to 10 |
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| 395 | Distance between the points (12,4,7) and (10,5,3) is A ( cdot sqrt{21} ) B. ( sqrt{5} ) c. ( sqrt{17} ) D. none of these |
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| 396 | The distances of the point ( boldsymbol{P}(mathbf{1}, mathbf{2}, mathbf{3}) ) from the coordinates axes are: A ( cdot sqrt{13}, sqrt{10}, sqrt{5} ) в. ( sqrt{11}, sqrt{10}, sqrt{5} ) c. ( sqrt{13}, sqrt{20}, sqrt{15} ) D. ( sqrt{23}, sqrt{10}, sqrt{5} ) |
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| 397 | If a line makes an angle of ( pi / 4 ) with the positive direction of each of ( x ) -axis and ( boldsymbol{y} ) -axis, then the angle that the line makes with the positive direction of the ( z ) -axis is A ( cdot frac{pi}{6} ) в. c. D. |
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| 398 | Find the vector equation of the line passing through the point (3,1,2) and perpendicular to the plane ( vec{r} cdot(2 hat{i}-hat{j}+widehat{k})=8 ) Also find the point of intersection of line and plane. |
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| 399 | Which of the following is true for a plane? This question has multiple correct options A. A locus is called a plane if the line joining any two arbitrary points on the locus is also a part of the locus. B. Value of ( y ) in a ( z x ) plane is non-zero. C. Value of ( z ) in a ( x y ) plane is zero. D. None of the above |
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| 400 | The point which divides the line joining the points (1,3,4) and (4,3,1) internally in the ratio ( 2: 1, ) is A. (2,-3,3) в. (2,3,3) c. ( left(frac{5}{2}, 3, frac{5}{2}right) ) D. (-3,3,2) E ( .(3,3,2) ) |
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| 401 | If ( G ) is centroid of ( triangle A B C, ) then A ( cdot vec{G}=vec{a}+vec{b}+vec{c} ) B. ( vec{G}=frac{vec{a}+vec{b}+vec{c}}{2} ) c. ( 3 vec{G}=vec{a}+vec{b}+vec{c} ) ” ( quad 3 vec{G}=frac{vec{a}+vec{b}+vec{c}}{2} ) |
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| 402 | A line OP where ( 0=(0,0,0) ) makes equal angles with ox, oy, oz. The point on OP, which is at a distance of 6 units from 0 is: A. ( left(frac{6}{sqrt{3}}, frac{6}{sqrt{3}}, frac{6}{sqrt{3}}right) ) B . ( (2 sqrt{3},-2 sqrt{3}, 2 sqrt{3}) ) ( c cdot-(2 sqrt{3}, 2 sqrt{3}, 2 sqrt{3}) ) |
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| 403 | Show that the points ( boldsymbol{O}(mathbf{0}, mathbf{0}), boldsymbol{A}(mathbf{2},-mathbf{3}, mathbf{3}), boldsymbol{B}(-mathbf{2}, mathbf{3},-mathbf{3}) ) are collinear. Find the ratio in which each point divides the segment joining the other two. |
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| 404 | Find the equation of the line passing through the points ( A(3,2,-1) ) and ( boldsymbol{B}(mathbf{4},-mathbf{1}, mathbf{3}) ) |
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| 405 | A line makes the same angle ( theta ) with each of the ( X ) and ( Z ) -axes. If the angle ( beta ) which it makes with ( Y ) -axis, is such that ( sin ^{2} beta=3 sin ^{2} theta, ) then ( cos ^{2} theta ) equals A ( cdot frac{2}{5} ) в. ( frac{1}{5} ) ( c cdot frac{3}{5} ) D. ( frac{2}{3} ) |
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| 406 | The foot of the perpendicular from the point ( boldsymbol{A}(mathbf{7}, mathbf{1 4}, mathbf{5}) ) to the plane ( mathbf{2} boldsymbol{x}+mathbf{4} boldsymbol{y}- ) ( z=2 ) is? A. (3,1,8) в. (1,2,8) c. (3,-3,5) D. (5,-3,-4) |
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| 407 | The ratio in which the line segment joining the points whose position vectors are ( 2 hat{i}-4 hat{j}-7 hat{k} ) and ( -3 hat{i}+ ) ( mathbf{5} hat{boldsymbol{j}}-boldsymbol{8} hat{boldsymbol{k}} ) is divided by the plane whose equation is ( hat{r} cdot(hat{i}-2 hat{j}+3 hat{k})=13 ) is- A. 13: 12 internally B. 12: 25 externally c. 13: 25 internally D. 37: 25 internally |
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| 408 | Distance between plane ( 3 x+4 y- ) ( mathbf{2 0}=mathbf{0} ) and point ( (mathbf{0}, mathbf{0},-mathbf{7}) ) is A. 4 units B. 3 units c. 2 units D. 1 units |
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| 409 | The scalar product and the magnitude of vector products of two vectors are ( 48 sqrt{3} ) and 144 respectively. Then the angle between the two vectors is A. 54.7 B. 60 c. 90 D. 120 |
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| 410 | If a line has the direction ratios 4,-12,18 then find its direction cosines. A ( cdot-frac{2}{11},-frac{6}{11},-frac{9}{11} ) B. ( -frac{2}{11}, frac{6}{11},-frac{9}{11} ) c. ( frac{2}{11},-frac{6}{11}, frac{9}{11} ) D. ( frac{2}{11}, frac{6}{11}, frac{9}{11} ) |
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| 411 | The locus of a point ( P ) which moves such that ( P A^{2}-P B^{2}=2 k^{2} ) where ( A ) and ( B ) ( operatorname{are}(3,4,5) ) and (-1,3,-7) respectively is A. ( 8 x+2 y+24 z-9+2 k^{2}=0 ) B. ( 8 x+2 y+24 z-2 k^{2}=0 ) c. ( 8 x+2 y+24 z+9+2 k^{2}=0 ) D. ( 8 x-2 y+24 z-2 k^{2}=0 ) |
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| 412 | Assertion In each of the three planes determined by two of the lines ( O A, O B, O C ) ( ( O ) being the origin), a straight line is drawn through ( O ) perpendicular to the third line. The three lines so determined are coplanar. Reason ( (a times b) times c+(b times c) times a+(c times a) times ) ( b=0, ) where ( O A=a, O B=b ) and ( boldsymbol{O} boldsymbol{C}=boldsymbol{c} ) A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion c. Assertion is correct but Reason is incorrect D. Both Assertion and Reason are incorrect |
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| 413 | Prove that ( boldsymbol{A}(mathbf{1}, mathbf{2}, mathbf{0}), boldsymbol{B}(mathbf{3}, mathbf{1}, mathbf{1}), boldsymbol{C}(mathbf{7},-mathbf{1}, mathbf{3}) ) are collinear. |
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| 414 | The d.r. of normal to the plane through (1, 0, 0), (0, 1.0 which makes an angle /4 with plane x+y=3 are [20021 (a) 1,12,1 (b) 1,1, 2 (c) 1,1,2 (d) √2,1,1 |
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| 415 | If the projections of the line segment ( A B ) on the coordinate axes are 2,3,6 then the square of the sine of the angle made by ( boldsymbol{A B} ) with ( boldsymbol{x}=mathbf{0}, ) is A ( cdot frac{3}{7} ) B. ( frac{3}{49} ) ( c cdot frac{4}{7} ) D. ( frac{40}{49} ) |
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| 416 | The angle between any two faces is ( A cdot cos ^{-1}(1 / 3) ) B. ( cos ^{-1}(1 / 4) ) c. ( pi / 3 ) ( mathbf{D} cdot cos ^{-1}(1 / 2) ) |
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| 417 | The point equidistant from the point ( boldsymbol{O}(mathbf{0}, mathbf{0}, mathbf{0}), boldsymbol{A}(boldsymbol{a}, mathbf{0}, mathbf{0}), boldsymbol{B}(mathbf{0}, boldsymbol{b}, mathbf{0}) ) and ( C(0,0, c) ) has the coordinates ( mathbf{A} cdot(a, b, c) ) B. ( (a / 2, b / 2, c / 2) ) c. ( (a / 3, b / 3, c / 3) ) D. ( (a / 4, b / 4, c / 4) ) |
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| 418 | The distance between (5,1,3) and the line ( x=3, y=7+t, z=1+t ) is A . 4 B. 2 ( c cdot 6 ) D. 8 |
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| 419 | The point of intersection of the lines ( frac{x-5}{3}=frac{y-7}{-1}=frac{z+2}{1} ) and ( frac{x+3}{-36}= ) ( frac{y-3}{2}=frac{z-6}{4} ) is A ( cdotleft(21, frac{5}{3}, frac{10}{3}right) ) в. (2,10,4) c. (-3,3,6) D. (5,7,-2) |
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| 420 | Assertion(A): If centroid and circumcentre of a triangle are known its orthocentre can be found. Reason (R) : Centriod, orthocentre and |
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| 421 | ( mathbf{I f A}=(mathbf{1}, mathbf{2}, mathbf{3}), mathbf{B}=(mathbf{2}, mathbf{3}, mathbf{4}) ) and ( mathbf{C} ) is a point of trisection of AB such that ( mathbf{C}_{mathbf{x}}+ ) ( mathbf{C}_{mathbf{y}}=frac{mathbf{1 3}}{mathbf{3}} ) then ( mathbf{C}_{mathbf{z}}= ) A ( cdot frac{10}{3} ) B. ( frac{11}{3} ) c. ( frac{11}{2} ) D. 11 |
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| 422 | +1 33. x If the angel e between the line – 1 -2 22 – and the plane 2x-y+ Váz+4 = 0 is such that sin 0= then the value of 1 is [2005] |
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| 423 | Find the distance between the points ( R(-3,0), Sleft(0, frac{5}{2}right) ) | 12 |
| 424 | Find the angle between the two lines whose direction cosines are given by equations ( l+m+n=0 ) and ( l^{2}+ ) ( boldsymbol{m}^{2}-boldsymbol{n}^{2}=mathbf{0} ) A ( cdot frac{pi}{6} ) в. ( c cdot frac{pi}{2} ) D. |
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| 425 | The projection of the line segment joining (0,0,0) and (5,2,4) on the line whose direction ratios are 2,-3,6 is A . 28 B. 4 c. ( frac{40}{7} ) D. ( sqrt{45} ) |
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| 426 | Find the co-ordinates of the points of trisection of the line joining the points (-3,0) and (6,6) | 12 |
| 427 | If the lines ( frac{x-1}{2}=frac{y+1}{3}=frac{z-1}{4} ) and ( frac{boldsymbol{x}-mathbf{3}}{mathbf{1}}=frac{boldsymbol{y}-boldsymbol{k}}{mathbf{2}}=frac{boldsymbol{z}}{mathbf{1}} ) intersect, then the value of ( k ) is A ( cdot frac{3}{2} ) B. ( frac{9}{2} ) ( c cdot-frac{2}{9} ) D. ( -frac{3}{2} ) |
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| 428 | If the distance of the point ( P(4,3,5) ) from the Y-axis is ( lambda ), then the value of ( 7 lambda^{2} ) is A .287 B. ( 7 sqrt{41} ) c. 63 D. 21 |
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| 429 | The shortest distance between z-axis and the line ( boldsymbol{x}+boldsymbol{y}+mathbf{2} boldsymbol{z}-mathbf{3}=mathbf{0}=mathbf{2} boldsymbol{x}+mathbf{3} boldsymbol{y}+mathbf{4} boldsymbol{z} ) ( 4, ) is ( mathbf{A} cdot mathbf{1} ) B . 2 ( c cdot 4 ) D. 3 |
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| 430 | A line makes angles ( alpha, beta, gamma ) with the positive directions of the axes of reference. The value of ( cos 2 alpha+ ) ( cos 2 beta+cos 2 gamma ) is ( mathbf{A} cdot mathbf{1} ) B. 2 ( c cdot-1 ) D. 0 |
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| 431 | 17. The radius of the circle in which the sphere x2 + y2 + z2 + 2x – 2y – 42 – 19=0 is cut by the plane x +2y + 2z +7= 0 is [2003] (a) 4 (6) (c) 2 (d) 3 |
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| 432 | A line with direction cosines proportional to 2,1,2 meets each of the line ( boldsymbol{x}=boldsymbol{y}+boldsymbol{a}=boldsymbol{z} ) and ( boldsymbol{x}+boldsymbol{a}=boldsymbol{2} boldsymbol{y}= ) 2 ( z ). The co-ordinates of each of the points of intersection are given by: A. ( (3 a, 3 a, 3 a),(a, a, a) ) в. ( (3 a, 2 a, 3 a),(a, a, a) ) c. ( (3 a, 2 a, 3 a),(a, a, 2 a) ) D. ( (2 a, 3 a, 3 a),(2 a, a, a) ) |
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| 433 | If ( vec{A} times vec{B}=vec{C}, ) which of the following options is wrong? ( A cdot vec{C} ) is ( perp ) to ( vec{A} ) B cdot ( vec{C} ) is ( perp ) to ( vec{B} ) c. ( vec{C} ) is ( perp(vec{A}+vec{B}) ) D ( cdot vec{C} ) is ( perp ) to ( (vec{A} times vec{B}) ) |
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| 434 | Find the shortest distance between the following pair of lines. ( overline{boldsymbol{r}}=(overline{boldsymbol{i}}+mathbf{2} overline{boldsymbol{j}}+overline{boldsymbol{k}})+boldsymbol{lambda}(mathbf{2} overline{boldsymbol{i}}-overline{boldsymbol{j}}+mathbf{3} overline{boldsymbol{k}}) & ) ( overline{boldsymbol{r}}=(overline{boldsymbol{i}}-mathbf{3} overline{boldsymbol{j}}-overline{boldsymbol{k}})+boldsymbol{mu}(mathbf{3} overline{boldsymbol{i}}+mathbf{2} hat{boldsymbol{j}}-mathbf{5} overline{boldsymbol{k}}) ) |
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| 435 | Show that the points (2,-1,3),(4,3,1) and (3,1,2) are collinear. | 12 |
| 436 | f a point is in the ( X Z ) -plane. What can you say about its ( y ) -coordinate? |
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| 437 | The plane ( x+2 y-z=4 ) cuts the sphere ( boldsymbol{x}^{2}+boldsymbol{y}^{2}+boldsymbol{z}^{2}-boldsymbol{x}+boldsymbol{z}-boldsymbol{2}=mathbf{0} ) in a circle of radius A . 1 B. 3 ( c cdot sqrt{2} ) D. |
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| 438 | To find the vector and the Cartesian equation in symmetric form of line passing through the points, (2,0,-3) and ( (mathbf{7}, mathbf{3},-mathbf{1 0}) ) |
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| 439 | If ( theta ) is the angle between two lines whose d.c.s are ( l_{1}, m_{1}, n_{1} ) and ( l_{2}, m_{2}, n_{2} ) then the d.cs of one of the angular bisectors of the two lines are ( ^{text {A }} cdot frac{l_{1}+l_{2}}{2}, frac{m_{1}+m_{2}}{2}, frac{n_{1}+n_{2}}{2} ) B. ( frac{l_{1}+l_{2}}{2 cos left(frac{theta}{2}right)}, frac{m_{1}+m_{2}}{2 cos left(frac{theta}{2}right)}, frac{n_{1}+n_{2}}{2 cos left(frac{theta}{2}right)} ) c. ( frac{l_{1}+l_{2}}{cos left(frac{theta}{2}right)}, frac{m_{1}+m_{2}}{cos left(frac{theta}{2}right)}, frac{n_{1}+n_{2}}{cos left(frac{theta}{2}right)} ) D. ( frac{l_{1}+l_{2}}{2 sin left(frac{theta}{2}right)} frac{m_{1}+m_{2}}{2 sin left(frac{theta}{2}right)} frac{n_{1}+n_{2}}{2 sin left(frac{theta}{2}right)} ) |
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| 440 | The image of the point with position vector ( hat{i}+3 hat{k} ) in the plane ( r cdot(hat{i}+hat{j}+ ) ( hat{boldsymbol{k}})=mathbf{1} ) is ( mathbf{A} cdot hat{i}+2 hat{j}+hat{k} ) B ( cdot hat{i}-2 hat{j}+hat{k} ) c. ( -hat{i}-2 hat{j}+hat{k} ) D. ( hat{i}+2 hat{j}-hat{k} ) |
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| 441 | The vector form of the equation of the line passing through points (3,4,7) and (5,1,6) is ( mathbf{A} cdot vec{r}=(3 hat{i}+4 hat{j}-7 hat{k})+lambda(2 hat{i}-3 hat{j}+13 hat{k}) ) B ( cdot vec{r}=(3 hat{i}+4 hat{j}-7 hat{k})+lambda(8 hat{i}+5 hat{j}-hat{k}) ) C ( . vec{r}=(3 hat{i}+4 hat{j}+7 hat{k})+lambda(2 hat{i}-3 hat{j}-hat{k}) ) D ( cdot vec{r}=(3 hat{i}+4 hat{j}-7 hat{k})+lambda(2 hat{i}-3 hat{j}-13 hat{k}) ) |
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| 442 | The angle between any two diagonals of cube are: A ( cdot cos ^{-1}left(frac{1}{2}right) ) B. ( cos ^{-1}left(frac{1}{3}right) ) ( ^{mathbf{C}} cdot cos ^{-1}left(frac{1}{sqrt{3}}right) ) ( mathrm{D} cdot cos ^{-1}left(frac{1}{sqrt{2}}right) ) |
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| 443 | Find the point of intersection of the following pair of lines, assuming that the vectors ( vec{a} ) and ( vec{b} ) are not parallel. ( vec{r}=gamma(vec{b}+vec{a}), vec{r}=mu(vec{b}-vec{a}) ) A . origin в. ( vec{b}+vec{a} ) ( c cdot 2 b ) D. no intersection point |
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| 444 | Find the equation of the line in vector and in Cartesian form that passes through the point with position vector ( 2 hat{i}-hat{j}+4 hat{k} ) and is in the direction ( hat{i}+ ) ( mathbf{2} hat{boldsymbol{j}}-hat{boldsymbol{k}} ) | 12 |
| 445 | Find the equation of the plane passing through the points (0,0,0) and (3,-1,2) are parallel to the line ( frac{boldsymbol{x}-mathbf{4}}{mathbf{1}}=frac{boldsymbol{y}+mathbf{3}}{mathbf{- 4}}=frac{boldsymbol{z}+mathbf{1}}{mathbf{7}} ) | 12 |
| 446 | If a line has direction ratio 2,-1,-2 determine its direction cosines. |
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| 447 | 10 80 8 |
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| 448 | The angle between the lines whose direction cosines satisfy the equations ( l+m+n=0 ) and ( l^{2}+m^{2}+n^{2} ) is A ( cdot frac{pi}{2} ) в. c. D. ( frac{pi}{6} ) |
