We provide application of integrals practice exercises, instructions, and a learning material that allows learners to study outside of the classroom. We focus on application of integrals skills mastery so, below you will get all questions that are also asking in the competition exam beside that classroom.

#### List of application of integrals Questions

Question No | Questions | Class |
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1 | In an isosceles trapezium, the length of one of the parallel sides, and the lengths of the non-parallel sides are all equal to ( 30 . ) In order to maximize the area of the trapezium, the smallest angle should be A ( cdot frac{pi}{6} ) B. ( c cdot frac{pi}{3} ) D. | 12 |

2 | (i) Find the area bounded by the curve ( boldsymbol{y}=boldsymbol{x}(mathbf{1}-boldsymbol{x}) ) between the points where it crosses the x-axis. (ii) Find the area between the curves ( boldsymbol{y}=boldsymbol{x} ) and ( boldsymbol{y}=boldsymbol{x}^{3} ) | 12 |

3 | Find the area between the region of parabola ( boldsymbol{y}^{2}=mathbf{2} boldsymbol{x} ) and circle ( boldsymbol{x}^{2}+boldsymbol{y}^{2}= ) 8 | 12 |

4 | Find the area of the region bounded by the curves ( x^{2}+y^{2}=16 ) and ( x^{2}=6 y ) | 12 |

5 | The area of the region described by ( A=(x, y): x^{2}+y^{2} leq 1 ) and ( y^{2} leq 1-x ) is: A ( cdot frac{pi}{2}+frac{4}{3} ) В ( cdot frac{pi}{2}-frac{4}{3} ) c. ( frac{pi}{2}-frac{2}{3} ) D. ( frac{pi}{2}+frac{2}{3} ) | 12 |

6 | ( operatorname{Given} f(x)=left{begin{array}{l}x, 0 leq x<frac{1}{2} \ frac{1}{2}, x=frac{1}{2} \ 1-x, frac{1}{2}<x leq 1end{array} ) and right. ( g(x)=left(x-frac{1}{2}right)^{2}, x epsilon R, ) Then the area (in sq.units) of the region bounded by the curves ( y=f(x) ) and ( y=g(x) ) between the lines ( 2 x=1 ) and ( 2 x=sqrt{3} ) is ( ^{A} cdot frac{1}{3}+frac{sqrt{3}}{4} ) B. ( frac{1}{2}-frac{sqrt{3}}{4} ) c. ( frac{1}{2}+frac{sqrt{3}}{4} ) D. ( frac{sqrt{3}}{4}-frac{1}{3} ) | 12 |

7 | The area bounded by the curves ( y= ) ( sqrt{5-x^{2}} ) and ( y=|x-1| ) is ( ^{mathbf{A}} cdotleft(frac{5 pi}{4}-2right) ) square unit B. ( frac{(5 pi-2)}{4} ) square units c. ( frac{(5 pi-2)}{2} ) square units D ( cdotleft(frac{5 pi}{2}-2right) ) square unit | 12 |

8 | Find the area of the region. ( left{(x, y): x^{2}+y^{2} leq 8, x^{2} leq 2 yright} ) | 12 |

9 | 6. bounded by the curves y =lnx, y = ln x,y=| In x and y=| In pxis [2002] (a) 4er units (b) 6 sq. units (c) 10 sq. units (d) none of these | 12 |

10 | The total area enclosed by the lines ( y= ) ( |x|,|x|=1 ) and ( y=0 ) is: ( mathbf{A} cdot mathbf{1} ) B. 2 ( c cdot frac{1}{2} ) D. 4 | 12 |

11 | Show the region of feasible solution under the following constraints ( 2 x+ ) ( boldsymbol{y} leq mathbf{8}, boldsymbol{x} geq mathbf{0}, boldsymbol{y} geq mathbf{0} ) in answer book | 12 |

12 | A conic passes through the point (2,4) and is such that the segment of any of its tangents at any point contained between the coordinate axes is bisected at the point of tangency. Then equation of auxiliary circle of the conic is A ( cdot x^{2}+y^{2}=16 ) B . ( x^{2}+y^{2}=25 ) c. ( x^{2}+y^{2}=4 sqrt{2} ) D. none of these | 12 |

13 | The area bounded by the curve ( y= ) ( 2 x^{4}-x^{2}, x ) -axis and the two ordinates corresponding to the minima of the function is ( frac{8}{a} . ) Find ( a ) | 12 |

14 | The equation of tangent to ( x^{2}=y^{3} ) at (1,1) is | 12 |

15 | The area bounded by the curves ( x+ ) ( mathbf{2}|boldsymbol{y}|=mathbf{1} ) and ( boldsymbol{x}=mathbf{0} ) is? A ( cdot frac{1}{4} ) в. ( frac{1}{2} ) ( c .1 ) D. 2 | 12 |

16 | The area bounded by the curves y = f(x), the x-axis and the ordinates x=1 and x=b is (b-1) sin (35+4). Then f(x) is (a) (x – 1) cos (3x +4) (1982 – 2 Marks) (b) sin (3x+4) (c) sin (3x+4) +3 (x – 1) cos (3x+4) (d) none of these | 12 |

17 | The area of the region bounded by the curve ( y=2 x-x^{2} ) and the line ( y=x ) is square units. A ( cdot frac{1}{6} ) в. ( frac{1}{2} ) ( c cdot frac{1}{3} ) D. ( frac{7}{6} ) | 12 |

18 | Area bounded by ( y=2 x-x^{2} &(x- ) 1) ( ^{2}+y^{2}=1 ) in first quadrant, is: A ( cdot frac{pi}{2}-frac{4}{3} ) B ( cdot frac{pi}{2}-frac{2}{3} ) c. ( frac{pi}{2}+frac{4}{3} ) D. ( frac{pi}{2}+frac{2}{3} ) | 12 |

19 | 7. Find the area bounded by the x-axis, part of the curve y = 1+ and the ordir and the ordinates at x = 2 and x = 4. If the ordinate at x= a divides the area into two equal parts, find a. (1983 – 3 Marks) | 12 |

20 | Identify a possible graph for function ( boldsymbol{f} ) given by ( boldsymbol{f}(boldsymbol{x})=-boldsymbol{2}|boldsymbol{x}| ) A. graph a B. graph b c. graph c D. graph d | 12 |

21 | Identify the graph of the exponential function ( boldsymbol{f} ) ( boldsymbol{f}(boldsymbol{x})=boldsymbol{e}^{(boldsymbol{x}+mathbf{2})} ) | 12 |

22 | Find the area bounded by line ( y=3 x+ ) 2, x-axis and ordinates ( x=-1 ) and ( boldsymbol{x}=mathbf{1} ) | 12 |

23 | What is the area of the region bounded by the parabola ( boldsymbol{y}^{2}=mathbf{6}(boldsymbol{x}-mathbf{1}) ) and ( boldsymbol{y}^{2}=mathbf{3} boldsymbol{x} ) A ( cdot frac{sqrt{6}}{3} ) в. ( frac{2 sqrt{6}}{3} ) c. ( frac{4 sqrt{6}}{3} ) D. ( frac{5 sqrt{6}}{3} ) | 12 |

24 | In the given figure, a square ( boldsymbol{O A B C} ) has been inscribed in the quadrant OPB ( Q ) If ( boldsymbol{O} boldsymbol{A}=mathbf{2 0} boldsymbol{c m} ) then the area of the shaded region is ( [text {take } pi=mathbf{3 . 1 4}] ) A ( cdot 214 c m^{2} ) B. ( 228 c m^{2} ) ( c cdot 222 c m^{2} ) ( mathrm{D} cdot 242 mathrm{cm}^{2} ) | 12 |

25 | The area of the region bounded by ( x= ) ( mathbf{0}, boldsymbol{y}=mathbf{0}, boldsymbol{x}=mathbf{2}, boldsymbol{y}=mathbf{2}, boldsymbol{y} leq boldsymbol{e}^{boldsymbol{x}} ) and ( boldsymbol{y} geq ln boldsymbol{x}, ) is ( mathbf{A} cdot 6-4 ln 2 ) B. 4 ellп ( 2-2 ) ( c cdot 2 ln 2 ) D. ( 6-2 ) ellп 2 | 12 |

26 | Prove that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius ( R ) is ( frac{2 R}{sqrt{3}} . ) Also find the maximum volume | 12 |

27 | The line ( 3 x+2 y=13 ) divides the area enclosed by the curve ( 9 x^{2}+4 y^{2}- ) ( 18 x-16 y-11=0 ) in two parts Find the ratio of the larger area to the smaller area A ( cdot frac{3 pi+2}{pi-2} ) в. ( frac{3 pi-2}{pi+2} ) c. ( frac{pi+2}{pi-2} ) D. ( frac{pi-2}{pi+2} ) | 12 |

28 | Sketch the region ( left{(x, y): 9 x^{2}+4 y^{2}=36right} ) and find the area of the region enclosed by it, using integration. | 12 |

29 | Find the area of the region common to be circle ( x^{2}+y^{2}=9 ) and the parabola ( boldsymbol{y}^{2}=mathbf{8} boldsymbol{x} ) | 12 |

30 | fthe area of the region bounded by point ( (x, y) ) satisfying the condition ( left{(x, y) ; 0 leq y leq x^{2}+1,0 leq y leq x+1right. ) | 12 |

31 | Find area enclosed by ( |x|+|y| leq 3 ) and ( x y geq 2 ) ( mathbf{A} .5-2 log 4 ) B. ( 3-2 log 4 ) c. ( 3-log 4 ) D. ( 3-2 log 7 ) | 12 |

32 | For the curve ( f(x)=frac{1}{1+x^{2}}, ) let two points on it be ( boldsymbol{A}(boldsymbol{alpha}, boldsymbol{f}(boldsymbol{alpha})), boldsymbol{B}left(-frac{1}{boldsymbol{alpha}}, boldsymbol{f}left(-frac{1}{boldsymbol{alpha}}right)right)(boldsymbol{alpha}>mathbf{0}) ) Find the minimum area bounded by the line segments ( 0 A, O B ) and ( f(x) ) where ‘O’ is the origin. A ( cdot frac{(pi-1)}{2} ) в. ( frac{pi}{2} ) c. ( frac{(pi-2)}{2} ) D. Maximum area is always infinitt | 12 |

33 | ( * ) ( * ) ( * ) ( k ) | 12 |

34 | Which of the following is true regarding the symmetry of the function: ( f(x)= ) ( boldsymbol{x}^{boldsymbol{5}}+boldsymbol{x}^{boldsymbol{3}}+boldsymbol{3} ) A. ( f(x)=c ) B. Symmetric about x-axis c. Its an odd function D. None of these | 12 |

35 | The area common to the cardioids ( r= ) ( boldsymbol{a}(mathbf{1}+cos boldsymbol{theta}) ) and ( boldsymbol{r}=boldsymbol{a}(mathbf{1}-cos boldsymbol{theta}) ) is: ( ^{A} cdotleft(frac{3 pi}{2}+4right) a^{2} ) в. ( left(frac{3 pi}{2}-4right) a^{2} ) c. ( left(frac{pi}{2}+4right) a^{2} ) D ( cdotleft(frac{pi}{2}-4right) a^{2} ) | 12 |

36 | Area bounded by the curves satisfying the conditions ( frac{x^{2}}{25}+frac{y^{2}}{36} leq 1 leq frac{x}{5}+frac{y}{6} ) is given by A ( cdot 15left(frac{pi}{2}+1right) ) sq.units B. ( frac{15}{4}left(frac{pi}{2}-1right) ) sq.units c. ( 30(pi-1) ) sq.unit D. ( frac{15}{2}(pi-2) ) sq.unit | 12 |

37 | If the area enclosed by the parabolas ( y=a-x^{2} ) and ( y=x^{2} ) is ( 18 sqrt{2} ) sq. units Find the value of ‘a’ A ( . a=-9 ) B. ( a=6 ) ( mathbf{c} cdot a=9 ) D. ( a=-6 ) | 12 |

38 | Find the area of the region enclosed by the parabola ( x^{2}=y, ) the line ( y=x+2 ) and ( x ) -axis. | 12 |

39 | The area of the curve ( y^{2}=(7-x)(5+ ) ( x ) ) above ( x ) -axis and between the ordinates ( x=-5 ) and ( x=1 ) is ( n pi ). Find ( boldsymbol{n} ) | 12 |

40 | The area bounded by the parabola ( y^{2}= ) ( 4 x ) and the line ( y=2 x-4 ) A. 9 sq. units B. 5 sq. units c. 4 sq. units D. 2 sq. units | 12 |

41 | Find the area bounded by ( y=x+sin x ) and its inverse between ( boldsymbol{x}=mathbf{0} ) and ( boldsymbol{x}= ) ( 2 pi ) A .2 B. 4 ( c .6 ) D. 8 | 12 |

42 | Find the area bounded by the curve ( y= ) ( cos , x- ) axis and the ordinates ( x=0 ) and ( x=2 pi ) | 12 |

43 | Using integration, find the area bounded by the curve ( x^{2}=4 y ) and the line ( x=4 y-2 ) | 12 |

44 | Find area bounded by ( x^{2}+y^{2} leq ) ( 2 a x, y^{2} geq a x, x geq 0, y leq 0 ) | 12 |

45 | ( # ) ( * ) ( k ) ( k ) | 12 |

46 | Area bounded by the curves ( y=x^{2} ) and ( y=2-x^{2} ) is A ( cdot frac{8}{3} ) sq. units B. ( frac{3}{8} ) sq. units c. ( _{-5 text {funits }}^{text {s }} ) D. None of these | 12 |

47 | Example 2.7 Using the method of integration, show that the volume of a right circular cone of base radius r and height his V = -rah. 3 | 12 |

48 | The parabola ( y^{2}=4 x+1 ) divides the ( operatorname{disc} x^{2}+y^{2} leq 1 ) into two regions with ( operatorname{areas} A_{1} ) and ( A_{2} . ) Then ( left|A_{1}-A_{2}right| ) equals A ( cdot frac{1}{3} ) B. ( frac{2}{3} ) c. D. ( frac{pi}{3} ) | 12 |

49 | The area bounded by ( x^{2}=4 a y ) and ( y= ) ( 2 a ) is? A ( cdot frac{16 sqrt{2} a^{2}}{3} ) в. ( frac{16 a^{2}}{3} ) c. ( frac{8 a^{2}}{3} ) D. ( frac{8 sqrt{2} a^{2}}{3} ) | 12 |

50 | Find the area bounded on the right by the line ( x+y=2, ) on the left by the parabola ( y=x^{2} ) and below by the ( x- ) axis. | 12 |

51 | The area of the region bounded by the curve ( boldsymbol{x}=boldsymbol{y}^{2}-boldsymbol{2} ) and ( boldsymbol{x}=boldsymbol{y} ) is A ( cdot frac{9}{4} ) B. 9 c. ( frac{9}{2} ) D. | 12 |

52 | The area of the smaller part of the circle ( x^{2}+y^{2}=a^{2}, ) cut off by the line ( x=frac{a}{sqrt{2}} ) is given by: ( ^{text {A }} cdot frac{a^{2}}{2}left(frac{pi}{2}+1right) ) в. ( frac{a^{2}}{2}left(frac{pi}{2}-1right) ) c. ( a^{2}left(frac{pi}{2}-1right) ) D. None of these | 12 |

53 | Find the area between the two parabolas ( boldsymbol{y}^{2}=boldsymbol{x}, boldsymbol{X}^{2}=boldsymbol{y} ) | 12 |

54 | 41. Find the area of the region bounded by the curves y = x², y = 12 – xº|and y=2, which lies to the right of the line x=1. (2002 – 5 Marks) | 12 |

55 | Let ( boldsymbol{y}=boldsymbol{g}(boldsymbol{x}) ) be the inverse of ( mathbf{a} ) bijective mapping ( boldsymbol{f}: boldsymbol{R} rightarrow boldsymbol{R} quad boldsymbol{f}(boldsymbol{x})= ) ( 3 x^{3}+2 x . ) The area bounded by graph of ( g(x), ) the axis and the ordinate at ( x=5 ) is A ( cdot frac{5}{4} ) B. ( frac{7}{4} ) ( c cdot frac{9}{4} ) D. ( frac{13}{4} ) | 12 |

56 | If ( f(x) ) is monotonic in ( (a, b) ) then prove that the area bounded by the ordinates at ( boldsymbol{x}=boldsymbol{a}: boldsymbol{x}=boldsymbol{b}: boldsymbol{y}=boldsymbol{f}(boldsymbol{x}) ) and ( boldsymbol{y}= ) ( f(c), c epsilon(a, b) ) is minimum when ( c= ) ( frac{a+b}{2} ) Hence if the area bounded by the graph of ( f(x)=frac{x^{3}}{3}-x^{2}+a, ) the straight lines ( x=0, x=2 ) and the ( x ) -axis is minimum then find the value or ‘a’. ( A cdot frac{2}{3} ) B. ( frac{2}{5} ) ( c cdot frac{7}{3} ) D. ( frac{4}{3} ) | 12 |

57 | Find the area of the region bounded by the curves ( x=frac{1}{2}, x=2, y=log x ) and ( boldsymbol{y}=mathbf{2}^{boldsymbol{x}} ) ( ^{mathbf{A}} cdot frac{4-sqrt{2}}{log 2}-frac{5}{2} log 2+frac{3}{2} s q cdot u n i t s ) B. ( frac{4+sqrt{2}}{log 2}-frac{3}{2} log 2+frac{5}{2} ) sq.units ( ^{mathrm{c}} cdot frac{4-sqrt{2}}{log 2}-frac{3}{2} log 2+frac{5}{2} s q cdot u n i t s ) D. ( frac{4+sqrt{2}}{log 2}-frac{5}{2} log 2+frac{3}{2} ) sq.units | 12 |

58 | Let S be the area of the region enclosed by y = e* v=0, x=0 and x= 1; then (2012) (a) Sz- (6) 821-1 A V +- hot otiofith | 12 |

