We provide complex numbers and quadratic equations practice exercises, instructions, and a learning material that allows learners to study outside of the classroom. We focus on complex numbers and quadratic equations skills mastery so, below you will get all questions that are also asking in the competition exam beside that classroom.
List of complex numbers and quadratic equations Questions
| Question No | Questions | Class |
|---|---|---|
| 1 | If ( z_{1}, z_{2} ) are roots of equation ( z^{2}-a z+ ) ( a^{2}=0, ) then ( left|frac{z_{1}}{z_{2}}right|= ) |
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| 2 | ff ( z=sqrt{20 i-21}+sqrt{21+20 i} ) then principal value of arg z can not be |
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| 3 | If ( z_{1} ) and ( z_{2} ) are two non-zero complex number such that ( left|frac{z_{1}}{z_{2}}right|=2 ) and ( arg left(z_{1} z_{2}right)=frac{3 pi}{2}, ) then ( frac{overline{z_{1}}}{z_{2}} ) is equal to A. 2 B . – – ( c cdot-2 ) D. |
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| 4 | Number of ordered pair(s) ( (a, b) ) of real numbers such that ( (boldsymbol{a}+boldsymbol{i} boldsymbol{b})^{2008}=boldsymbol{a}- ) ib holds good, is? A . 2008 B. 2009 ( c .2010 ) D. 2011 |
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| 5 | Let ( z ) and ( w ) be two non-zero complex numbers such that ( |z|=|boldsymbol{w}| ) and ( arg (z)+arg (w)=pi, ) then ( z ) equals A. ( -w ) в. ( w ) ( c cdot bar{w} ) D. ( -bar{w} ) |
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| 6 | Find the argument of ( frac{1+sqrt{3} i}{sqrt{3}+i} ) A ( cdot frac{pi}{3} ) в. ( frac{pi}{6} ) c. ( frac{pi}{2} ) D. |
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| 7 | Evalaute ( :left[i^{18}+left(frac{1}{i}right)^{25}right]^{3} ) | 11 |
| 8 | 21. If the roots of the equation bx2 + cx + a=0 be imaginary, then for all real values of x, the expression 362×2 +6bcx +2c2 is: [2009] (a) less than 4ab (6) greater than -4ab (c) less than – 4ab (d) greater than 4ab |
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| 9 | If ( S(n)=i^{n}+i^{-n}, ) where ( i=sqrt{-1} ) and ( n ) is a positive integer, then the total number of distinct values of ( boldsymbol{S}(boldsymbol{n}) ) is: A . 1 B. 2 ( c .3 ) D. 4 |
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| 10 | Find the real and imaginary parts of the complex number ( z=frac{3 i^{20}-i^{19}}{2 i-1} ) | 11 |
| 11 | 24. To the equation 227 /cos-‘x – a+ 21/cos ‘* – a² = 0 has only one real root, then a. 1 Sas3 c. a -3 b. a21 d. a 23 |
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| 12 | For a positive integer ( n ) ( (1-i)^{n}left(1-frac{1}{i}right)^{n}=k^{n}, ) find the value of ( k ) |
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| 13 | State true or false: ( frac{3+2 i sin theta}{1-2 i sin theta}=frac{left(3-4 sin ^{2} thetaright)+8 i sin theta}{1+4 sin ^{2} theta} ) |
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| 14 | If ( z_{1} ) and ( z_{2} ) are complex numbers, prove that ( left|z_{1}+z_{2}right|^{2}=left|z_{1}right|^{2}+left|z_{2}right|^{2} ) if and only is ( z_{1} bar{z}_{2} ) is pure imaginary. |
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| 15 | The smallest integer n such that ( left(frac{1+i}{1-i}right)^{n}=1 ) is A . 16 B. 12 ( c cdot 8 ) D. |
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| 16 | f ( left|z_{1}right|=2,left|z_{2}right|=3,left|z_{3}right|=4 ) and ( left|2 z_{1}+3 z_{2}+4 z_{3}right|=4 ) then the absolute value of ( 8 z_{3} z_{2}+27 z_{3} z_{1}+64 z_{1} z_{2} ) equals A .24 B . 48 ( c cdot 72 ) D. 96 |
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| 17 | Represent follow complex no. in polar form. ( boldsymbol{z}=-mathbf{1}+sqrt{mathbf{3}} boldsymbol{i} ) |
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| 18 | Solve: ( left(frac{1}{1-2 i}+frac{3}{1+i}right)left(frac{3+4 i}{2-4 i}right) ) | 11 |
| 19 | Amplitude of ( frac{1+sqrt{3} i}{sqrt{3}+i} ) is A ( cdot frac{pi}{3} ) в. ( frac{pi}{2} ) ( c cdot 0 ) D. |
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| 20 | If ( z+sqrt{2}|z+1|+i=0 ) and ( z=x+i y ) then A. ( x=-2 ) B. ( x=2 ) c. ( y=-2 ) D. ( y=1 ) |
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| 21 | If ( a^{2}+b^{2}=1, ) then ( frac{(1+b+i a)}{(1+b-i a)} ) is A . B. 2 c. ( b+i a ) ( mathbf{D} cdot a+i b ) |
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| 22 | If ( (sqrt{3}-i)^{n}=2^{n}, n in N, ) then ( n ) is a multiple of ( mathbf{A} cdot mathbf{6} ) B. 10 ( c .9 ) D. 12 |
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| 23 | If ( bar{z} ) lies in the third quadrant then ( z ) lies in the A. First quadrant B. Second quadrant c. Third quadrant D. Fourth quadrant |
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| 24 | 2. [2002] If|z-410 (b) Re(z)3 (d) Re(z)>2 |
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| 25 | If ( |z-2+i| leq 2 ), then find the greatest value of ( |z| ) |
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| 26 | For ( boldsymbol{a}<mathbf{0}, ) arg ( boldsymbol{a}= ) A ( cdot frac{pi}{2} ) в. ( frac{-pi}{2} ) ( c . pi ) D. – ( pi ) |
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| 27 | Find the modulus and the principal value of the argument of the number ( 1-i ) A ( cdot sqrt{2}, pi / 4 ) B ( cdot sqrt{2},-pi / 4 ) c. ( sqrt{2},-pi / 3 ) D. ( sqrt{2}, 3 pi / 4 ) |
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| 28 | For each real ( boldsymbol{x}, ) let ( boldsymbol{f}(boldsymbol{x})= ) ( max left{x, x^{2}, x^{3}, x^{4}right}, ) then ( f(x) ) is A . ( x^{4} ) for ( x leq-1 ) B . ( x^{2} ) for ( -1<x leq 0 ) c. ( fleft(frac{1}{2}right)=frac{1}{2} ) D. ( fleft(frac{1}{2}right)=frac{1}{4} ) |
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| 29 | Complex number ( z ) satisfy the equation ( |z-(4 / z)|=2 . ) Locus of ( z ) if ( left|z-z_{1}right|= ) ( left|z-z_{2}right|, ) where ( z_{1} ) and ( z_{2} ) are complex numbers with the greatest and the least moduli, is A. line parallel to the real axis B. line parallel to the imaginary axis c. line having a positive slope D. line having a negative slope |
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| 30 | A number of two-digit numbers having the property that they are perfectly divided by the sum of their digits with quotient equal to ( 7, ) is: ( A cdot 2 ) B. 3 ( c cdot 4 ) ( D ) |
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| 31 | The value of ( sum_{k=0}^{n}left(i^{k}+i^{k+1}right), ) where ( i^{2}=-1, ) is equal to : ( mathbf{A} cdot i-i^{n} ) B. ( -i+i^{n+1} ) c. ( i-i^{n+1} ) D. ( i-i^{n+2} ) E ( .-i-i^{n} ) |
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| 32 | Simplify the multiplication of complex numbers: ( (boldsymbol{x}, boldsymbol{y}) times(mathbf{1}, mathbf{0}) ) A. ( (-x,-y) ) в. ( (y, x) ) c. ( (x, y) ) D. None of these |
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| 33 | Which of the given alternatives represent a point in Argand plane, equidistant from roots of the equation ( (z+1)^{4}=16 z^{4} ? ) B ( cdotleft(-frac{1}{3}, 0right) ) ( ^{c} cdotleft(frac{1}{3}, 0right) ) D ( cdotleft(0, frac{2}{sqrt{5}}right) ) |
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| 34 | If ( alpha neq beta ) and ( |beta|=1 ) then ( left|frac{alpha-beta}{1-alpha beta}right| ) equals A . – B. 0 ( c ) D. None of these |
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| 35 | Find ( arg (1+sqrt{2}+i) ) A ( . pi / 16 . ) в. ( pi / 8 ) c. ( pi / 12 ) D. ( pi / 10 ) |
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| 36 | Evaluate: ( i^{24}+left(frac{1}{i}right)^{26} ) ( A cdot 0 ) B. ( c cdot-1 ) D. |
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| 37 | ( boldsymbol{a}+boldsymbol{i} boldsymbol{b}=(mathbf{1}+boldsymbol{i} sqrt{boldsymbol{3}})^{300} ) then ( boldsymbol{a}= ) and ( b= ) A .0,1 B . ( 2^{300}, 0 ) ( c .0,0 ) D. none of these |
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| 38 | The argument of every complex number is A. Double valued B. single valued c. Many valued D. Triple valued |
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| 39 | Calculate ( sqrt[3]{-1} ) | 11 |
| 40 | If ( i z^{3}+z^{2}-z+i=0, ) then ( |z|=? ) B . |z| = 2 c. ( |z|=3 ) D. ( |z|=4 ) |
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| 41 | Locate the complex numbers ( z=x+ ) iy such that ( |z-i|=1, arg frac{z}{z+i}=frac{pi}{2} ) |
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| 42 | Let ( z ) and ( omega ) be the complex numbers.If ( boldsymbol{R} boldsymbol{s}(boldsymbol{z})=|boldsymbol{z}-boldsymbol{2}|, boldsymbol{R} e(boldsymbol{omega})=|boldsymbol{omega}-boldsymbol{2}| ) and ( arg (z-omega)=frac{pi}{3}, ) find the value of ( operatorname{Im}(z+omega) ) |
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| 43 | If ( z+frac{1}{z}=2 cos 6^{0}, ) then ( z^{1000}+frac{1}{z^{1000}}+1 ) is equal to A . в. ( c cdot-1 ) D. |
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| 44 | Express the following in the form of ( a+ ) ( boldsymbol{b} boldsymbol{i} ) (i) ( (-i)(2 i)left(-frac{1}{8} iright)^{3} ) |
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| 45 | ( mathbf{f} boldsymbol{omega}=frac{Z}{bar{Z}}, ) then ( |boldsymbol{omega}|= ) | 11 |
| 46 | Show that if ( left|frac{z-3 i}{z+3 i}right|=1, ) then ( z ) is a real number. |
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| 47 | Find the maximum value of ( |z| ) when ( left|z-frac{3}{z}right|=2, ) where ( z ) being a complex number. A ( .1+sqrt{3} ) B. 3 ( c cdot 1+sqrt{2} ) D. |
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| 48 | If ( z=a+i b ) then its conjugate is ( a-i b ) If ( 1, omega, omega^{2} ) are cube roots of unity then (i) ( 1+omega+omega^{2}=mathbf{0} ) (ii) ( omega^{3}=1 ) The conjugate of ( frac{mathbf{6}-mathbf{3} i}{mathbf{7}+mathbf{i}} ) is A ( cdot frac{39-27 i}{50} ) в. ( frac{-39+27 i}{50} ) c. ( frac{39+27 i}{50} ) D. ( frac{-39-27 i}{50} ) |
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| 49 | Let ( n ) be a positive integer. Then ( (i)^{4 n+1}+(-i)^{4 n+5}= ) ( A ) в. 2 ( c ) D. ( -i ) |
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| 50 | If ( operatorname{Arg}(z+i)-operatorname{Arg}(z-i)=frac{pi}{2}, ) then ( z ) lies on a circle. If statement is True, enter 1 , else enter 0 |
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| 51 | If ( z_{1}, z_{2} ) be two non zero complex numbers satisfying the equation ( left|frac{z_{1}+z_{2}}{z_{1}-z_{2}}right|=1 ) then ( frac{z_{1}}{z_{2}}+left(frac{z_{1}}{z_{2}}right) ) is A. zero B. c. purely imaginary D. |
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| 52 | ( sin x+i cos 2 x ) and ( cos x-i sin 2 x ) are conjugate to each other for A . ( x=n pi ) B. ( x=left(n+frac{1}{2}right) frac{pi}{2} ) c. ( x=0 ) D. No value of ( x ) |
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| 53 | Solve for ( z:(i-z)(1+i)=2 i ) | 11 |
| 54 | The simplified form of ( i^{n}+i^{n+1}+ ) ( boldsymbol{i}^{boldsymbol{n}+boldsymbol{2}}+boldsymbol{i}^{boldsymbol{n}+boldsymbol{3}} ) is ( mathbf{A} cdot mathbf{0} ) B. ( c cdot-1 ) D. |
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| 55 | Let ( z ) be a complex number and ( c ) be a real number ( geq 1 ) such that ( z+ ) ( boldsymbol{c}|boldsymbol{z}+mathbf{1}|+boldsymbol{i}=mathbf{0}, ) then ( c ) belongs to A . [2,3] B. (3,4) c. ( [1, sqrt{2}] ) D. None of these |
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| 56 | If ( operatorname{Re}left(frac{z+2 i}{z+4}right)=0 ) then ( z ) lies on a circle with center: A ( cdot(-2,-1) ) B. (-2,1) c. (2,-1) D. (2,1) |
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| 57 | Express the following in the form ( A+i B ) ( frac{1}{1-cos theta+2 i sin theta} ) |
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| 58 | ( i^{n} . i^{n+1} i^{n+3}= ) ( mathbf{A} cdot(i)^{n} ) B. ( -i ) ( mathbf{C} cdot(-i)^{n} ) D. 1 |
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| 59 | Express the following complex numbers in the form ( r(cos theta+i sin theta) ) ( 1+i tan alpha ) |
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| 60 | Find the modulus and the argument of the complex number ( z=-sqrt{3}+i ) | 11 |
| 61 | Represent the following complex number in trigonometric form: ( 3-4 i ) |
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| 62 | If ‘ ( omega^{prime} ) is a complex cube root of unity,then ( omegaleft(frac{1}{3}+frac{2}{9}+frac{4}{27} dots inftyright) ) ( omegaleft(frac{1}{2}+frac{3}{8}+frac{9}{32} dots inftyright)= ) ( mathbf{A} cdot mathbf{1} ) B. – ( c cdot omega ) D. |
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| 63 | 16. Let 2, and z, be nth roots of unity which subtend a right angle at the origin. Then n must be of the form (20015) (a) 4k+1 (b) 4k+2 (c) 4k+3 (d) 4k fi |
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| 64 | If ( Z_{r}=left(cos frac{r pi}{10}+i sin frac{r pi}{10}right) . ) Then find the value of ( Z_{1} cdot Z_{2} cdot Z_{3} cdot Z_{4} ) |
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| 65 | The value of ( (x- ) ( left(x+frac{1}{2}-frac{sqrt{3}}{2} iright)left(x+frac{1}{2}+frac{sqrt{3}}{2} iright) ) A. ( x^{3}+x^{2}+x 1 ) B . ( x^{3}-1 ) c. ( x^{3}+1 ) D. ( x^{3}-x^{2}+x+1 ) |
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| 66 | Suppose ( n ) is a natural number such that ( left|i+2 i^{2}+3 i^{3}+ldots+n i^{n}right|=18 sqrt{2} ) where ( i ) is the square root of ( -1 . ) Then ( n ) is. ( mathbf{A} cdot mathbf{9} ) B. 18 ( c cdot 36 ) D. 72 |
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| 67 | What is the value of the sum ( sum_{n=2}^{11}left(i^{n}+i^{n+1}right) ) where ( i=sqrt{-1} ? ) ( mathbf{A} cdot i ) в. ( 2 i ) c. ( -2 i ) D. ( 1+i ) |
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| 68 | Express ( frac{-1+i}{sqrt{2}} ) in the polar form | 11 |
| 69 | The value of ( sum_{n=0}^{100} i^{n !} ) equals ( (text { where } i= ) ( sqrt{-1}) ) A . -1 B. ( c cdot 2 i+95 ) D. ( 96+i ) |
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| 70 | If ( z(1+a)=b+i c ) and ( a^{2}+b^{2}+c^{2}= ) 1, then ( frac{1+i z}{1-i z}= ) A ( cdot frac{a+i b}{1+c} ) в. ( frac{b-i c}{1+a} ) c. ( frac{a+i c}{1+b} ) D. None of these |
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| 71 | If ( z_{1}, z_{2}, z_{3} ) be three unimodular complex numbers then ( boldsymbol{E}=left|boldsymbol{z}_{1}-boldsymbol{z}_{2}right|^{2}+ ) ( left|z_{2}-z_{3}right|^{2}+left|z_{3}-z_{1}right|^{2} ) then cannot exceed ( A cdot 6 ) B. ( c cdot 12 ) D. none |
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| 72 | 36. Let a and ß be the roots of x2 – 6x – 2 = 0, with a>.B. If 410 – Lag is an = a”-B” for n 2 1, then the value of “10 (2011) (a) 1 (b) 2 (©) 3 (d) 4 |