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| 449 | Find the vector and Cartesian equations of the plane passing through the points with position vectors ( 3 vec{i}+4 vec{jmath}+ ) ( 2 vec{k}, 2 vec{i}-2 vec{j}-vec{k} ) and ( 7 vec{i}+vec{k} ) |
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| 450 | A straight line ( L ) on the ( x y ) -plane bisects the angle between ( O X ) and ( O Y ) What are the direction cosines of ( L ? ) A ( cdot(1 / sqrt{2}, 1 / sqrt{2}, 0) ) в. ( (1 / 2, sqrt{3} / 2,0) ) c. (0,0,1) D. ( (2 / 3,2 / 3,1 / 3) ) |
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| 451 | ( mathbf{A}=(mathbf{1}, mathbf{1}, mathbf{4}) ) and ( mathbf{B}=(mathbf{5},-mathbf{3}, mathbf{4}) ) are two points. If the points ( P, Q ) are on the line ( A B ) such that ( A P=P Q=Q B ) then ( P Q= ) A ( cdot 2 sqrt{2} ) в. 4 c. ( sqrt{frac{32}{9}} ) D. ( sqrt{2} ) |
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| 452 | The angle between the lines, whose direction ratios are 1,1,2 and ( sqrt{3}- ) ( 1,-sqrt{3}-1,4, ) is A . ( 45^{circ} ) B. ( 30^{circ} ) ( c cdot 60^{circ} ) D. ( 90^{circ} ) |
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| 453 | Show that the points ( boldsymbol{A}(mathbf{1}, mathbf{1}, mathbf{1}), boldsymbol{B}(mathbf{1}, mathbf{2}, mathbf{3}) ) and ( boldsymbol{C}(mathbf{2},-mathbf{1}, mathbf{1}) ) are vertices of an isosceles triangle. |
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| 454 | Find the vector and Cartesian equation of the line that passes through the points (3,-2,-5) and (3,-2,6) | 12 |
| 455 | If two vertices of an equilateral triangle ( operatorname{are}(2,1,5) ) and ( (3,2,3), ) then its third vertex is: A. (1,2,4) в. (4,0,4) c. (0,-4,4) a 5 D. (4,4,1) |
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| 456 | Given that ( boldsymbol{P}(boldsymbol{3}, boldsymbol{2},-boldsymbol{4}), boldsymbol{Q}(boldsymbol{5}, boldsymbol{4},-boldsymbol{6}) ) and ( R(9,8,-10) ) are collinear. Find the ratio in which ( Q ) divides ( P R ) |
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| 457 | If the lines ( frac{x-0}{1}=frac{y+1}{2}=frac{z-1}{-1} ) and ( frac{x+1}{k}= ) ( frac{y-3}{-2}=frac{z-2}{1} ) are at right angles, then the value of k is ( mathbf{A} cdot mathbf{5} ) B. ( c cdot 3 ) D. – |
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| 458 | 2. The equation x +2y + 2z = 1 and 2x + 4y + 4z=9 have (a) Only one solution ono (1979) (b) Only two solutions (c) Infinite number of solutions i n the (d) None of these. |
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| 459 | Find the angles at which the normal vector to the plane ( 4 x+8 y+z=5 ) is inclined to the coordinate axes. |
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| 460 | Find ( a ), if the distance between the points ( P(11,-2) ) and ( Q(a, 1) ) is 5 units. |
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| 461 | In what ratio, the line joining (-1,1) and (5,7) is divided by the line ( x+y= ) ( 4 ? ) |
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| 462 | If the origin is the centroid of the triangle whose vertices are ( boldsymbol{A}(mathbf{2}, boldsymbol{p},-mathbf{3}), boldsymbol{B}(boldsymbol{q},-mathbf{2}, mathbf{5}) ) and ( boldsymbol{R}(-mathbf{5}, mathbf{1}, boldsymbol{r}) ) then find the values of ( boldsymbol{p}, boldsymbol{q}, boldsymbol{r} ) |
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| 463 | The equation of the plane passing through the point (1,1,1) and perpendicular to the planes ( 2 x+y- ) ( 2 z=5 ) and ( 3 x-6 y-2 z=7 ) is? A. ( 14 x+2 y-15 z=1 ) B. ( -14 x+2 y+15 z=3 ) c. ( 14 x-2 y+15 z=27 ) D. ( 14 x+2 y+15 z=31 ) |
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| 464 | f points (1,2),(3,5) and ( (0, b) ) are collinear the value of b is A ( cdot frac{1}{2} ) в. ( frac{7}{2} ) ( c cdot 2 ) D. – |
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| 465 | Two distinct lines in a plane A. always intersect B. always either intersect or are parallel C . always have two common points D. none of these |
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| 466 | h the origin. ant distance P.: 2x-y+z 17. In R’, let L be a straight line passing through the ori Suppose that all the points on L are at a constant dista from the two planes P, :x+2y-z+1=0 and P, : 2x – -1= 0. Let M be the locus of the feet of the perpendicula, drawn from the points on L to the plane P,. Which of th following points lie (s) on M? (JEE Adv. 2015) (b) |
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| 467 | ( boldsymbol{L} ) and ( boldsymbol{M} ) are two points with position vectors ( 2 bar{a}-bar{b} ) and ( a+2 bar{b} ) respectively. The position vector of the point ( N ) which divides the line segment ( L M ) in the ratio 2: 1 externally is ( mathbf{A} cdot 3 bar{b} ) в. ( 4 bar{b} ) ( c .5 bar{b} ) D. ( 3 bar{a}+4 bar{b} bar{hline} bar{b}+4+4 bar{a} bar{a} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} bar{b} ) |
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| 468 | A non-zero vector ( vec{a} ) is parallel to the line of intersection of plane ( boldsymbol{p}_{mathbf{1}} ) determined by ( hat{i}+hat{j} ) and ( hat{i} ) and plane ( P_{2} ) determined by vectors ( hat{mathbf{i}}-hat{mathbf{j}} ) and ( hat{mathbf{i}}+hat{boldsymbol{k}} ) then angle between a and vector ( hat{mathbf{i}}- ) ( 2 hat{j}+2 hat{k} ) is This question has multiple correct options A ( cdot frac{pi}{4} ) B. ( frac{pi}{2} ) c. ( frac{pi}{3} ) D. ( frac{3 pi}{4} ) |
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| 469 | Find the coordinates of the points of trisection of the line segment joining the points ( A(-4,3) ) and ( B(2,-1) ) | 12 |
| 470 | The position vectors of three points are ( mathbf{2} overrightarrow{boldsymbol{a}}-overrightarrow{boldsymbol{b}}+mathbf{3} overrightarrow{boldsymbol{c}}, overrightarrow{boldsymbol{a}}-mathbf{2} overrightarrow{boldsymbol{b}}+boldsymbol{lambda} overrightarrow{boldsymbol{c}} ) and ( boldsymbol{mu} overrightarrow{boldsymbol{a}}-mathbf{5} overrightarrow{mathbf{b}} ) where ( vec{a}, vec{b}, vec{c} ) are non coplanar vectors then the points are collinear when A ( cdot lambda=-2, mu=frac{9}{4} ) B. ( lambda=-frac{9}{4}, mu=2 ) c. ( lambda=frac{9}{4}, mu=-2 ) D. None of these |
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| 471 | If the projections ofthe line segment ( A B ) on the ( y z ) -plane, ( z x ) -plane, ( x y ) -plane are ( sqrt{160}, sqrt{153}, 5 ) respectively, then the projection of ( A B ) on the ( z ) -axis is A ( cdot sqrt{12} ) B. ( sqrt{13} ) c. 12 D. 144 |
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| 472 | Find the distance of the point (-1,-5,-10) from the point of intersection of the line ( boldsymbol{r}=mathbf{2} overline{boldsymbol{i}}-overline{boldsymbol{j}}+mathbf{2} overline{boldsymbol{k}}+overline{boldsymbol{lambda}}(mathbf{3} overline{boldsymbol{i}}+mathbf{4} overline{boldsymbol{j}}+mathbf{2} overline{boldsymbol{k}}) ) and the plane ( bar{r} .(bar{i}-bar{j}+bar{k})=5 ) |
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| 473 | The coordinates of a point which divides the line joining the points ( P(2,3,1) ) and ( Q(5,0,4) ) in the ratio 1: 2 are ( ^{mathbf{A}} cdotleft(frac{7}{3}, 1, frac{5}{3}right) ) в. (4,1,3) c. (3,2,2) D. (1,-1,1) |
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| 474 | Find the value of ( k ) if the pts ( A equiv ) ( (1,2,-1), B equiv(4,-2,4) ) and ( C equiv ) ( (0,0, k) ) form a triangle right angled at ( boldsymbol{C} ) |
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| 475 | If ( P(3,2,-4), Q(5,4,-6) ) and ( R ) (9,8,-10) are collinear, then ( R ) divides PQ in the ratio A. 3: 2 internally B. 3: 2 externally c. 2: 1 internally D. 2: 1 externally |
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| 476 | Are the points (1,1),(2,3) and (8,11) collinear? A. collinear B. Non collinear c. coplaner D. None of above |
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| 477 | The distance between the parallel planes given by the equations, ( vec{r} .(2 hat{i}- ) ( mathbf{2} hat{mathbf{j}}+hat{boldsymbol{k}})+mathbf{3}=mathbf{0} ) and ( vec{r} cdot(mathbf{4} hat{mathbf{i}}-mathbf{4} hat{mathbf{j}}+ ) ( mathbf{2} hat{boldsymbol{k}})+mathbf{5}=mathbf{0} ) is – A ( cdot 1 / 2 ) в. ( 1 / 3 ) c. ( 1 / 4 ) D. ( 1 / 6 ) |
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| 478 | The equation of a plane which passes through the point of intersection of lines ( frac{x-1}{3}=frac{y-2}{1}=frac{z-3}{2}, ) and ( frac{x-3}{1}= ) ( frac{boldsymbol{y}-mathbf{1}}{mathbf{2}}=frac{boldsymbol{z}-mathbf{2}}{mathbf{3}} ) and at greatest distance from point (0,0,0) is- A. ( 4 x+3 y+5 z=25 ) B. ( 4 x+3 y+5 z=50 ) c. ( 3 x+4 y+5 z=49 ) D. ( x+7 y-5 z=2 ) |
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| 479 | ( boldsymbol{A}=(mathbf{1},-mathbf{1}, mathbf{2}) ) and ( boldsymbol{B}=(mathbf{2}, mathbf{3}, mathbf{7}) ) are two points. If ( boldsymbol{P}, boldsymbol{O} ) divide ( boldsymbol{A B} ) in the ratios 2: 3,-2: 3 respectively then ( P_{x}+ ) ( boldsymbol{Q}_{boldsymbol{y}}= ) A ( cdot frac{-38}{5} ) в. ( frac{38}{5} ) c. ( frac{-2}{5} ) D. ( frac{-47}{6} ) |
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| 480 | A line makes the same angle ( theta ) with each of the ( x ) and ( z ) -axes. If the angle ( beta ) which it makes with ( y ) -axis, is such that ( sin ^{2} beta=3 sin ^{2} theta, ) then ( cos ^{2} theta ) is equal to A ( cdot frac{2}{3} ) B. ( frac{1}{5} ) ( c cdot frac{3}{5} ) D. |
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| 481 | If ( boldsymbol{A}(1,2,-1), B(4,0,-3), C(1,2,-1), D( ) find the distance between ( A B ) and ( C D ) |
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| 482 | 90. The equation of the line passing through (4,3,1), parallel to the plane x + 2y – Z-5 = 0 and intersecting the line x +1 y-3 2-2 [JEE M 2019-9 Jan (M) -3 2 – 1. is: x – 4 y +3 z+1 2 1 |
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| 483 | The image of the point ( P(1,2,3) ) in the plane ( 2 x-y+z+3=0 ) is A ( cdot(-3,4,1) ) в. (3,5,2) c. (-3,5,2) D. (3,-5,2) |
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| 484 | If ( A ) and ( B ) be the points (3,4,5) and (-1,3,-7) respectively. Find the equation of the set of points ( boldsymbol{P} ) such that ( boldsymbol{P} boldsymbol{A}^{2}+boldsymbol{P} boldsymbol{B}^{2}=boldsymbol{K}^{2}, ) where ( boldsymbol{K} ) is a constant |
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| 485 | If sum of the perpendicular distances of a variable point ( boldsymbol{P}(boldsymbol{x}, boldsymbol{y}) ) from the lines ( boldsymbol{x}+boldsymbol{y}-mathbf{5}=mathbf{0} ) and ( mathbf{3} boldsymbol{x}-mathbf{2} boldsymbol{y}+mathbf{7}=mathbf{0} ) is always ( 10 . ) Show that ( P ) must move on a line. |
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| 486 | 14. The image of the line *<!-Y-3-2-4 in the plane 3 2x – y +2+3= 0 is the line: [JEE M 2014) x-3 y + 5 z-2 X-3 y +5 2-2 -3 – 15 x +3 15 Z -2 x+3y-5z +2 -3 -1 3 -5 |
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| 487 | Two planes intersect each other to form ( mathbf{a}: ) A. plane B. point c. straight line D. angle |
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| 488 | Find the direction cosines of vector ( vec{r} ) which is equally inclined to ( O X, O Y ) and ( O Z ). Find total number of such vectors. A ( cdot frac{1}{sqrt{3}}, frac{1}{sqrt{3}}, frac{1}{sqrt{3}} ; 6 ) B. ( frac{1}{sqrt{3}}, pm frac{1}{sqrt{3}}, frac{1}{sqrt{3}} ; 8 ) ( ^{mathbf{C}} pm frac{1}{sqrt{3}}, pm frac{1}{sqrt{3}}, pm frac{1}{sqrt{3}} ; 8 ) D. None of these |
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| 489 | If the projection of a line segment on ( x, y ) and ( z ) axes are respectively 3,4 and ( 5, ) then the length of the line segment is begin{tabular}{l} A ( 3 sqrt{2} ) \ hline end{tabular} B. ( 5 sqrt{2} ) c. ( 6 sqrt{2} ) D. None of these |
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| 490 | Show that the points with position vectors ( vec{a}+vec{b}, vec{a}-vec{b} ) and ( vec{a}+k vec{b} ) are collinear for all values of ( k ) |
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| 491 | The equation of the plane through the points (2,3,1) and (4,-5,3) and parallel to ( x ) -axis is A. ( x-z-1=0 ) в. ( 4 x+y-11=0 ) c. ( y+4 z-7=0 ) D. None of these |
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| 492 | If the d.rs of ( O A ) and ( O B ) are 1,-1,-1 and ( 2,-1,1, ) then the d.cs of the line perpendicular to both ( boldsymbol{O} boldsymbol{A} ) and ( boldsymbol{O B} ) are в. -2,-3,1 c. ( frac{-2}{sqrt{14}}, frac{-3}{sqrt{14}}, frac{1}{sqrt{14}} ) D. ( frac{2}{sqrt{41}}, frac{3}{sqrt{41}}, frac{-1}{sqrt{41}} ) |
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| 493 | Point ( D ) has coordinates as (3,4,5) Find the coordinates of point ( G ) ( mathbf{A} cdot(0,3,5) ) B. (3,0,4) ( mathbf{C} cdot(3,5,4) ) ( mathbf{D} cdot(3,0,5) ) |
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| 494 | The number of straight lines that are equally inclined to the threedimensional coordinate axes, is A .2 B. 4 ( c .6 ) D. 8 |
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| 495 | ( frac{x-2}{1}=frac{y-3}{1}=frac{z-4}{-1} & frac{x-1}{k}= ) ( frac{boldsymbol{y}-boldsymbol{4}}{boldsymbol{2}}=frac{boldsymbol{z}-boldsymbol{5}}{boldsymbol{2}} ) are coplanar then ( mathbf{k}=? ) A. any value B. exactly one value c. exactly 2 values D. exactly 3 values |
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| 496 | Determine if the points (1,5)(2,3) and (-2,-11) are collinear. A. True B. False |
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| 497 | The direction cosines of a line whose equations are ( frac{x-1}{2}=frac{y+3}{4}=frac{z-2}{-3} ) A ( cdot frac{1}{sqrt{14}}, frac{-3}{sqrt{14}}, frac{2}{sqrt{14}} ) В ( cdot frac{2}{sqrt{29}}, frac{4}{sqrt{29}}, frac{-3}{sqrt{29}} ) c. ( frac{1}{sqrt{29}}, frac{-3}{sqrt{29}}, frac{2}{sqrt{29}} ) D. 2,4,-3 |
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| 498 | Solve the following differential equation ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=boldsymbol{x}-mathbf{1} ) A ( cdot y=x^{2}+x ) B . ( y=x^{2} ) c. ( y=x^{2}-x ) D. None of the above |
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| 499 | If the foot of the perpendicular from (0,0,0) to a plane is ( P(1,2,2) . ) Then, the equation of the plane is A. ( -x+2 y+8 z-9=0 ) B. ( x+2 y+2 z-9=0 ) c. ( x+y+z-5=0 ) D. ( x+2 y-3 z+3=0 ) |
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| 500 | The ratio in which ( x y- ) plane divides the line joining the points (1,0,-3) and (1,-5,7) is given by A . 7: 3 в. 3: 7 ( c .3: 4 ) D. 4: 7 |
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| 501 | ( vec{A} cdot(vec{A} times vec{B}) ) ( A cdot vec{A} cdot vec{B} ) ( mathbf{B} cdot underset{A}{longrightarrow} times underset{B}{longrightarrow} ) ( c cdot 0 ) D. 1 |
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| 502 | The distance of the point (1,-2,3) from the plane ( boldsymbol{x}-boldsymbol{y}+boldsymbol{z}=mathbf{5} ) measured parallel to the line. ( frac{x}{2}=frac{y}{3}=frac{z}{-6}, quad ) is : ( A ) B. 6/7 ( c cdot 7 / 6 ) D. ( 1 / 6 ) |
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| 503 | Find the vector equation of the line through ( boldsymbol{A}(boldsymbol{3}, boldsymbol{4},-boldsymbol{7}) ) and ( boldsymbol{B}(boldsymbol{6},-1,1) ) Also find the cartesian form. |
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| 504 | One of the rectangular components of a force of ( 40 N ) is ( 20 N . ) Find the angle it makes with this component and magnitude of other component. |