59 | The area bounded by the line ( x=1 ) and the curve ( sqrt{frac{boldsymbol{y}}{boldsymbol{x}}}+sqrt{frac{boldsymbol{x}}{boldsymbol{y}}}=mathbf{4} ) is A ( .2 sqrt{3} ) B. ( sqrt{3} ) c. ( 3 sqrt{2} ) D. ( 4 sqrt{3} ) | 12 |

60 | Area bounded by the curves ( frac{y}{x}=log x ) and ( frac{boldsymbol{y}}{mathbf{2}}=-boldsymbol{x}^{2}+boldsymbol{x} ) equals: A. 7/12 B. 12/7 ( c cdot 7 / 6 ) D. 6/7 | 12 |

61 | Find the area under the curve ( boldsymbol{y}= ) ( left(x^{2}+2right)^{2}+2 x ) between the lines ( x=0 ) ( x=2 ) and the ( X ) -axis | 12 |

62 | 34. Let the straight line x = b divide the area enclosed by y=(1-x)2 , y=0, andx=0 into two parts R, (0 < x < b) and Rz (b < x < 1) such that R1 – R2 = 2. Then b equals (2011) | 12 |

63 | 30. The area of the plane region bounded by the curves x+2y2 = 0 and x+3y2 = lis equal to [2008] | 12 |

64 | Determine the area of the region enclosed by the curve ( boldsymbol{y}= ) ( sqrt{x+1} ) in [0,4] and the ( x ) -axis and the lines ( boldsymbol{x}=mathbf{0}, boldsymbol{x}=mathbf{4} ) | 12 |

65 | The area bounded by ( y=cos x, y= ) ( boldsymbol{x}+mathbf{1}, boldsymbol{y}=mathbf{0} ) is A ( cdot frac{3}{2} ) B. ( frac{2}{3} ) ( c cdot frac{1}{2} ) D. ( frac{5}{2} ) | 12 |

66 | For ( x, t in R ) let ( boldsymbol{p}_{1}(boldsymbol{x})=(sin t) boldsymbol{x}^{2}-(2 cos boldsymbol{t}) boldsymbol{x}+sin boldsymbol{t} ) be a family of quadratic polynomials in ( x ) with variable co efficients. Let ( A(t)= ) ( int_{0}^{1} p_{t}(x) d x . ) Which of the following statements are true? (I) ( A(t)<0 ) for all ( t ) (II) ( A(t) ) has infinitely many critica points. (III) ( A(t)=0 ) for infinitely many t. (IV) ( A^{prime}(t)<0 ) for all ( t ) A. (I) and (II) only B. (II) and (III) only C. (III) and (IV) only ( D cdot(I V) ) and (1) only | 12 |

67 | The area bounded by the curve ( y= ) ( sin ^{-1} x ) and ( operatorname{lines} x=0,|y|=frac{pi}{2} ) is 2 sq.unit A . True B. False | 12 |

68 | Area bounded by the curves ( boldsymbol{y}= ) ( e^{x}, y=e^{-x} ) and the straight line ( x=1 ) is (in sq units) A ( cdot_{e+frac{1}{e}} ) B. ( e+frac{1}{e}+2 ) c. ( _{e+frac{1}{e}-2} ) D. ( _{e-frac{1}{e}+2} ) E ( e-frac{1}{e} ) | 12 |

69 | The area bounded by ( y=cos x, y= ) ( x+1 ) and ( y=0 ) in the second quadrant is A ( cdot frac{3}{2} ) sq. units B. 2 sq. units c. 1 sq. unit D. ( frac{1}{2} ) sq, units | 12 |

70 | The area bounded by the curves ( x^{2}+ ) ( y^{2} leq 8 ) and ( y^{2} geq 4 x ) lying in the first quadrant is not equal to A ( cdot 32left(frac{pi}{8}-frac{1}{3}right) ) в. ( frac{32}{3}left(frac{3 pi}{8}-1right) ) c. ( _{4 pi}-frac{32}{3} ) D. ( frac{1}{3}(12 pi-32) ) | 12 |

71 | The area bounded by the curve ( x^{2}=4 y ) and straight line ( x=4 y-2 ) is ( A cdot frac{3}{8} ) B. ( c cdot frac{7}{8} ) D. | 12 |

72 | The larger area bounded between the curve ( |boldsymbol{x}|^{2}+|boldsymbol{y}|^{2}=mathbf{1} ) and the line ( boldsymbol{x}+ ) ( y=1 ) is ( ^{mathrm{A}} cdotleft(frac{pi}{4}-frac{1}{2}right) ) sq. unit B. ( left(frac{pi}{4}+1right) ) sq. unit c. ( left(frac{3 pi}{4}+frac{1}{2}right) ) sq. unit D ( cdotleft(frac{3 pi}{4}-frac{1}{2}right) ) sq. unit | 12 |

73 | Find the area of the region bounded by the ellipse ( frac{x^{2}}{4}+frac{y^{2}}{9}=1 ) | 12 |

74 | If the area of the region bounded by the curves, ( boldsymbol{y}=boldsymbol{x}^{2}, boldsymbol{y}=frac{mathbf{1}}{boldsymbol{x}} ) and the lines ( boldsymbol{y}=mathbf{0} ) and ( boldsymbol{x}=boldsymbol{t}(boldsymbol{t}>mathbf{1}) ) is ( mathbf{1} ) sq. unit then ( t ) is equal to? A ( cdot frac{4}{3} ) В. ( e^{2 / 3} ) ( c cdot frac{3}{2} ) ( mathbf{D} cdot e^{3 / 2} ) | 12 |

75 | Find the area of the region bounded by the curve ( x y=c^{2}, ) the ( X ) -axis, and the ( operatorname{lines} x=c, x=2 c ) | 12 |

76 | Find the area bounded by the curve ( y= ) ( 2 x-x^{2} ) and the straight line ( y=-x ) | 12 |

77 | 1. The area (in sq. units) bounded by the parabola y=x2-1. the tangent at the point (2, 3) to it and the y-axis is: [JEE M 2019-9 Jan (M) (a) wloo (b) (c) 56 3 | 12 |

78 | The area of region bounded by curve ( y=cos 2 x, ) line ( x=0 ) and ( x=frac{pi}{3} ) is A ( cdot frac{2-sqrt{3}}{4} ) B. ( frac{sqrt{3}}{4} ) c. ( frac{4-sqrt{3}}{4} ) D. ( frac{sqrt{3}-4}{4} ) | 12 |

79 | The parabola ( y^{2}=4 x ) and ( x^{2}=4 y ) divide the square region bounded by the ( operatorname{lines} x=4, y=4 ) and the coordinate axes. If ( S_{1}, S_{2}, S_{3} ) are the areas of these parts numbered from top to bottom respectively, then This question has multiple correct options A ( . S_{1}: S_{2} equiv 1: 1 ) B. ( S_{2}: S_{3} equiv 1: 2 ) c. ( S_{1}: S_{3} equiv 1: 1 ) ( mathbf{D} cdot S_{1}:left(S_{1}+S_{2}right) equiv 1: 2 ) | 12 |

80 | The area bounded by ( boldsymbol{x}=boldsymbol{a} cos ^{3} boldsymbol{theta}, boldsymbol{y}= ) ( a sin ^{3} theta ) is: A ( cdot frac{3 pi a^{2}}{16} ) в. ( frac{3 pi^{2}}{8} ) c. ( frac{3 pi a^{2}}{32} ) D. ( 3 pi a^{2} ) | 12 |

81 | The points at which the tangents to the curve ( y=x^{3}-12 x+18 ) are parallel to x-axis are? A ( cdot(2,-2),(-2,-34) ) в. (2,34),(-2,0) c. (0,34),(-2,0) D. (2,2),(-2,34) | 12 |

82 | What is the smallest distance between the point (-2,-2) and a point on the circumference of the circle given by ( (x-1)^{2}+(y-2)^{2}=4 ? ) ( A cdot 3 ) B. 4 ( c .5 ) D. 6 E. 7 | 12 |

83 | Area bounded by the parabola ( y^{2}=x- ) 3 and ( x=5, ) is ( dots . s q ) units ( A cdot frac{8 sqrt{2}}{3} ) B. ( 4 sqrt{2} ) ( mathrm{c} cdot 3 sqrt{2} ) D. ( 5 sqrt{2} ) | 12 |

84 | The ratio in which the area bounded by the curves ( y^{2}=4 x ) and ( x^{2}=4 y ) is divided by the line ( x=1 ) is A. 64: 49 B. 15: 34 c. 15: 49 D. None o fthese | 12 |

85 | Prove that the area enclosed between two parabolas ( y^{2}=4 a x ) and ( x^{2}=4 a y ) is ( frac{16 a^{2}}{3} ) | 12 |

86 | 31. Consider a square with vertices at (1, 1), (-1, 1), (-1,-1) and (1, -1). Let S be the region consisting of all points inside the square which are nearer to the origin than to any edge. Sketch the region S and find its area. | 12 |

87 | Using the method of integraton find the area of the region bounded by lines: ( 2 x+y=4,3 x-2 y=6 ) and ( x-3 y+ ) ( 5=0 ) | 12 |

88 | A square is inscribed in a circle of radius ( 7 mathrm{cm} ). Find the maximum area of the square. ( mathbf{A} cdot 98 ) B. 89 c. 86 D. 96 | 12 |

89 | Find the area of the region bounded by the ellipse ( frac{x^{2}}{16}+frac{y^{2}}{9}=1 ) A . ( 6 pi ) в. ( 12 pi ) c. ( 18 pi ) D. ( 24 pi ) | 12 |

90 | The area of the plane region bounded by the curve ( x+2 y^{2}=0 ) and ( x+3 y^{2}=1 ) is equal to: A ( cdot-frac{4}{3} ) B. ( frac{4}{3} ) ( c cdot frac{2}{3} ) D. None of these | 12 |

91 | Area bounded by ( x^{2}=4 a y ) and ( y= ) ( frac{8 a^{3}}{x^{2}+4 a^{2}} ) is : | 12 |

92 | Show that the maximum volume of the cylinder which can be inscribed in a sphere of radius ( 5 sqrt{3} mathrm{cm} ) is ( 500 pi c m^{3} ? ) | 12 |

93 | If the area enclosed between the curves ( boldsymbol{y}=boldsymbol{k} boldsymbol{x}^{2} ) and ( boldsymbol{x}=boldsymbol{k} boldsymbol{y}^{2},(boldsymbol{k}>mathbf{0}), ) is ( mathbf{1} ) square unit. Then ( k ) is? A ( cdot frac{1}{sqrt{3}} ) в. ( frac{2}{sqrt{3}} ) c. ( frac{sqrt{3}}{2} ) D. ( sqrt{3} ) | 12 |

94 | Find the equation of the curve passing through the point ( (0,1), ) if the slope of tangent to the curve at each of its point is equal to the sum of the abscissa and the product of the abscissa and ordinate of the point. | 12 |

95 | Area of the region bounded by the curves, ( boldsymbol{y}=boldsymbol{e}^{boldsymbol{x}}, boldsymbol{y}=boldsymbol{e}^{-boldsymbol{x}} ) and the straight line ( x=1 ) is given by A ( cdotleft(e-e^{-1}+2right) ) sq.unit B ( cdotleft(e-e^{-1}-2right) ) sq.unit c. ( left(e+e^{-1}-2right) ) sq.unit D. None of the above | 12 |

96 | The value of the integral ( int_{1}^{2} sqrt{(2 x+3)left(3 x^{2}+4right)} d x ) cannot exceed ( A cdot sqrt{48} ) B. ( sqrt{66} ) c. ( sqrt{73} ) D. none of these | 12 |

97 | The area (in square units) bounded by ( boldsymbol{y}=boldsymbol{x} e^{|x|} ) and lines ( |boldsymbol{x}|=mathbf{1}, boldsymbol{y}=mathbf{0}, ) is A .4 B. 6 ( c .1 ) D. 2 | 12 |

98 | The smaller area enclosed by ( boldsymbol{y}=boldsymbol{f}(boldsymbol{x}) ) where ( f(x) ) is polynomial of least degree satisfying ( left[lim _{x rightarrow 0} 1+frac{f(x)}{x^{3}}right]^{frac{1}{x}}=e ) and the circle ( x^{2}+y^{2}=2 ) above the ( x- ) axis is A ( cdot frac{pi}{2}+frac{3}{5} ) в. ( frac{pi}{2}-frac{3}{5} ) c. ( frac{pi}{2}-frac{6}{5} ) D. None of these | 12 |

99 | The area enclosed between the parabolas ( boldsymbol{y}^{2}=mathbf{1 6} boldsymbol{x} ) and ( boldsymbol{x}^{2}=mathbf{1 6} boldsymbol{y} ) is A ( cdot frac{64}{3} ) sq.units в. ( frac{256}{3} ) sq.units c. ( frac{16}{3} ) sq.units D. None of these | 12 |

100 | Show that the height of a closed right circular cylinder of given surface and maximum volume, is equal to the diameter of its base. | 12 |

101 | Give possible expressions for the length and breadth of the rectangle whose area is ( 3 x^{2}-8 x+5 ) | 12 |

102 | A conic ( C ) passing through ( P(1,2) ) is such that the slope of its tangent at any point on the conic is inversely proportional to the ordinate of that point and conic ( C ) passes through origin. If a circle touches the conic ( C ) at the point ( boldsymbol{P}(1,2) ) and passes through the focus of the conic then its radius is- A. B. ( sqrt{2} ) ( c cdot 2 ) D. ( sqrt{3} ) E. ( sqrt{5} ) | 12 |

103 | The area of the region between the curve ( boldsymbol{y}=boldsymbol{x}^{3} ) and the lines ( boldsymbol{y}=-boldsymbol{x} ) and ( boldsymbol{y}=mathbf{1} ) is: A. 5 sq. units B. ( frac{4}{5} ) sq. units c. ( frac{5}{4} ) sq. units D. ( frac{3}{5} ) sq. units | 12 |

104 | Draw the graph of ( 2 x+y=6 ) and ( 2 x- ) ( boldsymbol{y}+mathbf{2}=mathbf{0} . ) Shade the region bounded by these lines and ( x-y . ) Find the area of the shaded region. | 12 |

105 | 12. (1983 – 2 Sketch the region bounded by the curves y=V5-X and y=x-1) and find its area. (1985 – 5 Marks) | 12 |

106 | Find the area of the region bounded by the curve ( y^{2}=4 x quad ) and the line ( x=3 ) | 12 |

107 | If the area enclosed by ( y^{2}=4 a x ) and line ( y=a x ) is ( frac{1}{3} . ) sq.unit, then the area enclosed by ( y=4 x ) with same parabola is ( frac{4}{3} ) squnit A. True B. False | 12 |

108 | Show that the line ( frac{x}{a}+frac{y}{b}=1, ) tocuhes the curve ( y=b . e^{frac{-x}{a}} ) at the point where the curve intersects the axis of ( y ) | 12 |

109 | Let ( f(x)=x^{frac{2}{3}}, x geq 0 . ) Then the area of the region enclosed by the curve ( y= ) ( f(x) ) and the three lines ( y=x, x=1 ) and ( x=8 ) A ( cdot frac{63}{2} ) в. ( frac{93}{5} ) c. ( frac{105}{7} ) D. ( frac{129}{10} ) | 12 |

110 | The area of the figure bounded by the ( operatorname{lines} x=0, x=frac{pi}{2}, f(x)=sin x ) and ( g(x)=cos x ) is A ( cdot 2(sqrt{2}-1) ) B. ( sqrt{3}-1 ) c. ( 2(sqrt{3}-1) ) D ( cdot 2(sqrt{2}+1) ) | 12 |

111 | Find the equations corresponding to following graph: A. ( y=-4 x-6 ) B. ( y=4 x-6 ) ( c cdot y=4 x ) D. none of these | 12 |

112 | The area bounded by the curve ( y=sqrt{x} ) the line ( 2 y+3=x ) and the ( x ) -axis in the first quadrant is ( mathbf{A} cdot mathbf{9} ) в. ( frac{27}{4} ) ( c .36 ) D. 18 | 12 |

113 | Find the common area enclosed by the parabolas ( boldsymbol{y}^{2}=boldsymbol{x} ) and ( boldsymbol{x}^{2}=boldsymbol{y} ) | 12 |

114 | Area of the region bounded by ( y^{2} leq ) ( mathbf{4} boldsymbol{x}, boldsymbol{x}+boldsymbol{y} leq mathbf{1}, boldsymbol{x} geq mathbf{0}, boldsymbol{y} geq mathbf{0} ) is ( boldsymbol{a} sqrt{mathbf{2}}+boldsymbol{b} ) then value of ( a-b ) is? A .4 B. 6 ( c cdot 8 ) D. 12 | 12 |

115 | ( UJU 16. The area of the region bounded by the curves y = x-2), x = 1, x = 3 and the x-axis is (a) 4 (6) 2 (c) 3 (d) [2004] 1 | 12 |

116 | Find the area of the region bounded by the curve ( y^{2}=x ) and the lines ( x= ) ( 1, x=4 ) and the ( x ) -axis. | 12 |

117 | Area enclosed by the curves ( y= ) ( ln x ; y=ln |x| ; y=|ln x| ) and ( y= ) ( |ln | x|| ) is equal to ( A cdot 2 ) B. 4 ( c cdot 8 ) D. cannot be determined | 12 |

118 | A tangent to the curve ( y=x^{2}+3 x ) passes through a point (0,-9) if it drawn at the point- A. (-3,0) в. (1,4) c. (0,0) (年. ( 0,0,0,0) ) D. (-4,4) | 12 |

119 | Using integration find the area of the following region ( (x, y):|x+2| leq y leq ) ( sqrt{20-x^{2}} ) | 12 |

120 | Area of the region bounded by the ( operatorname{curves} boldsymbol{y}|boldsymbol{y}| pm boldsymbol{x}|boldsymbol{x}|=mathbf{1} ) and ( boldsymbol{y}=|boldsymbol{x}| ) is: A ( cdot frac{pi}{8} ) sq.unit B . ( frac{pi}{4} ) sq.unit c. ( frac{pi}{2} ) sq.unit D. ( pi ) sq.unit | 12 |