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| 73 | Represent the following complex number in trigonometric form: ( (-sqrt{3}+i)^{3} ) |
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| 74 | If ( (cos theta+i sin theta)(cos 2 theta+i sin theta) ) ( (cos n theta+i sin n theta)=1, ) then the value of ( boldsymbol{theta} ) is A ( cdot frac{2 m pi}{n(n+1)} ) B. ( 4 m pi ) c. ( frac{4 m pi}{n(n+1)} ) D. ( frac{m pi}{n(n+1)} ) |
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| 75 | If ( z=x+i y ) is a complex number such that ( bar{z}^{frac{1}{3}}=a+i b, ) then the value of ( frac{1}{a^{2}+b^{2}}left(frac{x}{a}+frac{y}{b}right)= ) A . -1 B. – ( c cdot 0 ) D. |
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| 76 | Find the arguments of each of the complex numbers. 1. ( boldsymbol{z}=-mathbf{1}-boldsymbol{i} sqrt{mathbf{3}} ) 2. ( z=-sqrt{3}+i ) 3. ( boldsymbol{z}=mathbf{1}+boldsymbol{i} sqrt{mathbf{3}} ) |
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| 77 | Let ( z=cos theta+i sin theta . ) Then the value of ( sum_{m=1}^{1} 5 operatorname{Im}left(z^{2 m-1}right) operatorname{at} theta=2^{0} ) is A ( cdot frac{1}{sin 2^{0}} ) в. ( frac{1}{3 sin 2^{circ}} ) c. ( frac{1}{2 sin 2^{circ}} ) D. ( frac{1}{4 sin 2^{circ}} ) |
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| 78 | Express the following expression in the form of ( boldsymbol{a}+boldsymbol{i} boldsymbol{b} ) ( frac{(3+i sqrt{5})(3-i sqrt{5})}{(sqrt{3}+sqrt{2} i)-(sqrt{3}-i sqrt{2})} ) |
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| 79 | Find the real numbers ( x ) and ( y, ) if ( (x- ) ( i y)(1+i) ) is the conjugate of ( -3-2 i ) | 11 |
| 80 | 2. Ifx=a+b, y =ay + bB and z=aß + by where y and B are the complex cube roots of unity, show that xyz = a + b3. |
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| 81 | Given ( : boldsymbol{u}=mathbf{1}+boldsymbol{i} sqrt{mathbf{3}} ) and ( boldsymbol{v}=sqrt{mathbf{3}}+boldsymbol{i} ) Calculate ( frac{u^{3}}{v^{4}} ) A ( cdot(1 / 4)-i sqrt{1 / 4} ) B. ( (3 / 4)-i sqrt{3} / 4 ) c. ( (1 / 4)-i sqrt{3} / 4 ) D. none of these |
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| 82 | If ( z=frac{sqrt{3}+i}{2}, ) then the value of ( z^{69} ) is A . ( -i ) B. ( c ) D. |
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| 83 | Let ( z_{1}=2-i, z_{2}=-2+i ) Find (i) ( operatorname{Re}left(frac{z_{1} z_{2}}{overline{z_{1}}}right) ) (ii) ( operatorname{Im}left(frac{1}{z_{1} overline{z_{1}}}right) ) |
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| 84 | The area of the triangle whose vertices are represented by ( 0, z, z^{i alpha}(0<alpha<pi) ) equals ( mathbf{A} cdot frac{1}{2}|z|^{2} cos alpha ) B . ( frac{1}{2}|z|^{2} sin alpha ) C ( cdot frac{1}{2}|z|^{2} sin alpha cos alpha ) D cdot ( frac{1}{2}|z|^{2} ) |
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| 85 | What is the value of ( i^{i} ) Where ( i=sqrt{-1} ) |
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| 86 | If ( z_{1} ) and ( z_{2} ) are two complex number such that ( operatorname{lm}left(z_{1}=z_{2}right)=0=operatorname{lm}left(z_{1} z_{2}right) ) then B. ( z_{1}=bar{z}_{2} ) ( mathbf{c} cdot z_{1}=-z_{2} ) D. ( z_{1}=-bar{z}_{2} ) |
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| 87 | If ( z ) be a complex number satisfying ( z^{4}+z^{3}+2 z^{2}+z+1=0, ) then find the value of ( |vec{z}| ) |
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| 88 | If the square of ( (a+i b) ) is real, then ( boldsymbol{a} boldsymbol{b}= ) ( mathbf{A} cdot mathbf{0} ) B. c. -1 D. 2 |
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| 89 | Represent the following complex number in trigonometric form: ( 1+i tan alpha ) |
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| 90 | If ( frac{(1+i)^{2}}{2-i}=x-i y, ) then find the value of ( boldsymbol{x}+boldsymbol{y} ) | 11 |
| 91 | The locus of complex number z such that ( z ) is purely real and real part is equal to -2 is A. Negative y-axis B. Negative x-axis c. The point (-2,0) D. The point ( 2, 0) |
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| 92 | Find the value of ( theta ) if ( frac{(3+2 i sin theta)}{(1-2 i sin theta)} ) is purely real or purely imaginary. A ( cdot theta=n pi pm frac{pi}{6}, n in Z ) в. ( theta=n pi pm frac{pi}{2}, n in Z ) c. ( theta=n pi pm frac{pi}{3}, n in Z ) D. ( theta=n pi pm frac{pi}{4}, n in Z ) |
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| 93 | Evaluate ( left(frac{1+cos frac{pi}{8}-i sin frac{pi}{8}}{1+cos frac{pi}{8}+i sin frac{8}{8}}right)^{8}= ) A . B. – – ( c cdot 2 ) D. |
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| 94 | The value of the sum ( sum_{n=1}^{13}left(i^{n}+i^{n+1}right) ) where ( i=sqrt{-1}, ) equals ( mathbf{A} cdot i ) B . ( i-1 ) ( c cdot-i ) D. |
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| 95 | Find the value of ( (-1+sqrt{-3})^{2}+ ) ( (-1-sqrt{-3})^{2} ) | 11 |
| 96 | If ( (x+i y)(2-3 i)=4+i ) then ( (x, y)= ) A. ( left(1, frac{1}{13}right) ) B ( cdotleft(-frac{5}{13}, frac{14}{13}right) ) c. ( left(frac{5}{13}, frac{14}{13}right) ) D. ( left(-frac{5}{13},-frac{14}{13}right) ) |
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| 97 | V3+i – and P = {w.n=1,2,3,…}. Further H, = {zeC:Rezand Hz={zeC: Rez<}, where c is the set of all complex numbers. If zi PCH,z2 | PCH2 and O represents the origin, then 22,0z2 = (JEE Adv. 2013) |
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| 98 | Verify the following: ( left|z_{1}-z_{2}right|^{2}=left|z_{1}right|^{2}+left|z_{2}right|^{2}-2 operatorname{Re}left(z_{1} overline{z_{2}}right) ) |
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| 99 | 27. If the equations x2 + 2x + 3 = 0 and ax2+bx+c=0, a,b,c e R, have a common root, then a :b:cis JEEM 2013] (a) 1:2:3 (b) 3:2:1 (c) 1:3:2 (d) 3:1:2 |
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| 100 | If ( (1+i)^{2 n}+(1-i)^{2 n}=-2^{n+1} ) where ( i=sqrt{-1} ) for all those ( n, ) which are A. even B. odd c. multiple of 3 D. None of these |
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| 101 | f ( pi / 2 ) and ( pi / 4 ) are respectively the arguments nof ( Z_{1} ) and ( overline{Z_{2}} ), what is the value of ( arg left(z_{1} / z_{2}right) ) |
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| 102 | 31. The sum of all real values of x satisfying the equation (x2- 5x+5)*%+4X-60 = 1 is : [JEE M 2016 (a) 6 (b) 5 c) 3 (d) – 4 o |
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| 103 | 6. Prove that the complex numbers z., z, and the origin form an equilateral triangle only if 2,2 +2,2-2,72=0. (1983 – 3 Marks) |
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| 104 | The greatest value of ( |z+1| ) if ( |z+4| leq ) 3 is A . 4 B. 5 ( c cdot 6 ) D. None of these |
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| 105 | 18) 3. If 2 and 22 are two nonzero complex numbers such that 14 +22 l=12 +1 22 l, then Arg 21 – Arg 22 is equal to (1987-2 Marks) (a) (b) – © o d I (e) 7 |
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| 106 | If ( i^{2}=-1, ) then ( (5+6 i)^{2}= ) A . -11 B . ( -11+11 i ) c. ( -11+30 i ) D. ( -11+60 i ) E . 61 |
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| 107 | Solve the equation ( |z|=z+1+2 i ) A ( cdot frac{3}{2}+2 i ) в. ( frac{3}{2}-i ) c. ( frac{3}{2}+i ) D. ( frac{3}{2}-2 i ) |
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| 108 | Find the modulus and argument of the complex number ( frac{1+2 i}{1-3 i} ) | 11 |
| 109 | ( boldsymbol{i}^{5 boldsymbol{7}}+frac{mathbf{1}}{boldsymbol{i}^{mathbf{1 2 5}}}= ) A. B. ( 2 i ) c. ( -2 i ) D. |
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| 110 | Find the conjugate of ( frac{(3-2 i)(2+3 i)}{(1+2 i)(2-i)} ) | 11 |
| 111 | locate the point representing the complex numbers ( z ) on the Argand diagram for which ( |z-1|=|z-3|=|z-i| ) |
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| 112 | In the complex numbers, where ( i^{2}=-1 ) what is the value of ( 5+6 i ) multiplied by ( 3-2 i ? ) A . 27 в. 27 c. ( 27+8 i ) D. ( 15+8 i ) E . ( 15-18 i ) |
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| 113 | Prove that for two complex numbers ( z_{1}, z_{2},left(z_{1}-z_{2}right)^{2}=z_{1}^{2}-2 z_{1} z_{2}+z_{2}^{2} ) | 11 |
| 114 | If ( boldsymbol{alpha}, boldsymbol{beta}, gamma ) are modulus of the complex number ( 3+4 i,-5+12 i, 1-i, ) then the increasing order for ( alpha, beta ) and ( gamma ) is A ( cdot alpha, gamma, beta ) в. ( alpha, beta, gamma ) ( mathrm{c} . gamma, alpha, beta ) D. can’t be determined |
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| 115 | What is the value of ( (1+i)^{5}+(1-i)^{5} ) where ( i=sqrt{-1} ? ) A . -8 B. 8 c. ( 8 i ) D. ( -8 i ) |
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| 116 | ( operatorname{Let} z=left(a-frac{i}{2}right) ; in mathrm{R} . ) Then ( |i+z|^{2} ) ( |i-z|^{2} ) is equal to |
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| 117 | If ( boldsymbol{a}=boldsymbol{e}^{boldsymbol{i} boldsymbol{alpha}}, boldsymbol{b}=boldsymbol{e}^{boldsymbol{i} boldsymbol{beta}}, boldsymbol{c}=boldsymbol{e}^{boldsymbol{i} gamma} ) and ( cos boldsymbol{alpha}+ ) ( cos beta+cos gamma=0=sin alpha+sin beta+ ) ( sin gamma, ) then prove the following ( sum cos 2 alpha=0=sum sin 2 alpha ) |
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| 118 | What is the square of the modulus of the complex number ( 2+3 i ) |
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| 119 | Find the modulus and the principal argument of the complex number ( (tan 1-i)^{2} ) ( mathbf{A} cdot|z|=(tan 1)^{2}+1, z ) lies in 4 rd quadrant, ( arg (z)=2- ) ( pi / 2 ) B ( cdot|z|=(tan 1)^{2}+1, z ) lies in 4 rd quadrant, ( arg (z)=2-pi ) ( mathbf{C} cdot|z|=(tan 1)^{2}+1, mathbf{z} ) lies in ( 3 mathrm{rd} ) quadrant, ( arg (z)=2- ) ( pi / 2 ) D ( cdot|z|=(tan 1)^{2}+1, z ) lies in 3rd quadrant, ( arg (z)=2-pi ) |
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| 120 | If the complex numbers ( z_{1}, z_{2} ) and ( z_{3} ) denote the vertices of an isosceles triangle, right angled at ( z_{1}, ) then ( left(z_{1}-right. ) ( left.z_{2}right)^{2}+left(z_{1}-z_{3}right)^{2} ) is equal to ( mathbf{A} cdot mathbf{0} ) B. ( left(z_{2}+z_{3}right)^{2} ) ( c cdot 2 ) D. 3 E ( cdotleft(z_{2}-z_{3}right)^{2} ) |
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| 121 | When ( a ) is real number then ( (z+ ) ( boldsymbol{a})(overline{boldsymbol{z}}+boldsymbol{a})= ) A. ( |z-a| ) B. ( z^{2}+a^{2} ) c. ( |z+a|^{2} ) D. ( z^{2}-a^{2} ) |
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| 122 | Represent the following complex number in trigonometric form: ( frac{3}{2}-i frac{sqrt{3}}{2} ) |
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| 123 | Represent following complex numbers ( z_{1}=1+2 i ) and ( z_{2}=5-7 i ) by points in Argand’s diagram and determine their amplitudes approximately. |
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| 124 | If ( log _{e}left(frac{1}{left(1+x+x^{2}+x^{3}right)}right) ) be expanded in a series of ascending powers of ( boldsymbol{x} ) the coefficient of ( x^{n} ) is ( -frac{b}{n} ) if ( n ) be odd or of the form ( 4 m+2 ) and ( frac{a}{n} ) if ( n ) be of the form ( 4 m . ) Find the value of ( a+b^{2} ) |
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| 125 | For ( boldsymbol{z}=boldsymbol{x}+boldsymbol{i} boldsymbol{y}, ) then for ( boldsymbol{e}^{|boldsymbol{z}|} ) A ( cdot e^{|x|} leq e^{|z|} ) B ( cdot e^{x} geq e^{|z|} ) ( mathbf{c} cdot e^{|x|}=e^{|z|} ) D. none of these |
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| 126 | Let ( z neq 1 ) be a complex number and let ( omega=x+i y neq 0 . ) If ( frac{omega-omega z}{1-z} ) is purely real, then | ( z mid ) is equal to : ( A cdot|omega| ) B . |omega| ( ^{2} ) c. ( frac{1}{|omega|^{2}} ) D. ( frac{1}{|omega|} ) ( E ) |
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| 127 | 29. Let a and B be the roots of equation px? +qx+r=0, p*0. If p, q, r are in A.P. and –+ – = 4, then the value of Q B [JEEM 2014 (a) V34 6 ) 2013 |
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| 128 | Simplify the following ( : i^{457} ) | 11 |
| 129 | If ( z_{1} ) and ( z_{2} ) two complex numbers satisfying the equation ( left|frac{z_{1}+i z_{2}}{z_{1} i z_{2}}right|=1 ) then ( frac{z_{1}}{z_{2}} ) is a A . purely real B. of unit modulus c. purely imaginary D. none of these |
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| 130 | Consider ( boldsymbol{a} boldsymbol{z}^{2}+boldsymbol{b} boldsymbol{z}+boldsymbol{c}=mathbf{0}, ) where ( a, b, c in R ) and ( 4 a c>b^{2} ) In the argand’s plane. if ( A ) is the point represnting ( z_{1} . ) B is the point representing ( z_{2} ) and ( z=frac{overrightarrow{O A}}{partial B} ) then z is: A. z is purely real B. z is purely imaginary c ( cdot|z|=1 ) D. ( Delta A O B ) is a scalene triangle |
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| 131 | If ( z=frac{1+i sqrt{3}}{sqrt{3}+i}, ) then ( (bar{z})^{100} ) lies in A. ( I ) quadrant B. II quadrant c. ( I I I ) quadrant D. ( I V ) quadrant |
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| 132 | Evaluate: ( left(frac{cos frac{pi}{8}-i sin frac{pi}{8}}{cos frac{pi}{8}+i sin frac{pi}{8}}right)^{4} ) ( A ) B. – ( c cdot 2 ) D. |
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| 133 | The number of solutions of ( z^{2}+|z|=0 ) is ( mathbf{A} cdot mathbf{1} ) B . 2 ( c .3 ) D. Infinite |
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| 134 | In the complex numbers, where ( i= ) ( sqrt{-1}, ) the conjugate of any value ( a+b i ) is ( a- ) ib. What is the result when you multiply ( 2+7 i ) by its conjugate? A . 45 в. -45 ( c .45 i ) D. 53 E. 53 |
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| 135 | Let ( boldsymbol{z}=frac{cos boldsymbol{theta}+boldsymbol{i} sin boldsymbol{theta}}{cos boldsymbol{theta}-boldsymbol{i} sin boldsymbol{theta}}, frac{boldsymbol{pi}}{boldsymbol{4}}<mathbf{0}<frac{boldsymbol{pi}}{boldsymbol{2}} ) Then arg z is ( A cdot 2 theta ) в. ( 2 theta-pi ) ( c . pi+2 theta ) D. None of these |
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| 136 | 18. If the difference between the roots of the equation x2 + ax + 1 = 0 is less than 5, then the set of possible values of a is [2007] (a) (3,00) (b) (-00,-3) C) (-3,3) (d) (-3,0). |
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| 137 | The complex number z satisfies the equation ( z+|z|=2+8 i . ) Then the value of ( |z| ) is A . 15 B. 16 ( c cdot 17 ) D. 18 |
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| 138 | If ( x ) and ( y ) are complex numbers, then system of equations ( (1+i) x+(1- ) ( boldsymbol{i}) boldsymbol{y}=1,2 i boldsymbol{x}+2 boldsymbol{y}=1+boldsymbol{i} ) has A. unique solution B. no solution c. infinite numbers of solution D. none of these |