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| 505 | Find the distance of the point (1,2,-1) from the plane ( boldsymbol{x}-mathbf{2} boldsymbol{y}+mathbf{4} boldsymbol{z}-mathbf{1 0}= ) ( mathbf{0} ) |
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| 506 | Consider three vectors ( overrightarrow{boldsymbol{p}}=boldsymbol{i}+boldsymbol{j}+ ) ( boldsymbol{k}, overrightarrow{boldsymbol{q}}=2 boldsymbol{i}+boldsymbol{4} boldsymbol{j}-boldsymbol{k} ) and ( overrightarrow{boldsymbol{r}}=boldsymbol{i}+boldsymbol{j}+boldsymbol{3} boldsymbol{k} . ) ( p, q ) and ( r ) denotes the position vector of three non-collinear points, then the equation of the plane containing these points is A. ( 2 x-3 y+1=0 ) B. ( x-3 y+2 z=0 ) c. ( 3 x-y+z-3=0 ) D. ( 3 x-y-2=0 ) |
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| 507 | A line makes equal angles with the coordinate axis. The direction cosines of this line are A ( cdotleft(frac{1}{3}, frac{1}{3}, frac{1}{3}right) ) B ( cdotleft(frac{1}{sqrt{3}}, frac{1}{sqrt{3}}, frac{1}{sqrt{3}}right) ) ( ^{mathbf{c}} cdotleft(frac{1}{sqrt{3}}, frac{1}{3}, frac{1}{3}right) ) D ( cdotleft(frac{1}{sqrt{2}}, frac{1}{sqrt{2}}, frac{1}{sqrt{2}}right) ) |
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| 508 | [2004] 25. If the straight lines 1+t, z=2-t, x=1+s, y=-3-as,z=1+as and x with parameters s and t respectively, are co-planar, then a equals. (a) 0 (6) 1 (c) — 1 (d) -2 1. 1 . |
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| 509 | The expression in the vector form for the point ( vec{r}_{1} ) of intersection of the plane ( vec{r}_{1} cdot vec{n}=d ) and the perpendicular line ( vec{r}=vec{r}_{0}+t vec{n} ) where ( t ) is a parameter given by A ( cdot overrightarrow{r_{1}}=overrightarrow{r_{0}}+left(frac{d-overrightarrow{r_{0}} cdot vec{n}}{vec{n}^{2}}right) vec{n} ) B ( cdot overrightarrow{r_{1}}=overrightarrow{r_{0}}+left(frac{overrightarrow{r_{0}} cdot vec{n}}{vec{n}^{2}}right) vec{n} ) C・ ( _{overrightarrow{r_{1}}}=overrightarrow{r_{0}}+left(frac{overrightarrow{r_{0}} cdot vec{n}-d}{|vec{n}|}right) vec{n} ) D ( overrightarrow{r_{1}}=overrightarrow{r_{0}}+left(frac{overrightarrow{r_{0}} cdot vec{n}}{|vec{n}|}right) vec{n} ) |
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| 510 | what is the distance of the point (p,q,r) from the ( x ) -axis ? | 12 |
| 511 | ( A B C D ) is a parallelogram. ( L ) is a point on ( B C ) which divides ( B C ) in the ratio ( mathbf{1}: mathbf{2} . boldsymbol{A} boldsymbol{L} ) intersects ( boldsymbol{B} boldsymbol{D} ) at ( boldsymbol{P} . boldsymbol{M} ) is a point on ( D C ) which divides ( D C ) in the ratio 1: 2 and ( A M ) intersects ( B D ) in ( Q ) Point ( P ) divides ( A L ) in the ratio A . 1: 2 B. 1: 3 c. 3: 1 D. 2: 1 |
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| 512 | For waht value of ( lambda ), the three numbers ( 2 lambda-1, frac{1}{4}, lambda-frac{1}{2} ) can be the direction cosines of a straight line? A ( cdot frac{1}{2} pm frac{sqrt{3}}{4} ) B. ( frac{3}{4} ) ( c cdot pm frac{3}{4} ) D. ( frac{sqrt{3}}{2} pm frac{1}{4} ) |
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| 513 | If the sum of the squares of the distance of a point from the three coordinate axes be ( 36, ) then its distance from the origin is A. 6 units B. ( 3 sqrt{2} ) units c. ( 2 sqrt{3} ) units D. none of these |
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| 514 | If (1,-1,0),(-2,1,8) and (-1,2,7) are three consecutive vertices of a parallelogram then the fourth vertex is A. (2,0,-1) В. (1,0,-1) c. (1,-2,0) a 5 D. (0,-2,1) |
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| 515 | In ( triangle A B C ) the mid points of the sides ( A B, B C ) and ( C A ) are respectively ( (l, 0,0),(0, m, 0) ) and ( (0,0, n) . ) Then ( frac{A B^{2}+B C^{2}+C A^{2}}{l^{2}+m^{2}+n^{2}} ) is equal to ( A cdot 2 ) B. 4 c. 8 D. 16 |
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| 516 | A point at which all the three perpendicular coordinate axes meets is known as A. Meeting point B. Origin c. Triple point D. None of these |
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| 517 | Find the coordinate of the point ( boldsymbol{P} ) where the line through ( A(3,-4,-5) ) and ( B(2,-3,1) ) crosses the plane passing through three points ( boldsymbol{L}(mathbf{2}, mathbf{2}, mathbf{1}), boldsymbol{M}(mathbf{3}, mathbf{0}, mathbf{1}) ) and ( boldsymbol{N}(mathbf{4},-mathbf{1}, mathbf{0}) ) Also, find the ratio in which ( P ) divides the line segment ( boldsymbol{A B} ) |
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| 518 | 35. If the plane 2ax – 3ay + 4az + 6 = 0 passes through the midpoint of the line joining the centres of the spheres x2 + y2 +z2 + 6x – 8y – 2z = 13 and x2 + y2 +22 – 10x + 4y – 2z = 8 then a equals [2005] (a) -1 (b) 1 (c) – (d) 2 |
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| 519 | The ratio in which the plane ( vec{r} .(hat{i}-2 hat{j}+ ) ( mathbf{3} hat{k})=17 ) divides the line joining the points ( (-2 hat{i}+4 hat{j}+7 hat{k}) ) and ( (3 hat{i}-5 hat{j}+ ) ( mathbf{8} hat{boldsymbol{k}}) ) is A .1: 5 B. 1: 10 ( c .3: 5 ) D. 3: 10 |
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| 520 | Find the equation of line of intersection of planes ( vec{r} cdot(3 vec{i}-vec{j}+vec{k})=1 ) and ( vec{r} ) ( (3 vec{i}+4 vec{j}-2 vec{k})=2 ) |
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| 521 | Find he equation of the line passing through (1,2,3) and perpendicular to the two lines ( frac{x}{1}=frac{y}{2}=frac{z}{-1} ) and ( frac{x-1}{3}=frac{y}{2}=frac{z}{6} ) | 12 |
| 522 | Plane passing through the points ( A(2,1 ) 3), ( mathrm{B}(-1,2,4) ) and ( mathrm{C}(0,2,1) ). Determine its point of intersection with the line ( r= ) ( boldsymbol{j}+boldsymbol{k}+boldsymbol{t}(boldsymbol{2} boldsymbol{i}+boldsymbol{k}) ) A ( cdot(7,+1,4) ) B. (9,+1,-2) c. (7,-1,4) D. (9,-1,2) |
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| 523 | If ( theta ) is the angle between two lines whose d.cs are ( l_{1}, m_{1}, n_{1} ) and ( l_{2}, m_{2}, n_{2} ) then ( frac{Sigmaleft(l_{1}+l_{2}right)^{2}}{4 cos ^{2}left(frac{theta}{2}right)}+frac{Sigmaleft(l_{mathrm{I}}-l_{2}right)^{2}}{4 sin ^{2}left(frac{theta}{2}right)}= ) ( A ) B. c. -1 ( D ) |
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| 524 | ( mathbf{2} boldsymbol{x}-boldsymbol{y}+mathbf{3} boldsymbol{z}-mathbf{1}=mathbf{0} ) and ( mathbf{2} boldsymbol{x}-boldsymbol{y}+ ) ( mathbf{3} boldsymbol{z}+mathbf{3}=mathbf{0} ) |
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| 525 | Find the direction cosines of a line which is perpendicular to the lines whose direction ratios are (1,-1,2) and (2,1,-1) |
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| 526 | The line passes through the points ( (5,1, a) &(3, b, 1) ) crosses the ( y z ) plane at the point ( left(0, frac{17}{2},-frac{13}{2}right), ) then A ( . a=4, b=6 ) B. ( a=6, b=4 ) c. ( a=8, b=2 ) D. ( a=2, b=8 ) 8 |
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| 527 | The point in the ( x y- ) plane which is equidistant from (2,0,3),(0,3,2) and (0,0,1) is A ( .(1,2,3) ) В. (-3,2,0) c. (3,-2.0) D. (3,2,0) E ( .(3,2,1) ) |
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| 528 | The equation of the plane, which bisects the line joining the points (1,2,3) and (3,4,5) at right angles is? A. ( x+y+z=0 ) B. ( x+y-z=9 ) c. ( x+y+z=9 ) D. ( x+y-z+9=0 ) |
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| 529 | Find vector equation of line passing through the point whose position vector is ( mathbf{3} hat{mathbf{i}}-mathbf{4} hat{mathbf{j}}+hat{boldsymbol{k}} ) and parallel to the vector ( 2 hat{i}+hat{j}-3 hat{k} . ) Also write the equation in Cartesian form. |
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| 530 | Find the coordinates of the point ( boldsymbol{P} ) where the line through ( boldsymbol{A}(mathbf{3},-mathbf{4},-mathbf{5}) ) and ( B(2,-3,1) ) crosses the plane passing through three points ( boldsymbol{L}(mathbf{2}, mathbf{2}, mathbf{1}), boldsymbol{M}(mathbf{3}, mathbf{0}, mathbf{1}) ) and ( boldsymbol{N}(mathbf{4},-mathbf{1}, mathbf{0}) ) Also, find the ratio in which ( P ) divides the line segment ( boldsymbol{A B} ) |
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| 531 | The distance between two points (1,1) and ( left(frac{2 t^{2}}{1+t^{2}}, frac{(1-t)^{2}}{1+t^{2}}right) ) is ( A cdot 4 t ) B. 3t ( c ) D. none of these |
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| 532 | The coordinates of the foot of the perpendicular drawn from the point ( A(1,0,3) ) to the join of the points ( B(4,7,1) ) and ( C(3,5,3) ) are A ( cdot(5,7,17) ) B ( cdotleft(frac{-5}{7}, frac{7}{3}, frac{-17}{3}right) ) ( ^{mathbf{c}} cdotleft(frac{5}{7}, frac{-7}{3}, frac{17}{3}right) ) D. ( left(frac{5}{7}, frac{7}{3}, frac{17}{3}right) ) |
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| 533 | 1. The angles which a vector i + j + 2 k makes with X, Y and Z axes respectively are (a) 60°, 60°, 60° (b) 45°, 45°, 45° (c) 60°, 60°, 45° (d) 45°, 45°, 60° |
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| 534 | The angle between the line ( 2 x=3 y= ) ( -z ) and ( 6 x=-y=-4 z ) is A ( .90^{circ} ) B . ( 0^{circ} ) ( c cdot 30^{0} ) D. ( 45^{circ} ) |
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| 535 | The equation of a line passing through (2,-3) and inclined at an angle of ( 135^{circ} ) with the positive direction of ( x ) -axis is. A. ( x+y-1=0 ) в. ( x+y+1=0 ) c. ( x-y-1=0 ) D. ( -x+y+1=0 ) |
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| 536 | A ray makes angles ( frac{pi}{3}, frac{pi}{3} ) with ( overrightarrow{O X} ) and ( boldsymbol{O Y} ) respectively. Find the angle made by it with ( overrightarrow{O Z} ) | 12 |
| 537 | 49. If (2,3,5) is one end of a diameter of the sphere x2 + y2 +22 – 6x – 12y – 2z + 20 = 0, then the cooordinates of the other end of the diameter are [2007] (a) (4,3,5) (b) (4,3,-3) (c) (4,9, -3) (d) (4, -3,3). |
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| 538 | If ( boldsymbol{A} times boldsymbol{B}=boldsymbol{B} times boldsymbol{A}, ) then the angle between ( A ) and ( B ) is A. ( pi ) в. ( pi / 3 ) c. ( pi / 2 ) D. ( pi / 4 ) |
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| 539 | The chord of contact of tangents from a point ( boldsymbol{P} ) to a circle passes through ( boldsymbol{q} ). If ( l_{1} ) and ( l_{2} ) are the lengths of the tangents from ( boldsymbol{P} ) and ( boldsymbol{Q} ) to the circle, then ( boldsymbol{P} boldsymbol{Q} ) is equal to A. ( frac{l_{1}+l_{2}}{2} ) в. ( frac{l_{1}-l_{2}}{2} ) C ( . sqrt{left|l_{1}^{2}-l_{2}^{2}right|} ) D. ( sqrt[2]{l_{1}^{2}+l_{2}^{2}} ) |
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| 540 | In what ratio does the plane ( 2 x+y- ) ( z=3 ) divide line segment joining the point ( boldsymbol{a}=(mathbf{2}, mathbf{1}, mathbf{3}), boldsymbol{b}=(mathbf{9},-mathbf{2}, mathbf{5}) ) |
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| 541 | If (1,-2,-2) and (0,2,1) are direction ratios of two lines, then the direction cosines of a perpendicular to both the lines are ( ^{mathrm{A}} cdotleft(frac{1}{3},-frac{1}{3}, frac{2}{3}right) ) в. ( left(frac{2}{3},-frac{1}{3}, frac{2}{3}right) ) ( ^{c} cdotleft(-frac{2}{3},-frac{1}{3}, frac{2}{3}right) ) D. ( left(frac{2}{sqrt{14}},-frac{1}{sqrt{14}}, frac{3}{sqrt{14}}right) ) |
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| 542 | If the vectors ( 3 bar{p}+bar{q} ; 5 bar{p}-3 bar{q} ) and ( 2 bar{p}+ ) ( bar{q} ; 4 bar{p}-2 bar{q} ) are pairs of mutually perpendicular vectors then ( sin (theta) ) is ( (theta ) is the angle between ( overline{boldsymbol{p}} ) and ( overline{boldsymbol{q}} ) A ( cdot sqrt{55} / 4 ) B . ( sqrt{55} / 8 ) c. ( 3 / 16 ) D. ( sqrt{247} / 16 ) |
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| 543 | A hall has dimensions ( 24 m times 8 m times ) 6 ( m ). The length of the longest pole which can be accommodated in the hall is A. 26 B. 28 m ( c cdot 30 m ) D. 36 m |
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| 544 | Let ( vec{a}=x^{2} hat{i}+2 hat{j}-2 hat{k}, vec{b}=hat{i}-hat{j}+hat{k} ) and ( vec{c}=x^{2} hat{i}+5 hat{j}-4 hat{k} ) be three vectors. Find the values of ( x ) for which the angle between ( vec{a} ) and ( vec{b} ) is acute and the angle between ( vec{b} ) and ( vec{c} ) is obtuse. A ( cdot(-3,-2) cup(2,3) ) B . (-3,-1)( cup(1,3) ) c. (-3,-1)( cap(1,3) ) D. (-3,-2)( cap(2,3) ) |
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| 545 | 91. The plane through the intersection of the planes x+y+z=1 and 2x+3y-z+4= 0 and parallel to y-axis also passes through the point: JEEM 2019-9 Jan (MI (a) (-3,0,-1) (b) (-3,1,1) (c) (3,3,-1) (d) (3, 2, 1) |
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| 546 | Find ( x ) so that the point (6,5,-3) is at a distance of 13 from the point ( (x,-7,0) ) | 12 |
| 547 | Find the locus of the point, the sum of the squares of whose distances from the planes ( boldsymbol{x}+boldsymbol{y}+boldsymbol{z}=mathbf{0}, boldsymbol{x}-boldsymbol{y}=mathbf{0} ) ( x+y-2 z=0 ) is 7 |
12 |
| 548 | Find the equation of line passing through ( (5,0,5) &(2,1,3) . ) Also show that ( (5,0,5),(2,1,3) &(-4,3,-1) ) are collinear. |
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| 549 | Let the equation of the plane through the points (-1,1,1) and (1,-1,1) and perpendicular to the plane ( boldsymbol{x}+mathbf{2} boldsymbol{y}+ ) ( mathbf{2} z=mathbf{7} ) be ( boldsymbol{k} boldsymbol{x}+boldsymbol{m} boldsymbol{y}-boldsymbol{n} boldsymbol{z}+boldsymbol{p}=mathbf{0} . ) Find ( boldsymbol{k}+boldsymbol{m}+boldsymbol{n}+boldsymbol{p} ? ) |
12 |
| 550 | ( boldsymbol{x}+boldsymbol{y}+boldsymbol{z}=mathbf{1} ) | 12 |
| 551 | The distance of the point (1,-5,9) from the planer. ( (hat{mathbf{i}}-hat{mathbf{j}}+hat{boldsymbol{k}})=mathbf{5} ) measured long the line ( r=hat{i}+hat{j}+hat{k} ) is A ( 3 sqrt{5} ) 5 B. ( 10 sqrt{3} ) ( c cdot 5 sqrt{3} ) D. ( 3 sqrt{10} ) |
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| 552 | Find the direction cosines of the unit vector perpendicular to the plane ( vec{r} cdot(6 hat{i}-3 hat{j}-2 hat{k})+1=0 ) |
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| 553 | The coordinates of any point, which lies in ( boldsymbol{y} boldsymbol{z} ) plane, are This question has multiple correct options A ( .(x, y, y) ) в. ( (0, y, y) ) c. ( (0, y, x) ) D. ( (x, y, z) ) |
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| 554 | 34. The line passing through the points (5, 1, a) and (3, b, 1) crosses the yz-plane at the point (0,7 -13). Then (a) a=2, b=8 (b) a=4, b=6 (c) a=6, b=4 (d) a=8, b=2 |
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| 555 | A line passes through the point (6,-7,-1) and ( (2,-3,1) . ) if the angle ( alpha ) which the line makes with the positive direction of ( x ) -axis is acute, the direction cosines of the line are, A ( .2 / 3,-2 / 3,-1 / 3 ) в. ( 2 / 3,2 / 3,-1 / 3 ) c. ( 2 / 3,-2 / 3,1 / 3 ) D. ( 2 / 3,2 / 3,1 / 3 ) |
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| 556 | Find the equations to the straight lines which are conjugate to the coordinate axes with respect to the conic ( A x^{2}+ ) ( 2 H x y+B y^{2}=1 ) Find the condition that they may coincide, and interpret the result. |
12 |
| 557 | 80. Z+4 3 lies in the plane, kx +my-z=9, x-3 y +2 If the line, 2 -1 then 12 + m2 is equal to : (a) 5 (b) 2 (c) 26 [JEE M 2016] (d) 18 |
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| 558 | x – 1 V -4 z 5 1 1 -k 15. The lines *-2 =973 – 3-4 and **= coplanar if (a) k=3 or-2 b) k=0 or -1 (c) k=1 or-1 1) k=0 or-3 =-7 are [2003] |