121 | Find the area of the figure contained between the parabola ( x^{2}=4 y ) and the curve ( boldsymbol{y}=frac{boldsymbol{8}}{boldsymbol{x}^{2}+boldsymbol{4}} ) | 12 |

122 | Consider the curves ( y=sin x ) and ( y= ) ( cos x ) What is the area of the region bounded by the above two curves and the lines ( x=frac{pi}{4} ) and ( x=frac{pi}{2} ? ) B. ( sqrt{2}+1 ) ( c cdot 2 sqrt{2} ) ( D ) | 12 |

123 | Area enclosed between the curves ( y= ) ( 8-x^{2} ) and ( y=x^{2}, ) is: A . ( 32 / 3 ) B. ( 64 / 3 ) c. ( 30 / 4 ) D. | 12 |

124 | f ( boldsymbol{y}=boldsymbol{x}^{2}+boldsymbol{x}, ) then find ( frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}} ) | 12 |

125 | 36. Let C, and C, be the graphs of the functions y = x2 and y = 2x, 0 < x < 1 respectively. Let C, be the graph of a function y=f(x), 0 SX S1,f(0) = 0. For a point P on C, let the lines through P, parallel to the axes, meet C, and C, at Q and R respectively (see figure.) If for every position of P (on (), the areas of the shaded regions OPQ and ORP are | 12 |

126 | The area between the curves ( y=sqrt{x} ) and ( boldsymbol{y}=boldsymbol{x}^{3} ) is A ( cdot frac{1}{12} ) sq. units в. ( frac{5}{12} ) sq. units c. ( frac{3}{5} ) sq. units D. ( frac{4}{5} ) sq. units | 12 |

127 | The subnormal at any point on the curve ( x y^{n}=a^{n+1} ) is constant for ( mathbf{A} cdot n=0 ) B . ( n=1 ) c. ( n=-2 ) D. ( n=2 ) | 12 |

128 | Find area between: ( boldsymbol{x}=mathbf{2} boldsymbol{y}-boldsymbol{y}^{2} boldsymbol{a} boldsymbol{n} boldsymbol{d} boldsymbol{y}=boldsymbol{2}+boldsymbol{x} ) | 12 |

129 | toppr Q Type your question_ or ( x ) between the oralnates ( x=a ) メ ( x= ) ( b, ) is given by definite integral ( int_{a}^{b} y d x ) or ( int_{a}^{b} f(x) d x ) and the area bounded by the curve ( x=f(y), ) the axis of ( y & ) two abscissae ( y=c & y=d ) is given by ( int_{c}^{d} x d y ) or ( int_{c}^{d} f(x) d y . ) Again if we consider two curves ( boldsymbol{y}=boldsymbol{f}(boldsymbol{x}), boldsymbol{y}= ) ( g(x) ) where ( f(x) geq g(x) ) in the interval ( [a, b] ) where ( x=a & x=b ) are the points of intersection of these two curves Shown by the graph given Then area bounded by these two curves is given by ( int_{a}^{b}[boldsymbol{f}(boldsymbol{x})-boldsymbol{g}(boldsymbol{x})] boldsymbol{d} boldsymbol{x} ) On the basis of above information answer the following questions. The area bounded by parabolas ( boldsymbol{y}= ) ( x^{2}+2 x+1 & y=x^{2}-2 x+1 ) and the line ( y=frac{1}{4} ) is equal to ( mathbf{A} ) square unit B. ( overline{3} ) square unit ( c ) ( overline{2} ) square unit D. 2 square unit | 12 |

130 | Find the smaller area enclosed by the circle ( x^{2}+y^{2} ) and the line ( x+y=2 ) | 12 |

131 | Draw a rough sketch of the curve ( y^{2}= ) ( 4 x ) and find the area of the region enclosed by the curve and the line ( y=x ) | 12 |

132 | The area between the curves ( y=x^{2} ) and ( boldsymbol{y}=frac{2}{1+x^{2}} ) is A ( cdot pi-frac{1}{3} ) B . ( pi-2 ) c. ( pi-frac{2}{3} ) D. ( pi+frac{2}{3} ) | 12 |

133 | Set up an equation of a tangent to the graph of the following function. A cone is circumscribed about a sphere of radius R. The vertex angle in the axial section of the cone is ( 2 alpha ). Find the area of the axial section of the cone. At what value of ( alpha ) is the area of the cone the least? | 12 |

134 | The area (in sq.units) of the region ( left{(x, y): y^{2} geq 2 x text { and } x^{2}+y^{2} leqright. ) ( 4 x, x geq 0, y geq 0 ) is A ( cdot pi-frac{8}{3} ) B. ( _{pi-frac{4 sqrt{2}}{3}} ) ( ^{mathrm{C}} cdot frac{pi}{2}-frac{2 sqrt{2}}{3} ) D. ( pi-frac{4}{3} ) | 12 |

135 | The area in the first quadrant bounded by the curves ( x^{2}=2 y, y^{2}=2 x ) and ( x^{2}+y^{2}=3 ) is | 12 |

136 | Which of the following equations shows the shaded region in the above figure? A ( cdot xleft(y-frac{2}{3} xright) geq 0 ) B. ( xleft(y-frac{3}{2} xright) geq 0 ) c. ( xleft(y+frac{3}{2} xright) geq 0 ) D ( cdot xleft(y+frac{2}{3} xright) geq 0 ) E ( cdot xleft(y+frac{3}{2} xright) leq 0 ) | 12 |

137 | Show that all rectangles inscribed in a fixed circle square has maximum area. | 12 |

138 | An equilateral triangle has area ( A mathrm{cm}^{2} ) A regular hexagon of maximum area is cut off from the triangle. If the area of the hexagon is ( 320 mathrm{cm}^{2} ), then the area ( A ) is A ( cdot 640 mathrm{cm}^{2} ) в. ( 480 mathrm{cm}^{2} ) ( c cdot 600 c m^{2} ) D. ( 400 mathrm{cm}^{2} ) | 12 |

139 | If the area bounded by the curve ( |boldsymbol{y}|= ) ( sin ^{-1}|x| ) and ( x=1 ) is ( a(pi+b), ) then the value ( a-b ) is: A ( . ) B. 2 ( c cdot 3 ) D. 4 | 12 |

140 | The area of the region, bounded by the curves ( boldsymbol{y}=sin ^{-1} boldsymbol{x}+boldsymbol{x}(mathbf{1}-boldsymbol{x}) ) and ( boldsymbol{y}= ) ( sin ^{-1} x-x(1-x) ) in the first quadrant is ( A cdot 1 ) в. ( frac{1}{2} ) ( c cdot frac{1}{3} ) D. | 12 |

141 | Find the area of the region bounded by the following curves, the ( X ) -axis and the given lines. ( boldsymbol{y}=boldsymbol{x}^{4}, boldsymbol{x}=mathbf{1}, boldsymbol{x}=mathbf{5} ) | 12 |

142 | The triangle formed by the tangent to the curve ( f(x)=x^{2}+b x-b ) at the point (1,1) and the co-ordinate axes lies in the first quadrant.lf its area is 2 sq.unit, then the value of ( b ) is: A. -3 sq.unit B. – 2 sq.unit c. -1 sq.unit D. 0 sq.unit | 12 |

143 | Maximum area of rectangle whose two sides are ( boldsymbol{x}=boldsymbol{x}_{0}, boldsymbol{x}=boldsymbol{pi}-boldsymbol{x}_{0} ) and which is inscribed in a region bounded by ( boldsymbol{y}= ) ( sin x ) and ( x- ) axis is obtained, when ( x_{0} in ) | 12 |

144 | Show that the differential equation of the family of circles having their centre at the origin and radius ( a ) is [ boldsymbol{x}+boldsymbol{y} frac{boldsymbol{d} boldsymbol{y}}{boldsymbol{d} boldsymbol{x}}=mathbf{0} ] | 12 |

145 | The area bounded by the curves ( y^{2}=4 x ) and ( x^{2}=4 y ) is : A ( cdot frac{32}{3} ) B. ( frac{16}{3} ) ( c cdot frac{8}{3} ) D. | 12 |

146 | Find the area of the region enclosed between the two circles ( x^{2}+y^{2}=1 & ) ( (x-1)^{2}+y^{2}=1 ) A ( cdot frac{pi}{6}-frac{sqrt{3}}{2} ) squnits B. ( frac{pi}{3}-frac{sqrt{3}}{2} ) sq.units c. ( frac{pi}{6}-frac{sqrt{3}}{4} ) sq.units D. ( frac{pi}{3}-frac{sqrt{3}}{4} ) sq.units | 12 |

147 | 24. Sketch the region bounded by the curves y = x- and 2 y=- 9. Find the area. 1+x2 (1992 – 4 Marks) | 12 |

148 | The angle of intersection between the curves ( boldsymbol{x}^{2}=mathbf{4}(boldsymbol{y}+mathbf{1}) ) and ( boldsymbol{x}^{2}=-mathbf{4}(boldsymbol{y}+ ) 1) is. A ( cdot frac{pi}{6} ) B. ( frac{pi}{4} ) c. 0 D. | 12 |

149 | Area common to the curves ( y^{2}=a x ) and ( x^{2}+y^{2}=4 a x ) is equal to A ( cdot(9 sqrt{3}+4 pi) frac{a^{2}}{3} ) B . ( (9 sqrt{3}+4 pi) a^{2} ) c. ( (9 sqrt{3}-4 pi) frac{a^{2}}{3} ) D. None of these | 12 |

150 | The area enclosed between the curves ( x^{2}=y ) and ( y^{2}=x ) is equal to This question has multiple correct options A ( cdot frac{1}{3} cdot ) sq.unit B ( cdot 2 int_{0}^{1}left(x-x^{2}right) d x ) C cdot area of the region ( left{(x, y): x^{2} leq y leq|x|right} ) D. none of the above | 12 |

151 | The area of the smaller part bounded by the semicircle ( boldsymbol{y}=sqrt{mathbf{4}-boldsymbol{x}^{2}}, boldsymbol{y}=boldsymbol{x} sqrt{mathbf{3}} ) and x-axis is A ( cdot frac{pi}{3} ) в. ( frac{2 pi}{3} ) c. ( frac{4 pi}{3} ) D. | 12 |

152 | The area bounded by parabola ( boldsymbol{y}^{2}=boldsymbol{x} ) straight line ( y=4 ) and ( y-a x i s ) is- A ( cdot frac{16}{3} ) B. ( 7 sqrt{2} ) c. ( frac{32}{3} ) D. ( frac{64}{3} ) | 12 |

153 | The ratio in which the area bounded by the curves ( y^{2}=4 x ) and ( x^{2}=4 y ) is divided by the line ( x=1 ) is A. 64: 49 B. 15: 34 c. 15: 49 D. None o fthese | 12 |

154 | An edge of a variable cube is increasing at the rate of ( 3 mathrm{cm} / mathrm{s} ). How fast is the volume of the cube increasing when the edge is ( 10 mathrm{cm} ) long? A. ( 900 mathrm{cm}^{3} / mathrm{s} ) B. ( 920 mathrm{cm}^{3} / mathrm{s} ) ( mathbf{c} .850 mathrm{cm}^{3} / mathrm{s} ) D. ( 950 mathrm{cm}^{3} / mathrm{s} ) | 12 |

155 | Sketch the graph for ( y=1+ ) ( mathbf{3}(log |sin boldsymbol{x}|+log |mathbf{c s c} boldsymbol{x}|) ) | 12 |

156 | The area bounded by the tangent and normal to the curve ( y(6-x)=x^{2} ) at (3,3) and the ( x ) -axis is ( mathbf{A} cdot mathbf{5} ) B. 6 c. 15 D. 3 | 12 |

157 | The area bounded by the circle ( x^{2}+ ) ( y^{2}=8, ) the parabola ( x^{2}=2 y ) and the line ( y=x ) in ( y geq 0 ) is A ( cdot frac{2}{3}+2 pi ) в. ( frac{2}{3}-2 pi ) c. ( frac{2}{3}+pi ) D. ( frac{2}{3}-pi ) | 12 |

158 | Two vertices of a rectangle are on the positive ( mathbf{x} ) -axis. The other two vertices lie on the lines ( y=4 x ) and ( y=-5 x+6 ) Then the maximum area of the rectangle is? ( A cdot frac{2}{3} ) B. ( frac{2}{4} ) ( c cdot frac{1}{3} ) D. ( frac{4}{3} ) | 12 |

159 | Find a continuous function ( f, ) where ( left(x^{4}-4 x^{2}right) leq f(x) leqleft(2 x^{2}-x^{3}right) ) such that the area bounded by ( boldsymbol{y}=boldsymbol{f}(boldsymbol{x}), boldsymbol{y}= ) ( x^{4}-4 x^{2}, y ) -axis, and the line ( x=t ) where ( (0 leq t leq 2) ) is ( k ) times the area bounded by ( boldsymbol{y}=boldsymbol{f}(boldsymbol{x}), boldsymbol{y}=mathbf{2} boldsymbol{x}^{2}-boldsymbol{x}^{3}, mathbf{y} ) axis, and line ( boldsymbol{x}=boldsymbol{t} text { (where } mathbf{0} leq boldsymbol{t} leq mathbf{2}) ) | 12 |

160 | Find area of circle ( 4 x^{2}+4 y^{2}=9 ) which is interior to the parabola ( x^{2}=4 y ) | 12 |

161 | 28. The area area bounded by the parabolas y = (x + 1)2 and = (x – 1)2 and the line y = 1/4 is (2005) (b) 1/6 sq. units (d) 1/3 sq. units (a) 4 sq. units (c) 4/3 sq. units | 12 |

162 | Of all the closed right circular cylindrical cans of volume ( 128 pi c m^{3} ) find the dimensions of the can which has minimum surface area. | 12 |

163 | 38. L →R (the set of all real number) be a po number) be a positive, non-constant and differentiable function such that f'(x) <2f(x) and f 1. Then the value of f(x) dx lies – 1/2 (JEE Adv. 2013) (b) (e-1, 2e-1) in the interval (a) (2e-1,2e) | 12 |

164 | he line y = mx For which of the following values of m, is the area of region bounded by the curve y = x – x and the line y = equals 9/2? (1999 – 3 Marko (a) – 4 (6) – 2 (c) 2 (d) 4 | 12 |

165 | Let ‘a’ be a positive constant number. Consider two curves ( C_{1}: y=e^{x}, C_{2} ) ( y=e^{a-x} . ) Let ( S ) be the area of the part surrounded by ( C_{1}, C_{2} ) and the ( y ) -axis, then A ( cdot lim _{a rightarrow infty} S=1 ) B. ( lim _{a rightarrow 0} frac{S}{a^{2}}=frac{1}{4} ) C. Range of ( operatorname{sis}(0, infty) ) D. S(a) is neither odd nor even | 12 |

166 | What is the following parabola’s axis of symmetry? ( boldsymbol{y}=boldsymbol{x}^{2}-boldsymbol{4} boldsymbol{x}+mathbf{5} ) A. ( x=-2 ) B. ( x=2 ) c. ( y=-2 ) D. None of these | 12 |

167 | Draw the graph of the linear equation ( 4 x+y=6 . ) At what points the graph of the equation cuts the ( x ) -axis and the ( y ) -axis ? Find area bounded by this line and coordinate axes. | 12 |

168 | Find the area of the region ( {(x, y) ) ( left.boldsymbol{x}^{2}+boldsymbol{y}^{2} leq mathbf{4}, boldsymbol{x}+boldsymbol{y} geq mathbf{2}right} ) ( mathbf{A} cdot pi-2 ) B. ( pi-1 ) c. ( 2 pi-2 ) D. ( 4 pi-2 ) | 12 |

169 | 28. The area enclosed between the curves y2 = x and y = |x is [2007] (a) 1/6 (b) 1/3 (c) 2/3 (d) i | 12 |

170 | The area between the curves ( y=sqrt{x} ) and ( boldsymbol{y}=boldsymbol{x}^{3} ) is A ( cdot frac{1}{12} ) sq. units в. ( frac{5}{12} ) sq. units c. ( frac{3}{5} ) sq. units D. ( frac{4}{5} ) sq. units | 12 |

171 | The area of the region bounded by the curve ( boldsymbol{y}=boldsymbol{x}^{2}+mathbf{1} ) and ( boldsymbol{y}=mathbf{2} boldsymbol{x}-mathbf{2} ) between ( x=-1 ) and ( x=2 ) is: A. 9 sq . units B. 12sq. units c. 15 sq. units D. 14sq. units | 12 |

172 | n Fig ( 31 A B C D ) is a rectangle with diameter ( 32 m ) by ( 18 m . A D E ) is a triangle such that ( E F perp A D ) and ( E F= ) 14 ( c m . ) Calculate the area of the shaded region. | 12 |

173 | Area bounded by the curve ( y=x^{3}, ) the x-axis and the ordinates ( x=-2 ) and ( x=1 ) is : | 12 |

174 | The area bounded by ( y=frac{3 x^{2}}{4} ) and the line ( 3 x-2 y+12=0 ) is: ( mathbf{A} cdot mathbf{9} ) B. 18 c. 27 D. None of these | 12 |

175 | Find the area of region lying between parabolas ( y^{2}=4 a x & x^{2}=4 a y ) where ( a> ) 0 | 12 |

176 | A balloon in the form of a right circular cone surmounted by a hemisphere, having a diameter equal to the height of the cone, is being inflated. How fast is its volume changing with respect to its total height ( h, ) when ( h=9 mathrm{cm} ) | 12 |

177 | Area bounded by the curves ( y=x e^{x} ) and ( y=x e^{-x} ) and the line ( |x|=1 ) is ( mathbf{A} cdot mathbf{1} ) B. ( frac{4}{e} ) ( c ) D. – | 12 |

178 | Two vertices of a rectangle are on the positive ( mathbf{x} ) -axis. The other two vertices lie on the lines ( y=4 x ) and ( y=-5 x+6 ) Then the maximum area of the rectangle is? ( A cdot frac{2}{3} ) B. ( frac{2}{4} ) ( c cdot frac{1}{3} ) D. ( frac{4}{3} ) | 12 |