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| 139 | Evaluate : ( (-sqrt{-1})^{4 n+3}, n in N ) A . ( -i ) B. ( c cdot 1 ) D. – |
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| 140 | State whether the given statement is true or false ( overline{left(z^{-1}right)}=(bar{z})^{-1} ) | 11 |
| 141 | If ( z_{1} ) and ( z_{2} ) are two complex numbers, then ( R eleft(z_{1} z_{2}right) ) is: ( mathbf{A} cdot operatorname{Re}left(z_{1}right) operatorname{Re}left(z_{2}right) ) B ( cdot operatorname{Re}left(z_{1}right) cdot operatorname{Re}left(z_{2}right)-operatorname{Im}left(z_{1}right) cdot operatorname{Im}left(z_{2}right) ) ( mathbf{c} cdot operatorname{Im}left(z_{1}right) cdot operatorname{Re}left(z_{2}right) ) D. ( operatorname{Re}left(z_{1}right) . operatorname{Im}left(z_{2}right) ) |
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| 142 | ( (i)^{457} ) ( A cdot-1 ) B . ( -i ) ( mathbf{c} cdot i ) D. 1 |
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| 143 | If ( frac{z+1}{z+i} ) is purely imaginary, then z lies on a A. straight lone B. circle c. circle with radius 1 D. circle passing through (1,1) |
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| 144 | Find the multiplicative of ( 2-3 i a=2 ) ( boldsymbol{b}=-mathbf{3} ) |
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| 145 | In the complex plane, what is the distance of ( 4-2 i ) from the origin? ( A cdot 2 ) в. 3.46 c. 4.47 D. 6 E . 12 |
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| 146 | Find the minimum value of ( |z-1| ) if ||( z-3|-| z+||1=2 ) A ( cdot|z-1| geq 0 ) B ( cdot|z-1| geq 1 ) c. ( |z-1| geq 2 ) D. ( |z-1| geq 3 ) |
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| 147 | 27. A particle P starts from the point zo=1+2i, where i= It moves horizontally away from origin by 5 units and then -1. vertically away from origin by 3 units to reach a point z. From z, the particle moves 2 units in the direction of the vector î+and then it moves through an angle in anticlockwise direction on a circle with centre at origin, to reach a point z. The point z, is given by 1 (2008) (a) 6+7i (6) -7+6i c) 7+6i (d) 6+7i |
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| 148 | ( z ) is a complex number. If ( a=|x|+|y| ) and ( b=sqrt{2}|x+i y| ) |
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| 149 | ( frac{(1+i)^{3}}{2+i} ) is equal to A ( cdot frac{2}{5}-frac{6}{5} ) B. c. ( -frac{1}{5}+frac{6}{5} i ) D. ( -frac{2}{5}+frac{6}{5} ) |
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| 150 | Find the harmonic conjugate of the point ( R(5,1) ) with respect to points ( boldsymbol{P}(mathbf{2}, mathbf{1 0}) ) and ( boldsymbol{Q}(boldsymbol{6},-mathbf{2}) ) |
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| 151 | The two complex numbers satisfying the equation ( z bar{z}-(1+i) z- ) ( (3+2 i) bar{z}+(1+5 i)=0 ) are A ( .1+i, 3+2 i ) B. ( 1+i, 3-2 i ) c. ( 1-i, 3+2 i ) D. ( 1-i, 3-2 i ) |
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| 152 | Solve ( frac{1}{1+i} ) | 11 |
| 153 | If ( z ) satisfies ( |z+1|<|z-2|, ) and ( omega= ) ( mathbf{3} z+mathbf{2}+mathbf{i}, ) then ( mathbf{A} cdot|omega+1|<|omega-8| ) B . ( |omega+1|7 ) D ( cdot|omega+5|<mid omega-4 ) |
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| 154 | The value of ( 2 x^{4}+5 x^{3}+7 x^{2}-x+41 ) when ( boldsymbol{x}=-boldsymbol{2}-sqrt{mathbf{3} boldsymbol{i}} ) is: A . – B. 4 ( c cdot-6 ) D. 6 |
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| 155 | If z = x+iy and @=(1 – iz)/(z-i), then o=implies that, in the complex plane, (1983-1 Mark) (a) z lies on the imaginary axis (b) z lies on the real axis (c) z lies on the unit circle (d) None of these |
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| 156 | The minimum value of ( |Z-1+2 i|+ ) ( |4 i-3-Z| ) is A ( cdot sqrt{5} ) B. 5 c. ( 2 sqrt{13} ) D. ( sqrt{15} ) |
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| 157 | Find the arguments of ( z_{1}=5+ ) ( mathbf{5} i, z_{2}=-4+4 i, z_{3}=-3-3 i ) and ( z_{4}=2-2 i, ) where ( i=sqrt{-1} ) |
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| 158 | 20. If o (#1) be a cube root of unity and (1 + @2)” = (1 + 04)n then the least positive value of n is (2004) (2) 2 (b) 3 C) 5 (d) 6 |
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| 159 | What is value of ( (-i)^{12} ) | 11 |
| 160 | State true or false: The complex numbers ( boldsymbol{z}=boldsymbol{x}+boldsymbol{i} boldsymbol{y} ) which satisfy the equation ( left|frac{z-5 i}{z+5 i}right|=1 ) lie on the axis of ( Y ) |
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| 161 | If ( z ) is a complex number such that ( |z|=1, ) prove that ( frac{z-1}{z+1} ) is purely imaginary. What will be your conclusion, if ( z=1 ? ) | 11 |
| 162 | If a complex number ( z ) and ( z+frac{1}{z} ) have same argument then- A. z must be purely real B. z must be purely imaginary c. z cannot be imaginary D. z must be raal |
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| 163 | Assertion (A): The principal amplitude of complex number ( boldsymbol{x}+boldsymbol{i} boldsymbol{x} ) is ( frac{boldsymbol{pi}}{boldsymbol{4}} ) Reason (R): The principal amplitude of a complex number ( boldsymbol{x}+boldsymbol{i} boldsymbol{y} ) is ( frac{boldsymbol{pi}}{boldsymbol{4}} ) if ( boldsymbol{y}=boldsymbol{x} ) A. Both A and R are true and R is the correct explanation of A B. A is true R is false c. A is false, R is true D. Both A and R are false |
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| 164 | The argument of the complex number ( sin frac{6 pi}{5}+ileft(1+cos frac{6 pi}{5}right) ) is A ( cdot frac{6 pi}{5} ) в. ( frac{5 pi}{6} ) c. ( frac{9 pi}{10} ) D. ( frac{2 pi}{5} ) |
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| 165 | ( frac{1}{1-cos theta+2 i sin theta}=frac{1-2 i cot (theta / 2)}{5+3 cos theta} ) f this is true enter 1 , else enter 0 |
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| 166 | Indicate the point of the complex plane ( z ) which satisfy the following equation ( boldsymbol{operatorname { R e }} boldsymbol{z}^{2}=mathbf{0} ) |
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| 167 | The real part of ( (1-cos theta+2 i sin theta)^{-1} ) is A. ( frac{1}{3+5 cos theta} ) B. ( frac{1}{5-3 cos theta} ) C. ( frac{1}{3-5 cos theta} ) D. ( frac{1}{5+3 cos theta} ) |
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| 168 | 26. If z)= 1 and 27+1, then all the values of lie on P 1-22 (a) a line not passing through the origin (2007-3 marks) (b) 121= 2 (c) the x-axis (d) the y-axis |
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| 169 | Let ( z ) be a complex number such that ( left|frac{boldsymbol{z}-boldsymbol{i}}{boldsymbol{z}+mathbf{2} boldsymbol{i}}right|=mathbf{1} ) and ( |boldsymbol{z}|=frac{mathbf{5}}{mathbf{2}} . ) Then the value of ( |z+3 i| ) is? ( A cdot frac{7}{2} ) в. ( frac{15}{4} ) ( c cdot 2 sqrt{3} ) D. ( sqrt{10} ) |
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| 170 | Find the values of ( x ) and ( y ) which satisfy the given equations ( (x, y in R) ) ( frac{x-1}{1+i}+frac{y-1}{1-i}=i ) | 11 |
| 171 | ( operatorname{Given}|z|=4 ) and ( A r g z=frac{5 z}{6}, ) then ( z ) is A ( cdot 2 sqrt{3}+2 i ) B. ( 2 sqrt{3}-2 i ) c. ( -2 sqrt{3}+2 i ) D. ( -sqrt{3}+i ) |
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| 172 | Find the value of ( left(frac{2 i}{1+i}right)^{2} ) | 11 |
| 173 | ( left|z_{1}+z_{2}right|=left|z_{1}right|+left|z_{2}right| ) is possible if A ( cdot z_{2}=overline{z_{1}} ) в. ( _{z_{2}}=frac{1}{z_{1}} ) ( mathbf{c} cdot arg z_{1}=arg z_{2} ) D ( cdotleft|z_{1}right|=left|z_{2}right| ) |
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| 174 | 28. If a eR and the equation -3(x-[x])+2(x-[x])+a? = 0 (where [x] denotes the greatest integer <x) has no integral solution, then all possible values of a lie in the interval: [JEEM 2014) (a) (-2,-1) (6) (-0,-2) U (2,00) (c) (-1,0)(0,1) (d) (1,2) |
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| 175 | 25. A man walks a distance of 3 units from the origin towards the north-east (N 45° E) direction. From there, he walks a distance of 4 units towards the north-west (N 45° W) direction to reach a point P. Then the position of P in the Argand plane is (2007-3 marks) (a) 3eiT/4 + 4i (b) (3-4i)eitt/4 (c) (4+3i)eint/4 (d) (3+4i)ein/4 |
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| 176 | Find the value of ( sum_{n=0}^{100} i^{n !}(text {where}, i= ) ( sqrt{-1}) ) |
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| 177 | 37. A value of b for which the equations x2 + bx-1=0 x +x+b=0 have one root in common is (a) – 2 (6) -i13 (2011) © iss (d) V2 |
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| 178 | ( z_{1} ) and ( z_{2} ) are two non-zero complex numbers such that ( left|z_{1}right|=left|z_{2}right| ) and ( arg z_{1}+arg z_{2}=pi, ) then ( z_{2} ) equals ( A cdot z_{1} ) B. ( -overline{z_{1}} ) ( mathbf{c} cdot z_{1} ) D. ( -z_{1} ) |
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| 179 | The complex numbers ( sin x+i cos 2 x ) and ( cos x-i sin 2 x ) are conjugate to each other, for A . ( x=n pi ) B. ( x=left(n+frac{1}{2}right) pi ) c. ( x=0 ) D. No value of ( x ) |
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| 180 | Find the modulus and argument of ( z= ) ( frac{3+2 i}{-2+i} ) | 11 |
| 181 | Find the multiplicative inverse of ( sqrt{5}+ ) ( mathbf{3} i ) A ( . sqrt{5}-3 i ) B. ( frac{sqrt{5}-3 i}{14} ) c. ( -sqrt{5}+3 i ) D. ( frac{-sqrt{5}+3 i}{14} ) |
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| 182 | Find the value of the complex number ( left(i^{25}right)^{3} ) |
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| 183 | Given that ( i z^{2}=1+frac{2}{z}+frac{3}{z^{2}}+frac{4}{z^{3}}+frac{5}{z^{4}}+ ) ( ldots . ) and ( z=n pm sqrt{-i}, ) find ( lfloor 100 nrfloor ) |
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| 184 | If ( z=(i)^{(i)}^{(i)} ) where ( i=sqrt{-1}, ) then ( |z| ) is equal to A . 1 B . ( e^{-pi / 2} ) ( mathbf{c} cdot e^{-pi} ) D. none of these |
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| 185 | Find the conjugates of the following complex numbers: ( frac{(3-2 i)(2+3 i)}{(1+2 i)(2-i)} ) A ( cdot frac{63}{25}-frac{16}{25} i ) В ( cdot frac{62}{25}+frac{16}{25} i ) c. ( frac{63}{25}+frac{17}{25} i ) D. ( frac{63}{25}+frac{16}{25} i ) |
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| 186 | Find the multiplicative inverse of the complex number ( 4-3 i ) |
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| 187 | If ( i^{2}=-1, ) then find the odd one out of the following expressions. ( mathbf{A} cdot-i^{2} ) B ( cdot(-i)^{2} ) ( c cdot i^{4} ) D. ( (-i)^{4} ) ( E cdot-i^{6} ) |
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| 188 | If ( boldsymbol{z}_{1}=mathbf{9}+mathbf{5} boldsymbol{i} ) and ( boldsymbol{z}_{2}=mathbf{3}+mathbf{5} boldsymbol{i} ) and if ( arg left(frac{boldsymbol{z}-boldsymbol{z}_{1}}{boldsymbol{z}-boldsymbol{z}_{2}}right)=frac{boldsymbol{pi}}{boldsymbol{4}} ) then ( mid boldsymbol{z}-boldsymbol{6}- ) ( mathbf{8 i} mid=mathbf{3} sqrt{mathbf{2}} ) |
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| 189 | In the Argand’s plane, the locus of ( z(neq ) 1) such that ( arg left{frac{3}{2}left(frac{2 z^{2}-5 z+3}{3 z^{2}-z-2}right)right}=frac{2 pi}{3} i s ) A. a hyperbola with the directrices at ( z=-3 / 2 ) and ( z= ) ( -2 / 3 ) B. an ellipse with the directrices at ( z=3 / 2 ) and ( z=2 / 3 ) c. a segment of a circle subtending angle ( frac{2 pi}{3} ) on arc between points ( z=-3 / 2 ) and ( z=2 / 3 ) lying below real axis. D. a segment of a circle subtending angle ( frac{2 pi}{3} ) on arc between points ( z=3 / 2 ) and ( z=-2 / 3 ) lying above real axis. |
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| 190 | For any two complex numbers ( z_{1}, z_{2} ) and any two real numbers a, b show that ( left|boldsymbol{a} boldsymbol{z}_{1}-boldsymbol{b} boldsymbol{z}_{2}right|^{2}+left|boldsymbol{b} boldsymbol{z}_{1}+boldsymbol{a} boldsymbol{z}_{2}right|^{2}= ) ( left(a^{2}+b^{2}right)left(left|z_{1}right|^{2}+left|z_{2}right|^{2}right) ) |
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| 191 | Represent the complex number ( 2+3 i ) in argand plane |
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| 192 | The principal argument of ( frac{i-3}{i-1} ) is A ( cdot tan ^{-1} frac{1}{2} ) в. ( tan ^{-1} frac{3}{2} ) c. ( tan ^{-1} frac{5}{2} ) D. ( tan ^{-1} frac{7}{2} ) |
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| 193 | 22. If z is a complex number of unit modulus and argument e, then arg (1 ) equals: JJEE M 2013] Itz 1+2 (a) – (b) – © e (d) – 0 |
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| 194 | 9. (2004) If |22 – 11/22 +1, then z lies on (a) an ellipse (b) the imaginary axis (c) a circle (d) the real axis |
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| 195 | 2. Let z, and zz be complex numbers such that 21 7 22 and [21] = |22). Ifz, has positive real part and z, has negative may be (1986 – 2 Marks) imaginary part, then 21-22 (a) zero (c) real and negative (e) none of these. (b) real and positive (d) purely imaginary |
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| 196 | If ( z_{1}=2-i, z_{2}=-2+i ),find [ operatorname{Im}left(frac{1}{z_{1} overline{z_{2}}}right) ] |
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| 197 | Find the value of ( 2 i^{2}+6 i^{3}+3 i^{16}- ) ( mathbf{6} i^{19}+4 i^{25} ) |
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| 198 | 9. Let Z -10 + 61 and 2-4 +61. If Z is any complex number such that the argument of 2 ar (2-2). ( 222) 4 |
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| 199 | If ( i z^{3}+z^{2}-z+i=0 ) then the value of ( mathbf{7}|boldsymbol{z}| ) is ( A cdot 7 ) B. 14 ( c cdot 21 ) D. 28 |
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| 200 | If ( z=x+i y ) and ( x^{2}+y^{2}=16, ) then the range of ( |boldsymbol{x}|-| boldsymbol{y}| ) is A . [0,4] B. [0,2] c. [2, ( 4] ) D. none of these |
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| 201 | If ( boldsymbol{x}=mathbf{1}+mathbf{2} boldsymbol{i} ) and ( boldsymbol{A}=boldsymbol{x}^{mathbf{3}}+mathbf{7} boldsymbol{x}^{mathbf{2}}-boldsymbol{x}+ ) ( 26, ) then one of the value of ( sqrt{A} ) equals A ( .4-3 i ) B. ( 3-4 i ) ( c .-3+4 i ) D. ( 3+4 i ) |
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| 202 | Find the number of integral solution of ( (1-i)^{x}=2^{x} ) |
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| 203 | The number of complex numbers ( z ) satisfies ( boldsymbol{R e}left(boldsymbol{z}^{2}right)=mathbf{0},|boldsymbol{z}|=sqrt{mathbf{3}} ) |
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| 204 | If ( z_{1}=3+4 i, z_{2}=2-i ) find ( z_{2}-z_{1} ) A . -1-5 B. 2-5 ( c cdot 1+5 i ) D. 1-5 |