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| 559 | Let ( boldsymbol{A}=(mathbf{1}, mathbf{2}, mathbf{3}) boldsymbol{B}= ) ( (-1,-2,-1) C=(2,3,2) ) and ( D= ) ( (4,7,6) . ) Then ( A B C D ) is a A. rectangle B. square c. parallelogram D. none of these |
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| 560 | If the projections of the line segment ( A B ) on the coordinate axes are ( 12,3, k ) and ( A B=13, ) then ( k^{2}-2 k+3 ) is equal to: This question has multiple correct options A . 0 B. c. 11 D. 27 |
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| 561 | Find the coordinate of the points which trisect the line segment joining the points ( boldsymbol{A}(mathbf{2}, mathbf{1},-mathbf{3}) ) and ( boldsymbol{B}(mathbf{5},-mathbf{8}, mathbf{3}) ) | 12 |
| 562 | If ( mathbf{A}=(-mathbf{1}, mathbf{6}, mathbf{6}), mathbf{B}=(-mathbf{4}, mathbf{9}, mathbf{6}), mathbf{G}= ) ( frac{1}{3}(-5,22,22) ) and ( G ) is the centroid of the ( Delta A B C ) then the name of the triangle ( mathbf{A B C} ) is A. an isosceles triangle B. a right angled triangle c. an equilateral triangle D. a right-angled isosceles triangle |
12 |
| 563 | Find ( x, ) if ( triangle A B C ) is right-angled at ( A ) where ( boldsymbol{A} equiv(mathbf{4}, mathbf{2}, mathbf{3}), boldsymbol{B} equiv(mathbf{3}, mathbf{1}, mathbf{8}), boldsymbol{C} equiv ) ( (x,-1,2) ) |
12 |
| 564 | A plane ( pi ) makes intercepts 3 and 4 respectively on ( z- ) axis. If ( pi ) is parallel to ( boldsymbol{y}- ) axis, then its equation is ? A. ( 3 x+4 z=12 ) B. ( 3 z+4 x=12 ) c. ( 3 y+4 z=12 ) D. ( 3 z+4 y=12 ) |
12 |
| 565 | Determine whether the points are collinear. ( boldsymbol{P}(-mathbf{2}, mathbf{3}), boldsymbol{B}(mathbf{1}, mathbf{2}), boldsymbol{C}(mathbf{4}, mathbf{1}) ) |
12 |
| 566 | If ( P(x, y, z) ) is a point on the line segment joining ( boldsymbol{A}(mathbf{2}, mathbf{2}, mathbf{4}) ) and ( B(3,5,6) ) such that projection of ( overrightarrow{O P} ) on axes are ( frac{13}{5}, frac{19}{5}, frac{26}{5} ) respectively, then ( P ) divide AB in the ratio A .3: 2 B. 2: 3 c. 1: 2 D. 1: 3 |
12 |
| 567 | Find the direction cosines of the line passing through the two points (-2,4,-5) and (1,2,3) |
12 |
| 568 | Which one of the following is best condition for the plane ( a x+b y+c z+ ) ( d=0 ) to intersect the ( x ) and ( y ) axes at equal angle A ( cdot|a|=|b| ) ( b mid ) B . ( a=-b ) c. ( a=b ) D. ( a^{2}+b^{2}=1 ) |
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| 569 | The point of intersection of the line joining the points (-3,4,-8) and (5,-6,4) with the ( X Y ) -plane is ( mathbf{A} cdotleft(frac{7}{3},-frac{8}{3}, 0right) ) в. ( left(-frac{7}{3},-frac{8}{3}, 0right) ) ( ^{mathbf{c}} cdotleft(-frac{7}{3}, frac{8}{3}, 0right) ) D. ( left(frac{7}{3}, frac{8}{3}, 0right) ) |
12 |
| 570 | If ( P(x, y, x) ) is a point on the line segment joining ( Q(2,2,4) ) and ( R(3,5,6) ) such that the projection of ( O P ) on the axis are ( frac{13}{5}, frac{19}{5}, frac{26}{5} ) respectively, then ( P ) divides ( Q R ) in the ratio A . 1: 2 B. 3: 2 ( c cdot 2: 3 ) D. 1: 3 |
12 |
| 571 | The xy-plane divides the line joining the points (-1,3,4) and (2,-5,6) A. internally in the ratio 2: 3 B. externally in the ratio 2: 3 c. internally in the ratio 3: 2 D. externally in the ratio 3: 2 |
12 |
| 572 | Point ( (boldsymbol{alpha}, boldsymbol{beta}, boldsymbol{gamma}) ) lies on the plane ( boldsymbol{x}+boldsymbol{y}+ ) ( z=2 . ) Let ( vec{a}=alpha hat{i}+beta hat{j}+gamma hat{k} ) and ( hat{k} times ) ( (hat{k} times vec{a})=0 ) then ( gamma= ) A . B. 1 c. 2 D. |
12 |
| 573 | Write the abscissa of the following point (0,5) |
12 |
| 574 | The ratio in which the line joining (2,-4,3) and (-4,5,-6) is divided by the plane ( 3 x+2 y+z-4=0 ) is A .2: 1 B . 4: 3 ( c cdot-1: 4 ) D. 2: 3 |
12 |
| 575 | If the point ( P(a, b, c), ) with reference to Eq. ( (i) ) lies on the plane ( 2 x+y+z=1 ) then the value of ( 7 a+b+c ) is ( mathbf{A} cdot mathbf{0} ) B. 12 ( c cdot 7 ) D. 6 |
12 |
| 576 | The image of the point ( 3 hat{i}-2 hat{j}+hat{k} ) in the plane ( bar{r} .(3 hat{i}-hat{j}+4 hat{k})=2 ) ( mathbf{A} cdot-hat{j}+3 hat{k} ) B . ( hat{j}-3 hat{k} ) ( mathbf{c} .-hat{j}-3 hat{k} ) D ( .-2 hat{j}-3 hat{k} ) |
12 |
| 577 | Given planes are ( boldsymbol{P}_{1}: boldsymbol{c} boldsymbol{y}+boldsymbol{b} boldsymbol{z}=boldsymbol{x} quad boldsymbol{P}_{2}: boldsymbol{a} boldsymbol{z}+boldsymbol{c} boldsymbol{x}=boldsymbol{y} quad boldsymbol{P}_{3} ) ( P_{1}, P_{2} ) and ( P_{3} ) pass through one line, if A ( cdot a^{2}+b^{2}+c^{2}=a b+b c+c a ) B . ( a^{2}+b^{2}+c^{2}+2 a b c=1 ) ( mathbf{c} cdot a^{2}+b^{2}+c^{2}=1 ) D. ( a^{2}+b^{2}+c^{2}+2 a b+2 b c+2 c a+2 a b c=1 ) |
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| 578 | The plane which bisects the line segment joining the points (-3,-3,4) and (3,7,6) at right angles, passes through which one of the following points? A ( .(4,1,7) ) В. (4,1,-2) c. (2,3,5) D. (2,1,3) |
12 |
| 579 | The point (3,0,-4) lies on the A. Y-axis B. z-axis c. XY-plane D. xz-plane E. YZ-plane |
12 |
| 580 | The point of intersection of the line ( frac{x-1}{3}=frac{y+2}{4}=frac{z-3}{-2} ) and plane ( 2 x-y+3 z-1=0 ) is. B . (10,10,-3) c. (-10,10,3) D. None of these |
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| 581 | Show that the points ( boldsymbol{A}(mathbf{1}, mathbf{2}, mathbf{3}) ) ( boldsymbol{B}(-1,-2,-3), C(2,3,2) ) and ( D(4,7,3) ) are the vertices of a parallelogram. |
12 |
| 582 | The distance of the point (1,-2,3) from the plane ( boldsymbol{x}-boldsymbol{y}+boldsymbol{z}=mathbf{5} ) measured parallel to the line whose direction cosines are proportional to 2,3,-5 is A ( cdot frac{9}{7} ) B. ( frac{11}{7} ) c. ( frac{15}{7} ) D. None of these |
12 |
| 583 | Find the distance between the following pairs of points: (i) (2,3,5) and (4,3,1) ( (i i)(-3,7,2) ) and ((2,4,-1) (iii) (-1,3,-4) and (1,-3,4) (iv) (2,-1,3) and (-2,1,3) |
12 |
| 584 | In the ( Delta A B C, ) if ( A B=sqrt{2} ; A C= ) ( sqrt{mathbf{2 0}}, boldsymbol{B}=(mathbf{3}, mathbf{2}, mathbf{0}) ) and ( boldsymbol{C}=(mathbf{0}, mathbf{1}, mathbf{4}) ) then the length of the median passing through ( boldsymbol{A} ) is A ( cdot frac{3}{2} ) B. ( frac{9}{2} ) c. ( frac{3}{sqrt{2}} ) D. ( frac{sqrt{3}}{2} ) |
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| 585 | A point on the line ( frac{boldsymbol{x}+mathbf{2}}{mathbf{1}}=frac{boldsymbol{y}-mathbf{3}}{-mathbf{4}}= ) ( frac{z-1}{2 sqrt{2}} ) at a distance 6 from the point ( (2, ) 3, 1) is A ( cdot(4-21,1+12 sqrt{2}) ) В. ( left(frac{-4}{5}, frac{-9}{5}, 1right) ) C ( cdotleft(frac{-16}{5}, frac{39}{5}, frac{5-12 sqrt{2}}{5}right) ) D. ( left(frac{-16}{5},-21,1+12 sqrt{2}right) ) |
12 |
| 586 | The coordinates of the point where the line segment joining ( boldsymbol{A}(mathbf{5}, mathbf{1}, boldsymbol{6}) ) and ( B(3,4,1) ) crosses the yz plane are A ( cdotleft(0, frac{17}{2}, frac{13}{2}right) ) в. ( left(0,-frac{17}{2}, frac{13}{2}right) ) c. ( left(0, frac{17}{2},-frac{13}{2}right) ) D. ( left(0,-frac{17}{2},-frac{13}{2}right) ) |
12 |
| 587 | Show that the points ( boldsymbol{A}(boldsymbol{3}, boldsymbol{2},-boldsymbol{4}), boldsymbol{B}(boldsymbol{5}, boldsymbol{4},-boldsymbol{6}) ) and ( C(9,8,-10) ) are collinear, find the ratio in which ( B ) divides ( overline{A C} ). |
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| 588 | Find the direction cosines of the sides of the triangle whose vertices are (3,5,-4),(-1,1,2) and (-5,-5,-2) |
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| 589 | The distance between the X-axis and the point (3,12,5) is A. 3 B. 13 ( c cdot 14 ) D. 12 E. 5 |
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| 590 | The line ( frac{x-2}{3}=frac{y+1}{2}=frac{z-1}{-1} ) intersects the curve ( x y=c^{2}, z=0 ) if ( c ) is equal to: ( A cdot pm 1 ) B. ( pm frac{1}{3} ) ( mathrm{c} cdot pm sqrt{5} ) D. None of these |
12 |
| 591 | If the points ( (h, 3,-4),(0,-7,10) ) and ( (1, k, 3) ) are collinear, then ( h+k ) is ( mathbf{A} cdot mathbf{4} ) B. c. -4 D. 14 |
12 |
| 592 | The direction cosines of the line passing through ( mathbf{P}(mathbf{2}, mathbf{3},-mathbf{1}) ) and the origin are A ( cdot frac{2}{sqrt{14}}, frac{3}{sqrt{14}}, frac{1}{sqrt{14}} ) B. ( frac{2}{sqrt{14}}, frac{-3}{sqrt{14}}, frac{1}{sqrt{14}} ) c. ( frac{-2}{sqrt{14}}, frac{-3}{sqrt{14}}, frac{1}{sqrt{14}} ) D. ( frac{2}{sqrt{14}}, frac{-3}{sqrt{14}}, frac{-1}{sqrt{14}} ) |
12 |
| 593 | The vertices of a triangle are 2,3,5)( ,(-1,3,2),(3,5,-2), ) then the angles are ( mathbf{A} cdot 30^{circ}, 30^{circ}, 30^{circ} ) ( ^{mathrm{B}} cos ^{-1}left(frac{1}{sqrt{5}}right), 90^{circ}, cos ^{-1}left(frac{sqrt{5}}{sqrt{3}}right) ) c. ( 30^{circ}, 60^{circ}, 90^{circ} ) D ( cdot cos ^{-1}left(frac{1}{sqrt{3}}right), 90^{circ}, cos ^{-1}left(frac{sqrt{2}}{sqrt{3}}right) ) |
12 |
| 594 | Determine the three planes through the intersection of the planes ( boldsymbol{x}+boldsymbol{y}+boldsymbol{z}=mathbf{1} ) and ( 2 x+3 y-z+4=0 ) which are parallel to the three co – ordinate axes. Also find the equation of the plane perpendicular to the ( y z- ) plane and passing through the point (2,3,1) and (4,-5,3) A ( cdot y-3 z-6=0 ; x-4 z=7 ) and ( 3 x-4 y-3=0 ) ( y+4 z+7=0 ) B. ( y-3 z+6=0 ; x+4 z=7 ) and ( 3 x+4 y+3=0 ) ( y-4 z+7=0 ) c. ( y+3 z+6=0 ; x+4 z=7 ) and ( 3 x+4 y+3=0 ) ( y-4 z-7=0 ) D. ( y-3 z+6=0 ; x+4 z=7 ) and ( 3 x+4 y+3=0 ) ( y+4 z-7=0 ) |
12 |
| 595 | Find the angle between the planes whose vector equations are ( vec{r} cdot(2 hat{i}+2 hat{j}-3 hat{k})=5 ) and ( vec{r} cdot(3 hat{i}-3 hat{j}+5 hat{k})=3 ) |
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| 596 | If a line makes an angle ( theta_{1}, theta_{2}, theta_{3} ) which the axis respectively, then ( cos 2 theta_{1}+ ) ( cos 2 theta_{2}+cos 2 theta_{3}=? ) A . -4 B . 2 ( c .3 ) D. – |
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| 597 | The projection of the join of the two points (1,4,5),(6,7,2) on the line whose d.s’s are (4,5,6) is A ( cdot frac{17}{sqrt{77}} ) B. ( frac{7}{6} ) c. 21 D. ( frac{7}{9} ) |
12 |
| 598 | ( boldsymbol{A}(boldsymbol{3}, boldsymbol{2}, boldsymbol{0}), boldsymbol{B}(boldsymbol{5}, boldsymbol{3}, boldsymbol{2}), boldsymbol{C}(-boldsymbol{9}, boldsymbol{6},-boldsymbol{3}) ) are three points forming a triangle. If ( boldsymbol{A} boldsymbol{D} ) the bisector of ( angle B A C ) meets ( B C ) in ( D ) then coordinates of ( D ) are ( ^{mathrm{A}} cdotleft(-frac{19}{8}, frac{57}{16}, frac{17}{16}right) ) В ( cdotleft(frac{19}{8},-frac{57}{16}, frac{17}{16}right) ) ( ^{mathrm{C}} cdotleft(frac{19}{8}, frac{57}{16}, frac{17}{16}right) ) D. None of these |
12 |
| 599 | Let ( P(3,2,6) ) be point in space and ( Q ) be appoint on the line ( vec{r}=(hat{i}-hat{j}+2 hat{k})+ ) ( mu(-3 hat{i}+hat{j}+5 hat{k}) . ) Then the value of ( mu ) for which the vector ( overrightarrow{P Q} ) is parallel to the plane ( boldsymbol{x}-mathbf{4} boldsymbol{y}+mathbf{3} boldsymbol{z}=mathbf{1} ) is A. ( 1 / 4 ) B. – 1/4 c. ( 1 / 8 ) D. ( -1 / 8 ) |
12 |
| 600 | Find the direction cosines of the vector ( hat{mathbf{i}}+mathbf{2} hat{mathbf{j}}+mathbf{3} hat{boldsymbol{k}} ) |
12 |
| 601 | If the planes ( boldsymbol{x}-boldsymbol{c} boldsymbol{y}-boldsymbol{b} boldsymbol{z}=boldsymbol{0}, boldsymbol{c} boldsymbol{x}-boldsymbol{y}+ ) ( boldsymbol{a} boldsymbol{z}=mathbf{0} ) and ( boldsymbol{b} boldsymbol{x}+boldsymbol{a} boldsymbol{y}-boldsymbol{z}=mathbf{0} ) pass through a stright line,then the value of ( a^{2}+b^{2}+c^{2}+2 a b c ) is: A . 1 B. 2 ( c .3 ) D. none of these |
12 |
| 602 | The equations of two planes are ( P_{1} ) ( 2 x-y+z=2, ) and ( P_{2}: x+2 y-z= ) 3. The equation of the plane which passes through the point (-1,3,2) and is perpendicular to each of the planes ( P_{1} ) and ( P_{2} ) is A ( cdot x+3 y-5 z+2=0 ) B. ( x+3 y+5 z-18=0 ) c. ( x-3 y-5 z+20=0 ) D. ( x-3 y+5 z=0 ) |
12 |
| 603 | Let the equation of the plane through the intersection of the planes ( x+2 y+ ) ( mathbf{3} boldsymbol{z}-mathbf{4}=mathbf{0} ) and ( mathbf{2} boldsymbol{x}+boldsymbol{y}-boldsymbol{z}+mathbf{5}=mathbf{0} ) and perpendicular to the plane ( 5 x+3 y+ ) ( mathbf{6} z+mathbf{8}=mathbf{0} ) be ( boldsymbol{k} boldsymbol{x}+mathbf{1 5} boldsymbol{y}+boldsymbol{m} boldsymbol{z}+mathbf{1 7 3}= ) 0. Find ( k+m ) |
12 |
| 604 | The ( x ) -coordinate of a point on the line joining the points ( P(2,2,1) ) and ( Q(5,1,-2) ) is ( 4 . ) Find its z-coordinate. A . -1 B. -2 ( c .1 ) D. |
12 |
| 605 | The Cartesian equation of the plane ( overrightarrow{boldsymbol{r}}=(1+boldsymbol{lambda}-boldsymbol{mu}) hat{boldsymbol{i}}+(boldsymbol{2}-boldsymbol{lambda}) hat{boldsymbol{j}}+(boldsymbol{3}- ) ( 2 lambda+2 mu) hat{k} ) is- ( mathbf{A} cdot 2 x+y=5 ) В . ( 2 x-y=5 ) c. ( 2 x+z=5 ) D. ( 2 x-z=5 ) |
12 |
| 606 | The projection of the join of the points (3,4,2),(5,1,8) on the line whose d.c’s ( operatorname{are}left(frac{2}{7}, frac{3}{7}, frac{6}{7}right) ) is A. 7 B. ( frac{31}{71} ) ( mathbf{c} cdot frac{42}{13} ) ( D cdot frac{38}{138} ) |
12 |
| 607 | Find the value of ( p ) for which the points ( (-5,1),(1, p) ) and (4,-2) are collinear ( mathbf{A} cdot mathbf{1} ) B. ( c .-1 ) D. 2 |
12 |
| 608 | Find the shortest distance between the ( operatorname{lines} bar{r}=4 bar{i}-bar{j}+lambda(bar{i}+2 bar{j}-5 bar{k}) ) and ( overline{boldsymbol{r}}=overline{boldsymbol{i}}-overline{boldsymbol{j}}+2 overline{boldsymbol{k}}+boldsymbol{mu}(overline{boldsymbol{i}}+mathbf{2} overline{boldsymbol{j}}-mathbf{5} overline{boldsymbol{k}}) ) A. ( sqrt{220} ) B. ( frac{sqrt{221}}{sqrt{30}} ) c. 432 D. ( sqrt{33} ) |
12 |
| 609 | For what value of ( boldsymbol{m}, ) the points ( (boldsymbol{3}, mathbf{5}) ) ( (m, 6) ) and ( left(frac{1}{2}, frac{15}{2}right) ) are collinear? ( A cdot 9 ) B. 5 ( c .3 ) D. |
12 |
| 610 | The plane ( 2 x+3 y+k z-7=0 ) is parallel to the line whose direction ratios are (2,-3,1) then ( k= ) ( mathbf{A} cdot mathbf{5} ) B. 8 c. 1 ( D ) |
12 |
| 611 | Image of point ( mathrm{P}(1,2,3) ) with respect to plane ( boldsymbol{x}+boldsymbol{y}+boldsymbol{z}=mathbf{1 2}, ) is A. (5,4,3) ) B. (9,6,3) c. (5,6,7) D. (3,4,5) |
12 |
| 612 | The point of intersection of the lines ( overrightarrow{boldsymbol{r}}=(-overrightarrow{boldsymbol{i}}+mathbf{2} overrightarrow{boldsymbol{j}}+boldsymbol{3} overrightarrow{boldsymbol{k}})+boldsymbol{t}(-boldsymbol{2} overrightarrow{boldsymbol{i}}+overrightarrow{boldsymbol{j}}+overrightarrow{boldsymbol{k}}) ) and ( vec{r}=(2 vec{i}+3 vec{j}+5 vec{k})+s(vec{i}+2 vec{j}+ ) ( mathbf{3} overrightarrow{boldsymbol{k}}) ) is: A ( .(1,1,2) ) в. (2,1,1) c. (1,1,1) D. (1,2,1) |
12 |
| 613 | The equation of the plane which is parallel to the ( x y- ) plane is A. ( x=y ) B. ( z=c ) ( mathbf{c} cdot y=c ) D. ( z=x y ) |
12 |
| 614 | The sum of the intercepts on the coordinate axes of the plane passing through the point (-2,-2,2) and containing the line joining the points (1,-1,2) and ( (1,1,1), ) is A . 4 B. -4 c. 12 D. -8 |
12 |
| 615 | Determine if the points (5,-1,1),(7,-4,7),(1,-8,10) and (-1,-3,4) are the vertices of a rhombus or a square |
12 |
| 616 | Assertion ( (A) ). The direction ratios of the line joining origin and point ( (x, y, z) ) must be ( x, y, z )
Reason (R): If ( P(x, y, z) ) is a point in space and ( |O P|=r, ) then the direction cosines of ( O P ) are ( frac{x}{r}, frac{y}{r}, frac{z}{r} ) |
12 |
| 617 | A line separates a plane into three parts namely the two half-planes and the line itself A. True B. False |
12 |