179 | If ( f(x)=x^{2 / 3}, x geq 0 . ) Then,the area of the region enclosed by the curve ( y= ) ( f(x) ) and the three lines ( y=x, x=1 ) and ( x=8 ) is ( ^{mathrm{A}} cdot frac{63}{2} ) в. ( frac{93}{5} ) ( ^{c} cdot frac{105}{7} ) D. ( frac{129}{10} ) | 12 |

180 | Consider two curves ( C_{1}: y=frac{1}{x} ) and ( C_{2}: y=ln x ) on the ( x y ) plane. Let ( D_{1} ) denotes the region surrounded by ( C_{1}, C_{2} ) and the line ( x=1 ) and ( D_{2} ) denotes the region surrounded by ( C_{1}, C_{2} ) and the line ( x=a . ) If ( D_{1}=D_{2} ) then the sum of logarithm of possible values of ( a ) is: | 12 |

181 | The curves ( y=x^{2}-1, y=8 x-x^{2}-9 ) at A. Intersect at right angles at (2,3) B. Touch each other at (2,3) c. Do not intersect at (2,3) D. Intersect at an angle ( frac{pi}{3} ) | 12 |

182 | Find the area bounded by the curve ( y= ) ( x^{2}+x+1 ) and tangent to it at (1,3) from ( boldsymbol{x}=-mathbf{1} ) to ( boldsymbol{x}=mathbf{1} ) A ( cdot frac{2}{3} ) в. ( frac{5}{3} ) c. ( -frac{2}{3} ) D. ( frac{1}{3} ) | 12 |

183 | The area bounded by the curves ( y= ) ( log _{e} x ) and ( y=left(log _{e} xright)^{2} ) is ( mathbf{A} cdot 3-e ) B . ( e-3 ) C ( cdot frac{1}{2}(3-e) ) D. ( frac{1}{2}(e-3) ) | 12 |

184 | If the ( x ) -axis divide the area of the region bounded by the parabolas ( y=4 x-x^{2} ) and ( y=x^{2}-x ) in the ratio of ( a: b ) then ( a b ) is equal to | 12 |

185 | Find the area included between the parabolas ( boldsymbol{y}^{2}=boldsymbol{x} ) and ( boldsymbol{x}=boldsymbol{3}-boldsymbol{2} boldsymbol{y}^{2} ) ( mathbf{A} cdot mathbf{1} ) B . 2 ( c .3 ) D. | 12 |

186 | The area of the portion of the circle ( x^{2}+y^{2}=1, ) which lies inside the parabola ( boldsymbol{y}^{2}=mathbf{1}-boldsymbol{x}, ) is A ( cdot frac{pi}{2}-frac{2}{3} ) B. ( frac{pi}{2}+frac{2}{3} ) c. ( frac{pi}{2}-frac{4}{3} ) D. ( frac{pi}{2}+frac{4}{3} ) | 12 |

187 | Find the area cut off from the parabola ( 4 y=3 x^{2} ) by the straight line ( 2 y= ) ( 3 x+12 ) A. 25 sq.units B. 27sq.units c. ( 36 s q ).units D. 16 sq.units | 12 |

188 | Find the area enclosed between the boundry of circle ( x^{2}+y^{2}=1 ) and the line ( boldsymbol{x}+boldsymbol{y}=1 ) lying in the first quadrant. | 12 |

189 | If the area of the closed figure bounded by the following curves ( boldsymbol{y}=boldsymbol{x}^{2}, boldsymbol{y}=mathbf{2}- ) ( boldsymbol{x}, boldsymbol{y}=mathbf{0} ) is k. Find ( 6 mathbf{k} ) | 12 |

190 | The area enclosed by the curves ( y= ) ( sin x+cos x ) and ( y=|cos x-sin x| ) over the interval ( left[mathbf{0}, frac{pi}{2}right] ) is A. ( 4(sqrt{2}-1) ) B. ( 2 sqrt{2}(sqrt{2}-1) ) c. ( 2(sqrt{2}+1) ) D. ( 2 sqrt{2}(sqrt{2}+1) ) | 12 |

191 | 14. 0 — –ww van Find the area bounded by the curves, x2 + y2 = 25, 4y=14 – x2 | and x = 0 above the x-axis. (1987 – 6 Marks) | 12 |

192 | 48. Let g(x) = cos x?,f(x)= x, and a, Ba<B) be the roots of the quadratic equation 18×2 – 97x + 12 = 0 . Then the area (in sq. units) bounded by the curve y =(gof)(x) and the lines x = Q,x = B and y = 0, is: [JEE M 2018|| (a) }(V3+1) (0) 1 1 (2-1) (b) (V3 – V2) (d) 1 2 (3-1) | 12 |

193 | The area of the region bounded by the parabola ( (boldsymbol{y}-mathbf{2})^{2}=(boldsymbol{x}-mathbf{1}), ) the tangent to the parabola at the point (2,3) and the X-axis is? ( A cdot 3 ) B. 6 ( c .9 ) D. 12 | 12 |

194 | The area of the region bounded by ( y^{2}= ) ( x ) and ( x=36 ) is divided in the ratio 1: 7 by the line ( x=a ). then a equals- A. 7 B. 8 ( c cdot 9 ) D. | 12 |

195 | A wire of length ( 28 mathrm{cm} ) is to be cut into 2 pieces, 1 piece is to be made into a square ( & ) the other one into a circle what should be the length of 2 piece so that the combined area of the square ( & ) the circle is min? | 12 |

196 | Find the area of the shaded region | 12 |

197 | The area bounded by the curves ( x= ) ( a cos ^{3} t, y=a sin ^{3} t ) is A ( cdot frac{3 pi a^{2}}{8} ) в. ( frac{3 pi a^{2}}{16} ) c. ( frac{3 pi a^{2}}{32} ) D. None of the above | 12 |

198 | Let ( S(alpha)=left{(x, y): y^{2} leq x, 0 leq x leq alpharight} ) and ( A(alpha) ) is area of the region ( S(alpha) . ) If for a ( lambda, 0<lambda<4, A(lambda): A(4)=2: 5 ) then ( lambda ) equals ( ^{mathrm{A}} cdotleft(frac{4}{25}right)^{frac{1}{3}} ) ( ^{mathrm{B}} 4left(frac{4}{25}right)^{frac{1}{3}} ) ( ^{c} cdotleft(frac{2}{5}right)^{frac{1}{3}} ) ( ^{mathrm{D}} 4left(frac{2}{5}right)^{frac{1}{3}} ) | 12 |

199 | The area enclosed between the curves ( boldsymbol{y}=boldsymbol{a} boldsymbol{x}^{2} ) and ( boldsymbol{x}=boldsymbol{a} boldsymbol{y}^{2}(boldsymbol{a}>boldsymbol{0}) ) is 1 sq.unit. then ( a= ) A ( cdot frac{1}{sqrt{3}} ) в. ( frac{2}{sqrt{3}} ) c. ( frac{4}{sqrt{3}} ) D. ( sqrt{3} ) | 12 |

200 | The area bounded by the curve ( y= ) ( f(x), ) above the ( x ) -axis, between ( x=a ) and ( boldsymbol{x}=boldsymbol{b} ) is: ( ^{mathrm{A}} cdot int_{f(a)}^{b} y d y ) B. ( int_{a}^{f(b)} x d x ) ( ^{c} cdot int_{a}^{b} x d y ) D. ( int_{a}^{b} y d x ) | 12 |

201 | Find the area bounded by ( boldsymbol{y}= ) ( cos ^{-1} x, y=sin ^{-1} x ) and ( y- ) axis A ( cdot(2-sqrt{2}) ) sq. units B ( cdot(sqrt{2}-2) ) sq. units c. ( 2 sqrt{2} ) sq. units D. ( sqrt{2} ) sq. units | 12 |

202 | : The area bounded by ( x=2 cos theta, y= ) ( 3 sin theta ) is ( 36 pi ) sq. units. II: The area bounded by ( x=2 cos theta, y= ) ( 2 sin theta ) is ( 4 pi ) sq.units. Which of the above statement is correct? A. Onlyı B. Only II c. Both I and II D. Neither I nor II. | 12 |

203 | The value of ( c ) for which the area of the figure bounded by the curve ( y=8 x^{2}- ) ( x^{5}, ) the straight lines ( x=1 ) and ( x=c ) and the ( x- ) axis is equal to ( frac{16}{3} ) is ( A cdot 2 ) B . ( sqrt{8-sqrt{17}} ) ( c .3 ) D. – | 12 |

204 | The area of the region bounded by the curves ( boldsymbol{y}=|boldsymbol{x}-mathbf{1}| ) and ( boldsymbol{y}=mathbf{3}-|boldsymbol{x}| ) in square units is | 12 |

205 | 53. The area (in sq. units) of the region A={(x, y):xSy<x+2) is: (JEEM 2019-9 April (M)] 10 9 31 (a) ž (b) (c) (d) – 13 | 12 |

206 | Find the area enclosed by the curves ( boldsymbol{y}=bmod (boldsymbol{x}-mathbf{1}) ) and ( boldsymbol{y}=boldsymbol{operatorname { m o d }}(boldsymbol{x}- ) 1) +1 | 12 |

207 | I: The area bounded by the line ( y=x ) and the curve ( y=x^{3} ) is ( 1 / 2 ) sq. units. II: The area bounded by the curves ( boldsymbol{y}= ) ( x^{3} ) and ( y=x^{2} ) and the ordinates ( x=1 ) ( x=2 ) is ( frac{7}{12} ) sq. units. Which of the above statement is correct? A. onlyı c. Both I and II D. Neither I nor II. | 12 |

208 | The area bounded by ( boldsymbol{y}=mathbf{2}- ) ( |2-x|, y=frac{3}{|x|} ) is ( ^{text {A } cdot frac{5-4 ln 2}{3} . text { sq.unit }} ) B. ( frac{2-ln 3}{2} . ) sq.unit c. ( frac{4-3 ln 3}{2} ). sq.unit D. none of these | 12 |

209 | The area bounded by ( frac{|boldsymbol{x}|}{boldsymbol{a}}+frac{|boldsymbol{y}|}{boldsymbol{b}}=mathbf{1} ) where ( a>0 ) and ( b>0 ) is A ( cdot frac{1}{2} a b ) в. ( a b ) c. ( 2 a b ) D. ( 4 a b ) | 12 |

210 | 23. The area bounded by the curves y= Vx, 2y+3 = x and x-axis in the 1st quadrant is (2003) (a) 9 (6) 27/4 (c) 36 (d) 18 | 12 |

211 | 3. Area of the region bounded by the curve y = er and lines x= 0 and y= e is (2009) (a) e-1 (b) ſ In (e+1- y) dy (C) e-jedx (2) Singay | 12 |

212 | 33. The area bounded by the curves y = cos x and y = sin x between the ordinates x = 0 and x = – [2010] (a) 412+2 (6) 42-1 (c) 4V2+1 (d) 4/2–2 | 12 |

213 | The area bounded by the curves ( y= ) ( -sqrt{-x} ) and ( x=-sqrt{-y} ) were ( x, y leq 0 ) This question has multiple correct options A. Can not be determined B. is ( 1 / 3 ) ( c cdot ) is ( 2 / 3 ) D. is same as that of the figure by the curves ( y= ) ( sqrt{-x} ; x leq 0 ) and ( x=sqrt{-y} ; y leq 0 ) | 12 |

214 | Let ( f ) and ( g ) be continuous function on ( a leq x leq b ) and ( operatorname{set} p(x)=max ) ( {f(x), g(x)} ) and ( q(x)=min ) ( {f(x), g(x)}, ) the area bounded by the curves ( boldsymbol{y}=boldsymbol{p}(boldsymbol{x}), boldsymbol{y}=boldsymbol{q}(boldsymbol{x}) ) and the ordinates ( boldsymbol{x}=boldsymbol{a} ) and ( boldsymbol{x}=boldsymbol{b} ) is given by This question has multiple correct options A ( cdot int_{a}^{b}(f(x)-g(x)) d x ) B ( cdot int_{a}^{b}(p(x)-q(x)) d x ) c ( cdot int_{a}^{b}|p(x)-q(x)| d x ) D ( cdot int_{a}^{b}|f(x)-g(x)| d x ) | 12 |

215 | Draw the graph of the linear equations ( 4 x-3 y+4=0 ) and ( 4 x+3 y-20=0 ) Find the area bounded by these lines and x-axis. | 12 |

216 | The area bounded by the curves ( y= ) ( log x, y=log |x|, y=|log x| ) and ( y= ) ( |log | x|| ) A. 4 sq. units B. 6 sq. units c. 10 sq. units D. None of these | 12 |

217 | The area of the region(s) enclosed by the curves ( y=x^{2} ) and ( y=sqrt{|x|} ) is: ( A cdot 1 / 3 ) B. 2/3 ( c cdot 1 / 6 ) D. | 12 |

218 | The area bounded by the curves ( boldsymbol{y}= ) ( |x|-1 ) and ( y=-|x|+1 ) is ( mathbf{A} cdot mathbf{1} ) B . 2 c. ( 2 sqrt{2} ) D. | 12 |

219 | Area bounded by ( x^{2}=4 a y ) and ( y= ) ( frac{8 a^{3}}{x^{2}+4 a^{2}} ) is: A ( cdot frac{a^{2}}{3}(6 pi-4) ) в. ( frac{pi a^{2}}{3} ) c. ( frac{a^{2}}{3}(6 pi+4) ) D. | 12 |

220 | 28. In what ratio does the x-axis divide the area of the region bounded by the parabolas y = 4x – x2 and y = x2 – x? (1994 – 5 Marks) | 12 |

221 | A polynomial ( P ) is positive for ( x>0 ) and the area of the region bounded by ( boldsymbol{P}(boldsymbol{x}) ) the ( x- ) axis and the vertical lines ( x=0 ) and ( x=lambda ) is ( frac{lambda^{2}(lambda+3)}{3} ) squnit. Then polynomial ( boldsymbol{P}(boldsymbol{x}) ) is: A ( cdot x^{2}+2 x ) B. ( x^{2}+2 x+1 ) c. ( x^{2}+x+1 ) D. ( x^{3}+2 x^{2}+2 ) | 12 |

222 | If area bounded by ( boldsymbol{f}(boldsymbol{x})=boldsymbol{x}^{frac{1}{3}}(boldsymbol{x}-mathbf{1}) ) ( x- ) axis is ( A ) then find the value of ( 28 A ) A . 5 B. 6 ( c cdot 7 ) D. 9 | 12 |

223 | Area bounded by the curve ( y^{2}(2 a- ) ( x)=x^{3} ) and the line ( x=2 a ) is: ( A cdot 3 pi a^{2} ) B. ( 2 pi ) c. ( 2 pi a ) D. ( 3 pi ) | 12 |

224 | Find the common area (in sq. units) enclosed by the parabolas ( 4 y^{2}=9 x ) and ( 3 x^{2}=16 y ) | 12 |

225 | The area bounded by ( y=x^{2} ) and ( y= ) ( 1-x^{2} ) is A ( cdot frac{sqrt{8}}{3} ) в. ( frac{16}{3} ) c. ( frac{32}{3} ) D. ( frac{17}{3} ) | 12 |

226 | Area bounded by the curve ( mathbf{y}=mathbf{x}+sin mathbf{x} ) and its inverse function between the ordinates ( mathbf{x}=mathbf{0} ) and ( mathbf{x}=mathbf{2} pi ) is A. ( 8 pi ) sqp. units B. 4 ( pi ) sq. units c. 8 sq. units D. 3 ( pi ) sq. units | 12 |

227 | From a piece of cardboard, in the shape of a trapezium ( A B C D, ) and ( A B | C D ) and ( angle ) ( mathrm{BCD}=90^{circ}, ) quarter circle is removed Given ( A B=B C=3.5 mathrm{cm} ) and ( mathrm{DE}=2 mathrm{cm} ) Calculate the area of the remaining piece of the cardboard.(Take ( pi ) to be ( frac{22}{7} ) A ( .9 .625 mathrm{cm}^{3} ) в. ( 6.125 . ) ст ( mathbf{c} cdot 2.625 mathrm{cm}^{2} ) D. None of these | 12 |

228 | Let f(x)=Maximum {x’, (1-x), 2x(1 – x)}, where 0 <x< 1. Determine the area of the region bounded by the curves y=f(x), X-axis, x= 0 and x=1. (1997- 5 Marks) | 12 |

229 | Find the area of the region bounded by the curves ( boldsymbol{y}=boldsymbol{x}^{2}, boldsymbol{y}=mathbf{2}-boldsymbol{x} ) and ( boldsymbol{y}=mathbf{1} ) | 12 |

230 | Find the area enclosed between the curves ( boldsymbol{y}^{2}-mathbf{2} boldsymbol{y} e^{s i n^{-1} boldsymbol{x}}+boldsymbol{x}^{2}-mathbf{1}+[boldsymbol{x}]+ ) ( e^{2 sin ^{-1} x}=0 ) and line ( x=0 ) and ( x=frac{1}{2} ) is (where [.] denotes greatest integer function) A. ( frac{sqrt{3}}{4}+frac{pi}{6} ) B. ( frac{sqrt{3}}{2}+frac{pi}{6} ) c. ( frac{sqrt{3}}{4}-frac{pi}{6} ) D. ( frac{sqrt{3}}{2}-frac{pi}{6} ) | 12 |

231 | The equation of the curve through the point (3,2) and whose slope is ( frac{x^{2}}{y+1}, ) is ( ^{mathrm{A}} cdot frac{y^{2}}{2}+y=frac{x^{3}}{3}+5 ) B . ( y+y^{2}=x^{3}-21 ) c. ( y^{2}+2 y=frac{2 x^{3}}{3}-10 ) D. ( frac{y^{2}}{2}+y=frac{x^{3}}{3}-5 ) | 12 |

232 | Area of the region bounded by ( y= ) ( e^{x}, y=e^{-x}, x=0 ) and ( x=1 ) in sq. units is: ( ^{A} cdotleft(e+frac{1}{e}right)^{2} ) ( ^{text {B }}left(e-frac{1}{e}right)^{2} ) ( ^{c} cdotleft(sqrt{e}+frac{1}{sqrt{e}}right)^{2} ) D ( left(sqrt{e}-frac{1}{sqrt{e}}right)^{2} ) | 12 |