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| 205 | ( frac{(3-2 i)(2+3 i)}{(1+2 i)(2-i)}= ) A. ( -frac{63}{25}+frac{16}{25} i ) в. ( -frac{63}{25}-frac{16}{25} i ) c. ( frac{63}{25}+frac{16}{25} i . ) D. ( frac{63}{25}-frac{16}{25} i ) |
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| 206 | If ( a^{2}+b^{2}=1, ) then ( frac{1+b+i a}{1+b-i a}=? ) ( A ) B. 2 c. ( b+i a ) ( mathbf{D} cdot a+i b ) |
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| 207 | If ( left|z_{1}-1right| leq 1,left|z_{2}-2right| leq 2,left|z_{3}-3right| leq 3 ) then find the greatest value of ( left|z_{1}+z_{2}+z_{3}right| ) A. the greatest value is 6 B. the greatest value is 7 . c. the greatest value is ( 9 . ) D. the greatest value is 12 |
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| 208 | 25. Ifx2+(a-b)x+ (1 -a-b) =0 where a, b e Rthen find the values of a for which equation has unequal real roots for all values of b. (2003 – 4 Marks) |
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| 209 | 25. A value of for which 2+ Zi sine one is purely imaginary, is: JEEM 2016 Sin Blo |
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| 210 | If ( a=e^{i alpha}, b=e^{i beta}, c=e^{i gamma} ) and ( cos alpha+ ) ( cos beta+cos gamma=0=sin alpha+sin beta+ ) ( sin gamma, ) then prove the following ( a+b+c=0 ) |
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| 211 | If the roots of the equation x2 – bx + c = 0 be two consecutive integers, then b2 – 4c equals gers then 12 – Ac equals 12005 (a) -2 (6) 3 (C 2 (d) 1 if(n=6 T |
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| 212 | If ( A r gleft(frac{z+1}{z-1}right)=frac{pi}{6}, ) then find the locus of ( mathbf{z} ) |
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| 213 | The sequence ( boldsymbol{S}=boldsymbol{i}+boldsymbol{2} boldsymbol{i}^{2}+boldsymbol{3} boldsymbol{i}^{3}+ldots ldots ) upto 100 times simplifies to where ( i= ) ( sqrt{-1} ) A. ( 50(1-i) ) B . 25 c. ( 25(1+i) ) D. ( 100(1-i) ) |
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| 214 | Find the locus of complex number ( boldsymbol{z}=boldsymbol{x}+boldsymbol{i} boldsymbol{y} ) if ( |boldsymbol{z}+boldsymbol{4} boldsymbol{i}|+|boldsymbol{z}-boldsymbol{4} boldsymbol{i}|=mathbf{1 0} ) | 11 |
| 215 | If ( sqrt{5-12 i}+sqrt{-5-12 i}=z, ) then principal value of arg z can be This question has multiple correct options A ( cdot-frac{pi}{4} ) в. ( frac{pi}{4} ) c. ( frac{3 pi}{4} ) D. ( -frac{3 pi}{4} ) |
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| 216 | If ( cos left(log i^{4 i}right)=a+i b, ) then B . ( a=-1, b=1 ) c. ( a=1, b=0 ) D. ( a=1, b=2 ) |
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| 217 | The polynomial ( f(x)=x^{4}+a x^{3}+ ) ( b x^{2}+c x+d ) has real coefficients and ( (2 i)=f(2+i)=0 . ) The value of ( (a+ ) ( b+c+d ) ) equals to A . B. 4 ( c .9 ) D. 10 |
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| 218 | 15. If the roots of the quadratic equation [2006] x + px +9 = 0 are tan30° and tan 15°, respectively, then the value of 2 + q-p is (a) 2 (b) 3 (c) 0 (d) 1 heuotion |
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| 219 | If ( |z|=1 ) and ( w=frac{z-1}{z+1}(z neq-1) ) then ( operatorname{Re}(w) ) is ( mathbf{A} cdot mathbf{0} ) в. ( frac{-1}{|z+1|^{2}} ) c. ( frac{1}{|z+1|^{2}} ) D. ( frac{sqrt{2}}{|z+1|^{2}} ) |
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| 220 | 8. If a, b, c and u, v, ware complex numbers representing the vertices of two triangles such that c=(1-r) a + rb and w=(1-r)u + rv, where ris a complex number, then the two triangles (1985 – 2 Marks) (a) have the same area (b) are similar (c) are congruent (d) none of these |
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| 221 | f ( x=2+5 i(text { where } 1 i=sqrt{-1}) ) and ( 2left(frac{1}{1 ! 9 !}+frac{1}{3 ! 7 !}right)+frac{1}{5 ! 5 !}=frac{2^{a}}{b !} operatorname{then} x^{3}- ) ( 5 x^{2}+33 x-10= ) ( mathbf{A} cdot a+b ) в. ( b-a ) ( mathbf{c} cdot a-b ) D. ( -a-b ) E ( .(a-b)(a+b) ) |
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| 222 | If ( z=x+i y ) and ( w=frac{(1-i z)}{(z-i)}, ) then ( |boldsymbol{w}|=1 ) implies that, in the complex plane A. ( z ) lies on the imaginary axis B. ( z ) lies on the real axis c. ( z ) lies on the unit circle D. None of these |
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| 223 | If ( z=sqrt{frac{1-i}{1+i}}, ) then arg ( z= ) A. ( frac{pi}{4}, frac{pi}{2} ) B. ( -frac{pi}{4}, frac{pi}{2} ) c. ( frac{3 pi}{4}, pi ) D. ( -frac{pi}{4}, frac{3 pi}{4} ) |
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| 224 | If ( boldsymbol{x}-boldsymbol{i} boldsymbol{y}=sqrt{frac{boldsymbol{a}-boldsymbol{i} boldsymbol{b}}{boldsymbol{c}-boldsymbol{i} boldsymbol{d}}} ) prove that ( left(boldsymbol{x}^{2}+right. ) ( left.boldsymbol{y}^{2}right)^{2}=frac{a^{2}+b^{2}}{c^{2}+d^{2}} ) |
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| 225 | Inequality ( a+i b>c+i d ) can be explained only when : A. ( b=0, c=0 ) В. ( b=0, d=0 ) c. ( a=0, c=0 ) D. ( a=0, d=0 ) |
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| 226 | The argument of ( frac{(1-i sqrt{3})}{(1+i sqrt{3})} ) is A ( cdot 60^{circ} ) B . ( 120^{circ} ) ( c cdot 210^{circ} ) D. ( 240^{circ} ) |
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| 227 | 7. Let z and w be complex numbers such that z +iū= 0 and arg zw=1. Then arg z equals [2004] |
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| 228 | If ( (1-i) x+(1+i) y=1-3 i, ) then ( (x, y)= ) A. (2,-1) в. (-2,-1) c. (-2,1) D. (2,1) |
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| 229 | Represent the following complex number in trigonometric form: -1 |
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| 230 | The value of ( left(sin frac{pi}{8}+right. ) ( left.i cos frac{pi}{8}right)^{8}left(sin frac{pi}{8}-i cos frac{pi}{8}right)^{8} ) is ( A cdot-1 ) B. ( c ) ( D cdot 2 ) |
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| 231 | 11. If 21 and 22 are two non-zero complex numbers such that 121 +22 l= | 211 + 122 1, then arg zi – arg zz is equal to [2005] (a) (b) – () 0 (a) * |
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| 232 | Find the smallest natural number such that, ( left(frac{1+i}{1-i}right)^{n}=1 ) |
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| 233 | If ( frac{l z_{2}}{m z_{1}} ) is purely imaginary number then ( left|frac{boldsymbol{lambda} boldsymbol{z}_{1}+boldsymbol{mu} boldsymbol{z}_{2}}{boldsymbol{lambda} boldsymbol{z}_{1}-boldsymbol{mu} boldsymbol{z}_{2}}right| ) is equal to A ( cdot frac{l}{m} ) B. ( frac{lambda}{mu} ) ( c cdot frac{-lambda}{mu} ) ( D ) |
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| 234 | ( boldsymbol{i}^{2}=-mathbf{1}, ) then ( boldsymbol{i}^{2}+boldsymbol{i}^{4}+boldsymbol{i}^{boldsymbol{6}}+boldsymbol{i}^{boldsymbol{8}}+ldots+ ) ( (2 n) ) terms is: ( mathbf{A} cdot mathbf{0} ) B. – ( c cdot i ) D. ( -i ) |
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| 235 | If ( z=4+i sqrt{7}, ) then value of ( z^{3}-4 z^{2}- ) ( 9 z+91 ) equals ( mathbf{A} cdot mathbf{0} ) B. ( c cdot-1 ) D. 2 |
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| 236 | 32. Let S be the set of all complex numbers z satisfying 12-2+i V5. If the complex number zo is such that he maximum of the set , then the po ay is the maximum of these 15 – s} trenthe principal argument or * *, * S (JEE Adv. 2019) |
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| 237 | If ( left|z_{1}right|=left|z_{2}right|=left|z_{3}right|=ldots ldots . .=left|z_{n}right|=1 ) then ( left|z_{1}+z_{2}+z_{3}+ldots ldots .+z_{n}right|= ) A ( cdotleft|frac{1}{z_{1}}+frac{1}{z_{2}}+frac{1}{z_{3}}+ldots . .+frac{1}{z_{n}}right| ) в. ( left|frac{1}{z_{1}}-frac{1}{z_{2}}-frac{1}{z_{3}}-ldots . .+frac{1}{z_{n}}right| ) с ( cdotleft|frac{1}{z_{1}^{2}}+frac{1}{z_{2}^{2}}+frac{1}{z_{3}^{2}}+ldots . .+frac{1}{z_{n}^{2}}right| ) D ( cdotleft|frac{1}{z_{1}^{2}}-frac{1}{z_{2}^{2}}-frac{1}{z_{3}^{2}}-ldots . .+frac{1}{z_{n}^{2}}right| ) |
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| 238 | Complex number ( z ) satisfy the equation ( |z-(4 / z)|=2 . ) Then the value of ( arg left(z_{1} / z_{2}right), ) where ( z_{1} ) and ( z_{2} ) are complex numbers with the greatest and the least moduli, can be A . ( 2 pi ) в. ( pi ) ( c cdot frac{pi}{2} ) D. none of these |
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| 239 | If ( z=frac{1+i}{sqrt{2}}, ) then the value of ( z^{1929} ) is A. ( 1+i ) B. – c. ( frac{1+i}{2} ) D. ( frac{1+i}{sqrt{2}} ) |
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| 240 | The real part of ( (1-cos theta+2 i sin theta)^{-1} ) is? A ( cdot frac{1}{3+5 cos theta} ) в. ( frac{1}{5-3 cos theta} ) c. ( frac{1}{3-5 cos theta} ) D. ( frac{1}{5+3 cos theta} ) |
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| 241 | For positive integers ( n_{1}, n_{2} ) the value of the expression ( (1+i)^{n_{1}}+left(1+i^{3}right)^{n_{1}}+ ) ( left(1+i^{5}right)^{n_{2}}+left(1+i^{7}right)^{n_{2}}, ) where ( i=sqrt{-1} ) is a real number if A ( cdot n_{1}=n_{2}+1 ) В . ( n_{1}=n_{2}-1 ) c. ( n_{1}=n_{2} ) D ( . n_{1}>0, n_{2}>0 ) |
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| 242 | If ( boldsymbol{alpha}+boldsymbol{i} boldsymbol{beta}=tan ^{-1} boldsymbol{z}, boldsymbol{z}=boldsymbol{x}+boldsymbol{i} boldsymbol{y} ) and ( boldsymbol{alpha} ) is constant then the locus of ( z ) is A ( cdot x^{2}+y^{2}+2 x cot 2 alpha=1 ) B cdot ( cot 2 alphaleft(x^{2}+y^{2}right)=1+x ) c. ( x^{2}+y^{2}+2 y tan 2 alpha=1 ) D. ( x^{2}+y^{2}+2 x=1 ) |
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| 243 | Express the following in the form of a = ib, ( a, b in R i=sqrt{-1} . ) State the values of ( a ) and b. ( (1+i)(1-i)^{-1} ) |
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| 244 | If ( z ) is unimodular complex number then ( mathbf{z}=left(frac{mathbf{1}+mathbf{i} a}{mathbf{1}-mathbf{i} mathbf{a}}right)^{mathbf{4}} ) has A. 2 real 2 imaginary roots B. 4 real roots c. 4 imaginary roots D. 3 real and imaginary roots |
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| 245 | Find the locus of a complex number, ( z=x+i y, ) satisfying the relation ( left|frac{z-3 i}{z+3 i}right| leq sqrt{2} ) Illustrate the locus of ( z ) in the Argand plane. |
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| 246 | For any two complex numbers ( z_{1} ) and ( z_{2} ) with ( left|z_{1}right| neqleft|z_{2}right|,left|sqrt{2} z_{1}+i sqrt{3} overline{z_{2}}right|^{2}+ ) ( left|sqrt{3} overline{z_{1}}+i sqrt{2} z_{2}right|^{2} ) is A ( cdot ) less than ( 5left(left|z_{1}right|^{2}+left|z_{2}right|^{2}right) ) B. greater than ( 10 mid z_{1} z_{2} ) C . equal to ( 2left|z_{1}right|^{2}+3left|z_{2}right|^{2} ) D. zero |
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| 247 | The value of ( frac{1}{i}+frac{1}{i^{2}}+frac{1}{i^{3}}+ldots+frac{1}{i^{102}} ) is equal to A ( .-1-i ) в. ( -1+i ) ( mathrm{c} cdot 1-i ) D. ( 1+i ) |
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| 248 | If ( operatorname{Re}(a), operatorname{Re}(b)>0, ) and ( x=|a-b| ) ( |bar{a}+b|, ) then ( mathbf{A} cdot x0 ) c. ( x geq 1 ) D. |
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| 249 | If ( z_{1} ) and ( z_{2} ) be complex numbers such that ( z_{1}+i(overline{z_{2}})=0 ) and ( arg left(overline{z_{1}} z_{2}right)=frac{pi}{3} ) Then, ( arg (overline{z_{1}}) ) is equal to A ( cdot frac{pi}{3} ) B. ( pi ) c. D. ( frac{5 pi}{12} ) E ( cdot frac{5 pi}{6} ) |
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| 250 | Find real values of ( theta ) for which ( left(frac{4+3 i sin theta}{1-2 i sin theta}right) ) is purely real |
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| 251 | Write the additive inverse of the complex number ( 4-3 i ) |
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| 252 | ( |f| z_{1}|=2,| z_{2}|=3,| z_{3} mid=4 ) and ( mid z_{1}+ ) ( z_{2}+z_{3} mid=5, ) then ( mid 4 z_{2} z_{3}+9 z_{3} z_{1}+ ) ( mathbf{1 6 z}_{1 mathbf{z}_{2}} mid= ) A . 20 B. 24 ( c cdot 48 ) D. 120 |
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| 253 | ff ( z_{1}=1+i=sqrt{3}+i, ) then the principle ( arg left(frac{z_{1}}{z_{2}}right) ) | 11 |
| 254 | Number of roots of the equation ( z^{10}- ) ( z^{5}-992=0 ) where real parts are negative is ( mathbf{A} cdot mathbf{3} ) B. 4 c. 5 D. 6 |
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| 255 | If ( frac{a+3 l}{2+i b}=1-1, ) show that ( (5 a- ) ( mathbf{7 b})=mathbf{0} ) | 11 |
| 256 | If ( a>0 ) and ( z=frac{(1+i)^{2}}{a-i}, ) has magnitude ( sqrt{frac{2}{5}}, ) then ( bar{z} ) is equal to: A. ( -frac{3}{5}-frac{1}{5} i ) B. ( -frac{1}{5}+frac{3}{5} i ) c. ( -frac{1}{5}-frac{3}{5} i ) D. ( frac{1}{5}-frac{3}{5} i ) |
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| 257 | The simplified value of ( frac{1+i}{1-i} ) is ( mathbf{A} cdot mathbf{1} ) в. ( c cdot-i ) D. ( 2 i ) |
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| 258 | The principal argument of ( 1+sqrt{2}+i ) is A ( cdot frac{pi}{3} ) в. c. D. |
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| 259 | The value of ( (-i)^{-i} ) equals? A ( cdot e^{4 n-1 frac{pi}{2}}, n epsilon I ) B . ( e^{i 4 n-1 frac{pi}{2}}, n epsilon I ) C . ( e^{4 n+1 pi / 2}, n epsilon I ) D cdot ( e^{-i 4 n+1 frac{pi}{2}}, n epsilon I ) |
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| 260 | If ( z=frac{sqrt{3}+i}{sqrt{3}-i} ) then the fundamental amplitude of z is A. ( -frac{pi}{3} ) в. ( frac{pi}{3} ) c. ( frac{pi}{6} ) D. None of these |
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| 261 | 16. All the values of m for which both roots of the equation x2 – 2mx + m² -1=0 are greater than -2 but less then 4, lie in the interval [2006] (a) -2<m 3 © -1<m<3 (d) i<m<4 |
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| 262 | ( sqrt{-3} sqrt{-75}= ) A . 15 в. 15 c. -15 D. – -15 |
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| 263 | A complex number ( z ) is said to be unimodular if ( |z|=1 . ) Suppose ( z_{1} ) and ( z_{2} ) are complex numbers such that ( frac{z_{1}-2 z_{2}}{2-z_{1} bar{z}_{2}} ) is unimodular and ( z_{2} ) is not unimodular. Then the point ( z_{1} ) lies on a A. straight line parallel to x-axis B. straight line parallel to y-axis c. circle of radius 2 D. circle of radius ( sqrt{2} ) |
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| 264 | ( left(sqrt[3]{3}+left(begin{array}{c}5 \ 36 \ iend{array}right)^{3} ) is an integer where right. ( boldsymbol{i}=sqrt{-1} . ) The value of the integer is equal to. A .24 B . -24 c. -22 D. -21 |
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| 265 | ( left.left(frac{1+i}{1-i}right)^{2}+frac{1-i}{1+i}right)^{2} ) ) is equal to | 11 |
| 266 | ( frac{1+i}{1-i}-frac{1-i}{1+i} ) ( mathbf{A} cdot-2 i ) B. ( c cdot 2 i ) ( D ) |