| 618 | If the line passing through the origin makes angles ( theta_{1}, theta_{2}, theta_{3} ) with the planes ( X O Y, X O Z ) and ( Z O X ) respectively then prove that ( cos ^{2} theta_{1}+cos ^{2} theta_{2}+ ) ( cos ^{2} theta_{3}=2 ) |
12 |
| 619 | A triangle ( A B C ) is placed so that the mid-points of the sides are on the ( x, y, z ) axes. Lengths of the intercepts made by the plane containing the triangle on these axes are respectively ( boldsymbol{alpha}, boldsymbol{beta}, boldsymbol{gamma} ) Coordinates of the centroid of the triangle ( boldsymbol{A B C} ) are A ( cdot(-alpha / 3, beta / 3, gamma / 3) ) B. ( (alpha / 3,-beta / 3, gamma / 3) ) c. ( (alpha / 3, beta / 3,-gamma / 3) ) D. ( (alpha / 3, beta / 3, gamma / 3) ) |
12 |
| 620 | The point on the line ( frac{boldsymbol{x}-mathbf{1}}{mathbf{1}}=frac{boldsymbol{y}+mathbf{3}}{-mathbf{2}}= ) ( frac{z+5}{-2} ) at a distance of 6 from the point (1,-3,-5) is в. (3,-7,-9) D. (-3,5,3) |
12 |
| 621 | Find the direction cosines of the vector joining the points ( A(1,2,-3) ) and ( B(-1,-2,1) ) directed from ( A ) to ( B ) |
12 |
| 622 | The ordinate of the point which divides the lines joining the origin and the point (1,2) externally in the ratio of 3: 2 is A . -2 B. ( frac{3}{5} ) ( c cdot frac{2}{5} ) D. 6 |
12 |
| 623 | A tangent to the curve ( y=f(x) ) at ( boldsymbol{p}(boldsymbol{x}, boldsymbol{y}) ) meets ( boldsymbol{x}-boldsymbol{a} boldsymbol{x} boldsymbol{i} boldsymbol{s} ) at ( boldsymbol{A} ) and ( boldsymbol{y}- ) axis at ( B . ) If ( overline{A P}: overline{B P}=1: 3 ) and ( f(1)=1 ) then the curve also passes through the point. ( ^{mathbf{A}} cdotleft(frac{1}{2}, 4right) ) B ( cdotleft(frac{1}{3}, 24right) ) c. ( left(2, frac{1}{8}right) ) D. ( left(3, frac{1}{28}right) ) |
12 |
| 624 | The equation of the plane passing through ( (a, b, c) ) and parallel to the plane ( r cdot(hat{i}+hat{j}+hat{k})=2 ) is A. ( x+y+z=1 ) B . ( a x+b y+c z=1 ) c. ( x+y+z=a+b+c ) D. None of these |
12 |
| 625 | If a unit vector ( vec{a} ) makes angles ( frac{pi}{3} ) with ( hat{i} ) ( frac{pi}{4} ) with ( hat{j} ) and an acute angle ( theta ) with ( hat{k} ) then find ( theta ) and hence, the components of ( overrightarrow{boldsymbol{a}} ) |
12 |
| 626 | The three planes divides the space into A. four parts B. six parts c. eight parts D. sixteen parts |
12 |
| 627 | Assertion The points (1,1)( &(-1,-1) ) lie on the same side of the line ( boldsymbol{x}-boldsymbol{y}+mathbf{1}=mathbf{0} ) Reason The algebraic perpendicular distances from the given points to the line have same sign A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion c. Assertion is correct but Reason is incorrect D. Both Assertion and Reason are incorrect |
12 |
| 628 | 4. A vector Ă has components A,, A2, A, in a right-handed rectangular Cartesian coordinate system oxyz. The coordinate system is rotated about the x-axis through an T angle =. Find the components of A in the new coordinate system, in terms of A,, A2, Az. (1983 – 2 Marks) |
12 |
| 629 | The three point ( boldsymbol{A}(mathbf{0}, mathbf{0}, mathbf{0}), boldsymbol{B}(mathbf{2},-mathbf{3}, mathbf{3}), boldsymbol{C}(-mathbf{2}, mathbf{3},-mathbf{3}) ) are collinear. Find in what ratio each point divides the segment joining other two |
12 |
| 630 | The ratio in which the join of (1,-2,4) and (4,2,-1) divided by the ( X-Y ) plane is A . 1: 3 B. 3: 1 c. 4: 1 D. 1: 4 |
12 |
| 631 | The graph of the equation ( y^{2}+z^{2}=0 ) in three dimensional space is A. x- axis B. y- axis c. z- axis D. yz-plane |
12 |
| 632 | If ( frac{1}{2}, frac{1}{2}, n(n<0) ) are the dos of a line, then the angle made by that line with ( boldsymbol{O} boldsymbol{Z} ) where ( boldsymbol{O}=(mathbf{0}, mathbf{0}, mathbf{0}) ) is A ( cdot frac{-1}{sqrt{2}} ) B . ( 45^{circ} ) ( c cdot 60^{circ} ) D. ( 135^{circ} ) |
12 |
| 633 | Find the co-ordinates of the points on the join of (-3,7,-13) and (-6,1,-10) which is nearest to the intersection of the planes: ( 3 x-y- ) ( mathbf{3} z+mathbf{3 2}=mathbf{0} ) and ( mathbf{3} boldsymbol{x}+mathbf{2} boldsymbol{y}-mathbf{1 5 z}-mathbf{8}= ) 0 |
12 |
| 634 | The equation of plane passing through (-1,0,-1) parallel to ( x z ) plane is В. ( y=0 ) c. ( -x-z=0 ) D. None of the above |
12 |
| 635 | The condition that the line ( frac{x-alpha^{prime}}{l}= ) ( frac{boldsymbol{y}-boldsymbol{beta}^{prime}}{boldsymbol{m}}=frac{boldsymbol{z}-boldsymbol{gamma}^{prime}}{boldsymbol{n}} ) in the plane ( boldsymbol{A} boldsymbol{x}+ ) ( boldsymbol{B} boldsymbol{y}+boldsymbol{C} boldsymbol{z}+boldsymbol{D}=mathbf{0} ) is ( mathbf{A} cdot A alpha^{prime}+B beta^{prime}+C gamma^{prime}+D=0 ) and ( A l+B m+C n neq 0 ) B ( cdot A alpha^{prime}+B beta^{prime}+C gamma^{prime}+D neq 0 ) and ( A l+B m+C n=0 ) C ( cdot A alpha^{prime}+B beta^{prime}+C gamma^{prime}+D=0 ) and ( A l+B m+C n=0 ) D ( cdot A alpha^{prime}+B beta^{prime}+C gamma^{prime}=0 ) and ( A l+B m+C n=0 ) |
12 |
| 636 | A rectangular parallelopiped is formed by drawing planes through the point (-1,2,5) and (1,-1,-1) and paralle to the coordinates planes. The length of the diagonal of the parallelopiped is | 12 |
| 637 | 82. The distance of the point (1, -5,9) from the plane x-y+z=5 measured along the line x=y=zis: [JEE M 2016] 10 a) TT (b) (c) 3/10 (d) 103 |
12 |
| 638 | If the points (-1,3,2),(-4,2,-2) and ( (5,5, lambda) ) are collinear, then ( lambda ) is equal to A . -10 B. 5 ( c .-5 ) D. 10 |
12 |
| 639 | The equations of the line of intersection of the planes ( x+y+z=2 ) and ( 3 x- ) ( boldsymbol{y}+mathbf{2} z=mathbf{5} ) in symmetric form are A ( frac{x-frac{7}{4}}{4}=frac{y-frac{1}{4}}{-1}=frac{z}{-3} ) B. ( quad frac{x}{3}=frac{y+frac{1}{3}}{1}=frac{z-frac{7}{4}}{-4} ) c. ( frac{x}{1}=frac{3 y+1}{1}=frac{3 z-7}{-4} ) D. none of these |
12 |
| 640 | 23. (1) Find Find the equation of the plane passing through the points (2,1,0), (5,0, 1) and (4,1,1). If P is the point (2,1, 6) then find the point Q such that PQ is perpendicular to the plane in (i) and the mid point of PQ lies on it. (2003 – 4 Marks) |
12 |
| 641 | Find the angle between the following pair of lines: (i) ( vec{r}=2 hat{i}-5 hat{j}+hat{k}+lambda(3 hat{i}-2 hat{j}+6 hat{k}) ) and ( vec{r}=mathbf{7} hat{boldsymbol{i}}-boldsymbol{6} hat{boldsymbol{k}}+boldsymbol{mu}(hat{boldsymbol{i}}+mathbf{2} hat{boldsymbol{j}}+mathbf{2} hat{boldsymbol{k}}) ) (ii) ( vec{r}=3 hat{i}+hat{j}-2 hat{k}+lambda(hat{i}-hat{j}-2 hat{k}) ) and ( vec{r}=2 hat{i}-hat{j}-56 hat{k}+ ) ( mu(hat{mathbf{3}} mathbf{i}-mathbf{5} hat{mathbf{j}}-mathbf{4} hat{boldsymbol{k}}) ) |
12 |
| 642 | 2. Let : 2x + y – z = 3 and P2 : x +2y +z = 2 be two planes. Then, which of the following statement(s) is (are) TRUE? (JEE Adv. 2018) (a) The line of intersection of P, and P, has direction ratios 1, 2, -1 3x -4 1-3yZ (b) The line . -= = is perpendicular to the line of intersection of P, and P2 c) (d) The acute angle between P, and P, is 60°. If P, is the plane passing through the point (4, 2, -2) and perpendicular to the line of intersection of P, and P, then the distance of the point (2, 1, 1) from the plane Pz is 7 |
12 |
| 643 | 14 The equation of a plane passing through the line of intersection of the planes x + 2y + 3z=2 and x-y+z=3 and at a distance from the point (3,1,-1) is (2012) (a) 5x-1ly+z=17 (c) x+y+z= 13 (b) V2x+y = 3/2 – 1 (d) x-V2y =1-2 |
12 |
| 644 | Find the vector equation of the line joining (1,2,3) and (-3,4,3) and show pependicular to the z-axis | 12 |
| 645 | In the given figure, co-ordinates of the midpoint of ( boldsymbol{A B} ) are A . (0,2) в. (0,3) c. (1,2) D. (3,1) |
12 |
| 646 | The centroid of triangle ( boldsymbol{A}(mathbf{3}, mathbf{4}, mathbf{5}) ; boldsymbol{B}(mathbf{6}, mathbf{7}, mathbf{2}) ; boldsymbol{C}(mathbf{0},-mathbf{5}, mathbf{2}) ) is A ( .(3,2,3) ) в. (5,2,1) c. (2,5,1) D. (3,4,1) |
12 |
| 647 | Find the coordinates of point which divides the line joining the points (3,4) and (6,1) in the ratio of 1: 2 |
12 |
| 648 | ( bar{a}, bar{b}, bar{c} ) are three non-zero vectors such that any two of them are non-collinear. If ( bar{a}+bar{b} ) is collinear with ( bar{c} ) and ( bar{b}+bar{c} ) is collinear with ( bar{a} ), then what is their sum? A . -1 B. 0 c. 1 D. 2 |
12 |
| 649 | If a line makes angles ( boldsymbol{alpha}, boldsymbol{beta}, boldsymbol{gamma} ) with coordinate axes, find ( cos ^{2} alpha+cos ^{2} beta+cos ^{2} gamma+1 ) |
12 |
| 650 | The straight lines ( frac{boldsymbol{x}-mathbf{1}}{mathbf{1}}=frac{boldsymbol{y}-mathbf{2}}{mathbf{2}}= ) ( frac{z-3}{3} ) and ( frac{x-1}{1}=frac{y-2}{2}=frac{z-3}{3} ) are A. Parallel lines B. Intersecting at ( 60^{circ} ) c. skew lines D. Intersecting at right angle |
12 |
| 651 | Find the angles between the lines, whose direction cosines are give by the equation ( l^{2}-m^{2}+n^{2}=0, l+m+ ) ( boldsymbol{n}=mathbf{0} ) A. 0 в. ( frac{pi}{6} ) ( c cdot frac{pi}{4} ) D. |
12 |
| 652 | Find the equation of the line passing through (1,2,-4) and perpendicular to both the lines ( frac{x-1}{2}=frac{y+2}{-3}=frac{z-4}{4} ) and ( frac{boldsymbol{x}-mathbf{3}}{mathbf{5}}=frac{boldsymbol{y}+mathbf{6}}{mathbf{1}}=frac{boldsymbol{z}+mathbf{1 0}}{mathbf{2}} ) |
12 |
| 653 | If the ( z x ) -plane divides the line segment joining (1,-1,5) and (2,3,4) in the ratio ( p: 1, ) then ( p+1= ) A ( cdot frac{1}{3} ) B. 1: 3 ( c cdot frac{3}{4} ) D. ( frac{4}{3} ) |
12 |
| 654 | The image of the point (-1,3,4) in the plane ( boldsymbol{x}-mathbf{2} boldsymbol{y}=mathbf{0} ) is ( ^{mathbf{A}} cdotleft(-frac{17}{3},-frac{19}{3}, 4right) ) В. (15,11,4) ( ^{mathbf{C}} cdotleft(-frac{17}{3},-frac{19}{3}, 1right) ) D. ( left(frac{9}{5},-frac{13}{5}, 4right) ) |
12 |
| 655 | 75. The angle between the lines whose direction cosines satisfy the equations 1+m+n= 0 and 12 = m² +nis [JEE M 2014 wa |
12 |
| 656 | f a plane passes through the point (1,1,1) and is perpendicular to the line ( frac{x-1}{3}=frac{y-1}{0}=frac{z-1}{4} ) then its perpendicular distance from the origin is A ( cdot frac{3}{4} ) B. ( frac{4}{3} ) ( c cdot frac{7}{5} ) D. |
12 |
| 657 | If the lines ( frac{x-1}{2}=frac{y+1}{3}=frac{z-1}{4} ) and ( frac{boldsymbol{x}-mathbf{3}}{mathbf{1}}=frac{boldsymbol{y}-boldsymbol{k}}{mathbf{2}}=frac{boldsymbol{z}}{mathbf{1}} ) intersect, then ( boldsymbol{k}= ) A. 0 B. 3 ( c cdot frac{7}{2} ) D. – |
12 |
| 658 | If the vertices of a triangle are (-1,6,-4),(2,1,1) and (5,-1,0) then the centroid of the triangle is В. (2,2,-1) ( ^{mathbf{c}} cdotleft(3,3,-frac{3}{2}right) ) D. none of these |
12 |
| 659 | An equation of sphere with centre at origin and radius ( r ) can be represented as A ( cdot x^{2}+y^{2}+z^{2}=r ) B . ( x^{2}+y^{2}+z^{2}=r^{2} ) c. ( x^{2}+y^{2}+z^{2}=2 r^{2} ) D. None of the above |
12 |
| 660 | Obtain the equation of the line passing through (1,1,2) and (2,1,2) in the vector form. | 12 |
| 661 | The reflection of the plane ( 2 x+3 y+ ) ( 4 z-3=0 ) in the plane ( x-y+z- ) ( mathbf{3}=mathbf{0} ) is the plane A. ( 4 x-3 y+2 z-15=0 ) B. ( x-3 y+2 z-15=0 ) c. ( 4 x+3 y-2 z+15=0 ) D. none of these |
12 |
| 662 | A normal to the plane ( x=2 ) is… A ( .(0,1,1) ) в. (2,0,2) c. (1,0,0) D. (0,1,0) |
12 |
| 663 | An ordered triplet corresponds to in three dimensional space. A. three points B. a unique point c. a point in each octant D. infinite number of points |
12 |
| 664 | A point at a distance of ( sqrt{6} ) from the origin which lies on the straight line ( frac{x-1}{1}=frac{y-2}{2}=frac{z+1}{3} ) will be ( mathbf{A} cdot(1,-1,2) ) В. (1,2,-1) C ( cdotleft(frac{5}{7}, frac{10}{7}, frac{-13}{7}right) ) D. ( left(frac{5}{7}, frac{2}{7}, frac{-6}{7}right) ) |
12 |
| 665 | If ( (3, lambda, mu) ) is a point on the line then ( 2 x+y+z=0=x-2 y+z-1 ) then A ( cdot lambda=frac{-8}{3}, mu=-frac{1}{3} ) B. ( lambda=frac{-1}{3}, mu=-frac{8}{3} ) c. ( lambda=frac{-4}{3} mu=frac{-14}{3} ) D. ( lambda=-5, mu=-1 ) |
12 |
| 666 | Find the point of intersection of the plane ( bar{r} .(1,1,1)=2 ) and the line ( bar{r}= ) ( (4,5,3)+k(2,2,1), k in R ) | 12 |
| 667 | The equation of the plane which is equidistant from the two parallel planes ( 2 x-2 y+z+3=0 ) and ( 4 x-4 y+ ) ( mathbf{2} z+mathbf{9}=mathbf{0} ) is : A. ( 8 x-8 y+2 z+15=0 ) B. ( 8 x-8 y+4 z+15=0 ) c. ( 8 x-8 y+4 z+3=0 ) D. ( 8 x-8 y+4 z-3=0 ) E ( .8 x-8 y+4 z+4=0 ) |
12 |
| 668 | The distance between the line ( r=2 hat{i}- ) ( 2 hat{j}+3 hat{k}+lambda(hat{i}-hat{j}+4 hat{k}) ) and the plane ( r cdot(hat{i}+5 hat{j}+hat{k})=5 ) is A ( cdot frac{10}{9} ) в. ( frac{10}{3 sqrt{3}} ) c. ( frac{10}{3} ) D. None of these |
12 |
| 669 | If the distance between a point ( P ) and the point (1,1,1) on the line ( frac{x-1}{3}= ) ( frac{y-1}{4}=frac{z-1}{12} ) is ( 13, ) then the coordinates of ( P ) are A. (3,4,12) В. ( left(frac{3}{13}, frac{4}{13}, frac{12}{13}right) ) c. (4,5,12) D. (40, 53, 157) |
12 |
| 670 | The equation of the plane through the intersection of ( p_{1} & p_{2} ) containing the point (1,1,2) is A. ( 5 x-6 y+4 z=4 ) в. ( 5 x+6 y-4 z=3 ) c. ( 3 x-2 y+4 z=9 ) D. Nonoe of these |
12 |
| 671 | If a plane passes through a fixed point (2,3,4) and meets the axes of reference in ( A, B ) and ( C, ) the point of intersection of the planes through ( A, B, C ) parallel to the coordinate planes can be This question has multiple correct options A ( cdot(6,9,12) ) B ( cdot(4,12,16) ) ( mathbf{c} cdot(1,1,-1) ) D. (2,3,-4) |
12 |
| 672 | ( boldsymbol{L}_{1}: frac{boldsymbol{x}-mathbf{1}}{mathbf{2}}=frac{boldsymbol{y}-mathbf{2}}{mathbf{3}}=frac{boldsymbol{z}-mathbf{3}}{mathbf{4}} ) ( L_{2}: frac{x-2}{3}=frac{y-4}{2}=frac{z-5}{5} ) be two given lines, point P lies on ( L_{1} ) and Q lies on ( L_{2} ) then distance between ( P ) and ( Q ) can be This question has multiple correct options A ( cdot frac{1}{3} ) B. c. 15 D. 30 |
12 |
| 673 | Let ( X ) and ( Y ) be two related variables. The two regression lines are given by ( boldsymbol{x}-boldsymbol{y}+mathbf{1}=mathbf{0} ) and ( mathbf{2} boldsymbol{x}-boldsymbol{y}+mathbf{4}=mathbf{0} . ) The two regression lines pass through the point: A ( cdot(-4,-3) ) в. (-6,-5) c. (3,-2) D. (-3,-2) |
12 |
| 674 | The values of a for which ( (8,-7, a),(5,2,4) ) and (6,-1,2) are collinear, is given by? A .2 B. -2 c. -1 D. |
12 |
| 675 | If ( z_{1} ) and ( z_{2} ) are ( z ) co-ordinates of the points of trisection of the segment joining the points ( boldsymbol{A}(mathbf{2}, mathbf{1}, mathbf{4}), boldsymbol{B}(-mathbf{1}, mathbf{3}, mathbf{6}) ) then ( z_{1}+z_{2}= ) A . 1 B. 4 ( c .5 ) D. 10 |
12 |
| 676 | If ( boldsymbol{P}=(mathbf{0}, mathbf{0}, mathbf{0}), boldsymbol{Q}=(mathbf{3}, mathbf{6}, mathbf{9}) ) and ( boldsymbol{R} ) is a point of trisection of ( boldsymbol{P Q}, ) then ( boldsymbol{R}_{boldsymbol{y}}= ) ( A cdot frac{4}{3} ) B . 2 ( c .3 ) D. |
12 |
| 677 | Find the image of (1,5,1) in the plane ( boldsymbol{x}-mathbf{2} boldsymbol{y}+boldsymbol{z}+mathbf{5}=mathbf{0} ) |
12 |
| 678 | If the centroid of tetrahedron ( O A B C ) where ( A, B, C ) are given by ( (a, 2,3),(1, b, 2) ) and ( (2,1, c) ) respectively is ( (1,2,-2), ) then distance of ( boldsymbol{P}(boldsymbol{a}, boldsymbol{b}, boldsymbol{c}) ) from origin is ( mathbf{A} cdot sqrt{195} ) в. ( sqrt{14} ) c. ( sqrt{frac{107}{14}} ) D. ( sqrt{13} ) |
12 |
| 679 | 1. From a point O inside a triangle ABC, perpendiculars OD, OE, OF are drawn to the sides BC, CA, AB respectively. Prove that the perpendiculars from A, B, C to the sides EF, FD, DE are concurrent. (1978) hone |
12 |
| 680 | A parallelepiped is formed by planes drawn through the point ( P(6,8,10) ) and ( Q(3,4,8) ) parallel to the coordinate planes. Find the length of edges and edges and diagonals of the parallelepiped. |
12 |
| 681 | The angle between the lines whose direction cosines satisfy the equations ( l+m+n=0 ) and ( l^{2}+m^{2}+n^{2} ) is A ( cdot frac{pi}{2} ) в. c. D. ( frac{pi}{6} ) |