233 | 42. The area of the region {(x, y: xy S8,15 y 5x?} is (JEE Adv. 2018) (a) Blog 2-14 (b) 16log. 2-14 (c) Blog 2-5 (d) 16log, 2-6 | 12 |

234 | The area of the quadrilateral formed by the tangents at the endpoints of the latus recta to the ellipse, ( frac{x^{2}}{9}+frac{y^{2}}{5}=1 ) is A ( cdot frac{27}{4} ) B. 18 c. ( frac{27}{2} ) D. 27 | 12 |

235 | 21. Sketch the curves and identify the region bounded by x= x=2, y=In x and y=2*. Find the area of this region | 12 |

236 | Find the area of the closed figure bounded by the following curve ( boldsymbol{y}=boldsymbol{x}, boldsymbol{y}=mathbf{2} boldsymbol{x}-boldsymbol{x}^{2} ) | 12 |

237 | 46. The area (in sq. units) of the region {(x, y): x20,x+y<3, x2 < 4y and y < 1+ 1x } is : [JEEM 2017 59 | 12 |

238 | Consider the line ( x=sqrt{3} y ) and the circle ( x^{2}+y^{2}=4 ) What is the area of the region in the first quadrant enclosed by the ( x ) -axis, the line ( x=sqrt{3} y ) and the circle? A ( cdot frac{pi}{3} ) в. ( ^{C} cdot frac{pi}{3}-frac{sqrt{3}}{2} ) D. None of the above | 12 |

239 | The area (in sq. units) of the region described by ( boldsymbol{A}= ) ( left{(x, y) mid y geq x^{2}-5 x+4, x+y geq 1, y leqright. ) is: A ( cdot frac{17}{6} ) в. ( frac{13}{6} ) c. ( frac{19}{6} ) D. ( frac{7}{2} ) | 12 |

240 | Find the area of the region bounded by the curves ( x^{2}+y^{2}=36 ) and ( y^{2}=9 x ) | 12 |

241 | The area of the figure bounded by ( y^{2}= ) ( 2 x+1 ) and ( x-y-1=0 ) is: ( A cdot 2 / 3 ) B . ( 4 / 3 ) ( c cdot 8 / 3 ) D. ( 11 / 3 ) | 12 |

242 | 20. Compute the area of the region bounded by the curves In x y=ex In x and y= — where In e=1. (1990 – 4 Mark ex | 12 |

243 | The area bounded by ( y=x e^{|x|} ) and lines ( |boldsymbol{x}|=mathbf{1}, boldsymbol{y}=mathbf{0} ) is A. 4 sq units B. 6 sq units c. 1 sq units D. 2 sq units | 12 |

244 | An open tank with a square base and vertical side is to be constructed a metal sheet so as to hold a given quantity of water. Show that the cost of the material will be least when depth of the tank is half of its width. | 12 |

245 | The curve ( y=frac{x^{2}}{2} ) and the line ( y=frac{x}{2} ) intersect at the origin and at the point ( (a, b), ) as shown in the figure above. Find the value of ( b ) ( A ) ( overline{8} ) B. ( frac{1}{4} ) ( c cdot frac{1}{2} ) D. ( E ) | 12 |

246 | An open box with a square base is to be made out of a given quantity of cardboard of area ( c^{2} ) square units. Show that the maximum volume of the box is ( frac{c^{3}}{6 sqrt{3}} ) cubic units. | 12 |

247 | 17. The area bounded by the curves y=[xl-1 and y=-x+13 (20025 (a) 1 (6) 2 (c) 2/2 (d) 4 | 12 |

248 | The tangent to the curve ( y=x^{2}+6 ) at ( a ) point (1,7) touches the circle ( x^{2}+ ) ( y^{2}+16 x+12 y+c=0 ) at a point ( Q ) then the coordinate of ( Q ) then the coordinate of ( Q ) are. A. (-6,-11) B. (-9,-13) begin{tabular}{l} c. (-10,-15) \ hline end{tabular} D. (-6,-7) | 12 |

249 | Find the area of the closed figure bounded by the following curve ( boldsymbol{y}=boldsymbol{x}^{2}-boldsymbol{2} boldsymbol{x}+boldsymbol{2}, boldsymbol{y}=boldsymbol{2}+boldsymbol{4} boldsymbol{x}-boldsymbol{x}^{2} ) | 12 |

250 | Consider two curves ( C_{1}: y=frac{1}{x} ) and ( C_{2} ) ( boldsymbol{y}=ln x ) on the xy plane Let ( boldsymbol{D}_{1} ) denotes the region surrounded by ( C_{1}, C_{2} ) and the line ( x=1 ) and ( D_{2} ) denotes the region surrounded by ( C_{1}, C_{2} ) and the line ( x=a ) If ( D_{1}=D_{2} ) then the value of ‘a’: A ( cdot frac{e}{2} ) в. ( mathbf{c} cdot e-1 ) D. ( 2(e-1) ) | 12 |

251 | The area bounded by two branches of the curve ( (boldsymbol{y}-boldsymbol{x})^{2}=boldsymbol{x}^{3} & boldsymbol{x}=mathbf{1} ) equals A . ( 3 / 5 ) B. ( 5 / 4 ) ( c cdot 6 / 5 ) D. ( 4 / 5 ) | 12 |

252 | The area between the curve ( y^{2}=9 x ) and the line ( y=3 x ) is A ( cdot frac{1}{3} ) sq. units B. ( frac{8}{3} ) sq. units c. ( frac{1}{2} ) sq, units D. ( frac{1}{5} ) sq. units | 12 |

253 | 1. Find the area bounded by the curve xú= 4y line x=4y-2. e area bounded by the curve x2=4y and the straight (1981 – 4 Marks) | 12 |

254 | Find the area of the region bounded by the parabola ( y^{2}=2 x ) and the line ( x- ) ( boldsymbol{y}=mathbf{4} . ) And ( mathbf{1 8} boldsymbol{s} boldsymbol{q} ) unit. | 12 |

255 | The number of solutions for ( sin left(frac{pi x}{2}right)=frac{99 x}{500} ) is: ( A cdot 3 ) B. 5 ( c cdot 7 ) ( D ) | 12 |

256 | If ( A ) is the area of the figure bounded by the straight lines ( x=0 ) and ( x=2, ) and the curves ( y=2^{x} ) and ( y=2 x-x^{2} ) then the value of ( 672left(frac{3}{log 2}-Aright) ) is | 12 |

257 | Area of the region bounded by ( boldsymbol{y}=|boldsymbol{x}| ) and ( boldsymbol{y}=mathbf{1}-|boldsymbol{x}| ) is A ( cdot frac{1}{3} ) sq. units B. 1 sq. units c. ( frac{1}{2} ) sq. unit D. 2 sq. units | 12 |

258 | A curve passes through the point ( (2 a, a) ) and is such that sum of subtangent and abscissa is equal to a. Its equation is A ( cdot(x-a) y^{2}=a^{3} ) в. ( (x-a)^{2} y=a^{3} ) c. ( (x-a) y=a^{2} ) D. ( (x+a) y=a^{2} ) | 12 |

259 | The area bounded by curves ( 3 x^{2}+ ) ( mathbf{5} boldsymbol{y}=mathbf{3} 2 ) and ( boldsymbol{y}=|boldsymbol{x}-mathbf{2}| ) is A . 25 B. 17/2 c. ( 33 / 2 ) D. 33 | 12 |

260 | Set up an equation of a tangent to the graph of the following function. A sector with a central angle ( alpha ) is cut off | 12 |

261 | The area bounded by ( |y|=1-x^{2} ) is ( A cdot 8 / 3 ) в. ( 4 / 3 ) ( mathrm{c} cdot 16 / 3 ) D. None of these | 12 |

262 | Tangents are drawn to the ellipse ( frac{x^{2}}{9}+ ) ( frac{y^{2}}{5}=1 ) at the ends of both latus rectum. The area of the quadrilateral so formed is A. 27 sq.units B. ( frac{13}{2} ) sq.units c. ( frac{15}{4} ) sq.units D. 45 sq.units | 12 |

263 | If ( boldsymbol{f}(boldsymbol{x}+boldsymbol{y})=boldsymbol{f}(boldsymbol{x})+boldsymbol{f}(boldsymbol{y})-boldsymbol{x} boldsymbol{y} ) for all ( boldsymbol{x}, boldsymbol{y} in boldsymbol{R} ) and ( lim _{boldsymbol{h} rightarrow mathbf{0}} frac{boldsymbol{f}(boldsymbol{h})}{boldsymbol{h}}=mathbf{3}, ) then the area bounded by the curves ( boldsymbol{y}=boldsymbol{f}(boldsymbol{x}) ) and ( boldsymbol{y}=boldsymbol{x}^{2} ) is: A . 1 B. 2 ( c cdot 3 ) D. 4 | 12 |

264 | The area of the region ( left{(x, y): x y leq 8,1 leq y leq x^{2}right} ) is ( mathbf{A} cdot 16 log _{6} 2-6 ) в. ( 8 log _{6} 2-frac{7}{3} ) ( mathrm{c} cdot_{16 log _{6} 2-frac{14}{3}} ) D. ( 8 log _{6} 2-frac{14}{3} ) | 12 |

265 | 38. Let f(x) be a continuous function given by (x) = { 2x lxs11 (1999 – 10 Marks) (x2 + ax + b, | x | >1) Find the area of the region in the third quadrant bounded by the curves x=-2y2 and y=f(x) lying on the left of the line 8x+1=0. | 12 |

266 | The function ( f(x) ) whose graph passes through the point ( left(0, frac{7}{3}right) ) and whose derivatives is ( x sqrt{1-x^{2}} ) is given by | 12 |

267 | The area (in square units) bounded by the curves ( x=-2 y^{2} ) and ( x=1-3 y^{2} ) is A ( cdot frac{2}{3} ) B. ( c cdot frac{4}{3} ) D. | 12 |

268 | The area(in sq. units) of the smaller portion enclosed between the curves, ( x^{2}+y^{2}=4 ) and ( y^{2}=3 x, ) is A ( cdot frac{1}{sqrt{3}}+frac{4 pi}{3} ) в. ( frac{1}{2 sqrt{3}}+frac{pi}{3} ) c. ( frac{1}{2 sqrt{3}}+frac{2 pi}{3} ) D. ( frac{1}{sqrt{3}}+frac{2 pi}{3} ) | 12 |

269 | Let ( f(x)=x^{2}-3 x+2 ) then area bounded by the curve ( f(|x|) ) (in square units) and ( x ) -axis is A ( cdot frac{1}{3} ) в. ( frac{5}{6} ) c. ( frac{5}{3} ) D. None of thes | 12 |

270 | The area bounded by the ( y ) -axis, ( y= ) ( cos x ) and ( y=sin x ) when ( 0 leq x leq frac{pi}{2} ) A ( cdot 2(sqrt{2}-1) ) B. ( sqrt{2}-1 ) ( c cdot sqrt{2}+1 ) D. ( sqrt{2} ) | 12 |

271 | State the following statement is True or False The area bounded by the circle ( x^{2}+ ) ( boldsymbol{y}^{2}=mathbf{1}, boldsymbol{x}^{2}+boldsymbol{y}^{2}=boldsymbol{4} ) and the pair of lines ( sqrt{3}left(x^{2}+y^{2}right)=4 x y, ) is equal to ( frac{pi}{2} . ) The statement is true or false. A. True B. False | 12 |

272 | Assertion If ( boldsymbol{f}(boldsymbol{x})=(boldsymbol{x}-mathbf{1})(boldsymbol{x}-mathbf{2})(boldsymbol{x}-mathbf{3}), ) then area enclosed by ( |boldsymbol{f}(boldsymbol{x})| ) between the lines ( x=2.2, x=2.8 ) and ( x- ) axis is equal to ( int_{2.2}^{2.8}(x-1)(x-2)(x-3) d x ) Reason ( (x-1)(x-2)(x-3) leq 0 ) for all ( x in ) [2.2,2.8] 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. Assertion is incorrect but Reason is correct | 12 |

273 | The positive value of ( k ) for which ( mathbf{k e}^{mathbf{x}}- ) ( mathbf{x}=mathbf{0} ) has only one root is A . ( 1 / e ) B. 1 ( c ) D. ( log _{mathrm{e}} 2 ) | 12 |

274 | The area bounded by ( y=f(x), x-a x i s ) and the line ( y=1, ) where ( f(x)=1+ ) ( frac{1}{x} int_{1}^{x} f(t) d t ) is A ( cdot 2(e+1) ) B ( cdotleft(1-frac{1}{e}right) ) c. ( 2(e-1) ) D. None of these | 12 |

275 | ( A: ) The area bounded by ( x=3, y^{2}=3 x ) ( B: ) The area bounded by ( y=1-|x| ) and X-axis ( C: ) The area enclosed between the curve ( y=x^{2} ) and the line ( y=sqrt{3} x ) The descending order of ( boldsymbol{A}, boldsymbol{B}, boldsymbol{C} ) is A. ( A, C, B ) в. ( C, B, A ) c. ( A, B, C ) D. ( C, A, B ) | 12 |

276 | Find the area of that part of the circle ( x^{2}+y^{2}=16 ) which is exterior to the parabola ( boldsymbol{y}^{2}=mathbf{6} boldsymbol{x} ) | 12 |

277 | The common area between the curve ( x^{2}+y^{2}=8 ) and ( y^{2}=2 x ) is A ( cdot frac{4}{3}+2 pi ) B. ( (2 sqrt{2}+pi-1) ) c. ( (sqrt{2}+pi-1) ) D. None of these | 12 |

278 | The area bounded by the curves ( y=x^{2} ) and line ( boldsymbol{y}=boldsymbol{x} ) ( A cdot sin 1 ) B. ( 1-sin 1 ) c. ( 1+sin 1 ) D. None of these | 12 |

279 | The area of the region bounded by the curves ( boldsymbol{y}=boldsymbol{x}^{2} ) and ( boldsymbol{x}=boldsymbol{y}^{2} ) is A ( cdot frac{1}{3} ) в. ( frac{1}{2} ) ( c cdot frac{1}{4} ) D. 3 | 12 |

280 | For a quadratic function in standard form, ( boldsymbol{y}=boldsymbol{a} boldsymbol{x}^{2}+boldsymbol{b} boldsymbol{x}+boldsymbol{c}, ) Find the axis of symmetry. A ( cdot y=frac{-b}{2 a} ) в. ( x=frac{-b}{2 a} ) c. ( y=frac{b^{2}-4 a c}{2 a} ) D. None of these | 12 |

281 | Find the area of the closed figure bounded by the following curves ( boldsymbol{y}=sqrt{boldsymbol{x}}, boldsymbol{y}=sqrt{mathbf{4}-mathbf{3} boldsymbol{x}}, mathbf{y}=mathbf{0} ) A ( cdot frac{8}{9} ) B. ( frac{7}{9} ) ( c cdot frac{5}{9} ) D. | 12 |

282 | Find the point(s) on the curve ( y^{3}+ ) ( 3 x^{2}=12 y ) where the tangent is vertical (parallel to ( y-a x i s) ) | 12 |

283 | Find the area enclosed between the parabola ( 4 y=3 x^{2} ) and the straight line ( 3 x-2 y+12=0 ) | 12 |

284 | Find the area of one of the curvilinear triangles formed by ( boldsymbol{y}=sin boldsymbol{x}, boldsymbol{y}= ) ( cos x ) and ( x ) axis | 12 |

285 | The area bounded by the parabola ( boldsymbol{x}= ) ( y^{2} ) and the line ( y=x-6 ) is A ( cdot frac{125}{3} ) sq. units B. ( frac{125}{6} ) sq. units c. ( frac{125}{4} ) sq. units D. ( frac{115}{3} ) sq. units | 12 |

286 | Find the point on the curve ( y=x^{2} ) where rate of change of ( x ) -co ordinate is equal to the rate of change of ( y-c o ) ordinate. | 12 |

287 | If the area enclosed by the curves ( y^{2}= ) ( 4 lambda x ) and ( y=lambda x ) is ( frac{1}{9} ) square units then value of ( lambda ) is A .24 B. 37 c. 48 D. 38 | 12 |

288 | A farmer ( F_{1} ) has a land in the shape of a triangle with vertices at ( boldsymbol{P}(mathbf{0}, mathbf{0}), boldsymbol{Q}(mathbf{1}, mathbf{1}) ) and ( R(2,0) . ) From this land, a neighbouring farmer ( F_{2} ) takes away the region which lies between the side PQ and a curve of the form ( boldsymbol{y}=boldsymbol{x}^{n}(boldsymbol{n}>1) ) If the area of the region taken away by the farmer ( F_{2} ) is exactly ( 30 % ) of the area of ( Delta P Q R, ) then the value of ( n ) is | 12 |

289 | The area in square units bounded by the curves ( boldsymbol{y}=boldsymbol{x}^{3}, boldsymbol{y}=boldsymbol{x}^{2} ) and the ordinates ( boldsymbol{x}=mathbf{1}, boldsymbol{x}=mathbf{2} ) is A ( cdot frac{17}{12} ) B. ( frac{12}{13} ) ( c cdot frac{2}{7} ) ( D cdot frac{7}{2} ) | 12 |

290 | The area bounded by curve ( y= ) ( sin 2 x(x=0 quad text { to } quad x=pi) ) and ( X ) -axis is A .4 B. 2 c. D. | 12 |

291 | The area (in sq. units) of the region bounded by the curves ( y=2^{x} ) and ( y= ) ( |x+1|, ) in the first quadrant is: ( ^{mathbf{A}} cdot frac{3}{2}-frac{1}{log _{e} 2} ) в. ( frac{1}{2} ) ( ^{mathrm{C}} cdot log _{e} 2+frac{3}{2} ) D. | 12 |