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| 267 | The amplitude of ( frac{1+sqrt{3 i}}{sqrt{3}+1} ) is ( ^{A} cdot frac{pi}{3} ) в. ( -frac{pi}{3} ) ( c cdot frac{pi}{6} ) D. ( -frac{pi}{6} ) |
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| 268 | ( sqrt{i}+sqrt{-i}=? ) A ( cdot sqrt{2} ) B. ( -sqrt{2} ) ( c cdot pm frac{1}{sqrt{2}} ) D. ( pm sqrt{2} ) |
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| 269 | Let ( z ) be a complex number such that the imaginary part of z is nonzero and a ( =z^{2}+z+1 ) is real. Then a cannot take the value A . -1 B. ( frac{1}{3} ) ( c cdot frac{1}{2} ) D. ( frac{3}{4} ) |
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| 270 | The value of the sum ( sum_{n=1}^{10}left(i^{n}+i^{n+1}right) ) where ( i=sqrt{-1}, ) equals ( mathbf{A} cdot i ) B. – c. ( -i ) D. |
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| 271 | Given: ( z_{1}+z_{2}+z_{3}=A ; z_{1}+z_{2} w+ ) ( z_{3} w^{2}=B ; z_{1}+z_{2} w^{2}+z_{3} w=C ) where ( boldsymbol{w} ) is cube rott of unity Prove: ( |boldsymbol{A}|^{2}+|boldsymbol{B}|^{2}+|boldsymbol{C}|^{2}= ) ( left(left|z_{1}right|^{2}+left|z_{2}right|^{2}+left|z_{3}right|^{2}right) ) |
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| 272 | Solve: ( left(frac{2 i}{1+i}right)^{2} ) ( mathbf{A} cdot-i ) B. ( i ) ( mathbf{c} cdot 2 i ) D. ( 1-i ) |
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| 273 | The value of ( i^{2}+i^{4}+i^{6}+ldots i^{2(2 n+1)}=? ) A . -1 B. ( c cdot-i ) ( D ) |
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| 274 | What is the smallest positive integer for which ( (1+i)^{2 n}=(1-i)^{2 n} ? ) | 11 |
| 275 | Let ( z=x+ ) iy and ( v=frac{1-i z}{z-i}, ) show that if ( |boldsymbol{v}|=1, ) then ( boldsymbol{z} ) is purely real |
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| 276 | If ( z_{1} ) and ( z_{2} ) are two complex numbers such that ( left|z_{1}+z_{2}right|^{2}=left|z_{1}right|^{2}+left|z_{2}right|^{2}, ) then This question has multiple correct options A ( cdot z_{1} overline{z_{2}} ) is purely imaginary B . ( z_{1} / z_{2} ) is purely imaginary c. ( z_{1} overline{z_{2}}+overline{z_{1}} z_{2}=0 ) D. ( O, z_{1}, z_{2} ) are vertices of a right triangle |
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| 277 | For any two complex numbers ( z_{1} ) and ( z_{2} ) prove that ( operatorname{Re}left(z_{1} z_{2}right)=operatorname{Re} z_{1} operatorname{Re} z_{2} ) ( operatorname{Im} z_{1} operatorname{Im} z_{2} ) |
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| 278 | Simplify the multiplication of complex numbers: ( boldsymbol{a} times(boldsymbol{c}, boldsymbol{d}) ) ( mathbf{A} cdot(a c, a d) ) B. ( (-a d, a c) ) c. ( (a d, a c) ) D. None of these |
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| 279 | ( tan ileft[log _{e}left(frac{a-i b}{a+i b}right)right] ) is equal to A ( cdot frac{a^{2}-b^{2}}{2 a b} ) в. ( frac{2 a b}{a^{2}+b^{2}} ) ( c cdot a b ) D. ( frac{2 a b}{a^{2}-b^{2}} ) |
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| 280 | The complex number ( x+i y ) whose modulus is unity, ( y neq 0, ) can be represented as ( boldsymbol{x}+boldsymbol{i} boldsymbol{y}=frac{boldsymbol{a}+boldsymbol{i}}{boldsymbol{a}-boldsymbol{i}}, ) where ( boldsymbol{a} ) is real number. A . True B. False |
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| 281 | 4. If z anda are two non-zero complex numbers such that |z0|=1 and Arg(z) – Arg(@) = then zo is equal to [2003] (a) —; (b) 1 C) -1 (d) i |
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| 282 | 33. Let p. e R. If2 – V x+px+q=0, then: (a)p-4q+12 – 0 (c) q?+4p+14-0 is a root of the quadratic equation, JEEM 2019-9 April (M) (b) q-4p – 16-0 (d) p. -4q-12-0 |
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| 283 | ( (5 i)left(-frac{3}{5} iright) ) | 11 |
| 284 | Find the real number ( boldsymbol{x} ) if ( (boldsymbol{x}-mathbf{2} boldsymbol{i})(mathbf{1}+ ) ( i) ) is purely imaginary. ( A cdot 2 ) B. -2 ( c cdot 4 ) D. -4 |
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| 285 | What is ( i^{1000}+i^{1001}+i^{1002}+i^{1003} ) equal to (where ( boldsymbol{i}=sqrt{-1} ) )? ( mathbf{A} cdot mathbf{0} ) B. ( c cdot-i ) ( D ) |
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| 286 | Evaluate (i) ( boldsymbol{i}^{998} ) | 11 |
| 287 | The amplitude of ( 1+cos x-i sin x ) is A ( cdot frac{x}{2} ) B. ( c cdot-frac{x}{2} ) D. ( 2 x ) |
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| 288 | 30. Let z be a complex number such that the imaginary part of z is non-zero and a=z2+z+1 is real. Then a cannot take the value (2012) (a) 1 (b) (2) A |
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| 289 | Two complex numbers are represented by ordered pairs ( z_{1}:(a, b) & z_{2}:(c, d) ) when these two complex numbers are equal? A. If and only if ( a=-c, b=-d ) B. If and only if ( a=d, b=c ) c. If and only if ( a=c, b=d ) D. None of these |
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| 290 | 23. If a, b are the roots of ax2 + bx + c = 0, (a +0) and a +8, B+ are the roots of Ax2 + Bx +C=0, (A+0) for B – 4ac B² -40C some constant 8, then prove that 2 . 2 (2002 farks) |
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| 291 | Find the greatest value assumed by the function ( boldsymbol{w}=left|boldsymbol{z}-frac{boldsymbol{4}}{boldsymbol{z}}right|=mathbf{2} ) where ( mathbf{z} ) is a complex variable. |
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| 292 | Find the multiplicative inverse of the complex numbers given the following: ( 4-3 i ) |
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| 293 | Let ( z ) be a complex number such that the principal value of argument, arg ( z<0 . ) Then ( arg (-z)-arg (z) ) is A ( cdot frac{pi}{2} ) B. ( pm pi ) ( c . pi ) D. – |
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| 294 | Find the argument of ( -1-i sqrt{3} ) A. ( theta=-2 pi / 3 ) В. ( theta=2 pi / 3 ) c. ( theta=-4 pi / 3 ) D. ( theta=4 pi / 3 ) |
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| 295 | The figure formed by four points ( 1+ ) ( 0 i ;-1+0 i, 3+4 i ) and ( frac{25}{-3-4 i} ) on the argand plane is A. parallelogram but not a rectangle B. a trapesium which is not equilateral c. cyclic quadrilateral D. none of these |
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| 296 | Simplify the following: ( frac{3-i}{2+i}+frac{3+i}{2-i} ) |
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| 297 | ( left(frac{1}{1-2 i}+frac{3}{1+i}right)left(frac{3+4 i}{2-4 i}right)= ) A. ( frac{1}{2}+frac{9}{2} i ) B ( cdot frac{1}{2}-frac{9}{2} i ) c. ( frac{1}{4}-frac{9}{4} i ) D. ( frac{1}{4}+frac{9}{4} i ) |
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| 298 | If ( (x+i y)^{3}=u+i v, ) then prove that ( frac{u}{x}+frac{v}{y}=4left(x^{2}-y^{2}right) ) | 11 |
| 299 | If ( z ) is a complex number such that ( |z|=1, ) then ( left|frac{1}{z}right| ) is ( mathbf{A} cdot mathbf{0} ) в. -1 ( c cdot sqrt{2} ) D. |
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| 300 | The value of the sum ( sum_{n=1}^{13}left(i^{n}+i^{n+1}right) ) where ( i=sqrt{-1}, ) is? ( A ) B . ( i-1 ) ( c cdot-i ) D. |
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| 301 | If ( z_{1}, z_{2}, dots, z_{n} ) lie on ( |z|=r ) and ( operatorname{Re}left(sum_{j=1}^{n} sum_{k=1}^{n} frac{z_{j}}{z_{k}}right)=0, ) then This question has multiple correct options ( ^{mathbf{A}} cdot sum_{j=1}^{n} z_{j}=0 ) ( ^{mathbf{B}} cdotleft|sum_{j=1}^{n} z_{j}right|=0 ) ( ^{mathrm{c}} cdot sum_{j=1}^{n} frac{1}{z_{j}}=0 ) D. None of these |
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| 302 | 15. If I z + 4 ls 3, then the maximum value of 12+1| is [20071 (a) 6 (b) 0 C) 4 (d) 10 |
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| 303 | If ( left|mathbf{z}^{2}-mathbf{1}right|=|mathbf{z}|^{2}+mathbf{1}, ) then ( mathbf{z} ) lies on A. the real axis B. the imaginary axis c. a circle D. an ellipse |
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| 304 | Let ( z_{1} ) and ( z_{2} ) be roots of the equation ( z^{2}+p z+q=0, ) where the coefficients ( p ) and ( q ) may be complex numbers. Let ( A ) and ( B ) represents ( z_{1} ) and ( z_{2} ) in the complex plane. If ( angle A O B=alpha neq 0 ) and ( boldsymbol{O} boldsymbol{A}=boldsymbol{O} boldsymbol{B}, ) where ( boldsymbol{O} ) is the origin, then ( p^{2}=k cos ^{2} frac{alpha}{2}, ) where ( k= ) ( mathbf{A} cdot boldsymbol{q} ) B. ( 2 q ) c. ( 4 q ) D. None of these |
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| 305 | Conjugate surd of ( a+sqrt{6} ) is A ( .6-sqrt{a} ) B. ( 6+sqrt{a} ) c. ( sqrt{6}-a ) D. ( a-sqrt{6} ) |
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| 306 | The inequality ( |z-4|<|z-2| ) represents the region given by: A. ( operatorname{Re}(z) geq 0 ) в. ( operatorname{Re}(z)0 ) D. None of these |
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| 307 | If ( z ) is a unimodular complex number, then its multiplicative inverse is, ( mathbf{A} cdot bar{z} bar{z} ) B. ( c cdot-z ) D. – ( bar{z} ) |
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| 308 | The principal argument of ( z=-3+3 i ) is: A ( cdot frac{pi}{4} ) B. ( -frac{pi}{4} ) c. ( frac{3 pi}{4} ) D. ( -frac{3 pi}{4} ) |
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| 309 | If ( z=x+i y ) is a complex number such that ( |z|=operatorname{Re}(i z)+1, ) then the locus of ( z ) is A ( cdot x^{2}+y^{2}=1 ) в. ( x^{2}=2 y-1 ) C ( cdot y^{2}=2 x-1 ) D・ ( y^{2}=1-2 x ) E . ( x^{2}=1-2 y ) |
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| 310 | The real and imaginary parts of ( log (1+i)= ) ( ^{A} cdotleft(frac{1}{2}, frac{pi}{4}right) ) B ( cdotleft(log 2, frac{pi}{4}right) ) ( ^{mathbf{c}} cdotleft(log sqrt{2}, frac{pi}{4}right) ) ( ^{mathrm{D}} cdotleft(log frac{1}{2}, frac{pi}{4}right) ) |
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| 311 | The solution of the equation ( z(z-2 i)=2(2+i) ) are A. ( 3+i, 3-i ) B. ( 1+3 i, 1-3 i ) c. ( 1+3 i, 1-i ) D. ( 1-3 i, 1+i ) |
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| 312 | ( left(frac{1+i}{1-i}right)^{4}+left(frac{1-i}{1+i}right)^{4}= ) A. B. ( c cdot 2 ) ( D ) |
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| 313 | Find the multiplicative inverse of each of the complex numbers given in the Exercises 11 to 13 ( 4-3 i ) |
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| 314 | 4. If _ V3 , then (1982 – 2 Marks) 2 (a) Re(z)=0 (c) Re(z) >0, Im (z)>0 (b) Im(z)=0 (d) Re(z)>0, Im (z)< |
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| 315 | ( mathbf{f}|z+mathbf{1}|=z+mathbf{2}(mathbf{1}+i), ) then find ( z ) | 11 |
| 316 | If ( |z+5| leq 2 ) then the maximum value of ( |z+3| ) is ( where ( z ) is a complex number) A. zero B . 2 ( c cdot 4 ) D. 6 |
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| 317 | if ( z_{1}=3+4 i ) and ( operatorname{Im}left(z_{1} z_{2}right)=0 ) Find ( z_{2} ) A ( cdot z_{2}=3-4 i ) в. ( z_{2}=3+4 i ) c. ( z_{2}=3 pm 4 i ) D. None of these |
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| 318 | Find the greatest and the least values of the moduli of complex numbers ( z ) satisfying the equation ( |z-4 / z|=2 ) |
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| 319 | If ( Z_{1}=-1 ) and ( Z_{2}=i, ) then find ( boldsymbol{A} boldsymbol{r} boldsymbol{g}left(frac{boldsymbol{Z}_{1}}{boldsymbol{Z}_{2}}right) ) | 11 |
| 320 | If ( sum_{k=0}^{100} i^{k}=x+i y, ) then the values of ( x ) and y are A. ( x=-1, y=0 ) B. ( x=1, y=1 ) c. ( x=1, y=0 ) D. ( x=0, y=1 ) |
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| 321 | Find the amplitude of ( 1+i sqrt{3} ) | 11 |
| 322 | Prove that polynomial ( boldsymbol{x}^{boldsymbol{4} boldsymbol{p}}+boldsymbol{x}^{boldsymbol{4} boldsymbol{q}+1}+ ) ( x^{4 r+2}+x^{4 s+3} ) can be divided by ( x^{3}+ ) ( boldsymbol{x}^{2}+boldsymbol{x}+1, ) where ( boldsymbol{p}, boldsymbol{q}, boldsymbol{r}, boldsymbol{s} in boldsymbol{N} ) |
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| 323 | ( fleft(frac{1+i}{1-i}right)^{3}-left(frac{1-i}{1+i}right)^{3}=A+i B ) ( operatorname{then} A, B= ) A . 0,2 B. 0,-2 ( c .2,0 ) D. -2,0 |
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| 324 | If the biquadratic ( x^{4}+a x^{3}+b x^{2}+ ) ( c x+d=0(a, b, c, d epsilon R) ) has 4 non real roots, two with sum ( 3+4 i ) and the other two with product ( 13+i . ) Find the sum of the digits of value of ‘ ( b ) |
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| 325 | If ( alpha ) and ( beta ) are the roots of ( 4 x^{2}-16 x+ ) ( c=0, c>0 ) such that ( 1<alpha<2<beta<3 ) then the no.of integer values of ( c ) is A . 17 B . 14 c. 18 D. 15 |
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| 326 | Find the value of ( (1+i)^{3}+(1-i)^{6} ) | 11 |
| 327 | The real and imaginary parts of ( frac{a+i b}{a-i b} ) are: A ( cdot a^{2}-b^{2}, 2 a b ) B. ( frac{a^{2}+b^{2}}{a^{2}-b^{2}}, frac{2 a b}{a^{2}-b^{2}} ) c. ( frac{a^{2}-b^{2}}{a^{2}+b^{2}}, frac{2 a b}{a^{2}+b^{2}} ) D. ( frac{a^{2}+b^{2}}{a^{2}-b^{2}}, frac{2 a b}{a^{2}+b^{2}} ) |
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| 328 | Simplify: ( (-sqrt{3}+sqrt{-2})(2 sqrt{3}-i) ) | 11 |
| 329 | If ( left|frac{z_{1}}{z_{2}}right|=1 ) and ( arg left(z_{1} z_{2}right)=0, ) then A ( . z_{1}=z_{2} ) В ( cdotleft|z_{2}right|^{2}=z_{1} z_{2} ) ( mathbf{c} cdot z_{1} z_{2}=1 ) D. ( z_{1}=-z_{2} ) |
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| 330 | If ( a>0 ) and ( z|z|+a z+2 a=0, ) then ( z ) must be A. purely imaginary B. a positive real number c. a negative real number D. having positive imaginary part |
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| 331 | Evaluate the following ( left(i^{77}+i^{70}+i^{87}+i^{414}right)^{3} ) |
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| 332 | sum 6. The value of the sum 2 (+*+ **), where i = -1, equals (1998 – 2 Marks) (a) i 6 i-1 © -i (d) 0 |
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| 333 | 1. zand ware two nonzero complex numbers such that|z1=1W and Arg z + Arg w=it then z equals [2002] (a) 5 (b) – 5 (©) o (d) – 0 |
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| 334 | The area of the triangle formed by the three complex numbers ( 1+i, i-1,2 i ) in the Argand diagram is: A ( cdot frac{1}{2} ) B. ( c cdot sqrt{2} ) D. 2 |
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| 335 | The points of intersection of the curves ( |z-3|=2 ) and ( |z|=2 ) in an argand plane are ( mathbf{A} cdot frac{1}{2}(3 pm i sqrt{7}) ) B ( cdot frac{1}{2}(3 pm i sqrt{3}) ) c. ( frac{3}{2} pm i sqrt{frac{7}{2}} ) D ( cdot frac{7}{2} pm i sqrt{frac{3}{2}} ) ) |