12 |
| 682 | The line passing through the points ( 10 hat{i}+3 hat{j}, 12 hat{i}+5 hat{j} ) also passes through the point ( a hat{i}+11 hat{j}, ) then ( a= ) A . -8 B. 4 c. 18 D. 12 |
12 |
| 683 | Direction ratio of line given by ( frac{x-1}{3}= ) ( frac{6-2 y}{10}=frac{1-z}{-7} ) are: ( A cdot ) в. ( ) c. ( ) D. ( ) |
12 |
| 684 | Point ( D ) has coordinates as (3,4,5) Referring to the given figure, find the coordinates of point ( boldsymbol{B} ) A ( cdot(3,0,4) ) ( mathbf{B} cdot(4,3,0) ) C. (4,0,3) D. (3,4,0) |
12 |
| 685 | Which of the following are equations for the plane passing through the points ( P(1,1,-1), Q(3,0,2) ) and ( R(-2,1,0) ? ) A ( cdot(2 hat{i}-3 hat{j}+3 hat{k}) cdot((x+2) hat{i}+(y-1) hat{j}+z hat{k})=0 ) B . ( x=3-t, y=-11 t, z=2-3 t ) ( mathbf{c} cdot(x+2)+11(y-1)=3 x ) D. ( (2 hat{i}-hat{j}+3 hat{k}) times(-3 hat{i}+hat{k}) cdot((x+2) hat{i}+(y-1) hat{j}+z hat{k})= ) 0 |
12 |
| 686 | The direction ratios of a vector are ( 2,-3,4 . ) Find its direction cosines |
12 |
| 687 | If ( A=5 ) units, ( B=6 ) units and ( |vec{A} times vec{B}|= ) 15 units, then the angle between ( vec{A} ) and ( vec{B} ) is: This question has multiple correct options ( A cdot 30^{circ} ) B. ( 60^{circ} ) c. ( 90^{circ} ) D. ( 150^{circ} ) |
12 |
| 688 | Find the shortest distance between the skew lines: ( l_{1}: frac{x-1}{2}=frac{y+1}{1}=frac{z-2}{4} ) ( l_{2}: frac{x+2}{4}=frac{y-0}{-3}=frac{z+1}{1} ) |
12 |
| 689 | Consider the plane ( (boldsymbol{x}, boldsymbol{y}, boldsymbol{z})= ) ( (0,1,1)+lambda(1,-1,1)+mu(2,-1,0) . ) The distance of this plane from the origin is: A ( cdot frac{1}{3} ) B. ( frac{sqrt{3}}{2} ) ( c cdot sqrt{frac{3}{2}} ) D. ( frac{2}{sqrt{3}} ) |
12 |
| 690 | 71. Distance between two parallel planes 2x + y + 2z = 8 and 4x + 2y + 4z +5=0 is [JEE M 2013] |
12 |
| 691 | Vector equation of line ( frac{mathbf{3}-boldsymbol{x}}{mathbf{3}}= ) ( frac{2 y-3}{5}=frac{z}{2} ) is ( _{–}——-k in R ) A ( cdot bar{r}=(3,5,2)+k(3,3,0) ) B ( cdot quad bar{r}=left(3, frac{3}{2}, 0right)+k(-6,5,4) ) c. ( bar{r}=(3,3,0)+k(3,5,2) ) D ( cdot_{bar{r}}=(-6,5,4)+kleft(3, frac{3}{2}, 0right) ) |
12 |
| 692 | If ( (1,1, a) ) is the centroid of the triangle formed by the points ( (1,2,-3),(mathrm{b}, 0,1) ) and (-1,1,-4) then ( a-b= ) A . -5 B. -7 ( c .5 ) ( D ) |
12 |
| 693 | Show that the lines ( frac{x-1}{2}=frac{y-2}{3}= ) ( frac{z-3}{4} ) and ( 4 x-3 y+1=0=5 x- ) ( 3 z+2 ) are interesting lines. Also find point of intersection. |
12 |
| 694 | 88. If L, is the line of intersection of the planes 2x-2y+3z-2=0, x-y+z+1=0 and L, is the line of intersection of the planes x+2y-z-3=0, 3x-y+2z-1=0, then the distance of the origin from the plane, containing the lines L, and L,, is: [JEEM 2018] (a) ZNZ (6) 212 |
12 |
| 695 | What is the direction cosine of angle which the vector ( sqrt{2 hat{i}}+hat{j}+ ) ( hat{k} ) makes with ( y- ) axis ( ? ) |
12 |
| 696 | 2. If a vector P making angles a, b, and yrespectively with the X, Y and Z axes respectively. Then sin? a + sin2 B + sin2 y= (a) (b) 1 (c) 2 (d) 3 |
12 |
| 697 | Point ( D ) has coordinates as (3,4,5) Find the coordinates of the point ( boldsymbol{F} ) ( mathbf{A} cdot(0,4,0) ) B. (0,0,4) ( mathbf{C} cdot(0,0,5) ) ( mathbf{D} cdot(0,5,0) ) |
12 |
| 698 | If ( overline{O A}=3 bar{i}+bar{j}-bar{k},|overline{A B}|=2 sqrt{6} ) and ( A B ) has the direction ratios 1,-1,2 then ( |boldsymbol{O} boldsymbol{B}|= ) A . ( sqrt{35} ) B. ( sqrt{41} ) c. ( sqrt{26} ) D. ( sqrt{55} ) |
12 |
| 699 | Find the equation of the plane through the points ( boldsymbol{A}(mathbf{2}, mathbf{2}-mathbf{1}), boldsymbol{B}(mathbf{3}, mathbf{4}, mathbf{2}) ) and ( boldsymbol{C}(boldsymbol{7}, boldsymbol{0}, boldsymbol{6}) ) A. ( 5 x+2 y-3 z=17 ) B. ( 5 x+2 y+3 z=17 ) c. ( 5 x+y-3 z=7 ) D. ( 5 x+y+3 z=7 ) |
12 |
| 700 | Show that the three lines with direction ( operatorname{cosines} ) ( frac{12}{13}, frac{-3}{13}, frac{-4}{13}: frac{4}{13}, frac{12}{13}, frac{3}{13} ; frac{-4}{13}, frac{12}{13} ) are mutually perpendicular |
12 |
| 701 | If the projection of point ( boldsymbol{P}(overrightarrow{boldsymbol{p}}) ) on the plane ( vec{r} cdot vec{n}=q ) is the point ( S(vec{s}), ) then? A ( cdot vec{s}=frac{(q-vec{p} cdot vec{n}) vec{n}}{|vec{n}|^{2}} ) В ( cdot vec{s}=vec{p}+frac{(vec{p} cdot vec{n}-q)}{|vec{n}|^{2}} vec{n} ) C・ ( _{vec{s}}=vec{p}-frac{(vec{p} cdot vec{n}) vec{n}}{|vec{n}|^{2}} ) D ( vec{s}=vec{p}-frac{(q-vec{p} cdot vec{n}) vec{n}}{|vec{n}|^{2}} ) |
12 |
| 702 | Two systems of rectangular axes have the same origin. If a plane cuts them at distance ( a, b, c ) and ( d, b^{prime}, c^{prime} ) from the origin, then A ( cdot frac{1}{a^{2}}-frac{1}{b^{2}}-frac{1}{c^{2}}-frac{1}{a^{2}}-frac{1}{b^{2}}-frac{1}{c^{2}}=0 ) B ( cdot frac{1}{a^{2}}-frac{1}{b^{2}}-frac{1}{c^{2}}-frac{1}{a^{2}}-frac{1}{b^{2}}+frac{1}{c^{2}}=0 ) c. ( frac{1}{a^{2}}+frac{1}{b^{2}}+frac{1}{c^{2}}-frac{1}{a^{2}}-frac{1}{b^{2}}-frac{1}{c^{2}}=0 ) D ( frac{1}{a^{2}}+frac{1}{b^{2}}+frac{1}{c^{2}}+frac{1}{a^{2}}+frac{1}{b^{2}}+frac{1}{c^{2}}=0 ) |
12 |
| 703 | Find the equation of the line joining the points (-1,3) and (4,-2) | 12 |
| 704 | If the position vectors of the points ( boldsymbol{A}, boldsymbol{B} ) and ( C ) be ( i+j, i-j ) and ( a i+b j+c k ) respective;y, then the points ( A, B ) and ( C ) are collinear if: A ( . a=b=c=1 ) B. ( a=1, b ) and ( c ) are arbitrary scalars C. ( a=b=c=0 ) 0 |
12 |
| 705 | A point ( R ) with ( x ) -coordinate 4 lies on the line segment joining the points ( P(2,-3,4) ) and ( Q(8,0,10) . ) Find the coordinates of the point ( boldsymbol{R} ) |
12 |
| 706 | Fill in the blanks: (i) The ( x ) -axis and ( y ) -axis taken together determine a plane known as (ii) The coordinates of points in the ( boldsymbol{X} boldsymbol{Y} ) -plane are of the form (iii) Coordinate planes divide the space into octants |
12 |
| 707 | The Cartesian equation of a line is ( frac{boldsymbol{x}-mathbf{5}}{mathbf{3}}=frac{boldsymbol{y}+mathbf{4}}{mathbf{7}}=frac{boldsymbol{z}-mathbf{6}}{mathbf{2}} . ) Write its vector form. |
12 |
| 708 | The direction cosines of the lines bisecting the internal angle ( theta ) between the lines whose direction cosines are ( l_{1}, m_{1}, n_{1} ) and ( l_{2}, m_{2}, n_{2} ) are ( mathbf{A} cdot ) B. ( ) c. ( ) D. none of these |
12 |
| 709 | Find the distance of the point (-6,0,0) from the plane ( 2 x-3 y+6 z=2 ) |
12 |
| 710 | Let ( boldsymbol{A}=(mathbf{1}, mathbf{2}, mathbf{2}), boldsymbol{B}=(mathbf{2}, mathbf{3}, mathbf{6}) ) and ( boldsymbol{C}= ) ( (3,4,12) . ) The direction cosines of a line equally inclined with ( O A, O B ) and ( O C ) where ( boldsymbol{O} ) is the origin, are A ( cdot frac{1}{sqrt{2}}, frac{-1}{sqrt{2}}, 0 ) B. ( frac{1}{sqrt{2}}, frac{1}{sqrt{2}}, 0 ) c. ( frac{1}{sqrt{3}}, frac{-1}{sqrt{3}}, frac{1}{sqrt{3}} ) D. ( frac{1}{sqrt{3}}, frac{-1}{sqrt{3}}, frac{-1}{sqrt{3}} ) |
12 |
| 711 | Find the coordinates of the point on the ( x ) -axis that is equidistant from ( boldsymbol{P}(mathbf{4}, mathbf{3}, mathbf{1}) ) and ( boldsymbol{Q}(-mathbf{2},-mathbf{6},-mathbf{2}) ) A ( cdotleft(frac{3}{2}, 0,0right) ) B ( cdotleft(-frac{3}{2}, 0,0right) ) c. ( left(0,-frac{3}{2}, 0right) ) D. ( left(0, frac{3}{2}, 0right) ) |
12 |
| 712 | In the triangle with vertices ( boldsymbol{A}(mathbf{1},-mathbf{1}, mathbf{2}), boldsymbol{B}(mathbf{5},-mathbf{6}, mathbf{2}) ) and ( boldsymbol{C}(mathbf{1}, mathbf{3},-mathbf{1}) ) find the altitude ( n=|B D| ) ( A cdot 5 ) B. 10 c. ( 5 sqrt{2} ) D. ( frac{10}{sqrt{2}} ) |
12 |
| 713 | If the origin is the centroid of the triangle ( P Q R ) with vertices ( boldsymbol{P}(mathbf{2} boldsymbol{a}, mathbf{2}, mathbf{6}), boldsymbol{Q}(-mathbf{4}, mathbf{3} boldsymbol{b},-mathbf{1 0}) ) and ( boldsymbol{R}(mathbf{8}, mathbf{1 4}, mathbf{2 c}), ) then find the values of ( boldsymbol{a}, boldsymbol{b} ) and ( c ) |
12 |
| 714 | The points with the co-ordinates ( (2 a, 3 a),(3 b, 2 b) &(c, c) ) are collinear. |
12 |
| 715 | Using section formula show that the points ( boldsymbol{A}(mathbf{2},-mathbf{3}, mathbf{4}), boldsymbol{B}(-mathbf{1}, mathbf{2}, mathbf{1}) ) and ( Cleft(0, frac{1}{3}, 2right) ) are collinear. |
12 |
| 716 | If the line joining ( boldsymbol{A}(mathbf{1}, mathbf{3}, mathbf{4}) ) and ( boldsymbol{B} ) is divided by the point (-2,3,5) in the ratio ( 1: 3, ) then ( B ) is A. (-11,3,8) (年) (-1,3,8) в. (-11,3,-8) D. (13,6,-13) |
12 |
| 717 | 16. In R}, consider the planes P, :y=0 and P2: x+z=1. Let P be the plane, different from P, and P2, which passes through the intersection of P, and P,. If the distance of the point (0, 1, 0) from P, is 1 and the distance of a point (a, b, y) from Pz is 2, then which of the following relations is (are) true (a) 2a+3+2y+2=0 (c) 2a+B-2y-10=0 (JEE Adv. 2015) (b) 20-3+2y+4=0 (d) 2a-B+2y-8=0 |
12 |
| 718 | Find the distance between the points (3,4,-2),(1,0,7) |
12 |
| 719 | W J 84. If the image of the point P(1, -2, 3) in the plane, X Y Z 2x + 3y– 4z + 22=0 measured parallel to line, = = is Q, then PQ is equal to : JJEE M 2017] (a) 615 (b) 315 (c) 2742 (d) 142 |
12 |
| 720 | The points ( boldsymbol{A}(mathbf{1}, mathbf{2},-mathbf{1}), boldsymbol{B}(mathbf{2}, mathbf{5},-mathbf{2}), boldsymbol{C}(mathbf{4}, mathbf{4},-mathbf{3}) ) and ( D(3,1,-2) ) are A. collinear B. vertices of a rectangle c. vertices of a square D. vertices of a rhombus |
12 |
| 721 | Find the directions cosines of ( x, y ) and ( z ) axis. |
12 |
| 722 | The perpendicular distance from the point (3,1,1) on the plane passing through the point (1,2,3) and containing the line, ( vec{r}=hat{i}+hat{j}+ ) ( lambda(2 hat{i}+hat{j}+4 hat{k}), ) is: ( ^{A} cdot frac{1}{sqrt{11}} ) в. ( frac{4}{sqrt{41}} ) ( c .0 ) D. ( frac{3}{sqrt{11}} ) |
12 |
| 723 | If ( (0, b, 0) ) is the centroid of the triangle formed by the points (4,2,-3) ( (a,-5,1) ) and ( (2,-6,2) . ) If ( a, b ) are the roots of the quadratic equation ( x^{2}+ ) ( boldsymbol{p} boldsymbol{x}+boldsymbol{q}=mathbf{0}, ) then ( boldsymbol{p}, boldsymbol{q} ) are A .9,18 в. -9,-18 c. 3,-18 D. -3,18 |
12 |
| 724 | For two vectors ( overrightarrow{boldsymbol{A}} ) and ( overrightarrow{boldsymbol{B}}, overrightarrow{boldsymbol{A}}+overrightarrow{boldsymbol{B}}=overrightarrow{boldsymbol{C}} ) and ( |vec{A}|+|vec{B}|=|vec{C}| . ) The angle between two vectors is: A . в. ( pi / 3 ) c. ( pi / 2 ) D. |
12 |
| 725 | A plane meets the co-ordinate axes in A,B,C such that the centroid of the triangle ( A B C ) is the point ( (p, q, r) . ) The equation of the plane is A ( cdot frac{x}{p}+frac{y}{q}+frac{z}{r}=0 ) В ( cdot frac{x}{p}+frac{y}{q}+frac{z}{r}=1 ) c. ( frac{x}{p}+frac{y}{q}+frac{z}{r}=2 ) D. none of these |
12 |
| 726 | What is the sum of the squares of direction cosines of the line joining the points (1,2,-3) and (-2,3,1)( ? ) ( A cdot O ) B. ( c cdot 3 ) D. ( frac{2}{sqrt{26}} ) |
12 |
| 727 | The image of: (-4,0,0) in the ( x z ) – plane is (4,0,0) A. True B. False |
12 |
| 728 | The point which is equidistant from the points ( (boldsymbol{a}, boldsymbol{0}, boldsymbol{0}),(boldsymbol{0}, boldsymbol{b}, boldsymbol{0}),(boldsymbol{0}, boldsymbol{0}, boldsymbol{c}) ) and (0,0,0) is: ( mathbf{A} cdot(a, b, c) ) B . ( (sqrt{a}, sqrt{b}, sqrt{c}) ) c. ( (2 a, 2 b, 2 c) ) D. ( left(frac{a}{2}, frac{b}{2}, frac{c}{2}right) ) |
12 |
| 729 | Find distance of a point (3,4) from the origin. | 12 |
| 730 | 42. The plane x +2y-z=4 cuts the sphere x + y< +22-x+, – 2 = 0 in a circle of radius (a) 3 (6) 1 (c) 2 (d) & [2005] |
12 |
| 731 | 34. The angle between the lines 2x = 3y = – z and 6x=-y=-4z is [2005] (a) O ). (b) 90° (c) 45° (d) 30° |
12 |
| 732 | The number of lines which are equally inclined to the axes is ( A cdot 2 ) B. 4 ( c cdot 6 ) D. 8 |
12 |
| 733 | VEU 43. point P is the intersection of the straight line joining points (2,3,5) and R(1,-1, 4) with the plane 5x – 4y 1. If S is the foot of the perpendicular drawn from the point T(2, 1, 4) to QR, then the length of the line segment PS (2012) 7 (b) √2 (c) 2 (2) 252 |
12 |
| 734 | ( operatorname{lines} frac{x-1}{2}=frac{y-1}{2}=frac{z-2}{3} ) and ( frac{x-1}{2}=frac{y-2}{2}=frac{z-3}{-2} ) Check whether the lines are parallel, mutually perpendicular or intersecting in acute angle) |
12 |
| 735 | A line is perpendicular to the plane ( x+ ) ( 2 y+2 z=0 ) and passes through ( (0,1,0) . ) The perpendicular distance of this line from the origin is A ( frac{sqrt{5}}{3} ) B. ( frac{sqrt{7}}{3} ) ( c cdot frac{2}{3} ) D. 3 |
12 |
| 736 | The name of the figure formed by the points (0,0,0),(1,0,1) and (0,1,1) is A . a straight line B. an isosceles triangle c. an equilateral triangle D. a scalene triangle |
12 |
| 737 | The cartesian equation of the plane ( overline{boldsymbol{r}}=(mathbf{1}+boldsymbol{s}-boldsymbol{t}) hat{boldsymbol{i}}+(boldsymbol{2}-boldsymbol{s}) hat{boldsymbol{j}}+ ) ( (3-2 s+2 t) hat{k} ) A. ( 2 x-y=5 ) B. ( 2 x+z=5 ) c. ( 2 x+y=5 ) D. ( 2 x-z=5 ) |
12 |
| 738 | 27. A variable plane at a distance of the one unit from the origin cuts the coordinates axes at A, B and C. If the centroid D (x, y, z) of triangle ABC satisfies the relation 2 then the value k is (2005) y (a) 3 (b) 1 (c) (d) 9 |
12 |
| 739 | 21. Let ✓ = 2i +1 -k and W = i +3k . If Ū is a unit vector, then the maximum value of the scalar triple product |ŪVW | is (a) -1 (b) V10 + V6 (2002) (c) 159 (d) 160 |
12 |
| 740 | Two opposite vertices of a square are (2,-3,4) and ( (4,1,-2) . ) The length of the side of the square is A ( cdot sqrt{58} ) B. ( 2 sqrt{7} ) c. ( sqrt{14} ) D. ( sqrt{7} ) |
12 |
| 741 | If the angle between the planes ( boldsymbol{r} cdot(boldsymbol{m} hat{boldsymbol{i}}-hat{boldsymbol{j}}+boldsymbol{2} hat{boldsymbol{k}})+boldsymbol{3}=boldsymbol{0} ) and ( overline{boldsymbol{r}} cdot(boldsymbol{2} hat{boldsymbol{i}}- ) ( m hat{j}-hat{k})-5=0 ) is ( frac{pi}{3} ) then ( m= ) ( A cdot 2 ) B. ±3 ( c cdot 3 ) D. – – |
12 |
| 742 | Find the ratio in which (the plane) ( 2 x+ ) ( 3 y+5 z=1 ) divides the line joining the points (1,0,-3) and (1,-5,7) A .1: 2 B . 2: 3 ( c .3: 1 ) D. None of these |
12 |
| 743 | Let ( O ) be the origin and ( P ) be the point at a distance 3 units from origin. If d.x.s’ of OP are ( 1,-2,-2, ) then coordinates of ( P ) is given by A. 1,-2,-2 в. 3,-6,-6 c. ( frac{1}{3},-frac{2}{3},-frac{2}{3} ) D. ( frac{1}{9},-frac{2}{9},-frac{2}{9} ) |
12 |
| 744 | Write the direction cosines of ( x ) -axis | 12 |
| 745 | The product of the d.cs of the line which makes equal angles with ( o x, o y, o z ) is ( mathbf{A} cdot mathbf{1} ) B. ( sqrt{3} ) c. ( frac{1}{3 sqrt{3}} ) D. ( frac{1}{sqrt{3}} ) |
12 |
| 746 | If a line makes ( theta_{1}, theta_{2}, theta_{3} ) angles with the co-ordinates axes, then prove that ( cos 2 theta_{1}+cos 2 theta_{2}+cos 2 theta_{3}+1=0 ) |
12 |
| 747 | Verify the following: (0,7,-10),(1,6,-6) and (4,9,-6) are the vertices of an isosceles triangle. |
12 |
| 748 | If ( vec{A} times vec{B}=vec{B} times vec{A}, ) then the angle between ( vec{A} ) and ( vec{B} ) is A . ( pi ) в. ( c cdot frac{pi}{2} ) D. |
12 |
| 749 | Show that the lines whose d.c.s are given by ( l+m+n=0,2 m n+3 l n- ) ( 5 l m=0 ) are perpendicular to each other. |
12 |
| 750 | Show that the following set of point are collinear? (2,3,-4),(-1,0,5),(3,4,-7) |
12 |
| 751 | Equation of a plane making X-intercept 4, Y-intercept ( (-6), mathrm{Z} ) -intercept 3 is A. ( 3 x-4 y+6 z=12 ) B. ( 3 x-2 y+4 z=12 ) c. ( 4 x-6 y+3 z=1 ) D. ( 4 x-3 y+2 z=12 ) |
12 |
| 752 | Find the distance between (12,3,4) and (4,5,2) A ( cdot sqrt{72} ) B. ( sqrt{62} ) ( c cdot sqrt{64} ) D. None of these |
12 |