292 | The area bounded by the parabola ( y^{2}= ) ( 4 x ) and its latusrectum is: A ( cdot frac{8}{3} ) sq. units B. ( frac{3}{8} ) sq. units c. 12 sq. units D. ( frac{1}{3} ) sq. units | 12 |

293 | The radius of a circular garden is ( 90 mathrm{m} ) There is a road ( 9 mathrm{m} ) wide around it. Find total area of the road and the cost of levelling the road at Rs.5 per sq.m. | 12 |

294 | If the curve ( x+y=x^{2}(x+1) ) has two distinct horizontal tangents, then the distance between them is A ( cdot frac{32}{27} ) в. ( frac{27}{32} ) ( c cdot frac{4}{3} ) D. ( frac{22}{27} ) | 12 |

295 | If ( 0<A<frac{pi}{6} ) then ( A(csc A) ) is: ( A cdotfrac{pi}{3} ) ( c cdot=frac{pi}{3} ) D. ( =frac{pi}{6} ) | 12 |

296 | The area common to the circle ( x^{2}+ ) ( y^{2}=16 a^{2} ) and the parabola ( y^{2}=6 a x ) is A ( cdot 4 a^{2}(8 pi-sqrt{3}) ) B. ( frac{4 a^{2}(4 pi+sqrt{3})}{3} ) c. ( frac{8 a^{2}(4 pi-sqrt{3})}{5} ) D. none of these | 12 |

297 | The line ( 2 y=3 x+12 ) cuts the parabola ( 4 y=3 x^{2} . ) What is the area enclosed by the parabola and the line? A. 27 square unit B. 36 square unit c. 48 square unit D. 54 square unit | 12 |

298 | What is the greatest integer value of ( x ) for which ( 2 x-20<0 ? ) ( A cdot 8 ) B. c. 10 D. 1 | 12 |

299 | Sketch for ( boldsymbol{y}=sin ^{-1}left(frac{1+x^{2}}{2 x}right) ) | 12 |

300 | The area enclosed between the curves ( boldsymbol{y}=boldsymbol{x}^{3} ) and ( boldsymbol{y}=sqrt{boldsymbol{x}} ) is, (in square units): A ( cdot frac{5}{3} ) в. c. ( frac{5}{12} ) D. ( frac{12}{5} ) | 12 |

301 | Area enclosed between the curves ( |boldsymbol{y}|= ) ( 1-x^{2} ) and ( x^{2}+y^{2}=1 ) is A ( cdot frac{3 pi-14}{3} ) sq.units B. ( frac{pi-8}{3} ) sq.units c. ( frac{2 pi-8}{3} ) sq.units D. None of these | 12 |

302 | 26. The area enclosed between the curves y = ax? and x=ay2 (a > 0) is 1 sq. unit, then the value of a is (2004S) (a) 1/13 (b) 1/2 (c) 1 (d) 1/3 | 12 |

303 | The sum of the intercepts of the tangent to the curve ( sqrt{x}+sqrt{y}=3 ) on the coordinate axes where it meets the curve ( boldsymbol{y}=boldsymbol{e}^{boldsymbol{x}} ) ( A cdot S ) B. 10 ( c cdot 11 ) D. 12 | 12 |

304 | Radius of the circle that passes through origin and touches the parabola ( boldsymbol{y}^{2}= ) ( 4 a x ) at the point ( (a, 2 a) ) is A ( cdot frac{5}{sqrt{2}} a ) В. ( 2 sqrt{2} a ) c. ( sqrt{frac{5}{2} a} ) D. ( frac{3}{sqrt{2}} a ) | 12 |

305 | Find the area under the curve ( boldsymbol{y}= ) ( left(x^{2}+2right)^{2}+2 x ) between the lines ( x=0 ) ( x=2 ) and the ( X ) -axis | 12 |

306 | If area bounded by to curves ( boldsymbol{y}^{2}=4 a x ) and ( y=m x ) is ( frac{a^{2}}{3}, ) then the value of ( m ) is A .2 B. – 1 ( c cdot frac{1}{2} ) D. none of these | 12 |

307 | (2005 – 2 Marks) 46. Find the area bounded by the curves x2 = y, x = -y and p=4x-3. (2005 AM the curves 2005 22 Mark and | 12 |

308 | Find the area of the circle ( 4 x^{2}+4 y^{2}= ) 9 which is interior to the parabola ( x^{2}= ) ( 4 y ) | 12 |

309 | Find the area of the sector of a circle bounded by the circle ( x^{2}+y^{2}=16 ) and the line ( y=x ) in the first quadrant. | 12 |

310 | For a real number ( x ) let ( [x] ) denote the largest number less than or equal to ( x ) for ( boldsymbol{x} epsilon boldsymbol{R} ) let ( boldsymbol{f}(boldsymbol{x})=[boldsymbol{x}] sin pi boldsymbol{x} . ) Then A. fis differentiable on R B. fis symmetric about the line ( x=0 ) ( ^{mathbf{c}} cdot int_{-3}^{3} f(x) d x=0 ) D. For each real ( alpha ), the equation ( f(x)-alpha=0 ) has infinitely many roots. | 12 |

311 | Area enclosed between the curves ( y^{2}= ) ( x ) and ( x^{2}=y ) is equal to This question has multiple correct options A ( cdot_{2} int_{0}^{1}left(x-x^{2}right) d x ) B. c. area of region ( left{(x, y): x^{2} leq y leq|x|right} ) D. ( frac{2}{3} ) | 12 |

312 | Area enclosed by the graph of the function ( y=ln ^{2} x-1 ) lying in the ( 4 t h ) quadrant is ( A cdot frac{2}{e} ) B. ( frac{4}{e} ) c. ( _{2}left(e+frac{1}{e}right) ) D. ( 4left(e-frac{1}{e}right) ) | 12 |

313 | Area of region ( left{(x, y) in R^{2}: y geq sqrt{|x+3|}, 5 y leq x+9right. ) is equal to A ( cdot frac{1}{6} ) B. ( frac{4}{3} ) ( c cdot frac{3}{2} ) D. | 12 |

314 | Find the value(s) of the parameter’a’ (a 9) for each of which the area of the figure bounded by the straight line ( y= ) ( frac{a^{2}-a x}{1+a^{4}} & ) the parabola ( y= ) ( frac{x^{2}+2 a x+3 a^{2}}{1+a^{4}} ) is the greatest A ( cdot a=2^{1 / 4} ) В . ( a=5^{1 / 4} ) c. ( a=7^{1 / 4} ) | 12 |

315 | 19. The parabolas y2 = 4x and x2 = 4y divide the square region bounded by the lines x = 4, y = 4 and the coordinate axes. If S1, S2, Sz are respectively the areas of these parts numbered from top to bottom; then S: S2: S3 is [2005] (a) 1:2:1 (b) 1:2:3 (c) 2:1:2 (d) 1:1:1 | 12 |

316 | Draw the curve represented by ( sqrt{x}+ ) ( sqrt{boldsymbol{y}}=mathbf{1} ) | 12 |

317 | The area enclosed between the curves ( boldsymbol{y}=boldsymbol{x}^{3} ) and ( boldsymbol{y}=sqrt{boldsymbol{x}} ) is (in square units) A ( cdot frac{5}{3} ) B. ( frac{5}{4} ) c. ( frac{5}{12} ) D. ( frac{12}{5} ) | 12 |

318 | 16. If the line sx = a divides the area of region R = {(x,y) = R2 : x’ sysx,0sx51} into two equal parts, then (JEE Adv. 2017) (a) Ocasi (b) f<a<i (c) 2a4 -4a² +1=0 (d) a4 + 4a2 -1 = 0 | 12 |

319 | The area of the plane region bounded by the curves ( x+2 y^{2}=0 ) and ( x+3 y^{2}= ) 1 is equal to A ( cdot frac{5}{3}^{operatorname{sq} . u n i t} ) в. ( frac{1}{3} ) sq.unit c. ( frac{2}{3} ) squnit D ( cdot frac{4}{3} ) sq.unit | 12 |

320 | Find the area bounded between the curves ( boldsymbol{y}=boldsymbol{x}^{2}, boldsymbol{y}=sqrt{boldsymbol{x}} ) | 12 |

321 | The area bounded by the curve ( boldsymbol{y}=boldsymbol{e}^{boldsymbol{x}} ) and the lines ( boldsymbol{y}=|boldsymbol{x}-mathbf{1}|, boldsymbol{x}=mathbf{2} ) is given by ( mathbf{A} cdot e^{2}+1 ) B ( cdot e^{2}-1 ) c. ( e^{2}-2 ) D. ( e-2 ) | 12 |

322 | Find the area of the shaded region in fig:- if ( A B C D ) is a square of side ( 14 mathrm{cm} ) and ( A P D times B P C ) are semi circles. | 12 |

323 | Using the method of integration find the area bounded by the curve ( |boldsymbol{x}|+|boldsymbol{y}|=mathbf{1} ) | 12 |

324 | Area laying between the curves ( y= ) ( tan x, y=cot x ) and ( x- ) ( boldsymbol{a x i s}, boldsymbol{x}left[boldsymbol{o}, frac{pi}{2}right] boldsymbol{i s} ) | 12 |

325 | The closest distance of the origin from a curve given as ( boldsymbol{a} overline{boldsymbol{z}}+overline{boldsymbol{a}} boldsymbol{z}+boldsymbol{a} overline{boldsymbol{a}}=mathbf{0}(boldsymbol{a} ) is a complex number A . 1 в. ( frac{|a|}{2} ) c. ( frac{operatorname{Re}(a)}{|a|} ) D. ( frac{operatorname{Im}(a)}{|a|} ) | 12 |

326 | The area enclosed by the curves ( x^{2}= ) ( y, y=x+2 ) and ( x ) -axis is: A ( cdot frac{5}{6} ) B. ( frac{5}{4} ) ( c cdot frac{5}{2} ) D. | 12 |

327 | Find the area of the shaded portion in the given figure, where ( A B C D ) is a square of side ( 14 mathrm{cm} ) and semicircles are drawn with each side of square as diameter. | 12 |

328 | 33. Letf:/-1,2] → [0, 0 ) be a continuous function such that f(x)=f(1-x) for all x € (-1, 2] Let R1 = xf (x)dx , and R, be the area of the region -1 (2011) bounded by y=f(x),x=-1, x=2, and the x-axis. Then (a) R1 = 2R2 (b) R1 = 3R © 2R1 = R2 (d) 3R = R2 | 12 |

329 | The area bounded by ( y=sec ^{-1} x, y= ) ( operatorname{cosec}^{-1} x ) and the line ( x-1=0 ) is: ( mathbf{A} cdot ln (3+2 sqrt{2})-frac{pi}{2} ) В ( cdot frac{pi}{2}+ln (3+2 sqrt{2}) ) ( c cdot pi-ln 3 ) D . ( pi+ln 3 ) | 12 |

330 | A polynomial function ( f(x) ) satisfies the condition ( boldsymbol{f}(boldsymbol{x}+mathbf{1})=boldsymbol{f}(boldsymbol{x})+mathbf{2} boldsymbol{x}+mathbf{1} ) Find ( boldsymbol{f}(boldsymbol{x}) ) if ( boldsymbol{f}(mathbf{0})=1 . ) Find also the equations of the pair of tangents from the origin on the curve ( y=f(x) ) and compute the area enclosed by the curve and the pair of tangents. ( f(x)=x^{2}+1 ; y=pm 2 x ;, A=frac{2}{3} ) sq.units B. ( f(x)=x^{2}-1 ; y=pm 2 x ; A=frac{2}{3} ) sq.units c. ( f(x)=x^{2}+1 ; y=pm 2 x ; ), ( A=frac{3}{2} ) sq.units D. ( f(x)=x^{2}-1 ; y=pm 2 x ; A=frac{3}{2} ) sq.units | 12 |

331 | Find the area bounded by the curve ( y= ) ( sin x, 0 leq x leq pi ) and line ( y=frac{1}{sqrt{2}} ) | 12 |

332 | f ( boldsymbol{f}(boldsymbol{x})=sin boldsymbol{x} forall boldsymbol{x} in[mathbf{0}, boldsymbol{x} / 2], boldsymbol{f}(boldsymbol{x})+ ) ( boldsymbol{f}(boldsymbol{pi}-boldsymbol{x})=boldsymbol{2} forall boldsymbol{x} in[boldsymbol{x} / 2, boldsymbol{pi}] ) and ( boldsymbol{f}(boldsymbol{x})=boldsymbol{f}(boldsymbol{2} boldsymbol{pi}-boldsymbol{x}) forall boldsymbol{x} in[boldsymbol{pi}, boldsymbol{2} boldsymbol{pi}] ) then find the area bounded by ( y=f(x) ) and the ( x- ) axis. | 12 |

333 | The area of the region bounded by the curves ( boldsymbol{y}=mathbf{2}^{x}, boldsymbol{y}=mathbf{2} boldsymbol{x}-boldsymbol{x}^{2} ) and ( boldsymbol{x}=mathbf{2} ) is A ( cdot frac{3}{log 2}-frac{4}{3} ) в. ( frac{3}{log 2}-frac{4}{9} ) c. ( frac{3}{2}-frac{log 2}{9} ) D. None of these | 12 |

334 | 36. The area of the region enclosed by the curves 1201 y=x, x = e, y = – and the positive x-axis is square unit square units square units square unit | 12 |

335 | A stone is dropped into a quiet lake and waves move in circles at the sped of 5 cm/sec. At that instant, when radius of circular | 12 |

336 | A point P moves in xy-plane In such a way that ( [|x|]+[|y|]=1 ) were [.] denotes the greatest integer function. Area of the region representing all possible positions of the point ‘P’ is equal to A. 4 sq. units B. 16 sq. units c. ( 2 sqrt{2} ) sq. units D. 8 sq. units | 12 |

337 | Find the area of the region bounded by the x-axis and me curves defined by (1984 – 4 Marks) TT y = tan x, – ; y = cot x, <<3 | 12 |

338 | If ( A_{1} ) is the area bounded by ( y= ) ( cos x, y=sin x & x=0 ) and ( A_{2} ) the area bounded by ( boldsymbol{y}=cos boldsymbol{x}, boldsymbol{y}= ) ( sin x, y=0 ) in ( left(0, frac{pi}{2}right) ) then ( frac{A_{1}}{A_{2}} ) equals to: A ( cdot frac{1}{2} ) B. ( frac{1}{sqrt{2}} ) c. D. None of these | 12 |

339 | In the diagram ( square A B C D ) is the rectangular paper. If ( A B=20 mathrm{cm} ) and ( B C=14 mathrm{cm} ) then what is the area of the shaded region if the semicircle with diameter ( B C ) is cut from the paper. | 12 |

340 | The area under the curve ( y=2 sqrt{x} ) bounded by the lines ( x=0 ) and ( x=1 ) is ( A cdot frac{4}{3} ) B. ( frac{2}{3} ) c. 1 D. | 12 |

341 | Find the area of the closed figure bounded by the following curve ( y=x^{2}, y=2 x-x^{2} ) | 12 |

342 | Area of the region bounded by rays ( |x|+y=1 ) and ( X ) -axis is A ( cdot frac{1}{2} ) B. 2 c. D. | 12 |

343 | The parabolas ( y^{2}=4 x ) and ( x^{2}=4 y ) divide the square region bounded by the lines ( x=4, y=4 ) and the coordinate axes. f ( S_{1}, S_{2}, S_{3} ) are respectively the areas of these parts numbered from top to bottom(Example: ( S_{1} ) is the area bounded by ( y=4 ) and ( x^{2}=4 y ) ); then ( boldsymbol{S}_{1}, boldsymbol{S}_{2}, boldsymbol{S}_{3} ) is A . 1: 2: 1 B. 1: 2: 3 c. 2: 1: 2 D. 1: 1: 1 | 12 |

344 | The area included between the parabolas ( y^{2}=4 x ) and ( x^{2}=4 y ) is A ( cdot frac{8}{3} ) sq unit B. 8 sq unit c. ( frac{16}{3} ) sq unit D. 12 sq unit | 12 |

345 | The area of the plane region bounded by the curves ( x+2 y^{2}=0 ) and ( x+3 y^{2}= ) ( mathbf{1} ) is A ( cdot frac{1}{3} ) B. ( frac{2}{3} ) ( c cdot frac{4}{3} ) D. | 12 |

346 | Consider an ellipse ( frac{x^{2}}{a^{2}}+frac{y^{2}}{b^{2}}=1 ) What is the area included between the ellipse and the greatest rectangle inscribed in the ellipse? A ( . a b(pi-1) ) B. ( 2 a b(pi-1) ) c. ( a b(pi-2) ) D. None of the above | 12 |

347 | 18. The triangle formed by the tangent to the curve f(x)=x-T DX at the point (1, 1) and the coordinate axes, lies in the first quadrant. If its area is 2, then the value of bis (20015) (a) -1 (b) 3 (c) 3 (d) 1 | 12 |

348 | Find the area (sq.units) bounded by ( boldsymbol{y}=sin ^{-1} boldsymbol{x} ) and ( boldsymbol{y}=cos ^{-1} boldsymbol{x} ) and ( mathbf{x} ) -axis. | 12 |

349 | Find area curved by three circles ( mathbf{A} cdot(5 pi-3 sqrt{3}) ) units ( ^{2} ) ( mathbf{B} cdot(5 pi+4 sqrt{3}) ) units ( ^{2} ) ( mathbf{c} cdot(5 pi+3 sqrt{3}) ) units ( ^{2} ) ( mathbf{D} cdot(5 pi+3 sqrt{2}) ) units ( ^{2} ) | 12 |

350 | 9. The slope of the tangent to a curve y = f(x) at (x, f(x)] is 2x + 1. If the curve passes through the point (1,2), then the area bounded by the curve, the x-axis and the line x = 1 is (1995) (d) 6 | 12 |

351 | 5. Find the area of the region bounded by the curva π C:y=tan x, tangent drawn to Catx= ã and the x-axis. | 12 |