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| 336 | ( frac{mathbf{3}}{mathbf{1}+boldsymbol{i}}-frac{mathbf{2}}{mathbf{2}-boldsymbol{i}}+frac{mathbf{2}}{mathbf{1}-boldsymbol{i}} ) A ( cdot frac{1}{10}(17+9 i) ) в ( cdot frac{1}{5}(17-9 i) ) c ( cdot frac{1}{10}(17-9 i) ) D ( cdot frac{1}{5}(17+9 i) ) |
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| 337 | ( fleft(frac{1+i}{1-i}right)^{m}=1, ) then find the least positive integral value of ( mathrm{m} ) | 11 |
| 338 | Simplify the following: ( left(frac{1+i}{1-i}right)^{4 n+1} ) ( mathbf{A} cdot mathbf{1} ) B. c. D. None of these |
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| 339 | ( (1+i)^{-1} ) A ( cdot frac{1+i}{2} ) B. ( frac{1-i}{2} ) ( mathbf{C} cdot 1+i ) D. ( 1-i ) |
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| 340 | If ( alpha ) and ( beta ) are different complex number with ( |beta|=1 ) then find ( left|frac{beta-alpha}{1-bar{alpha} beta}right| ) | 11 |
| 341 | If ( n epsilon N>1 ), then the sum of real part of roots of ( z^{n}=(z+1)^{n} ) is equal to A ( cdot frac{n}{2} ) в. ( frac{(n-1)}{2} ) ( c cdot-frac{n}{2} ) D. ( frac{(1-n)}{2} ) |
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| 342 | ( operatorname{Let}left|frac{z_{1}-2 z_{2}}{2-z_{1} bar{z}_{2}}right|=1 ) and ( left|z_{2}right| neq 1, ) where ( z_{1} ) and ( z_{2} ) are complex numbers. Find the value of ( left|z_{1}right| ) |
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| 343 | f ( z_{1}=1+2 i, z_{2}=2+3 i, z_{3}=3+4 i ) ( operatorname{then} z_{1}, z_{2} ) and ( z_{3} ) are collinear. A . True B. False |
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| 344 | The principal value of arg z where ( z= ) ( 1+cos frac{6 pi}{5}+i sin frac{6 pi}{5} ) is given by A ( cdot frac{3 pi}{5} ) B. ( -frac{pi}{5} ) ( c cdot-frac{3 pi}{5} ) D. |
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| 345 | Two complex numbers are represented by ordered pairs ( z_{1}: a+i b & z_{2}: c+i d ) which of the following is correct representation for ( z_{1}-z_{2}=? ) A ( cdot(a-c)-i(b+d) ) B . ( (a-c)+i(b-d) ) c. ( (a+c)-i(b+d) ) D. None of these |
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| 346 | The value of ( left(1+i x+i^{2} x^{2}+i^{3} x^{3}+right. ) ( ldots text { to } infty) times ) ( left(1-i x+i^{2} x^{2}-i^{3} x^{3}+ldots . t o inftyright) ) is A. Imaginary B. a positive real c. a negative real D. equal to zero |
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| 347 | If ( a=frac{(3 i+1)}{10} ) and ( b=(2 i+3) times 10 ), then the value of ( (2 a b) ) is |
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| 348 | 3. Product of real roots of the equation tºx2+x+9=0 [2002] (a) is always positive (b) is always negative (c) does not exist (d) none of these |
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| 349 | 10. Let z and o be two non zero complex numbers such that Izl= 0 and Argz+ Argon, then z equals (1995) (a) (b) – (c) (d) – |
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| 350 | The principal amplitude of ( left(sin 40^{circ}+i cos 40^{circ}right)^{5} ) is A ( .70^{circ} ) B . ( -110^{circ} ) ( c cdot 110 ) D. ( -70^{circ} ) |
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| 351 | Two complex numbers are represented by ordered pairs ( z_{1}:(2,4) & z_{2}:(-4,5) ) which of the following is imaginary part of ( z_{1} times z_{2}=? ) A . -28 B. 6 ( c .-6 ) D. None of these |
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| 352 | ( boldsymbol{i}^{242}= ) ( mathbf{A} cdot i ) B. ( -i ) c. 1 D. – |
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| 353 | Prove that ( left(x^{2}+y^{2}right)^{4}=left(x^{4}-6 x^{2} y^{2}+right. ) ( left.y^{4}right)^{2}+left(4 x^{3} y-4 x y^{3}right)^{2} ) |
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| 354 | Let ( z neq-i ) be any complex number such that ( frac{z-i}{z+i} ) is a purely imaginary number. Then ( z+frac{1}{z} ) is : A. 0 B. Any non-zero real number other than 1. C. Any non-zero real number D. A purely imaginary number |
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| 355 | A complex number is represented by an ordered pair ( (a, b), ) which of the following is true for ( a ) and ( b ? ) A. ( a ) and ( b ) both are imaginary numbers B. ( a ) and ( b ) both are real numbers c. ( a ) is real and ( b ) is an imaginary number. D. None of these |
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| 356 | Represent ( frac{1}{1-cos theta+2 i sin theta} ) in the form ( boldsymbol{A}+boldsymbol{i} boldsymbol{B} ) |
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| 357 | If ( (a+i b)(c+i d)=A+i B ), then show ( operatorname{that}left(a^{2}+b^{2}right)left(c^{2}+d^{2}right)=A^{2}+B^{2} ) |
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| 358 | locate the point representing the complex numbers ( z ) on the Argand diagram for which ( |z+i|=|z-2| ) |
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| 359 | ( mathbf{f}|boldsymbol{z}-mathbf{1}+boldsymbol{i}|+|boldsymbol{z}+boldsymbol{i}|=mathbf{1} ) then find range of principle argument of z. |
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| 360 | Find the value of ( x ) such that ( frac{(x+alpha)^{2}-(x+beta)^{2}}{alpha+beta}=frac{sin 2 theta}{sin ^{2} theta} . ) when ( alpha ) and ( beta ) are the roots of ( t^{2}-2 t+2=0 ) A. ( x=i )cot ( theta-1 ) B. ( x=-(i text { cot } theta+1) ) ( mathbf{c} cdot x=i cot theta ) D. ( x= ) itan ( theta-1 ) |
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| 361 | The value of the sum ( sum_{n=1}^{13}left(i^{n}+i^{n+1}right) ) where ( i=sqrt{-1}, ) equals ( mathbf{A} cdot i ) B . ( i-1 ) ( c cdot-i ) D. |
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| 362 | ( f(a)=frac{-1+i sqrt{3}}{2}, b=frac{-1-i sqrt{3}}{2} ) then show that ( a^{2}=b ) and ( b^{2}=a ) |
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| 363 | In Argand diagram, O, P, Q represents the origin, ( z ) and ( z+i z ) respectively. then ( angle O P Q= ) A. в. ( c cdot frac{pi}{2} ) D. |
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| 364 | Find the value of ( (1+i)^{6}+(1-i)^{6} ) |
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| 365 | Find the principal argument of the complex number ( sin frac{6 pi}{5}+i(1+ ) ( left.cos frac{6 pi}{5}right) ) |
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| 366 | 9. If o (+1) is a cube root of unity and (1+)? = A+Bo then A and B are respectively (1995) (a) 0,1 B 11 © 1,0 (d) -1, í 10 Totod |
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| 367 | ( |f| Z mid=2 ) and ( arg (Z)=frac{pi}{4} ) then write ( Z ) | 11 |
| 368 | For ( z=x+i y ) find the real and imaginary part of ( e^{z} ) |
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| 369 | Simplify: ( (-sqrt{3}+sqrt{-2})(2 sqrt{3}-i)=(a+ ) ib) ( (-sqrt{3}+sqrt{-2}) . ) Find value of a and ( b ) |
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| 370 | The complex number ( frac{1+2 i}{1-i} ) lies in the quadrant : ( A ) B. I c. ॥॥ D. IV |
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| 371 | If ( Z=frac{1-sqrt{3} i}{1+sqrt{3} i} ) then find ( arg (z) ) ( mathbf{A} cdot-frac{2 pi}{3} ) B. ( frac{2 pi}{5} ) ( mathbf{C} cdot frac{pi}{3} ) D. ( frac{2 pi}{3} ) |
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| 372 | f ( left|x^{2}-7right| leq 9 ) then find the values of ( x ) | 11 |
| 373 | Two complex numbers are represented by ordered pairs ( z_{1}:(3,4) & z_{2}:(4,5) ) which of the following is correct representation for ( z_{1} times z_{2}=? ) A ( cdot(-3,31) ) в. (-8,31) c. (-1,21) D. None of these |
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| 374 | Assertion If ( z=i+2 i^{2}+3 i^{3}+ldots ldots ldots ldots+32 i^{32} ) then ( z, bar{z}, z & bar{z} ) forms the vertices of square on argand plane. Reason ( z, bar{z}, z, bar{z} ) are situated at the same distance from the origin on argand plane. 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 |
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| 375 | Find the multiplicative inverse of the complex numbers given the following: ( sqrt{5}+3 i ) |
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| 376 | Dividing ( f(z) ) by ( z-i, ) we obtain the remainder ( i ) and dividing it by ( z+i, ) we get the remainder ( 1+i, ) then remainder upon the division of ( f(z) ) by ( z^{2}+1 ) is A ( cdot frac{1}{2}(z+1)+i ) B ( cdot frac{1}{2}(i z+1)+i ) c. ( frac{1}{2}(i z-1)+i ) D ( cdot frac{1}{2}(z+i)+1 ) |
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| 377 | Write ( (1+2 i) cdot(1+3 i) cdot(2+i)^{-1} ) in the form ( (a+i b) ) and Find the square roots of the given complex number ( (7+24 i) ) |
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| 378 | Express each of the complex number given in the Exercises 1 to 10 in the form ( a+i b ) ( (mathbf{5} i)left(-frac{mathbf{3}}{mathbf{5}} boldsymbol{i}right) ) |
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| 379 | Find the modulus, argument and the principal argument of the complex numbers. ( (tan 1-i)^{2} ) A. Modulus( =sec ^{2} 1, operatorname{Arg}(z)=2 n pi+(2- ) ( pi ) ), Principal ( operatorname{Arg}(z)=(2-pi) ) B. Modulus = ( operatorname{cosec}^{2} 1, ) Arg( (z)=2 n pi-(2- ) ( pi), ) Principal ( operatorname{Arg}(z)=(-2-pi) ) C. Modulus( =sec ^{2} 1, operatorname{Arg}(z)=2 n pi-(2- ) ( pi), ) Principal ( operatorname{Arg}(z)=(-2-pi) ) D. Modulus ( =operatorname{cosec}^{2} 1, ) Arg( (z)=2 n pi+(2- ) ( pi), ) Principal ( operatorname{Arg}(z)=(2-pi) ) |
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| 380 | Find the additive inverse of ( frac{9}{2+i sqrt{5}} ) | 11 |
| 381 | Find the modulus and the argument of the complex number ( z=-1-i sqrt{3} ) | 11 |
| 382 | If ( z_{1}, z_{2} ) are two complex numbers ( left(z_{1} neqright. ) ( z_{2} ) ) satisfying ( left|z_{1}^{2}-z_{2}^{2}right|=mid overline{z_{1}^{2}}+overline{z_{2}^{2}}- ) ( mathbf{2} bar{z}_{1} bar{z}_{2} mid, ) then This question has multiple correct options A ( cdot frac{z_{1}}{z_{2}} ) is purely imaginary B . ( frac{z_{1}}{z_{2}} ) is purely real C ( cdotleft|a r g z_{1}-a r g z_{2}right|=pi ) D ( cdotleft|arg z_{1}-arg z_{2}right|=frac{pi}{2} ) |
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| 383 | Write the conjugate of complex number ( frac{5 i}{7+i} ) | 11 |
| 384 | Which of the following is true This question has multiple correct options ( mathbf{A} cdot(3+sqrt{-5})(3-sqrt{-5})=14 ) B ( cdot(-2+sqrt{-3})(-3+2 sqrt{-3})=-7 sqrt{3} i ) ( mathbf{c} cdot(2+3 i)^{2}=(-5+12 i) ) D. ( (sqrt{5}-7 i)^{2}=-44-14 sqrt{5} i ) |
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| 385 | ( boldsymbol{i}^{n}+boldsymbol{i}^{boldsymbol{n}+1}+boldsymbol{i}^{boldsymbol{n}+mathbf{2}}+boldsymbol{i}^{boldsymbol{n}+mathbf{3}}(boldsymbol{n} in boldsymbol{N}) ) is equal to ( mathbf{A} cdot mathbf{4} ) B. c. D. |
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| 386 | If ( frac{x-3}{3+i}+frac{y-3}{3-i}=i ) where ( x, y in R ) then A. ( x=2 ) & ( y=-8 ) B. ( x=-2 ) & ( y=8 ) c. ( x=-2 & y=-6 ) D. ( x=2 ) & ( y=8 ) |
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| 387 | The sum of two complex numbers ( a+ ) ib and ( c+i d ) is purely imaginary if A ( cdot a+c=0 ) B . ( a+d=0 ) c. ( b+d=0 ) D. ( b+c=0 ) |
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| 388 | Find the modulus and argument of the complex numbers. ( frac{5-i}{2-3 i} ) | 11 |
| 389 | Find the Modulas and argument of ( frac{1+i}{1-i} ) | 11 |
| 390 | 23. If a and B are the roots of the equation x2 – x + 1 = 0, then a2009 + ß2009 = [2010] (a) -1 (b) 1 (c) 2 (d) -2 |
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| 391 | Let z, and z, be two distinct complex numbers and let z=(1-1) 2, +tz, for some real number t with 0<t<1. IfArg (w) denotes the principal argument of a non-zero complex number w, then (2010) (a) 12-211 + 12-22 = 21 – 22 (b) Arg (2-2) = Arg (z-22) 2-7 Z-3 © 22-31 32-3 (d) Arg (2-2)= Arg (22-24) |
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| 392 | Find the value of ( sum_{n=1}^{13} i^{n}+i^{n+1} ) | 11 |
| 393 | Express ( left(frac{1-i}{1+i}right)^{1000} ) in the form of ( a+ ) ib. Find ( a+b ) |
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| 394 | The value of ( 5 sqrt{-8} ) is A. ( 10 i sqrt{4} ) В. ( 20 i sqrt{2} ) c. ( 10 i sqrt{2} ) D. None of these |
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| 395 | Indicate the point of the complex plane ( z ) which satisfy the following equation. ( z^{2}+|bar{z}|=0 ) |
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| 396 | If ( a, b, c ) are distincts ( & w(neq 1) ) is a cube of unity then minimum value of ( boldsymbol{x}=left|boldsymbol{a}+boldsymbol{b} boldsymbol{w}+boldsymbol{c} boldsymbol{w}^{2}right|+left|boldsymbol{a}+boldsymbol{b} boldsymbol{w}^{2}+boldsymbol{c} boldsymbol{w}right| ) A ( cdot 2 sqrt{3} ) B. 3 ( c cdot 4 sqrt{2} ) D. |
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| 397 | If z satisfies ( |z-1|<|z+3| ) then ( omega= ) ( 2 z+3-i ) satisfies This question has multiple correct options A ( .|omega-5-i|<|omega+3+1| ) B. ( |omega-5|1 ) D cdot ( |a r g(omega-1)|<frac{pi}{2} ) |
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| 398 | If ( frac{(1+i)^{2}}{2-i}=x+i y, ) find ( x+y ) A ( cdot frac{-2}{5} ) B. ( frac{2}{7} ) ( c cdot frac{2}{5} ) D. ( frac{-2}{7} ) |
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| 399 | The value of ( 1+i^{2}+i^{4}+i^{6}+i^{8}+ ) ( +i^{20} ) is : |
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| 400 | If ( m & M ) denotes the minimum and maximum value of ( |2 z+1| ) where ( mid z- ) ( mathbf{2} i mid leq mathbf{1} ) then ( (boldsymbol{m}+boldsymbol{M})^{2} ) is equal to A . 17 B. 34 c. 51 D. 68 |
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| 401 | The complex number ( z ) satisfying the equations ( |z|-mathbf{4}=|z-i|-|z+mathbf{5} i|= ) ( mathbf{0}, ) is This question has multiple correct options A. ( sqrt{3}-i ) В. ( 2 sqrt{3}-2 i ) c. ( -2 sqrt{3}-2 i ) D. 0 |
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| 402 | If ( z=frac{2-i}{i}, ) then ( quad R eleft(z^{2}right)+I mleft(z^{2}right) ) is equal to A . 1 B. – ( c cdot 2 ) D. – – ( E ) |
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| 403 | If ( z_{0}=frac{1-i}{2}, ) then ( left(mathbf{1}+boldsymbol{z}_{0}right)left(mathbf{1}+boldsymbol{z}_{mathbf{0}}^{mathbf{2}^{1}}right)left(mathbf{1}+boldsymbol{z}_{mathbf{0}}^{mathbf{2}^{2}}right) ldots ldots ldots(mathbf{1}+ ) must be A ( cdot(1-i)left(1+frac{1}{2^{2^{n}}}right) ) for ( n>1 ) B. ( (1-i)left(1-frac{1}{2^{2^{n}}}right) ) for ( n> ) c. ( frac{1+i}{2} ) for ( n>1 ) D. ( (1-i)left(1-frac{1}{2^{2 n+1}}right) ) for ( n>1 ) |
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| 404 | Solve ( sin 2 x+cos 4 x=2 ) | 11 |
| 405 | If ( left|z+frac{2}{z}right|=2, ) then the maximum value of ( |z| ) is ( sqrt{m}+1 . ) Find ( m ) | 11 |