| 753 | The general equation of plane which is parallel to x-axis is ( mathbf{A} cdot a x+b y+c z+d=0, a neq 0, b neq 0, c neq 0 ) B. ( b y+a x+d=0, a neq 0, b neq 0 ) c. ( a x+c z+d=0, a neq 0 . c neq 0 ) D. ( b y+c z+d=0, b neq 0, c neq 0 ) |
12 |
| 754 | If from the point ( boldsymbol{P}(boldsymbol{f}, boldsymbol{g}, boldsymbol{h}) ) perpendiculars ( P L, P M ) be drawn to ( y z ) and ( z x ) planes, then the equation to the plane ( boldsymbol{O} boldsymbol{L} boldsymbol{M} ) is A ( cdot frac{x}{f}+frac{y}{g}-frac{z}{h}=0 ) В ( cdot frac{x}{f}+frac{y}{g}+frac{z}{h}=0 ) c. ( frac{x}{f}-frac{y}{g}+frac{z}{h}=0 ) D. ( -frac{x}{f}+frac{y}{g}+frac{z}{h}=0 ) |
12 |
| 755 | The position vectors of point ( A ) and ( B ) ( operatorname{are} hat{boldsymbol{i}}-hat{boldsymbol{j}}+boldsymbol{3} hat{boldsymbol{k}} ) and ( boldsymbol{3} hat{boldsymbol{i}}+boldsymbol{3} hat{boldsymbol{j}}+boldsymbol{3} hat{boldsymbol{k}} ) respectively. The equation of a plane is ( r cdot(5 hat{i}+2 hat{j}-7 hat{k})+9=0 . ) The point ( A ) and ( B ) A. lie on the plane B. are on the same side of the plane c. are on the opposite side of the plane D. None of these |
12 |
| 756 | Find the distance between the following pairs of points (-5,7) and (-1,3) |
12 |
| 757 | The equation of the plane which is parallel to ( x y ) plane and cuts intercept of length 3 from the z-axis ( mathbf{A} cdot x=3 ) B. ( y=3 ) ( mathbf{c} cdot z=3 ) D. ( x+y+z=3 ) |
12 |
| 758 | The coordinate of any point, which lies in ( boldsymbol{x} boldsymbol{y} ) plane, is ( mathbf{A} cdot(x, 0, y) ) в. ( (x, x, 0) ) c. ( (x, 0, x) ) D. ( (y, 0, x) ) |
12 |
| 759 | What is the angle between the lines ( frac{boldsymbol{x}-mathbf{2}}{mathbf{1}}=frac{boldsymbol{y}+mathbf{1}}{-mathbf{2}}=frac{boldsymbol{z}+mathbf{2}}{mathbf{1}} ) and ( frac{boldsymbol{x}-mathbf{1}}{mathbf{1}}= ) ( frac{2 y+3}{3}=frac{z+5}{2}=? ) ( ^{A} cdot frac{pi}{2} ) в. c. ( frac{pi}{6} ) D. None of the above |
12 |
| 760 | The distance between the origin and the centroid of the tetrahedron whose vertices are ( (mathbf{0}, mathbf{0}, mathbf{0}) ) ( (a, 0,0),(0, b, 0),(0,0, c) ) is? A ( cdot sqrt{a^{2}+b^{2}+c^{2}} ) B. ( frac{sqrt{a^{2}+b^{2}+c^{2}}}{2} ) c. ( frac{sqrt{a^{2}+b^{2}+c^{2}}}{4} ) D. ( 4 sqrt{a^{2}+b^{2}+c^{2}} ) |
12 |
| 761 | A parallelepiped is formed by planes drawn through the points ( boldsymbol{P}(boldsymbol{6}, boldsymbol{8}, boldsymbol{1} boldsymbol{0}) ) and ( Q(3,4,8) ) parallel to the coordinate planes. Find the length of edges and diagonals of the parallelepiped. |
12 |
| 762 | Find the equation of the plane passing through the points ( A=(2.3,-1), B=(4,5, ) 2), ( C=(3,6,5) ) |
12 |
| 763 | The cartesian equation of plane ( bar{r} cdot(2,-3,4)=5 ) is A. ( 3 y-2 x-4 z+5=0 ) B. ( 2 x-3 y+4 z=0 ) c. ( 2 x-3 y+4 z+5=0 ) D. ( frac{x-1}{2}=frac{y-1}{-3}=frac{z-1}{4} ) |
12 |
| 764 | A plane which passes through the point (3,2,0) and the line ( frac{x-3}{1}=frac{y-7}{5}= ) ( frac{z-4}{4} ) is? A. ( x-y+z=1 ) в. ( x+y+z=5 ) c. ( x+2 y-z=1 ) D. ( 2 x-y+z=5 ) |
12 |
| 765 | 55. If the straight lines *;?v=2=27 and k 2 =intersect at a point, then the integer k 2 = 3 k is equal to (a) 5 [2008] (b) 5 (c) 2 (d) -2 |
12 |
| 766 | The line ( frac{x-3}{2}=frac{y-4}{5}=frac{z-6}{7} ) A . lies in ( 3 x+2 y+4 z-6=0 ) B. is parallel to ( 2 x-5 y+3 z=0 ) ( mathbf{c} cdot ) is ( perp ) to ( 2 x-5 y+3 z=0 ) D. passing through (1,2,3) |
12 |
| 767 | Direction cosines of ray from ( boldsymbol{P}(mathbf{1},-mathbf{2}, mathbf{4}) ) to ( boldsymbol{Q}(-mathbf{1}, mathbf{1},-mathbf{2}) ) are в. 2,-3,6 ( mathbf{c} cdot 2,3,6 ) D. ( frac{-2}{7}, frac{3}{7}, frac{-6}{7} ) |
12 |
| 768 | If the line, ( frac{x-1}{2}=frac{y+1}{3}=frac{z-1}{4} ) and ( frac{boldsymbol{x}-mathbf{3}}{mathbf{1}}=frac{boldsymbol{y}-boldsymbol{k}}{mathbf{2}}=frac{boldsymbol{z}}{mathbf{1}} ) intersect, then find the value of ( k ) |
12 |
| 769 | Point of intersection of the point (2,4,5)(3,6,-4) |
12 |
| 770 | Find the position vector of a point ( boldsymbol{R} ) which divides the line joining two points ( P ) and ( Q ) whose position vectors ( operatorname{are}(2 vec{a}+vec{b}) ) and ( (vec{a}-3 vec{b}) ) externally in the ratio ( 1: 2 . ) Also, show that ( P ) is the mid point of the line segment ( boldsymbol{R} boldsymbol{Q} ) |
12 |
| 771 | 13. Two lines L, : x=5, 3a = – and Ly :x=a, -2-a are coplanar. Then a can take value(s) (JEE Adv. 2013) (a) 1 (b) 2 (c) 3 (d) 4 |
12 |
| 772 | The distance between the points ( (cos theta, sin theta) ) and ( (sin theta-cos theta) ) is A. ( sqrt{3} ) B. ( sqrt{2} ) ( c cdot 2 ) D. |
12 |
| 773 | Find the coordinates of those points on the line ( frac{boldsymbol{x}+mathbf{1}}{mathbf{2}}=frac{boldsymbol{y}+mathbf{2}}{mathbf{3}}=frac{boldsymbol{z}-mathbf{3}}{mathbf{6}} ) which is at a distance of 3 units from the point (1,-2,3) |
12 |
| 774 | Find ( m ) if the point on the ( x ) -axis which is equidistant from (7,6) and (3,4) is ( left(frac{m}{2}, 0right) ) |
12 |
| 775 | 93. A plane passing through the points (0,–1, 0) and (0,0,1) and making an angle with the plane y-z+5 = 0, also passes through the point: [JEEM 2019-9 April (M) (a) (-,1,-4) (b) (12,-1,4) (C) (12,-1,-4) (d) (V2,1,4) |
12 |
| 776 | The distance between the line ( vec{r}=2 vec{i}- ) ( mathbf{2} overrightarrow{boldsymbol{j}}+boldsymbol{3} overrightarrow{boldsymbol{k}}+boldsymbol{lambda}(overrightarrow{boldsymbol{i}}-overrightarrow{boldsymbol{j}}+boldsymbol{4} overrightarrow{boldsymbol{k}}) ) and the plane ( vec{r} cdot(vec{i}+5 vec{j}+vec{k})=5 ) is A ( cdot frac{10}{3 sqrt{3}} ) в. ( frac{10}{9} ) c. ( frac{10}{3} ) D. |
12 |
| 777 | If ( vec{a}, vec{b} ) and ( vec{c} ) are mutually perpendicular vectors of equal magnitudes, If the angles which the vector ( 2 vec{a}+vec{b}+2 vec{c} ) makes with the vectors ( vec{a} ) is ( cos ^{-1} sqrt{frac{2}{m}} ) Find ( boldsymbol{m} ) | 12 |
| 778 | Find the equation of following planes: | 12 |
| 779 | The plane ( a x+b y+c z+d=0 ) divides the line joining the points ( left(x_{1}, y_{1}, z_{1}right) ) and ( left(x_{2}, y_{2}, z_{2}right) ) in the ratio A ( cdot frac{-left(a x_{1}+b y_{1}+c z_{1}+dright)}{left(a x_{2}+b y_{2}+c z_{2}+dright)} ) B. ( frac{left(a x_{1}+b y_{1}+c z_{1}+dright)}{left(a x_{2}+b y_{2}+c z_{2}+dright)} ) c. ( frac{a x_{1} x_{2}+b y_{1} y_{2}+c z_{1} z_{2}}{d^{2}} ) D. None of these |
12 |
| 780 | The vector equation ( boldsymbol{r}=boldsymbol{i}-boldsymbol{2} boldsymbol{j}-boldsymbol{k}+ ) ( t(6 j-k) ) represents a straight line passing through the points: A. (0,6,-1) and (1,-2,-1) в. (0,6,-1) and (-1,-4,-2) c. (1,-2,-1) and (1,4,-2) D. (1,-2,-1) and (0,-6,1) |
12 |
| 781 | If the distance of a point ( (a, a, a) ) from the origin is ( sqrt{108}, ) then the value of ( a ) is This question has multiple correct options ( A cdot 9 ) B. 6 ( c .-9 ) D. – 6 |
12 |
| 782 | Equation of a plane | 12 |
| 783 | In which ratio the plane ( Y Z ) divides the lines joining the points (2,1,2) and (-6,3,4) |
12 |
| 784 | A straight line passes through (1,-2,3) and perpendicular to the plane ( 2 x+ ) ( 3 y-z=7 . ) Find the direction ratios of normal to plane ( A cdot ) В. ( ) c. ( ) D. None of the above |
12 |
| 785 | ( X O Z ) plane divides the join of (2,3,1) and (6,7,1) in the ratio A . 3: 7 B. 2: 7 c. -3: 7 D. -2: 7 |
12 |
| 786 | 85. The distance of the point (1,3, -7) from the plane passing through the point (1, -1, -1), having normal perpendicular to both the lines Z-4 X-2 y+1 Z+7 , X-1 1 y +2 – 2 and 1-2 3 2 -1 , is: 2 JEE M 2017] S |
12 |
| 787 | If ( P(x, y, z) ) is point in the space at a distance ( r ) from the origin ( O ), then direction cosines of the line ( O P ) are |
12 |
| 788 | If the extremities of a diagonal of a square are (1,-2,3) and (2,-3,5) then area of the square is ( A cdot 6 ) B. 3 ( c cdot frac{3}{2} ) D. ( sqrt{3} ) |
12 |
| 789 | The equation of the plane passing through the intersection of the planes ( boldsymbol{x}+boldsymbol{y}+boldsymbol{z}=boldsymbol{6} ) and ( boldsymbol{2} boldsymbol{x}+boldsymbol{3} boldsymbol{y}+boldsymbol{4} boldsymbol{z}+boldsymbol{5}= ) ( 0, ) and the point (1,1,1) is A. ( 20 x+23 y+26 z-69=0 ) B. ( 20 x+23 y+26 z+69=0 ) c. ( 23 x+20 y+26 z-69=0 ) D. None of these |
12 |
| 790 | What are the direction ratios of the line if it passes through the intersection of the planes ( x=3 z+4 ) and ( y=2 z-3 ? ) A ( .(1,2,3) ) в. (2,1,3) c. (3,2,1) D. (1,3,2) |
12 |
| 791 | The ratio in which the joint of (2,1,5),(3,4,3) is divided by the plane ( 2 x+2 y-2 z-1=0 ) ( mathbf{A} cdot 5: 12 ) B. 12: 5 ( c .5: 7 ) D. 7: 5 |
12 |
| 792 | Three vertices of a tetrahedron are (0,0,0),(6,-5,-1) and ( (-4,1,3) . ) the centroid of the tetrahedron be (1,-2,5) then the fourth vertex is A. (2,-4,18) в. (1,-4,18) ( ^{c} cdotleft(frac{3}{2}, frac{-3}{2}, frac{7}{4}right) ) D. none of these |
12 |
| 793 | If ( boldsymbol{R} ) divides the line segment joining ( P(2,3,4) ) and ( Q(4,5,6) ) in the ratio ( -3: 2, ) then the parameter which represent ( boldsymbol{R} ) is ( A cdot 3 ) B . 2 ( c . ) D. – |
12 |
| 794 | If ( boldsymbol{alpha}, boldsymbol{beta}, boldsymbol{gamma} in[mathbf{0}, boldsymbol{2} boldsymbol{pi}], ) then the sum of all possible values of ( alpha, beta, gamma ) if ( sin alpha= ) ( -frac{1}{sqrt{2}}, cos beta=-frac{1}{2}, tan gamma=-sqrt{3}, ) is A ( cdot frac{22 pi}{3} ) B. ( frac{21 pi}{3} ) c. ( frac{20 pi}{3} ) D. ( 8 pi ) |
12 |
| 795 | A plane intersects the co ordinate axes at ( A, B, C . ) If ( O=(0,0,0) ) and (1,1,1) is the centroid of the tetrahedron ( O A B C ) then the sum of the reciprocals of the intercepts of the plane ( mathbf{A} cdot 12 ) B. ( frac{4}{3} ) c. 1 ( D cdot frac{3}{4} ) |
12 |
| 796 | The plane ( a x+b y+c z+(-3)=0 ) meet the co-ordinate axes in ( A, B, C ) The centroid of the triangle is B ( cdotleft(frac{3}{a} cdot frac{3}{b}, frac{3}{c}right) ) c. ( left(frac{a}{3} cdot frac{b}{3}, frac{c}{3}right) ) D. ( left(frac{1}{a} cdot frac{1}{b}, frac{1}{c}right) ) |
12 |
| 797 | The area of triangle whose vertices are (1,2,3),(2,5,-1) and (-1,1,2) is A . 150 sq.units B. 145 sq.units c. ( sqrt{155} / 2 ) sq.units D. ( 155 / 2 ) sq.units |
12 |
| 798 | Find the shortest distance between lines: ( frac{x-1}{1}=frac{y-2}{3}=frac{z-3}{2} ) and ( frac{x-4}{2}=frac{y-5}{3}=frac{z-6}{1} ) A . ( sqrt{6} ) B. ( sqrt{5} ) ( c cdot sqrt{3} ) D. 6 |
12 |
| 799 | Let the equation of the plane through the points (-2,-2,2),(1,1,1),(1,-1,2) be ( k x+ ) ( boldsymbol{m} boldsymbol{y}+boldsymbol{n} boldsymbol{z}+boldsymbol{p} . ) Find ( boldsymbol{k}+boldsymbol{m}+boldsymbol{n}+boldsymbol{p} ) ( A cdot 7 ) B. ( c cdot 4 ) D. 6 |
12 |
| 800 | Find the distance between the following pairs of points (-2,-3) and (3,2) | 12 |
| 801 | If ( C_{1}: x^{2}+y^{2}-20 x+64=0 ) and ( C_{2}: x^{2}+y^{2}+30 x+144=0 . ) Then the length of the shortest line segment ( boldsymbol{P Q} ) which touches ( C_{1} ) at ( P ) and to ( C_{2} ) at ( Q ) is A . 10 B. 15 ( c cdot 22 ) D. 27 |
12 |
| 802 | If ( x y- ) plane and ( y z- ) plane divides the line segment joining ( A(2,4,5) ) and ( B(3,5,-4) ) in the ratio ( a: b ) and ( p: q ) respectively then value of ( left(frac{a}{b}+frac{p}{q}right) ) may be A ( cdot frac{23}{12} ) в. ( frac{7}{5} ) ( c cdot frac{7}{12} ) D. ( frac{21}{10} ) |
12 |
| 803 | The image of the point (2,-1,1) by the plane ( 3 x+4 y-5 z=0 ) is A ( cdot(-2,1,-1) ) в. ( left(frac{2}{3}, frac{-1}{4}, frac{-1}{5}right) ) ( ^{mathbf{C}} cdotleft(frac{59}{25}, frac{-13}{25}, frac{2}{5}right) ) D. none of these |
12 |
| 804 | Distance between ( vec{r}=hat{i}+lambda(hat{j}+hat{k}) ) and ( vec{r}=hat{j}+mu(hat{j}+hat{k}) ) is equal to |
12 |
| 805 | ( operatorname{Let} boldsymbol{A}(mathbf{2} hat{boldsymbol{i}}+boldsymbol{3} hat{boldsymbol{j}}+mathbf{5} hat{boldsymbol{k}}) boldsymbol{B}(-hat{boldsymbol{i}}+boldsymbol{3} hat{boldsymbol{j}}+2 hat{boldsymbol{k}}) ) and ( C(lambda hat{i}+5 hat{j}+mu hat{k}) ) are vertices of ( a ) triangle and its median through ( A ) is equally inclined to the positive directions of the axes. The value of ( lambda+ ) ( mu ) is equal to A . -7 B . 2 c. 7 D. 17 |
12 |
| 806 | Direction cosines of the line ( frac{x+2}{2}= ) ( frac{2 y-5}{3}, z=-1 ) are A ( cdot frac{4}{5}, frac{3}{5}, 0 ) B. ( frac{3}{5^{prime}}, frac{4}{5^{prime}} frac{1}{5} ) c. ( quad-frac{3}{5}, frac{4}{5}, 0 ) D. ( frac{4}{5},-frac{2}{5}, frac{1}{5} ) |
12 |
| 807 | The plane through the intersection of the planes ( x+y+z=1 ) and ( 2 x+ ) ( 3 y-z+4=0 ) and parallel to ( y ) -axis also passes through the point. A. (-3,0,1) В. (3,3,-1) c. (3,2,1) D. (-3,1,1) |
12 |
| 808 | Find the equation of the plane passing through the points ( (mathbf{1}, mathbf{1}, mathbf{1}),(mathbf{3},-mathbf{1}, mathbf{2}),(-mathbf{3}, mathbf{5},-mathbf{4}) ) |
12 |
| 809 | Find the equation of the plane bisecting the line segment joining the points (-3,-2,1) and (1,6,-5) perpendicularly. |
12 |
| 810 | is 46. Let P be the image of the point (3,1,7) with respect to the plane x-y+z=3. Then the equation of the plane passing x y z . through P and containing the straight line = 1 z 1 (JEE Adv. 2016) (a) x+y-3z=0 (b) 3x+z=0 (c) X-4y+z=0 (d) 2x-y=0 hub the point (1 1 1 |
12 |
| 811 | The plane ( a x+b y+c z+(-3)=0 ) meet the co-ordinate axes in ( A, B, C . ) Then centroid of the triangle is A. ( (3 a, 3 b, 3 c) ) в. ( left(frac{3}{a} frac{3}{b}, frac{3}{c}right) ) ( ^{mathbf{C}} cdotleft(frac{a}{3}, frac{b}{3}, frac{c}{3}right) ) D. ( left(frac{1}{a}, frac{1}{b}, frac{1}{c}right) ) |
12 |
| 812 | If the points ( boldsymbol{A}(mathbf{3},-mathbf{2}, mathbf{4}), boldsymbol{B}(mathbf{1}, mathbf{1}, mathbf{1}) ) and ( C(-1,4,-2) ) are collinear, then the ratio in which ( C ) divides ( A B ) is A .1: 2 B . -2: 1 c. -1: 2 D. 4: 0 |
12 |
| 813 | Find the direction cosines (d.cs) of directed line ( O P ) if coordinates of ( P ) is ( (2,3,7), O ) being the origin. |
12 |
| 814 | Find the co-ordinates of a point lying on the line ( frac{boldsymbol{x}-mathbf{2}}{mathbf{3}}=frac{boldsymbol{y}+mathbf{3}}{mathbf{4}}=frac{boldsymbol{z}-mathbf{1}}{mathbf{7}} ) which is at a distance 10 units from (2,-3,1) begin{tabular}{l} A. (32,37,71) \ hline end{tabular} в. (-28,-43,-69) c. (-32,-37,-71) D. None of these |
12 |
| 815 | The direction angles of the line ( x= ) ( mathbf{4} z+mathbf{3}, boldsymbol{y}=mathbf{2}-mathbf{3} z ) are ( boldsymbol{alpha}, boldsymbol{beta} ) and ( gamma, ) then ( cos alpha+cos beta+cos gamma= ) A ( cdot frac{2}{sqrt{26}} ) B. ( frac{8}{sqrt{26}} ) c. 1 D. 2 |
12 |
| 816 | The equation of a plane passing through the point ( A(2,-3,7) ) and making equal intercepts on the axes, is? A. ( x+y+z=3 ) в. ( x+y+z=6 ) c. ( x+y+z=9 ) D. ( x+y+z=4 ) |
12 |
| 817 | If the dr’s the line are ( (1+lambda, 1-lambda, 2) ) and it makes an angle ( 60^{circ} ) with the ( Y ) – axis then ( lambda ) is A ( .1 pm sqrt{3} ) B. ( 4 pm sqrt{5} ) c. ( 2 pm 2 sqrt{3} ) D. ( 2 pm sqrt{5} ) |
12 |
| 818 | If direction ratios of the normal of the plane which contains the lines ( frac{x-2}{3}= ) ( frac{boldsymbol{y}-boldsymbol{4}}{boldsymbol{2}}=frac{boldsymbol{z}-boldsymbol{1}}{boldsymbol{1}} & frac{boldsymbol{x}-boldsymbol{6}}{boldsymbol{3}}=frac{boldsymbol{y}+boldsymbol{2}}{boldsymbol{2}}= ) ( frac{z-2}{1} ) are ( (a, 1,-26), ) then ( a ) is equal to A . 5 B. 6 ( c cdot 7 ) D. |
12 |
| 819 | What is the distance of the point ( (p, q, r) ) from the ( x- ) axis. |
12 |
| 820 | What is the angle between ( vec{A} ) and the resultant of ( (overrightarrow{boldsymbol{A}}+overrightarrow{boldsymbol{B}}) ) and ( (overrightarrow{boldsymbol{A}}-overrightarrow{boldsymbol{B}}) ) ( mathbf{A} cdot 0^{circ} ) B ( cdot tan ^{-1}left(frac{A}{B}right) ) ( ^{mathbf{c}} cdot tan ^{-1}left(frac{B}{A}right) ) D. ( tan ^{-1}left(frac{A-B}{A+B}right) ) |
12 |