352 | A hemispherical tank of radius ( boldsymbol{R} ) is completely filled with water. Now an orifice of small area ‘a’ is made at the bottom of tank. The time required to empty the tank is ( ^{mathrm{A}} cdot frac{14 pi R^{frac{5}{2}}}{15 sqrt{2} a sqrt{g}} ) ( ^{mathrm{B}} cdot frac{14 pi R^{frac{3}{2}}}{sqrt{2} a sqrt{g}} ) c. ( frac{14 pi R^{frac{5}{2}}}{5 sqrt{2} a sqrt{g}} ) ( frac{4 pi R^{frac{5}{2}}}{15 sqrt{2} a sqrt{g}} ) | 12 |

353 | The area bounded by the curve ( y= ) ( (x+1)^{2}, y=(x-1)^{2} ) and the line ( y= ) 0 is A ( cdot frac{1}{6} ) в. ( frac{2}{3} ) ( c cdot frac{1}{4} ) D. | 12 |

354 | The area bounded by ( y^{2}=4 x ) and ( x^{2}= ) ( 4 y ) is A ( -frac{20}{3} ) sq. units B. ( frac{16}{3} ) sq. units ( ^{mathrm{c}} cdot frac{14}{3} mathrm{sq} . ) units D. ( frac{10}{3} ) sq. units | 12 |

355 | What is the area of the region bounded by the lines ( x=y, y=0 ) and ( x=4 ? ) A. 4 square units B. 8 square units c. 12 square units D. 16 square units | 12 |

356 | The area lying in the first quadrant inside the circle ( x^{2}+y^{2}=12 ) and bounded by the parabolas ( boldsymbol{y}^{2}= ) ( 4 x, x^{2}=4 y ) is: A ( cdotleft(frac{sqrt{2}}{3}+frac{3}{2} sin ^{-1} frac{1}{3}right) ) B ( cdot 4left(frac{sqrt{2}}{3}+frac{3}{2} sin ^{-1} frac{1}{3}right) ) ( ^{c} cdotleft(frac{sqrt{2}}{3}+frac{3}{2} sin ^{-1} frac{1}{3}right) ) D. none of these | 12 |

357 | The area (in sq. units) of the region ( A=left{(x, y): x^{2} leq y leq x+2right} ) is? A ( cdot frac{10}{3} ) B. ( frac{9}{2} ) c. ( frac{31}{6} ) D. ( frac{13}{6} ) | 12 |

358 | The whole area of the curves ( x= ) ( a cos ^{3} t, y=b sin ^{3} t ) is given by? A ( cdot frac{3}{8} pi a b ) в. ( frac{5}{8} pi a b ) c. ( frac{1}{8} pi a b ) D. None of these | 12 |

359 | Show that the right circular cylinder of given surface and maximum volume is such that its height is equal to the diameter of the base. | 12 |

360 | If ( f(x)=max left{sin x, cos x, frac{1}{2}right}, ) then the area of the region bounded by the curves ( boldsymbol{y}=boldsymbol{f}(boldsymbol{x}), boldsymbol{x}- ) axis ( boldsymbol{y}- ) axis and ( boldsymbol{x}=2 pi ) is ( ^{mathbf{A}} cdotleft(frac{5 pi}{12}+3right) cdot ) sq.unit в. ( left(frac{5 pi}{12}+sqrt{2}right) ). sq.unit c. ( left(frac{5 pi}{12}+sqrt{3}right) cdot ) sq.unit D ( cdotleft(frac{5 pi}{12}+sqrt{2}+sqrt{3}right) cdot ) sq.unit | 12 |

361 | Find the area between the ( x ) -axis and the curve ( y=sin x ) from ( x=0 ) to ( x=2 pi ) | 12 |

362 | For which of the following values of ( m ) the area of the region bounded by the curve ( y=x-x^{2} ) and the line ( y=m x ) equals ( frac{mathbf{9}}{mathbf{2}} ) A . -4 B. – c. 2 D. 4 | 12 |

363 | The area between the curve ( y^{2}=9 x ) and the line ( y=3 x ) is A ( cdot frac{1}{3} ) sq units B. ( frac{8}{3} ) sq. units c. ( frac{1}{2} ) sq. units D. ( frac{1}{5} ) sq. units | 12 |

364 | The area of the smaller part bounded by the semi-circle ( y=sqrt{4-x^{2}}, y=x sqrt{3} ) and ( x ) -axis is A ( cdot frac{pi}{3} ) в. ( frac{2 pi}{3} ) c. ( frac{4 pi}{3} ) D. none of these | 12 |

365 | Find the area of the region bounded by the curves ( y=log _{e} x, y=sin ^{4} pi x, x= ) ( mathbf{0} ) A ( cdot frac{11}{8} ) sq.units B. ( frac{9}{8} ) sq.units c. ( frac{13}{8} ) sq.units D. ( frac{15}{8} ) sq.units | 12 |

366 | Assertion Let ( boldsymbol{f}(boldsymbol{x})=boldsymbol{m} boldsymbol{i} boldsymbol{n} cdot(boldsymbol{x}+1, sqrt{1-boldsymbol{x}}), ) then area bounded by ( y=f(X) & x ) -axis is ( mathbf{7} ) ( overline{6} ) square units. Reason ( min ) of ( boldsymbol{f}(boldsymbol{x})= ) ( left{begin{array}{ll}x+1 & text { for }-1 leq x<0 \ sqrt{1-x} & , 0<x leq 1end{array}right. ) 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 |

367 | The area enclosed between the curve ( boldsymbol{y}^{2}=boldsymbol{x} ) and ( boldsymbol{x}^{2}=boldsymbol{y} ) is equal to | 12 |

368 | The area bounded by the curve ( y= ) ( x^{4}-2 x^{3}+x^{2}+3 ) with ( x ) -axis and ordinates corresponding to the minima of y is : ( mathbf{A} cdot mathbf{1} ) B . ( frac{91}{30} ) ( c cdot frac{30}{9} ) D. 4 | 12 |

369 | Draw the curve ( y=sin ^{4} x ) between ( x= ) 0 and ( x=2 pi ) | 12 |

370 | ( A B ) and ( C D ) are two perpendicular diameters of a circle with centre ( 0,0 D ) is the diameter of smaller circle.If ( O A=7 ) ( mathrm{cm} ),then the area of the shaded region in figure,is ( A cdot 66.5 c m^{2} ) В. ( 66 c m^{2} ) ( c cdot 65 c m^{2} ) D. ( 67.5 mathrm{cm}^{2} ) | 12 |

371 | Use the test for symmetry to determine if the graph of ( y-5 x^{2}=4 ) is symmetric about the x-axis. A. Symmetrical about x-axis B. Not symmetrical about x-axis c. Can’t be determined D. None of these | 12 |

372 | The area of the figure bounded by two branches of the curve ( (boldsymbol{y}-boldsymbol{x})^{2}=boldsymbol{x}^{3} ) and the straight line ( x=1 ) is: A ( cdot frac{1}{3} ) sq.unit B ( cdot frac{4}{5} ) sq.unit c. ( frac{5}{4} ) sq.unit D. 3 sq.unit | 12 |

373 | 7. [200 The area of the region bounded by the curves y = x – 1 and y = 3-|xis (a) 6 sq. units (b) 2 sq. units (c) 3 sq. units (d) 4 sq. units. | 12 |

374 | The area bounded by the curves ( y= ) ( x(x-3)^{2} ) and ( y=x ) is (in sq.units): A . 28 B. 32 ( c cdot 4 ) D. 8 | 12 |

375 | Tangent is drawn from (1,0) to ( y=e^{x} ) then the area bounded between the coordinate axes and the tangent is equal to – A ( cdot frac{e}{2} ) B. ( e ) ( ^{c} cdot frac{e^{2}}{2} ) ( mathbf{D} cdot e^{2} ) | 12 |

376 | Smaller area enclosed by the circle ( x^{2}+y^{2}=4 ) and the line ( x+y=2 ) is A. ( 2(pi-2) ) B. ( pi-2 ) c. ( 2 pi-1 ) D. ( 2(pi+2) ) | 12 |

377 | A stone is dropped into a quiet lake an waves move in circles at the speed of 5 ( mathrm{cm} / mathrm{sec} . ) At the instant when the radius of the circular wave is ( 8 mathrm{cm}, ) how fast is the enclosed area increasing? | 12 |

378 | The area bounded by ( y=3 x ) and ( y=x^{2} ) is (in square units) A . 10 B. 5 c. 4.5 D. | 12 |

379 | Find the area of the region bounded by the curve ( y=4-x^{2}, x ) -axis and the line ( x=0 ) and ( x=2 ) A ( cdot frac{11}{3} ) в. ( frac{16}{3} ) ( c cdot frac{16}{5} ) D. | 12 |

380 | Form the differential equation of the family of curves represented by the equation(a being the parameter). ( (x-a)^{2}+2 y^{2}=a^{2} ) | 12 |

381 | The area (in square units) bounded by the curves ( boldsymbol{y}=sqrt{boldsymbol{x}} ) ( 2 y-x+3=0, ) and lying in the first quadrant is | 12 |

382 | 12. If y = x², then area of curve y v/s x from x = 0 to 2 will be: (a) 1/3 (b) 8/3 (c) 4/3 (d) 2/3 | 12 |

383 | Rewrite the equation ( |boldsymbol{y}|=|boldsymbol{x}| ) as two equations of two lines. | 12 |

384 | The area of the region lying between the line ( boldsymbol{x}-boldsymbol{y}+mathbf{2}=mathbf{0} ) and the curve ( boldsymbol{x}= ) ( sqrt{boldsymbol{y}} ) is ( mathbf{A} cdot mathbf{9} ) B. ( frac{9}{2} ) c. ( frac{10}{3} ) D. none | 12 |

385 | The area of the region bounded by the curves ( boldsymbol{f}(boldsymbol{x})=operatorname{maximum}left{|boldsymbol{x}|, boldsymbol{x}^{2}, sqrt{boldsymbol{x}}right} ) where ( boldsymbol{x}>mathbf{0}, boldsymbol{g}(boldsymbol{x})=|boldsymbol{x}| forall boldsymbol{x} leq mathbf{0} ) and ( h(x)=4 ) is- ( A cdot frac{7}{3} ) B. 9 c. 13 D. ( frac{40}{3} ) | 12 |

386 | Find the area of bounded by ( y=sin x ) from ( boldsymbol{x}=frac{boldsymbol{pi}}{boldsymbol{4}} ) to ( boldsymbol{x}=frac{boldsymbol{pi}}{boldsymbol{2}} ) A ( cdot frac{sqrt{2}-1}{sqrt{2}} ) B. ( frac{1}{2} ) ( c cdot frac{1}{4} ) D. None of these | 12 |

387 | The area of the region between the curve ( boldsymbol{y}=mathbf{4} boldsymbol{x}^{2} ) and the line ( boldsymbol{y}=mathbf{6} boldsymbol{x}-mathbf{2} ) is: A ( cdot frac{1}{9} ) sq. units B. ( frac{1}{12} ) sq. units c. ( frac{3}{2} ) sq. units D. ( frac{1}{5} ) sq. units | 12 |

388 | Find the area of the region in the first quadrant enclosed by x-axis, the line ( x=sqrt{3} y ) and the circle ( x^{2}+y^{2}=4 ) | 12 |

389 | Sketch for ( boldsymbol{y}=cos ^{-1}left(frac{1-x^{2}}{1+x^{2}}right) ) | 12 |

390 | Suppose ( g(x)=2 x+1 ) and ( h(x)= ) ( 4 x^{2}+4 x+5 ) and ( h(x)=(f o g)(x) ) The area enclosed by the graph of the function ( boldsymbol{y}=boldsymbol{f}(boldsymbol{x}) ) and the pair of tangents drawn to it from the origin is ( A cdot 8 / 3 ) B. ( 16 / 3 ) c. ( 32 / 3 ) D. none | 12 |

391 | Find the area of the closed figure bounded by the following curve ( boldsymbol{y}=boldsymbol{x}^{2}-boldsymbol{2} boldsymbol{x}+boldsymbol{3}, boldsymbol{y}=mathbf{3} boldsymbol{x}-mathbf{1} ) | 12 |

392 | The area of the region bounded by the parabola ( (boldsymbol{y}-mathbf{2})^{2}=(boldsymbol{x}-mathbf{1}), ) the tangent to the parabola at the point (2,3) and the X-axis is? ( A cdot 3 ) B. 6 ( c .9 ) D. 12 | 12 |

393 | Find the area of the region bounded by the curve ( y^{2}=4 x, y-a x i s & ) the line ( boldsymbol{y}=mathbf{3} ) | 12 |

394 | Find the area of the region bounded by the curve ( y^{2}=4 x ) and the line ( x=3 ) begin{tabular}{l} A. ( 4 sqrt{3} ) \ hline end{tabular} B. ( 8 sqrt{3} ) ( c cdot 6 ) D. ( 2 sqrt{3} ) | 12 |

395 | Find the area bounded by the curve ( y= ) ( 2 x-x^{2}, ) and the line ( y=x ) | 12 |

396 | If ( boldsymbol{A}+boldsymbol{B}+boldsymbol{C}=boldsymbol{pi} ) and ( boldsymbol{A}, boldsymbol{B}, boldsymbol{C} ) are angles of ( triangle ; ) then ( sin A+sin B+sin C ) is ( ^{A}=frac{3 sqrt{3}}{2} ) B. ( frac{3 sqrt{3}}{2} ) D. none of these | 12 |

397 | The area of the region, enclosed by the circle ( x^{2}+y^{2}=2 ) which is not common to the region bounded by the parabola ( y^{2}=x ) and the straight line ( y=x, ) is: A ( cdot frac{1}{3}(15 pi-1) ) в. ( frac{1}{6}(24 pi-1) ) c. ( frac{1}{6}(12 pi-1) ) D. ( frac{1}{3}(6 pi-1) ) | 12 |

398 | The area of the region bounded by ( y= ) ( -1, y=2, x=y^{3} ) and ( x=0 ) is A ( cdot frac{17}{4} ) sq. units B. ( frac{1}{4} ) sq. units c. 4 sq. units D. None of these | 12 |

399 | The area of the region bounded by the curve ( boldsymbol{y}=boldsymbol{x}^{3}, ) its tangent at ( (mathbf{1}, mathbf{1}) ) and ( boldsymbol{x} ) -axis, is A ( cdot frac{1}{12} ) sq unit B ( cdot frac{1}{6} ) sq unit c. ( frac{2}{17} ) sq unit D. ( frac{2}{15} ) sq unit | 12 |

400 | Find the area bounded by the curves ( boldsymbol{y}=sqrt{1-boldsymbol{x}^{2}} ) and ( boldsymbol{y}=boldsymbol{x}^{3}-boldsymbol{x} . ) Also find the ratio in which the y-axis divide this area A ( cdot frac{pi}{2} ; frac{pi-1}{pi+1} ) В. ( frac{pi}{4} ; frac{pi-1}{pi+1} ) c. ( frac{pi}{2} ; frac{pi+1}{pi-1} ) D. None of these | 12 |

401 | Find the area bounded by the parabola ( x^{2}=4 y ) and the straight line ( x=4 y- ) 2 | 12 |

402 | The area lying in the first quadrant between the curves ( x^{2}+y^{2}=pi^{2} ) and ( y=sin x ) and ( y- ) axis is ( ^{A} cdot frac{pi^{3}-8}{4} ) sq. units B. ( frac{pi^{3}+8}{4} ) sq. units c. ( 4left(pi^{3}-8right) ) sq. units D. ( frac{pi-8}{4} ) sq. units | 12 |

403 | The area of the region bounded by the ( operatorname{lines} y=2 x+1, y=3 x+1 ) and ( x=4 ) is A. 16 sq.unit в. ( frac{121}{3} ) sq.unit c. ( frac{121}{6} ) sq.unit D. 8 sq.unit | 12 |

404 | 37. The area enclosed by the curves y=sin x + cos x and y= |cos X-sin x over the interval (JEE Adv. 2013) (a) 4(2-1) (b) 2.12 (12-1) (C) 26/2 + 1) (d) 2/2(√2+1) | 12 |

405 | Find the area of the region bounded by the curve ( y=x^{2} ) and the line ( y=2 ) | 12 |

406 | The area bounded by curves | 12 |

407 | Area bounded by curve ( xleft(x^{2}+pright)= ) ( boldsymbol{y}-mathbf{1} ) with ( boldsymbol{y}=mathbf{1} boldsymbol{p}<mathbf{0} ) is A ( cdot frac{p^{2}}{4} ) в. ( frac{p}{2} ) c. ( frac{p^{2}}{2} ) D. ( frac{p}{4} ) | 12 |

408 | Which of the following equation shows the above graph? A ( cdot f(x)=x^{2}+9 ) B. ( f(x)=(x-9)^{2} ) ( mathbf{c} cdot f(x)=9-x^{2} ) D. ( f(x)=-left|x^{2}+9right| ) E ( cdot f(x)=mid-x^{2}+9 ) | 12 |

409 | The ratio of the area’s bounded by the curves ( y^{2}=12 x ) and ( x^{2}=12 y ) is divided by the line ( x=3 ) is A .15: 49 B. 9: 15 ( c cdot 7: 15 ) D. 7:5 | 12 |

410 | Find the area of the closed figure bounded by the following curve ( boldsymbol{y}=mathbf{1}+boldsymbol{x}^{2}, boldsymbol{y}=mathbf{2} ) | 12 |

411 | The area bounded by the two curves ( boldsymbol{y}=sin boldsymbol{x}, boldsymbol{y}=cos boldsymbol{x} ) and the X-axis in the first quadrant ( left[0, frac{pi}{2}right] ) is A ( .2-sqrt{2} ) sq. units B . ( 2+sqrt{2} ) sq. units c. ( 2(sqrt{2}-1) ) sq. units D. 4 sq. units | 12 |

412 | The normals to the curve ( y=x^{2}-x+ ) 1, drawn at the points with the abscissa ( boldsymbol{x}_{1}=mathbf{0}, boldsymbol{x}_{2}=-mathbf{1} ) and ( boldsymbol{x}_{3}=frac{mathbf{5}}{mathbf{2}} ) A. are parallel to each other B. are pair wise perpendicular c. are concurrent D. are not concurrent | 12 |