| 406 | A value of ( theta ) for which ( frac{2+3 i sin theta}{1-2 i sin theta} ) is purely imaginary, is: ( ^{mathrm{A}} cdot sin ^{-1}left(frac{1}{sqrt{3}}right) ) в. ( frac{pi}{3} ) ( mathbf{c} cdot cos ^{-1} sqrt{-} 1 ) D. Noneofthese |
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| 407 | If ( z_{1} ) and ( z_{2} ) are two complex numbers satisfying the equation ( left|frac{z_{1}+z_{2}}{z_{1}-z_{2}}right|=1 ) then ( frac{z_{1}}{z_{2}} ) is a number which is A . Positive real B. Negative real c. zero or purely imaginary D. none of these |
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| 408 | ( fleft|begin{array}{ccc}mathbf{6} i & -mathbf{3} i & mathbf{1} \ mathbf{4} & mathbf{3} i & -mathbf{1} \ mathbf{2 0} & mathbf{3} & boldsymbol{i}end{array}right|=boldsymbol{x}+boldsymbol{i} boldsymbol{y}, ) then A ( . x=3, y=0 ) B. ( x=1, y=3 ) c. ( x=0, y=3 ) D. ( x=0, y=0 ) |
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| 409 | 20. If o(+1) is a cube root of unity, and (1+0) = A+Bw. Then (A,B) equals [2011] (a) (1,1) (b) (1,0) © (-1,1) (d) (0,1) |
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| 410 | In the argand diagram, the complex number z is in the fourth quadrant, then ( bar{z},-z, overline{-z} ) are respectively are in quardrants В . 1,2,3 c. 3,2,1 D. 2,1, |
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| 411 | The conjugate complex number of ( frac{2-i}{(1-2 i)^{2}} ) is A ( cdot frac{2}{25}+frac{11}{25} i ) в. ( frac{2}{25}-frac{11}{25} i ) c. ( -frac{2}{25}+frac{11}{25} i ) D. ( -frac{2}{25}-frac{11}{25} i ) |
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| 412 | If ( |z|=3, ) then ( frac{9+z}{1+bar{z}} ) equals ( A ) B. ( c cdot 3 z ) D. ( z+bar{z} ) |
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| 413 | The complex number ( z ) is such that ( |z|=1, z neq-1 ) and ( omega=left(frac{z-1}{z+1}right) . ) Then the real part of ( omega ) is |
11 |
| 414 | 12. If w and 1-1, then z lies on 120051 (a) (c) an ellipse a straight line (b) a circle (d) a parabola |
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| 415 | 12. Find all non-zero complex numbers Z satisfying Z = iZ. (1996 – 2 Marks) |
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| 416 | locate the point representing the complex numbers ( z ) on the Argand diagram for which ( |z| geq 3 ) |
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| 417 | For ( boldsymbol{a}>mathbf{0}, ) arg ( (boldsymbol{i} boldsymbol{a})= ) A ( cdot frac{pi}{2} ) B. ( -frac{pi}{2} ) ( c . pi ) D. – ( pi ) |
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| 418 | The value of the sum ( sum_{n=1}^{13}left(i^{n}+i^{n+1}right) ) where ( i=sqrt{-1}, ) equals ( A ) B. i- ( c .- ) D. |
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| 419 | ( left|frac{1}{2}left(z_{1}+z_{2}right)+sqrt{z_{1} z_{2}}right|+ ) ( left|frac{1}{2}left(z_{1}+z_{2}right)-sqrt{z_{1} z_{2}}right|= ) A ( cdot mid z_{1}+z_{2} ) B . ( left|z_{1}-z_{2}right| ) ( mathbf{c} cdotleft|z_{1}right|+left|z_{2}right| ) D. ( left|z_{1}right|-mid z_{2} ) |
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| 420 | The value of ( (i+sqrt{3})^{100}+ ) ( (i-sqrt{3})^{100}+2^{100}= ) A . B. – – ( c cdot 0 ) D. |
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| 421 | If ( z_{1}=2+5 i, z_{2}=3-i ) find ( (a) z_{1} cdot z_{2} ) (b) ( z_{1} times z_{2} ) ( (c) z_{2} cdot z_{1} ) (d) ( z_{2} times z_{1} ) (e) acute angle between ( z_{1} & z_{2} ) Projection of ( z_{1} ) on ( z_{2} ) A ( cdot 1,17,1,17, cos ^{-1} frac{1}{sqrt{290}}, frac{1}{sqrt{10}} ) в. ( _{1,-17,1,17, cos ^{-1}} frac{1}{sqrt{290}}, frac{1}{sqrt{10}} ) c. ( _{1,17,1,-17, cos ^{-1}} frac{1}{sqrt{290}}, frac{1}{sqrt{10}} ) D. ( 1,-17,1,-17, cos ^{-1} frac{1}{sqrt{290}}, frac{1}{sqrt{10}} ) |
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| 422 | If ( left(frac{1+i}{1-i}right)^{x}=1, ) then A. ( x=2 n ), where ( n ) is any positive integer B. ( x=4 n+1 ), where ( n ) is any positive integer C. ( x=2 n+1, ) where ( n ) is any positive integer D. ( x=4 n ), where ( n ) is any positive integer |
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| 423 | If ( n ) is any positive integer, then the value of ( frac{i^{4 n+1}-i^{4 n-1}}{2} ) equals: ( mathbf{A} cdot mathbf{1} ) B. – ( c cdot i ) D. ( -i ) |
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| 424 | If ( boldsymbol{k}>mathbf{0},|boldsymbol{z}|=|boldsymbol{w}|=boldsymbol{k} ) and ( boldsymbol{alpha}=frac{boldsymbol{z}-overline{boldsymbol{w}}}{boldsymbol{k}^{2}+boldsymbol{z} overline{boldsymbol{w}}} ) then ( operatorname{Re}(alpha) ) equal A . B. ( c . k ) D. none of these |
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| 425 | 17. Prove that there exists no complex number z such that < and a,z' = 1 where la <2. (2003 – 2 Marks) r=1 |
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| 426 | In the Argand plane, the vector ( O P ) where ( O ) is the origin and ( P ) represents the complex number ( z=4-3 i, ) is turned in the clockwise sense through ( 180^{circ} ) and streched 3 times. the complex number represented by the new vector is |
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| 427 | Find ( (-5 i)left(frac{3}{5} iright) ) | 11 |
| 428 | If ( z ) is purely real and ( R e(z)<0, ) then ( arg (x) ) is A . 0 в. ( pi ) c. ( -pi ) D. |
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| 429 | 17. Ifx is real, the maximum value of – 3×2 +9x+17 3×2 + 9x +7 is [2006] (b) 41 (d) 17 © 1 |
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| 430 | 38. Which of the following is true for z=(3 + 2i sin )/(1-2 i sin o), where i = 1-1 ? a. z is purely real for 0= nt Tr/3, ne Z b. z is purely imaginary for 0=nt = n/2, ne z c. Z is purely real for O=nn, ne z d. none of these |
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| 431 | If a complex number z satisfies the equation ( z+sqrt{mathbf{2}}|z+mathbf{1}|+mathbf{i}=mathbf{0}, ) where ( boldsymbol{i}=sqrt{-mathbf{1}}, ) then ( |boldsymbol{z}| ) is equal to. A . B. 2 ( c cdot sqrt{3} ) D. ( sqrt{5} ) |
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| 432 | Represent ( (-1-sqrt{3} i) ) in the polar form. | 11 |
| 433 | ( f(a>0,|z|=a, ) then find the real part of ( left(frac{boldsymbol{z}-boldsymbol{a}}{boldsymbol{z}+boldsymbol{a}}right) ) | 11 |
| 434 | If ( f(z)=frac{1-z^{3}}{1-z}, ) where ( z=x+i y ) with ( z neq 1, ) then ( R e overline{{f(z)}}=0 ) reduces to A ( cdot x^{2}+y^{2}+x+1=0 ) B . ( x^{2}-y^{2}+x-1=0 ) c. ( x^{2}-y^{2}-x+1=0 ) D. ( x^{2}-y^{2}+x+1=0 ) E ( cdot x^{2}-y^{2}+x+2=0 ) |
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| 435 | If ( boldsymbol{w}=frac{boldsymbol{z}}{boldsymbol{z}-frac{1}{mathbf{3}} boldsymbol{i}} ) and ( |boldsymbol{w}|=1 ) then ( boldsymbol{z} ) lies on A . a circle B. an ellipse c. a parabola D. a straight line |
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| 436 | uw 6. 28. All the points in the set s={a+:a er (i= ) lie on a: lai [JEEM 2019-9 April (M) (a) straight line whose slope is 1. (b) circle whose radius is 1. (c) circle whose radius is v2. (d) straight line whose slope is -1. |
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| 437 | Show that: ( left(frac{1+i}{sqrt{2}}right)^{8}+left(frac{1-i}{sqrt{2}}right)^{8}=2 ) |
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| 438 | Find the real numbers ( x ) and ( y ) if ( (x- ) ( i y)(3+5 i) ) is the conjugate of ( -6- ) ( mathbf{2 4 i . 1}+boldsymbol{i} mathbf{1}-boldsymbol{i} ) |
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| 439 | ( operatorname{Let} boldsymbol{alpha}=frac{-mathbf{1}+boldsymbol{i} sqrt{mathbf{3}}}{mathbf{2}} . ) If ( boldsymbol{a}= ) ( (1+alpha) sum_{k=0}^{100} alpha^{2 k} ; b=sum_{k=0}^{100} alpha^{3 k}, ) then ( a ) and ( b ) are the roots of the quadratic equation: A ( cdot x^{2}-102 x+101=0 ) B. ( x^{2}-101 x+100=0 ) c. ( x^{2}+102 x+101=0 ) D. ( x^{2}+101 x+100=0 ) |
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| 440 | Find ‘x’ and ‘y’ if ( x^{2}-y^{2}-i(2 x+y)= ) ( mathbf{2} i ) |
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| 441 | If ( S=i+2 i^{2}+3 i^{3}+ldots ) up to 200 term then S equals A . ( 200 i ) B . ( 100(1+i) ) c. ( 100(1-i) ) D. ( 200(1-i) ) |
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| 442 | Modulus of nonzero complex number z satisfying ( bar{z}+z=0 ) and ( |z|^{2}-4 z i= ) ( z^{2} ) is |
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| 443 | The argument of the complex number ( sin frac{6 pi}{5}+ileft(1+cos frac{6 pi}{5}right) ) is A ( cdot frac{6 pi}{5} ) в. ( frac{5 pi}{6} ) c. ( frac{9 pi}{10} ) D. ( frac{2 pi}{5} ) |
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| 444 | Evaluate: ( left(begin{array}{c}1+cos frac{pi}{6}-i sin frac{pi}{6} \ 1+cos frac{pi}{6}+i sin frac{pi}{6}end{array}right) ) ( A ) B. -1 ( c cdot 2 ) ( D cdot underline{1} ) |
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| 445 | Find the value of ( [4+3 sqrt{-20}]^{1 / 2}+[4-3 sqrt{-20}]^{1 / 2} ) ( mathbf{A} cdot mathbf{6} ) B ( cdot 2(3+sqrt{5}) ) ( mathbf{c} cdot 2(3-sqrt{5}) ) D. ( 2 sqrt{5} ) |
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| 446 | If ( left|frac{z-5 i}{z+5 i}right|=1, ) prove that ( z ) is real. | 11 |
| 447 | 21. (a) 2 The locus of z which lies in shaded region (excluding the boundaries) is best represented by (2005) (-1+ v2,v2) arg (2) W arg (2) (+1+ 12,-12) (a) z:/z+1|>2 and larg (2+1)]2 and larg (2-1)]<1/4 (c) 2:12+11<2 and arg (z+1)]<2 (d) z:12-11<2 and larg (z+1)/<72 |
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| 448 | The amplitude of ( sin frac{pi}{5}+ileft(1-cos frac{pi}{5}right) ) A. в. ( frac{2 pi}{5} ) c. ( frac{pi}{10} ) D. ( frac{pi}{15} ) |
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| 449 | If for the complex numbers ( z_{1}, z_{2}, dots ., z_{n} ) ( left|z_{1}right|=left|z_{2}right|=ldots . .=left|z_{n}right|=1 . ) Then prove that ( |overline{z_{1}+z_{2}+ldots . .+z_{n}}|= ) ( left|frac{mathbf{1}}{z_{1}}+frac{1}{z_{2}}+ldots ldots+frac{1}{z_{n}}right| ) |
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| 450 | ff ( z=-5+2 sqrt{-4}, ) then the value of ( z^{2}+10 z+41 ) is equal to ( A cdot 2 ) B . – – ( c cdot 0 ) D. None of these |
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| 451 | Find the multiplicative inverse of ( frac{sqrt{3}}{2}-frac{1}{2} i ) |
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| 452 | The value of ( -3 sqrt{-10} ) is equal to A. ( -3 sqrt{10} ) 00 B. ( 3 sqrt{10} ) c. ( -3 i sqrt{10} ) D. None of these |
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| 453 | 1. If 21 = a +ib and z2 = c+id are complex numbers such that 2,1 = 122=1 and Re(z, 22)=0, then the pair of complex numbers wi = a +ic and wz =b+id satisfies – (1985 – 2 Marks) (a) wl=1 (b) |w2 = 1 (c) Re(ww.) = 0 (d) none of these |
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| 454 | ( i^{2}+i^{3}+i^{4}+i^{3} ) is equal to | 11 |
| 455 | Which of the the following is correct representation of the complex number: ( (a, b) ) в. ( (a, 0) times(0, b) ) c. ( (a, 0)+(0, b) ) D. None of these |
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| 456 | If ( sin alpha+sin beta+sin gamma=0=cos alpha+ ) ( cos beta+cos gamma ) then ( sin ^{2} alpha+sin ^{2} beta+ ) ( sin ^{2} gamma= ) A. ( -frac{3}{2} ) B. ( frac{3}{2} ) ( c cdot frac{2}{3} ) D. none of these |
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| 457 | Ifa+b+c=0, then the quadratic equation 3ax2 + 2bx+c=0 has (1983 – 1 Mark) (a) at least one root in [0, 1] (b) one root in [2, 3] and the other in [-2,-11 (c) imaginary roots (d) none of these |
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| 458 | ( (1+i)^{8}+(1-i)^{8}= ) A . 16 B. -16 ( c .32 ) D. -32 |
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| 459 | Find the multiplicative inverse of the complex number ( sqrt{5}+3 i ) | 11 |
| 460 | Find the value of ( arg left((1+i)^{i}right) ) A ( cdot frac{1}{4} ln (2) ) в. ( frac{1}{2} ln (2) ) ( ^{text {c. }} frac{1}{2} ln left(frac{1}{2}right) ) D ( cdot ln (2) ) |
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| 461 | Evaluate in standard form: ( frac{(2-3 i)}{(2-2 i)} ) where ( i^{2}=-1 ) A ( cdot frac{5}{4}-frac{i}{4} ) в. ( frac{5}{4}+frac{i}{4} ) c. ( -frac{5}{4}-frac{i}{4} ) D. ( -frac{5}{4}+frac{i}{4} ) |
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| 462 | (i) Find the real values of ( x ) and ( y ) for which ( z_{1}=9 y^{2}-4-10 i x ) is complex conjugate of each other. (ii) Find the value of ( x^{4}-x^{3}+x^{2}+ ) ( 3 x-5 ) if ( x=2+3 i ) |
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| 463 | Let ( bar{z}, bar{w} ) be complex numbers such that ( z+w ) purely imaginary and ( z-w ) is purely real ( mathbf{A} cdot z=w ) в. ( z=-w ) ( mathbf{c} cdot z=bar{w} ) D. ( z=-bar{w} ) |
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| 464 | Prove that ( x^{2}+y^{2}=9 ) where ( z=x+ ) ( i y ) and ( |z+6|=|2 z+3| ) |
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| 465 | f ( z_{1}=6+i z_{2}=3-4 i ) then find ( z_{1} z_{2} ) | 11 |
| 466 | Difference between the corresponding roots of x2+ax+b=0 and x2+bx+a=0 is same and a +b, then [2002] (a) a+b+4=0 (b) a+b-4=0 © a-6-4=0 (d) a-5+4=0 |
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| 467 | If ( alpha ) and ( beta ) are different complex numbers with ( |beta|=1 ) then ( left|frac{beta-alpha}{1-bar{alpha} beta}right| ) is equal to A . B. ( frac{1}{2} ) ( c cdot 1 ) D. |
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| 468 | Find the real values of ( x ) and ( y ) for which the following equation is satisfied [ frac{(1+i) x-2 i}{3+i}+frac{(2-3 i) y+i}{3-i}=i ] A. ( x=3, y=-1 ) B. ( x=-3, y=-1 ) c. ( x=3, y=1 ) D. ( x=-3, y=1 ) |
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| 469 | ( z_{1} ) and ( z_{2} ) are two distinct points in an Argand plane. If ( aleft|z_{1}right|=bleft|z_{2}right| ) (where ( a, b ) ( epsilon mathrm{R} ) ), then the point ( left(a z_{1} / b z_{2}right)+ ) ( left(b z_{2} / a z_{1}right) ) is a point on the A. Line segment [-2, 2] of the real axis B. Line segment [-2, 2] of the imaginary axis c. Unit circle ( |z|=1 ) D. The line with arg ( z=tan ^{-1} 2 ) |
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| 470 | The real part of ( (i-sqrt{3})^{13} ) is A ( cdot 2^{-10} sqrt{3} ) B. ( -2^{12} sqrt{3} ) ( mathrm{c} cdot 2^{-12} sqrt{3} ) D. ( -2^{-12} sqrt{3} ) E ( .-2^{10} sqrt{3} ) |
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| 471 | The value of ( sqrt{-36} ) is ( A cdot 6 ) в. -6 ( c cdot 6 i ) D. None of these |
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| 472 | 24. If is purely real where w= a +iB, B+0 and 2+1 1-Z then the set of the values of z is (2006 – 3M, -7 (a) {z: z=1} (6) {z:z=z} (c) {z:2+1} (d) {z: 121=1, 2+1} |