| 821 | Show that angles between any two ( operatorname{diagonals} boldsymbol{theta}=cos ^{-1}left(frac{1}{3}right) ) | 12 |
| 822 | A triangle ( A B C ) is placed so that the midpoints of its sides are on the ( boldsymbol{x}, boldsymbol{y} ) and ( z ) axes respectively. Lengths of the intercepts made by the plane containing the triangle on these axes ( operatorname{are} ) respectively ( boldsymbol{alpha}, boldsymbol{beta}, boldsymbol{gamma}, ) then the coordinates of the centroid of the triangle ( A B C ) are A ( cdotleft(-frac{alpha}{3}, frac{beta}{3}, frac{gamma}{3}right) ) В ( cdotleft(frac{alpha}{3},-frac{beta}{3}, frac{gamma}{3}right) ) ( ^{mathbf{C}} cdotleft(frac{alpha}{3}, frac{beta}{3},-frac{gamma}{3}right) ) D ( cdotleft(frac{alpha}{3}, frac{beta}{3}, frac{gamma}{3}right) ) |
12 |
| 823 | Using vectors, find the value of ( lambda ) such that the points ( (boldsymbol{lambda},-mathbf{1 0}, mathbf{3}),(mathbf{1},-mathbf{1}, mathbf{3}) ) and (3,5,3) are collinear. |
12 |
| 824 | Find the equation of the plane through the intersection of the planes ( 3 x- ) ( 4 y+5 z=10 ) and ( 2 x+2 y-3 z=4 ) and parallel to the line ( boldsymbol{x}=mathbf{2} boldsymbol{y}=mathbf{3} boldsymbol{z} ) |
12 |
| 825 | Find the direction cosines of the line PQ joining the points ( P(2,3,4) ) and ( Q(2,1,1) ) |
12 |
| 826 | A symmetrical form of the line of intersection of the planes ( boldsymbol{x}=boldsymbol{a} boldsymbol{y}+boldsymbol{b} ) and ( boldsymbol{z}=boldsymbol{c} boldsymbol{y}+boldsymbol{d} ) is : A. ( frac{x-b}{a}=frac{y-1}{1}=frac{z-d}{c} ) ( ^{text {В }} cdot frac{x-b-a}{a}=frac{y-1}{1}=frac{z-d-c}{c} ) c. ( frac{x-a}{b}=frac{y-0}{1}=frac{z-c}{d} ) D. ( frac{x-b-a}{b}=frac{y-1}{0}=frac{z-d-c}{d} ) |
12 |
| 827 | Find the distance of the point ( boldsymbol{P}(boldsymbol{3}, boldsymbol{4}, boldsymbol{4},) ) from the point, where the line joining the points ( boldsymbol{A}(boldsymbol{3},-boldsymbol{4},-boldsymbol{5}) ) and ( boldsymbol{B}(boldsymbol{2},-boldsymbol{3}, boldsymbol{1}) ) intersected the plane ( 2 x+y+z=7 ) |
12 |
| 828 | The distance of the point (1,-5,9) from the plane ( boldsymbol{x}-boldsymbol{y}+boldsymbol{z}=mathbf{5} ) measured along the line ( x=y=z ) is: ( A cdot 3 sqrt{10} ) в. ( 10 sqrt{3} ) c. ( frac{10}{sqrt{3}} ) D. ( frac{20}{3} ) |
12 |
| 829 | Find the equation of the plane passing through the intresection of the planes ( x-2 y+z=1 ) and ( 2 x+y+z=8 ) and parallel to the line with direction ratio proportional to ( 1,2,1, ) find also the perpendicular distance of (1,1,1) from this plane. |
12 |
| 830 | A rectangular parallelopiped is formed by drawing planes through the points (-1,2,5) and (1,-1,-1) and paralle to the coordinate planes. the length of the diagonal of the parallelopiped is ( A cdot 2 ) B. 3 c. 6 D. |
12 |
| 831 | Derive the equation of the locus of a point twice as far from (-2,3,4) as from (3,-1,-2) |
12 |
| 832 | Prove that 1,1,1 cannot be direction cosines of a straight line | 12 |
| 833 | The acute angle between two lines such that the direction cosines ( I, m, n ) of each of them satisfy the equations ( l+m+ ) ( boldsymbol{n}=mathbf{0} ) and ( l^{2}+boldsymbol{m}^{2}-boldsymbol{n}^{2}=mathbf{0} ) is : A . 30 B . 45 c. 60 D. 15 |
12 |
| 834 | Vector components of the vector with initial points (2,1) and terminal point (-5,7) are ( mathbf{A} cdot-6 hat{i}+7 widehat{j} ) B. ( -7 hat{i}+6 hat{j} ) c. ( -6 hat{i}-7 widehat{j} ) D. None |
12 |
| 835 | A mirror and a source of light are situated at the origin ( mathrm{O} ) and at a point on ( mathrm{O} X, ) respectively. A ray of light from the source strikes the mirror and is reflected. If the direction ratios of the normal to the plane are ( 1,-1,1, ) then find the DCs of the reflected ray. A ( cdot frac{1}{3}, frac{2}{3}, frac{2}{3} ) в. ( -frac{1}{3}, frac{2}{3}, frac{2}{3} ) ( c cdot-frac{1}{3},-frac{2}{3},-frac{2}{3} ) D. ( -frac{1}{3},-frac{2}{3}, frac{2}{3} ) |
12 |
| 836 | The points (2,5) and (5,1) are the two opposite vertices of a rectangle. If the other two vertices are points on the straight line ( y=2 x+k, ) then the value of k is ( A cdot 4 ) B. 3 ( c cdot-4 ) ( D cdot-3 ) ( E ) |
12 |
| 837 | Let ( vec{A}, vec{B} ) and ( vec{C} ) be unit vectors. Suppose that ( vec{A} cdot vec{B}=vec{A} cdot vec{C}=0 ) and that the angle between ( vec{B} ) and ( vec{C} ) is ( frac{pi}{6} ) then ( overrightarrow{boldsymbol{A}}= ) A ( cdot pm 2(vec{B} times vec{C}) ) B ( cdot pm(vec{B} times vec{C}) ) c. ( pm 2(vec{B}+vec{C}) ) D. ( pm(vec{B}+vec{C}) ) |
12 |
| 838 | If the lines through the points (4,1,2) and ( (5, k, 0) ) is parallel to the line through the points (2,1,1) and (3,3,1) find ( k ) |
12 |
| 839 | The coordinates of the foot of the perpendicular from the point (1,-2,1) on the plane containing the lines, ( frac{x+1}{6}=frac{y-1}{7}=frac{z-3}{8} ) and ( frac{x-1}{3}= ) ( frac{boldsymbol{y}-boldsymbol{2}}{mathbf{5}}=frac{boldsymbol{z}-boldsymbol{3}}{boldsymbol{7}}, ) is : A ( cdot(2,-4,2) ) в. (-1,2,-1 c. (0,0,0) D. (1,1,1) |
12 |
| 840 | If the points ( boldsymbol{A}(1,2,-1), B(2,6,2) ) and ( C(lambda,-2,-4) ) are collinear, then ( lambda ) is ( mathbf{A} cdot mathbf{0} ) B. ( c .-2 ) ( D ) |
12 |
| 841 | Find the centroid of a triangle, midpoints of whose sides are (1,2,-3),(3,0,1) and (-1,1,-4) |
12 |
| 842 | If the points ( (a, 1),(1,2) ) and ( (0, b+1) ) are collinear, then show that ( frac{1}{a}+frac{1}{b}=1 ) |
12 |
| 843 | The ratio of ( y z ) -plane divide the line joining the points ( A(3,1,-5), B(1,4,-6) ) is A . 3: 1 B. -1: 3 c. 1: 3 D ( .-3: 1 ) |
12 |
| 844 | ( P(1,1,1) ) and ( Q(lambda, lambda, lambda) ) are two points in the space such that ( P Q=sqrt{27} ), then the value(s) of ( lambda ) can be A . -4 в. -2,4 ( c cdot 2 ) D. 4,3 |
12 |
| 845 | The line ( boldsymbol{x}-mathbf{2} boldsymbol{y}+mathbf{4} boldsymbol{z}+mathbf{4}=mathbf{0}, boldsymbol{x}+boldsymbol{y}+ ) ( z-8=0 ) intersects the plane ( x-y+ ) ( mathbf{2} z+mathbf{1}=mathbf{0} ) at the point A ( cdot(3,2,3) ) B . (5,2,1) c. (2,5,1) D . (3,4,1) |
12 |
| 846 | In which ratio does the ( Y Z ) plane divide the line joining the points (-2,4,7) and (3,-5,8) |
12 |
| 847 | A sphere of constant radius ( 2 k ) passes through the origin and meets the axes in ( A, B, C . ) The locus of the centroid of the tetrahedron ( boldsymbol{O} boldsymbol{A} boldsymbol{B} boldsymbol{C} ) is A ( cdot x^{2}+y^{2}+z^{2}=4 k^{2} ) B cdot ( 9left(x^{2}+y^{2}+z^{2}right)=4 k^{2} ) C . ( x^{2}+y^{2}+z^{2}=k^{2} ) D. None of these |
12 |
| 848 | The point lying on angle bisector of the planes ( boldsymbol{x}+mathbf{2} boldsymbol{y}+mathbf{2} boldsymbol{z}-mathbf{6}=mathbf{0} ) and ( mathbf{2} boldsymbol{x}- ) ( boldsymbol{y}+boldsymbol{4}=mathbf{0} ) is A. (2,4,0) B. (-1,3,2) D. (-2,4,0) |
12 |
| 849 | Find the values of ( a ) and ( b ) so that the points ( (boldsymbol{a}, boldsymbol{b}, mathbf{3}),(mathbf{2}, mathbf{0},-mathbf{1}) ) and (1,-1,-3) are collinear. |
12 |
| 850 | If ( |overrightarrow{boldsymbol{A}} times overrightarrow{boldsymbol{B}}|=overrightarrow{boldsymbol{A}} cdot overrightarrow{boldsymbol{B}}, ) then ( |overrightarrow{boldsymbol{A}}+overrightarrow{boldsymbol{B}}| ) is: ( mathbf{A} cdot|vec{A}|+|vec{B}| ) B. ( sqrt{|vec{A}|^{2}+|vec{B}|^{2}} ) ( ^{mathrm{c}} cdot sqrt{|vec{A}|^{2}+|vec{B}|^{2}+frac{|vec{A}||vec{B}|}{sqrt{2}}} ) D cdot ( sqrt{|vec{A}|^{2}+|vec{B}|^{2}+sqrt{2}|vec{A}||vec{B}|} ) |
12 |
| 851 | 61. Statement-1: The point A(3, 1, 6) is the mirror image of the point B(1, 3, 4) in the plane x-y +z=5. Statement-2: The plane x-y+z=5 bisects the line segment joining A(3, 1,6) and B(1,3,4). [2010] (a) Statement -1 is true, Statement -2 is true; Statement-2 is not a correct explanation for Statement-1. (6) Statement -1 is true, Statement -2 is false. (c) Statement-1 is false, Statement -2 is true. (d) Statement – 1 is true, Statement 2 is true; Statement-2 is a correct explanation for Statement -1. |
12 |
| 852 | Write the vector equation of the line equation ( frac{x-1}{2}=frac{y-2}{3}=frac{z-3}{4} ) | 12 |
| 853 | The name of the figure formed by the points (3,-5,1),(-1,0,8) and (7,-10,-6) is A. a triangle B. a straight line c. an isosceles triangle D. an equilateral triangle |
12 |
| 854 | The lines ( frac{x-1}{2}=frac{y+1}{3}=frac{z-1}{4} ) and ( frac{boldsymbol{x}-mathbf{3}}{mathbf{1}}=frac{boldsymbol{y}-boldsymbol{k}}{mathbf{2}}=frac{boldsymbol{z}}{mathbf{1}} ) intersect if ( boldsymbol{K} ) equals ( A cdot frac{3}{2} ) в. ( frac{9}{2} ) c. ( frac{-2}{9} ) D. ( frac{-3}{2} ) |
12 |
| 855 | Find unit vector perpendicular to the plane passing through the points (1,2,3),(2,-1,1) and (1,2,-4) | 12 |
| 856 | If ( P ) is a point ( (x, y) ) on the line ( y=-3 x ) such that ( boldsymbol{P} ) and ( boldsymbol{Q}(boldsymbol{3}, boldsymbol{4}) ) are on opposite side of the line ( 3 x-4 y=8, ) then: A ( cdot x>frac{8}{5}, yfrac{8}{15}, y<-frac{8}{5} ) c. ( x=frac{8}{15}, y=frac{-8}{5} ) D. ( x=2, y=-2 ) |
12 |
| 857 | ( P ) is a variable points which moves such that ( 3 P A=2 P B . ) If ( A= ) (-2,2,3) and ( B=(13,-3,13) ) prove that ( P ) satisfies the equation ( x^{2}+y^{2}+ ) ( z^{2}+28 x-12 y+10 z-247=0 ) |
12 |
| 858 | For what value of ( mathrm{m}, ) the points (3,5) ( (m, 6) ) and ( left(frac{1}{2}, frac{15}{2}right) ) are collinear? |
12 |
| 859 | Find the direction cosines of the line ( frac{boldsymbol{x}+mathbf{2}}{mathbf{2}}=frac{mathbf{2} boldsymbol{y}-mathbf{5}}{mathbf{3}} ; boldsymbol{z}=-mathbf{1} ) |
12 |
| 860 | Find the co-ordinates of a point, which is at a distance of 21 units from the point ( boldsymbol{A}=(mathbf{1},-mathbf{3}, mathbf{4}) ) in the direction of vectors ( 2 hat{i}-3 hat{j}-6 hat{k} ) |
12 |
| 861 | 72. 22. If the lines *-72 173 If the lines Z y-3 1 -4 -ka and -1 and – k z-5 – are coplanar, then k can have [JEE M 2013] (a) any value (c) exactly two values (b) exactly one value (d) exactly three values |
12 |
| 862 | If two vertices of a triangle ( A B C ) are ( A(-1,2,4) ) and ( B(2,-3,0), ) and the centroid is (2,0,2) then the vertex ( C ) has the coordinates A. (5,1,2) в. ( left(1,-frac{1}{3}, frac{7}{3}right) ) ( ^{mathbf{C}} cdotleft(3,-frac{2}{3}, frac{5}{3}right) ) D. none of these |
12 |
| 863 | Find the distance of the point (2,12,5) from the point of intersection of the line ( overrightarrow{boldsymbol{r}}=2 hat{hat{boldsymbol{i}}}-mathbf{4} hat{boldsymbol{j}}+mathbf{2} hat{boldsymbol{k}}+boldsymbol{lambda}(boldsymbol{3} hat{boldsymbol{i}}+boldsymbol{4} hat{boldsymbol{j}}+boldsymbol{2} hat{boldsymbol{k}}) ) and the plane ( vec{r} cdot(hat{boldsymbol{i}}-boldsymbol{2} hat{boldsymbol{j}}+hat{boldsymbol{k}})=mathbf{0} ) |
12 |
| 864 | Find the direction ratio of the line ( frac{x-1}{2}=3 y=frac{2 z+3}{4} ) |
12 |
| 865 | The equation of line passing through (1,-2,3) and having ( operatorname{drs}(2,3,1) ) is ( ^{text {A }} cdot frac{x-1}{2}=frac{y+2}{3}=frac{z-3}{1} ) B. ( frac{x+1}{2}=frac{y+2}{3}=frac{z-3}{1} ) c. ( frac{x-1}{2}=frac{y-2}{3}=frac{z-3}{1} ) D. none of these |
12 |
| 866 | If a point ( P ) in the space such that ( overline{O P} ) is inclined to ( O X ) at 45 and ( O Z ) to 60 then ( overline{O P} ) inclined to ( O Y ) is A ( .75^{circ} ) B . ( 75^{circ} ) or ( 105^{circ} ) C. ( 60^{circ} ) or ( 120^{circ} ) D. None of these |
12 |
| 867 | Find the root of the perpendicular from point (2,3,2) to the line ( frac{4-x}{2}=frac{y}{6}= ) ( frac{1-z}{3} ) also find perpendicular distance from the point to the line. |
12 |
| 868 | 11. – and 25 x-1 y+1 x +1 Z. Ifthe straight lines 4+1 Z 1 2 k 5 2 k are coplanar, then the plane (s) containing these two lines is (are) (2012) (a) y + 2z=-1 (b) y+z=-1 (c) y-z=-1 . (d) y-2z=-1 |
12 |
| 869 | The projection of a directed line segment on the co-ordinate axes are ( 12,4,3, ) then the direction cosines of the line are A ( cdot frac{-12}{13}, frac{-4}{13}, frac{-3}{13} ) В. ( frac{12}{13}, frac{4}{13}, frac{3}{13} ) c. ( frac{12}{13}, frac{-4}{13}, frac{3}{13} ) D. ( frac{12}{13}, frac{4}{13}, frac{-3}{13} ) |
12 |
| 870 | If the point ( (x, y) ) is equidistant from the points ( (a+b, b-a) ) and ( (a- ) ( b, a+b), ) then ( b x=a y ) A. True B. False |
12 |
| 871 | If the points ( a(1,2,-1), B(2,6,2) ) and ( c(lambda,-2,-4) ) are collinear then ( lambda ) is ( mathbf{A} cdot mathbf{0} ) B . 2 ( c .-2 ) D. |
12 |
| 872 | The image of the point (1,2,3) through the plane ( boldsymbol{x}+boldsymbol{y}+boldsymbol{4} boldsymbol{z}=boldsymbol{0} ) is A ( cdotleft(frac{-2}{3}, frac{1}{3}, frac{-11}{3}right) ) в. ( left(frac{3}{7},-frac{6}{7}, frac{19}{7}right) ) ( ^{mathbf{c}} cdotleft(frac{2}{3}, frac{1}{3}, frac{11}{3}right) ) D. ( left(frac{-2}{3}, frac{-1}{3}, frac{11}{3}right) ) |
12 |
| 873 | Show that the points ( (3,3),(h, 0) ) and ( (0, k) ) are collinear, if ( frac{1}{n}+frac{1}{k}=frac{1}{3} ) |
12 |
| 874 | Angle between lines whose direction ( operatorname{cosine} operatorname{satisfy} l+m+n=0, l^{2}+ ) ( boldsymbol{m}^{2}-boldsymbol{n}^{2}=mathbf{0} ) |
12 |
| 875 | a b c a b c 23. Distance between two parallel planes 2x+y+2z=8 and 4x + 2y +4z +5=0 is [2004] (a) 2 min |
12 |
| 876 | ( operatorname{can} frac{2}{sqrt{3}}, frac{-2}{sqrt{3}}, frac{-1}{sqrt{3}} ) be the direction ratios of any directed line? Justify your answer |
12 |
| 877 | A vector equation of the line of intersection of the planes ( boldsymbol{r}=boldsymbol{b}+ ) ( boldsymbol{lambda}_{1}(boldsymbol{b}-boldsymbol{a})+boldsymbol{mu}_{1}(boldsymbol{a}+boldsymbol{c}) ) ( boldsymbol{r}=boldsymbol{c}+boldsymbol{lambda}_{2}(boldsymbol{b}-boldsymbol{c})+boldsymbol{mu}_{1}(boldsymbol{a}+boldsymbol{b}) boldsymbol{a}, boldsymbol{b}, boldsymbol{c} ) being non-coplanar vectors is. A ( cdot r=a+mu_{1}(b+c) ) B . ( r=b+mu_{1}(a+2 c) ) C . ( r=a+mu_{1}(b+2 c) ) D. ( r=b+mu_{1}(a+c) ) |
12 |
| 878 | The length of the perpendicular from the origin to the plane passing through the point ( a ) and containing the line ( r=bar{b}+ ) ( lambda bar{c} ) is A ( cdot frac{[a b c]}{|a times b+b times c+c times a|} ) В. ( frac{[a b c]}{|a times b+b times c|} ) c. ( frac{[a b c]}{|b times c+c times a|} ) D. ( frac{[a b c]}{|a times b+c times a|} ) |
12 |
| 879 | Let the equation of the plane which contains the line ( boldsymbol{x}=frac{boldsymbol{y}-boldsymbol{3}}{boldsymbol{2}}=frac{boldsymbol{z}-boldsymbol{5}}{boldsymbol{3}} ) and which is perpendicular to the plane ( 2 x+7 y-3 z=1 . ) be ( k x-m y-z+ ) ( boldsymbol{p}=mathbf{0} . ) Find ( boldsymbol{p}-boldsymbol{k}-boldsymbol{m} ? ) |
12 |
| 880 | Find direction cosine line ( boldsymbol{x}=mathbf{3} boldsymbol{z}+ ) ( mathbf{2}, boldsymbol{y}=mathbf{2}-mathbf{5} boldsymbol{z} ) |
12 |
| 881 | Find the coordinates of the points which divides the line joining the points (2,-4,3),(-4,5,-6) in the ratio ( (i) 1:-4 ) ( (i i) 2: 1 ) |
12 |
| 882 | Direction ratio of two lines are ( l_{1}, boldsymbol{m}_{1}, boldsymbol{n}_{1} ) and ( l_{2}, m_{2}, n_{2} ) then direction ratios of the line perpendicular to both the lines are A ( . l_{1}-l_{2}, m_{1}-m_{2}, n_{1}-n_{2} ) в. ( l_{1}+l_{2}, m_{1}+m_{2}, n_{1}+n_{2} ) c. ( m_{1} n_{2}-n_{1} m_{2}, n_{1} l_{2}-n_{2} l_{1}, l_{1} m_{2}-m_{1} l_{2} ) D. ( m_{1} n_{2}-n_{1} m_{2}, n_{1} l_{2}-n_{1} l_{1}, l_{1} m_{2}-m_{1} l_{2} ) |
12 |
| 883 | The points (3,2,0),(5,3,2) and ( (-9,6,-3), ) are the vertices of a triangle ( A B C . A D ) is the internal bisector of ( angle B A C ) which meets ( B C ) at D. Then the co-ordinates of ( D ), are A ( cdotleft[frac{17}{16}, frac{57}{16}, frac{19}{8}right] ) В. ( left[frac{19}{8}, frac{57}{16}, frac{17}{16}right] ) ( ^{mathbf{c}} cdotleft[0,0, frac{17}{16}right] ) D. ( left[frac{17}{16}, 0,0right] ) |
12 |
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