413 | If ( A_{n} ) is the area bounded by ( y= ) ( left(1-x^{2}right)^{n} ) and coordinate axes, ( n ) is in set of natural numbers, then A. ( A_{n}=A_{n-1} ) В. ( A_{n}A_{n-1} ) D. ( A_{n}=2 A_{n-1} ) | 12 |

414 | Find the point on the curve ( 9 y^{2}=x^{3} ) where normal to the curve has non zero x-intercept and both x-intercept and ( y ) intercept are equal | 12 |

415 | The figure shows a portions of the graph ( y=2 x-4 x^{3} . ) The line ( y=c ) is such that the areas of the regions marked I and II are equal. If a,b are the ( x ) -coordinates of A,B respectively, then a + b equals ( A cdot frac{2}{sqrt{7}} ) B. ( frac{3}{sqrt{7}} ) ( c cdot frac{4}{sqrt{7}} ) ( D cdot frac{5}{sqrt{7}} ) | 12 |

416 | If ( C_{1}={x: 1<x<2} ) and ( C_{2}={x: ) ( 4<x<5}, ) find ( Pleft(C_{1} cup C_{2}right) ) | 12 |

417 | Area bounded by curve ( y=x^{2} ) and ( y= ) ( 2-x^{2} ) is ( ? ) A ( cdot frac{8}{3} ) sq units B. ( frac{3}{8} ) sq units c. ( frac{3}{2} ) sq units D. None of these | 12 |

418 | Show that the height of the cylinder of maximum volume that can be inscribed ( boldsymbol{h} ) in a cone of height his | 12 |

419 | Estimating the absolute value of the ntegral ( int_{10}^{19} frac{sin x}{1+x^{8}} d x ) we get ( 10^{-m} ).Find ( boldsymbol{m} ? ) | 12 |

420 | Find the area bounded by curves ( left{(x, y): y geq x^{2} text { and } y=|x|right} ) | 12 |

421 | Semicircles are drawn outside by taking every side of regular hexagon as a diameter. The perimeter of hexagon is ( 60 mathrm{cm} . ) Find the area of complete figure formed as such. ( (pi=3.14)(sqrt{3}=1.73) ) A ( .495 mathrm{cm}^{2} ) B. 259.5cm ( ^{2} ) c. ( 235.5 mathrm{cm}^{2} ) D. ( 695.5 mathrm{cm}^{2} ) | 12 |

422 | 70. The area of the region between the curves y = 1+ sinx COS X 1-sin x and y= 1 A bounded by the lines x = 0 and x = 7 V cos x (2008) √2-1 √2-1 t. Ő (1+t2)/1-42 ” (b) [ V2+1 27 – dt (1+2 WI-42″ (d) (d) I (©) (+12) 1-2 *** | 12 |

423 | The area enclosed between the curves ( mathbf{y}=mathbf{a x}^{2} ) and ( mathbf{x}=mathbf{a y}^{2}(mathbf{a}>mathbf{0}) ) is 1 sq. unit, then the value of a is A. ( 1 / sqrt{3} ) в. ( 1 / 2 ) c. 1 D. ( 1 / 3 ) | 12 |

424 | OLI 10 IUIDUOLLON 40. The area (in square units) bounded by the curves y= Vx, 2y-x+3=0, x-axis, and lying in the first quadrant [JEEM 2013] 27 (a) 9 (6) 36 (c) 18 (d) – is: | 12 |

425 | The area bounded by ( boldsymbol{y}=boldsymbol{x}^{2}, boldsymbol{y}=[boldsymbol{x}+ ) 1], ( x leq 1 ) and the ( y ) -axis is A ( cdot frac{1}{3} ) B. ( frac{2}{3} ) ( c cdot 1 ) D. | 12 |

426 | Parabolas ( boldsymbol{y}^{2}=mathbf{4} boldsymbol{a}left(boldsymbol{x}-boldsymbol{c}_{1}right) ) and ( boldsymbol{x}^{2}= ) ( 4 aleft(y-c_{2}right), ) where ( c_{1} ) and ( c_{2} ) are variable are such that they touch each other. Locus of their point of contact is A ( . x y=2 a^{2} ) В . ( x y=4 a^{2} ) c. ( x y=a^{2} ) D. none of these | 12 |

427 | Area lying between the curve ( y^{2}=4 x ) and ( boldsymbol{y}=mathbf{2} boldsymbol{x} ) is : A ( cdot frac{2}{3} ) B. ( frac{1}{3} ) ( c cdot frac{1}{4} ) D. ( frac{3}{4} ) | 12 |

428 | Let ( A_{n} ) be the constant number such that ( c>1 . ) If the least area of the figure given by the line passing through the point (1,c) with gradient ‘m’ and the parabola ( boldsymbol{y}=boldsymbol{x}^{2} ) is 36 sq.units find the value of ( left(c^{2}+m^{2}right) ) A ( cdot 104 ) B. 105 ( c cdot 10 ) D. 100 | 12 |

429 | Area of the circle ( (x-2)^{2}+(y-3)^{2}= ) 32 which lies below the line ( y=x+1 ) is ( int_{-2}^{6}[(x+1)+sqrt{32-(x-2)^{2}}+3] d x ) | 12 |

430 | The area of the region bounded by ( y= ) ( boldsymbol{x}^{2}+mathbf{2}, boldsymbol{y}=-boldsymbol{x}, boldsymbol{x}=mathbf{0} ) and ( boldsymbol{x}=mathbf{1} ) is A ( cdot frac{17}{6} ) в. ( frac{17}{3} ) c. ( frac{10}{3} ) D. ( frac{8}{3} ) | 12 |

431 | The area of the region bounded by the curves ( mathbf{y}=sqrt{boldsymbol{x}} ) and ( boldsymbol{y}=sqrt{mathbf{4}-mathbf{3} boldsymbol{x}} ) and ( mathbf{y}=mathbf{0} ) is: A . ( 4 / 9 ) B. 16/9 ( c cdot 8 / 9 ) D. 9/2 | 12 |

432 | The area bounded by the ( x- ) axis, the curve ( y=f(x) ) and the lines ( x=1 ) and ( x=b ) is equal to ( (sqrt{b^{2}+1}-sqrt{2}) ) for all ( boldsymbol{b}>1, ) then ( boldsymbol{f}(boldsymbol{x}) ) is A. ( sqrt{x-1} ) B. ( sqrt{x+1} ) c. ( sqrt{x^{2}+1} ) D. ( frac{x}{sqrt{x^{2}+1}} ) | 12 |

433 | Circular arc of radius ( 7 mathrm{cm} ) has been drawn with vertex ( A ) of an equilateral triangle ( A B C ) of side ( 14 mathrm{cm} ) at center Find the area of shaded region. | 12 |

434 | The value of ( a ) for which the area between the curves ( y^{2}=4 a x ) and ( x^{2}= ) ( 4 a y ) is ( 1 s q . ) unit, is- A. ( sqrt{3} ) B. 4 c. ( 4 sqrt{3} ) D. ( frac{sqrt{3}}{4} ) | 12 |

435 | The area of the region bounded by ( 3 x pm ) ( 4 y pm 6=0 ) in sq.units is ( A cdot 3 ) в. 1.5 c. 4.5 D. 6 | 12 |

436 | The area bounded by curve ( y=|x-1| ) and ( y=1 ) is A . B. 2 c. ( 1 / 2 ) D. None of these | 12 |

437 | Sketch the curve for ( sin y=sin x ) | 12 |

438 | Find the area of the smaller region bounded by the ellipse ( frac{x^{2}}{9}+frac{y^{2}}{4}=1 ) and the line ( frac{x}{3}+frac{y}{2}=1 ) | 12 |

439 | The area of the region bounded by the curve ( y=x^{2} ) and the line ( y=16 ) is A ( cdot frac{128}{3} ) sq.units в. ( frac{64}{3} ) sq.units c. ( frac{32}{3} ) sq.units D. ( frac{256}{3} ) sq.units | 12 |

440 | Area of ( triangle A B C=68.2 s q m . ) Find the area of shaded region | 12 |

441 | The point intersection of the tangents drawn to the curve ( x^{2} y=1-y ) at the points where it is met by the curve ( x y=1-y ) is given by A ( .(0,1) ) в. (1,1) c. (1,0) (i) 5 D. ( (0, infty) ) | 12 |

442 | Area of the region bounded by the curve ( boldsymbol{y}=mathbf{2 5}^{boldsymbol{x}}+mathbf{1 6} ) and curve ( boldsymbol{y}=boldsymbol{b} . mathbf{5}^{boldsymbol{x}}+mathbf{4} ) whose tangent at the point ( boldsymbol{x}=mathbf{1} ) makes an angle ( tan ^{-1}(40 log 5) ) with the ( x- ) axis is: ( ^{mathbf{A}} cdot_{2 log _{5}}left(frac{e^{4}}{27}right) ) ( ^{mathbf{B}} cdot_{4 log _{5}}left(frac{e^{4}}{27}right) ) ( mathbf{c} cdot_{3 log _{5}}left(frac{e^{4}}{27}right) ) D. None of these | 12 |

443 | 3. Sketch the region whose area is represented by the definite integral and evaluate the integral using an appropriate formula form geometry. (a) f2dx (b) } (x + 2) dx (c) | (x – 1) dx | 12 |

444 | The area (in sq. units) of the region described by ( left{(x, y) ; y^{2} leq 2 x text { and } y geq 4 x-1right} ) is ( A cdot frac{7}{32} ) в. ( frac{5}{64} ) c. ( frac{15}{64} ) D. ( frac{9}{32} ) | 12 |

445 | 37. The area between the parabolas straight line y=2 is: las r? = and r2 =9y and the [2012] () 1012 @ 2012 (6) 102 () 2012 | 12 |

446 | Sketch the graph for ( boldsymbol{y}= ) ( log _{frac{1}{4}}left(x-frac{1}{4}right)+ ) ( frac{1}{2} log _{4}left(16 x^{2}-8 x+1right) ) | 12 |

447 | The curve whose subtangent is twice the abscissa of the point of contact and passing through (1,2) is A ( cdot y^{2}=4 x ) B ( cdot y^{2}=-4 x ) c. ( x^{2}=4 y ) D. ( x^{2}=-4 y ) | 12 |

448 | Find the area bounded by curves ( (x-1)^{2}+y^{2}=1 ) and ( x^{2}+y^{2}=1 ) | 12 |

449 | Find the axis of symmetry of the parabola shown A ( . x=1 ) в. ( x=2.5 ) ( mathbf{c} cdot x=3 ) D. None of these | 12 |

450 | Let ( boldsymbol{f}(boldsymbol{x})=-boldsymbol{x}^{2} / 2 . ) If the graph of ( boldsymbol{f}(boldsymbol{x}) ) is translated 2 units up and 3 units left and the resulting graph is that of ( g(x) ) ( operatorname{then} g(1 / 2)= ) A . B . ( -1 / 8 ) c. ( -2 / 8 ) D. ( -33 / 8 ) E . ( 13 / 8 ) | 12 |

451 | 41. Area of the region {(x,y) eR?:y>x +3], 5y 5x+9515} is equal to (JEE Adv. 2016) | 12 |

452 | If ( theta leq x leq pi ; ) then the area bounded by the curve ( boldsymbol{y}=boldsymbol{x} ) and ( boldsymbol{y}=boldsymbol{x}+sin boldsymbol{x} ) is A . 2 B. 4 ( c cdot 2 pi ) D. ( 4 pi ) | 12 |

453 | If the area (in sq. units) bounded by the parabola ( y^{2}=4 lambda x ) and the line ( y= ) ( lambda x, lambda>0, ) is ( frac{1}{9}, ) then ( lambda ) is equal to A .24 B . 48 ( c cdot 4 sqrt{3} ) D. ( 2 sqrt{6} ) | 12 |

454 | é tet 5. For any real t, x=9 et – et – is a point on the hyperbola x2 – y2 = 1. Show that the area bounded by this hyperbola and the lines joining its centre to the points corresponding to t, and -t, is t, (1982 – 3 Marks) | 12 |

455 | The area (in square units) of the region bounded by ( boldsymbol{x}=-mathbf{1}, boldsymbol{x}=mathbf{2}, boldsymbol{y}=boldsymbol{x}^{2}+mathbf{1} ) and ( y=2 x-2 ) is A . 10 B. 7 ( c cdot 8 ) D. | 12 |

456 | 40. Let b = 0 and for j = 0, 1, 2, …, n, let S, be the area of the region bounded by the y-axis and the curve xeay = sin by, jrcc (+1). Show that S, S,, S., …, S, are in ь – у – ь geometric progression. Also, find their sum for a=-1 and b=n. (2001 – 5 Marks) | 12 |

457 | The positive value of parameter’ ( a ) ‘ for which the area bounded by parabolas ( y=x-a x^{2} & a y=x^{2} ) attains its the maximum value is | 12 |

458 | Area lying in the first quadrant and bounded by the circle ( x^{2}+y^{2}=4 ) and the line ( boldsymbol{x}=boldsymbol{y} sqrt{mathbf{3}} ) is: A . ( pi ) в. ( frac{pi}{2} ) c. ( frac{pi}{3} ) D. None of these | 12 |

459 | 4. Find the area under the curve y = cos x over (a) [0, 7/2] (b) [0, 1] curve y = cos x over the interval AY y = cos x -1+ | 12 |

460 | Find the area of the region ( left{(boldsymbol{x}, boldsymbol{y}): boldsymbol{x}^{2}+boldsymbol{y}^{2} leq mathbf{4}, boldsymbol{x}+boldsymbol{y} geq mathbf{2}right} ) | 12 |

461 | Find the area of the region enclosed by the curves; ( boldsymbol{y}=cos boldsymbol{x}, boldsymbol{y}=1-frac{2 x}{pi} ) A ( cdot 2-frac{pi}{2} ) в. ( pi ) c. ( frac{pi}{2} ) D. | 12 |

462 | Sketch the graph for ( y=cos ^{-1}(cos x) ) | 12 |

463 | The area bounded by the curves ( y= ) ( cos x ) and ( y=sin x ) between the ordinates ( boldsymbol{x}=mathbf{0} ) and ( boldsymbol{x}=frac{mathbf{3} boldsymbol{pi}}{mathbf{2}} ) is A. ( (4 sqrt{2}-2) ) sq units B . ( (4 sqrt{2}+2) ) sq units c. ( (4 sqrt{2}-1) ) sq units D. ( (4 sqrt{2}+1) ) sq units | 12 |

464 | Find the area of the region bounded by the curve ( y^{2}=x ) and the lines ( x= ) ( 1, x=4 ) and the ( x ) -axis ( A cdot frac{8}{3} ) в. ( frac{14}{3} ) ( c cdot frac{7}{3} ) D. ( frac{1}{3} ) | 12 |

465 | The area of the plane region bounded by the curves ( x+2 y^{2}=0 ) and ( x+3 y^{2}= ) ( mathbf{1} ) A ( cdot frac{4}{3} ) B. ( c cdot frac{2}{3} ) D. | 12 |

466 | Let ( P(x, y) ) be a moving point in the ( x- ) ( boldsymbol{y} ) plane such that ( [boldsymbol{x}] .[boldsymbol{y}]=2, ) where ( [.] ) denotes the greatest integer function, then area of the region containing the points ( boldsymbol{P}(boldsymbol{x}, boldsymbol{y}) ) is equal to: A. 1 sq. units B. 2 sq. units c. 4 sq. units D. None of these | 12 |

467 | If a curve passes through the point ( left(2, frac{7}{2}right) ) and has slope ( left(1-frac{1}{x^{2}}right) ) at any point ( (x, y) ) on it, then the abscissa of the point on the curve whose ordinate is ( frac{-3}{2} ) is ( A cdot 2 ) B. -2 c. 1 D. – | 12 |

468 | The area enclosed between the curve ( boldsymbol{y}^{2}=boldsymbol{x} boldsymbol{a} boldsymbol{n} boldsymbol{d} boldsymbol{y}=|boldsymbol{x}| ) is A ( cdot frac{1}{6} ) в. ( frac{1}{3} ) ( c cdot frac{2}{3} ) D. 1 | 12 |

469 | Find the area bounded by the curves ( boldsymbol{x}=boldsymbol{a} cos boldsymbol{t}, boldsymbol{y}=boldsymbol{b} sin t ) in the first quadrant A ( cdot frac{pi a b}{4} ) B. ( frac{pi a^{2} b}{4} ) c. None of these D. ( frac{pi a b^{2}}{4} ) | 12 |

470 | Area bounded by the lines ( y=x, x= ) ( -1, x=2 ) and ( x ) -axis is A ( cdot frac{5}{2} ) sq. units B. ( frac{3}{2} ) squnits c. ( frac{1}{2} s q ) units D. None of these | 12 |

471 | ( * ) ( vdots ) ( * ) ( * ) | 12 |

472 | The ratio in which the area bounded by the curves ( y^{2}=12 x ) and ( x^{2}=12 y ) is divided by the line ( x=3 ) is A . 15: 16 B. 15: 49 ( c cdot 1: 2 ) D. None of these | 12 |

473 | The area bounded by the parabolas ( y^{2}=4 a(x+a) ) and ( y^{2}=-4 a(x-a) ) is A ( cdot frac{16}{3} a^{2} ) sq units B. ( frac{8}{3} ) sq units c. ( frac{4}{3} a^{2} ) sq units D. None of these | 12 |

474 | Compute the area of the figure which lies in the first quadrant inside the curve ( x^{2}+y^{2}=3 a^{2} & ) is bounded by the parabola ( x^{2}=2 a y & y^{2}= ) ( 2 a x(a>0) ) | 12 |

475 | If the curves ( y=x^{3}+a x ) and ( y= ) ( b x^{2}+c ) pass through the point (-1,0) and have common tangent line at this point, then the value of ( a+b ) is? ( mathbf{A} cdot mathbf{0} ) в. -2 c. -3 D. – | 12 |

476 | What is the area of a plane figure bounded by the points of the lines max ( (x, y)=1 ) and ( x^{2}+y^{2}=1 ? ) | 12 |

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