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| 473 | ( i+frac{1}{i}= ) A. B. – c. D. 2 |
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| 474 | Solve the problem:( left(frac{1}{5}+i frac{2}{5}right)-left(4+i frac{5}{2}right) ) | 11 |
| 475 | 8. If z = x-i y and z3 = p +iq, then +97) is 91 [2004] equal to (2) -2 (6)-1 (0) 2 (1) 1 |
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| 476 | 32. If aße are the distinct roots, of the equation x2-x+1=0, then q101 +8107 is equal to : [JEEM 2018] (a) O full (6) 1 (C) 2 (d) -1 |
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| 477 | Find the value of ( frac{i^{6}+i^{7}+i^{8}+i^{9}}{i^{2}+i^{3}} ) ( A cdot 0 ) B. ( c .-1 ) D. None |
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| 478 | Complex conjugate of ( 3 mathrm{i}-4 ) is ( A cdot 3 i+4 ) B. – 3i- 4 ( c cdot-3 i+4 ) D. None of these |
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| 479 | Consider the complex numbers ( z= ) ( frac{(1-i sin theta)}{(1+i cos theta)} . ) The value of ( theta ) for which ( z ) is purely imaginary are- A ( cdot n pi-frac{pi}{4}, n epsilon I ) В ( cdot n pi+frac{pi}{4}, n epsilon I ) c. ( n pi, n epsilon I ) D. No real values of ( theta ) |
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| 480 | The real part of ( left(frac{1+i}{3-i}right)^{2}= ) ( mathbf{A} cdot mathbf{1} ) B . 16 ( c cdot 16 omega^{2} ) D. ( frac{-3}{25} ) |
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| 481 | If ( z=x+i y ) and ( w=frac{1-i z}{z-i}, ) show that ( |boldsymbol{w}|=mathbf{1} Longrightarrow boldsymbol{z} ) is purely real. |
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| 482 | ( frac{(1+i)^{2011}}{(1-i)^{2009}}= ) A . -1 B. ( c cdot 2 ) D. – |
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| 483 | If the conjugate of ( (x+i y)(1-2 i) ) is ( 1+i, ) then This question has multiple correct options A ( cdot x-i y=frac{1+i}{1-2 i} ) B. ( x+i y=frac{1-i}{1-2 i} ) c. ( _{y}=frac{1}{5} ) D. ( x=-frac{1}{5} ) |
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| 484 | If ( boldsymbol{z}(mathbf{2}-boldsymbol{i})=mathbf{3}+boldsymbol{i}, quad boldsymbol{z}^{20}= ) A. ( 1-i ) B. -1024 ( c cdot 1024 ) D. ( 1+i ) |
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| 485 | If ( frac{4 z_{1}}{9 z_{2}}+frac{4 overline{z_{1}}}{9 overline{z_{2}}}=0, ) then the value of ( left|frac{z_{1}-z_{2}}{z_{1}+z_{2}}right| ) is A ( cdot frac{4}{9} ) в. ( frac{9}{4} ) ( c cdot 1 ) ( D ) |
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| 486 | ( $ $ x|,+|, ) i ( left|s q r t x^{wedge}right|, 4left|,+, x^{wedge} 2right|,+mid, 1 $ $ ) | 11 |
| 487 | 23. Ifz is a complex number such that 2 > 2, then the minimum value of 2 + value of z+: : [JEE M 2014 (a) is strictly greater than (b) is strictly greater than but less than (c) is equal to (d) lie in the interval (1,2) |
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| 488 | Write the argument of ( (1+sqrt{3})(1+ ) ( i)(cos theta+i sin theta) ) |
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| 489 | If ( z_{1}, z_{2}, z_{3} ) are the solutions of ( z^{2}+ ) ( bar{z}=z, ) then ( z_{1}+z_{2}+z_{3} ) is equal to (where ( z ) is a complex number on the argand plane and ( i=sqrt{-1}) ) ( mathbf{A} cdot 2+2 i ) B . ( 2-2 i ) c. 0 D. 2 |
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| 490 | Find the value of ( 1+i^{2}+i^{4}+i^{6}+ldots+ ) ( i^{2 n} ) ( mathbf{A} cdot mathbf{1} ) B. – c. 0 D. it cannot be determined |
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| 491 | Letf(x) be a quadratic expression which is positive for all the real values of x. Ifg(x)=f(x) + f(x) +S”(x), then for any real x, (199 Tarks) (a) g(x) 0 (c) g(x)=0 (d) g(x) > 0 |
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| 492 | If ( z=left(frac{sqrt{3}}{2}+i frac{1}{2}right)^{5}+left(frac{sqrt{3}}{2}-i frac{1}{2}right)^{5} ) then ( operatorname{lm}(z)= ) |
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| 493 | Find the modulus, argument and the principal argument of the complex numbers. ( z=frac{sqrt{5+12 i}+sqrt{5-12 i}}{sqrt{5+12 i}-sqrt{5-12 i}} ) |
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| 494 | If ( arg z<0 ) then ( arg (-z)-arg z ) is equal to A . ( pi ) в. ( -pi ) ( c cdot-frac{pi}{2} ) D. |
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| 495 | Write principal argument of ( frac{-sqrt{11} i}{17} ) | 11 |
| 496 | If ( 0 leq arg z leq frac{pi}{4}, ) then the least value of ( sqrt{mathbf{2}}|mathbf{2} z-mathbf{4}| ) is A. 6 B. ( c cdot 4 ) D. |
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| 497 | Simplify: ( (14+2 i)(7+12 i) ) where ( i= ) ( sqrt{-1} ) |
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| 498 | Find the modulus, argument and the principal argument of the complex numbers. ( z=1+cos frac{10 pi}{9}+i sin left(frac{10 pi}{9}right) ) A . Principal Arg ( z=-frac{4 pi}{9} ;|z|=2 cos frac{4 pi}{9} ; operatorname{Arg} z=2 k pi- ) ( frac{4 pi}{9} k epsilon l ) B. Principal Arg ( z=-frac{10 pi}{9} ;|z|=2 cos frac{10 pi}{9} ; ) Argz( =2 k pi- ) ( frac{10 pi}{9} k epsilon l ) C . Principal Arg ( z=-frac{-10 pi}{9} ;|z|=2 cos frac{-10 pi}{9} ; ) Argz( = ) ( 2 k pi-frac{4 pi}{9} k epsilon l ) D. Principal Arg ( z=-frac{-4 pi}{9} ;|z|=2 cos frac{-4 pi}{9} ; ) Argz( =2 k pi- ) ( frac{4 pi}{9} k epsilon l ) |
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| 499 | The conjugate surd of ( 2-sqrt{3} ) is | 11 |
| 500 | The number of solutions of the system of equations ( operatorname{Re}left(z^{2}right)=0 ;|z|=2 ) is ( A cdot 4 ) B. 3 ( c cdot 2 ) ( D ) |
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| 501 | If ( z=i^{9}+i^{19}, ) then ( z ) is equal to ( mathbf{A} cdot 0+0 i ) B. ( 1+0 i ) ( c cdot 0+i ) D. ( 1+2 i ) |
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| 502 | C ONCU O The smallest positive integer n for which (1980) (1+i)” =lis (-; (a) n=8 (c) n=12 (6) n=16 (d) none of these |
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| 503 | If ( boldsymbol{alpha}=cos left(frac{boldsymbol{8} boldsymbol{pi}}{mathbf{1 1}}right)+boldsymbol{i} sin left(frac{boldsymbol{8} boldsymbol{pi}}{11}right), ) then ( boldsymbol{R} eleft(boldsymbol{alpha}+boldsymbol{alpha}^{2}+boldsymbol{alpha}^{3}+boldsymbol{alpha}^{4}+boldsymbol{alpha}^{5}right) ) is A ( cdot frac{1}{2} ) B. ( -frac{1}{2} ) ( c cdot 0 ) D. None of the above |
11 |
| 504 | Solve ( :-frac{i^{592}+i^{590}+i^{588}+i^{586}+i^{584}}{i^{582}+i^{580}+i^{578}+i^{576}+i^{574}} ) | 11 |
| 505 | 17. The complex numbers 2, 2, and zz satisfying 21-23 1-i3 are the vertices of a triangle which is 22-232 (a) of area zero (20015) (b) right-angled isosceles (c) equilateral (d) obtuse-angled isosceles |
11 |
| 506 | ( mathbf{2}^{i}=e^{i(l n x)} ) | 11 |
| 507 | If ( z^{2}+z+1=0, ) then ( sum_{r=1}^{6}left(z_{1}+frac{1}{z_{2}}right)^{2} ) is equal to ( A cdot 6 ) B. 12 ( c cdot 18 ) D. 24 |
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| 508 | 8. The number of real solutions of the equation *2-3x/+2 = 0 is (a) 3 (b) 2 (c) 4 (d) 1 |
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| 509 | For ( i^{2}=-1 ) find the value of ( i^{253} ) | 11 |
| 510 | ( frac{3+2 i sin theta}{1-2 i sin theta} ) will be purely imaginary, if ( boldsymbol{theta}= ) A ( cdot 2 n pi-frac{pi}{3} ) в. ( n pi+frac{pi}{3} ) c. ( n pi-frac{pi}{3} ) D. None of these |
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| 511 | If ( boldsymbol{x}=mathbf{1}+boldsymbol{i} tan boldsymbol{alpha}, ) where ( boldsymbol{pi}<boldsymbol{alpha}<frac{mathbf{3} boldsymbol{pi}}{mathbf{2}} ) then ( |z| ) is equal to? |
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| 512 | Solve the system of equations ( boldsymbol{operatorname { R e }}left(z^{2}right)=mathbf{0},|z|=mathbf{2} ) | 11 |
| 513 | Real part of ( frac{(1+i)^{2}}{3-i}= ) A. ( -1 / 5 ) в. ( 1 / 5 ) c. ( 1 / 10 ) D. ( -1 / 10 ) |
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| 514 | Simplify the multiplication of complex numbers: ( (boldsymbol{x}, boldsymbol{y}) times(mathbf{0}, mathbf{1}) ) A. ( (-y, x) ) В. ( (-y,-x) ) c. ( (x,-y) ) D. None of these |
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| 515 | 3. The complex numbers z = x+iy which satisfy the equation (1981 – 2 Marks) z- Si =1 lie on z+ 5i| (a) the x-axis (b) the straight line y=5 (c) a circle passing through the origin (d) none of these |
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| 516 | Evaluate ( : sqrt{-mathbf{1 6}}+mathbf{3} sqrt{-mathbf{2 5}}+sqrt{-mathbf{3 6}}- ) ( sqrt{-mathbf{6 2 5}} ) |
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| 517 | Let ( z_{1} ) and ( z_{2} ) be complex numbers, then ( left|z_{1}+z_{2}right|^{2}+left|z_{1}-z_{2}right|^{2} ) is equal to A ( cdotleft|z_{1}right|^{2}+left|z_{2}right|^{2} ) B ( cdot 2left(left|z_{1}right|^{2}+left|z_{2}right|^{2}right) ) c. ( 2left(z_{1}^{2}+z_{2}^{2}right) ) D. ( 4 z_{1} z_{2} ) |
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| 518 | ff ( i z^{3}+z^{2}-z+i=0, ) then ( |z| ) is equal to ( mathbf{A} cdot mathbf{0} ) B. ( c cdot 2 ) D. None of these |
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| 519 | Find the square root of ( 4 a b-2left(a^{2}-right. ) ( left.b^{2}right) i ) |
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| 520 | 26. Let A= 3+ 2isine is purely imaginary 1-2i sin Then the sum of the elements in A is: [JEEM 2019-9 Jan (M) (b) |
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| 521 | ( z_{1} ) and ( z_{2} ) are two non-zero complex numbers such that ( z_{1}=2+4 i ) ( z_{2}=5-6 i, ) then ( z_{2}-z_{1} ) equals A. ( 3-10 i ) B. ( 3+10 i ) c. ( 7-2 i ) D. ( 10-24 i ) |
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| 522 | Prove that ( left|z_{1}right|+left|z_{2}right|=left|frac{1}{2}left(z_{1}+z_{2}right)+sqrt{z_{1} z_{2}}right|+ ) ( left|frac{1}{2}left(z_{1}+z_{2}right)-sqrt{z_{1} z_{2}}right| ) |
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| 523 | If ( i=sqrt{-1}, ) then ( 1+i^{2}+i^{3}-i^{6}+i^{8} ) is equal to – A. 2- в. ( c cdot-3 ) D. – |
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| 524 | The real part of ( left[mathbf{1}+cos left(frac{pi}{5}right)+i sin left(frac{pi}{5}right)right]^{-1} ) is A . в. ( frac{1}{2} ) c. ( frac{1}{2} cos left(frac{pi}{10}right) ) D. ( frac{1}{2} cos left(frac{pi}{5}right) ) |
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| 525 | Topic-wise suiveu Tupun 30. Let a and ß be the roots of equation x2 – 6x – 2 = 0. If 10 – 20g is equal to: a=an-Br, for n 1, then the value of [JEE M 2015) (a) 3 (b) – 3 (c) 6 (d) -6 |
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| 526 | M-16 19. Let a, ß be real and z be a complex number. If z2 + az+B=0 has two distinct roots on the line Rez=1, then it is necessary that: [2011] (a) Be (-1,0) (b) BI=1 (c) BE(1,) (d) Be(0,1) |
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| 527 | Represent the following complex number in trigonometric form: ( sqrt{3} i ) |
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| 528 | If ( z(neq-1) ) is complex number such that ( frac{z-1}{z+1} ) is purely imaginary, then ( |z| ) is equal to ( mathbf{A} cdot mathbf{1} ) B. 2 ( c .3 ) D. |
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| 529 | Find real values of ( x ) and ( y ) if ( frac{x-1}{3+i}+frac{y-1}{3-i}=i ) |
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| 530 | 18. The number of complex numbers z such that 12-1] =]z + 1) = z – iſ equals (2) 1 (b) 2 (c) oo (d) 0. [2010 |
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| 531 | 15 28. Let z = cos e + i sin 0. Then the value of Im(zam ) m=1 at O = 2° is (2009) (a) sin 20 3 sin 20 2 sin 2° 4sin 22 Taybe |
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| 532 | 5. If a, b, care distinct +ve real numbers and a2+b2+c2=1 then ab + bc + ca is [2002] (a) less than 1 (b) equal to 1 (c) greater than 1 (d) any real no. |
11 |
| 533 | If ( frac{pi}{3} ) and ( frac{pi}{4} ) are the arguments of ( z_{1} ) and ( bar{z}_{2}, ) then the value of ( arg left(z_{1} z_{2}right) ) is A ( cdot frac{5 pi}{12} ) в. ( frac{pi}{12} ) c. ( frac{7 pi}{12} ) D. None of these |
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| 534 | If ( n=4 m+3, m ) is an integer, then ( i^{n} ) is equal to: A . ( -i ) B. ( c . i ) D. – |
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| 535 | If ( alpha ) and ( beta ) are complex conjugates to each other and ( boldsymbol{alpha}=-sqrt{mathbf{2}}+boldsymbol{i} ) then find ( boldsymbol{alpha}^{2}+boldsymbol{beta}^{2}-boldsymbol{alpha} boldsymbol{beta} ) |
11 |
| 536 | Let the complex number ( z_{1}, z_{2}, z_{3} ) be the vertices of an equilxeral triangle. let ( z z_{0} ) be the circumcentre of the triangle, then ( z_{1}^{2}+z_{2}^{2}+z_{3}^{2}- ) A. ( z_{0}^{2} ) an ( ^{2} cdot frac{z^{2}}{2}_{2}^{2} ) B. ( -z_{0}^{2} ) ( c cdot 3 z_{0}^{2} ) D. ( -3 z_{0}^{2} ) |
11 |
| 537 | Solve: ( boldsymbol{x}^{2}-(boldsymbol{3} sqrt{mathbf{2}}-mathbf{2} boldsymbol{i}) boldsymbol{x}-sqrt{mathbf{2}} boldsymbol{i}=mathbf{0} ) | 11 |
| 538 | Reduce ( left(frac{1}{1-4 i}-frac{2}{1+i}right)left(frac{3-4 i}{5+i}right) ) the standard form. |
11 |
| 539 | Find ( a ) and ( b, ) where ( a ) and ( b ) are real numbers so that ( a+i b=(2-i)^{2} ) A ( . a=3, b=-4 ) В. ( a=-3, b=-4 ) c. ( a=3, b=4 ) D. ( a=-3, b=4 ) |
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| 540 | The conjugates of a) ( -2+3 i ) b) ( 1-i ) c) 4 d) ( 4 i ) in order are: A . ( 2-3 i, 1+i, 4,-4 i ) B. ( +2-3 i, 1+i, 4,-4 i ) c. ( 2-3 i, 1+i, 4,4 i ) D. ( -2-3 i, 1+i, 4,-4 i ) |
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| 541 | The complex number ( frac{1+2 i}{1-i} ) lies in which quadrant of the complex plane. A. First B. second c. Third D. Fourth |
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| 542 | Two complex numbers are represented by ordered pairs ( z_{1}:(a, 0) & z_{2}:(c, d) ) which of the following is correct simplification for ( z_{1} times z_{2}=? ) ( mathbf{A} cdot(a c,-a d) ) B. ( (a d, a c) ) ( mathbf{c} cdot(a c, a d) ) D. None of these |
11 |
| 543 | ( z_{1} ) and ( z_{2} ) are the roots of the equation ( z^{2}-a z+b=0, ) where ( left|z_{1}right|=left|z_{2}right|=1 ) and ( a, b ) are non zero complex numbers, then This question has multiple correct options ( mathbf{A} cdot|a| leq 1 ) B. ( |a| leq 2 ) C ( cdot arg left(a^{2}right)=arg (b) ) ( mathbf{D} cdot ) arga ( =arg left(b^{2}right) ) |
11 |
| 544 | Find conjugate: ( -i(9+6 i)(2-i)^{-1} ) | 11 |
| 545 | Among the complex numbers ( z ) which satisfy the condition ( |z-25 i| leq 15 ) find the number having the least positive and greatest positive argument. |
11 |
| 546 | Let ( z_{1} ) and ( z_{2} ) are two complex numbers such that ( (1-i) z_{1}=2 z_{2} ) and ( arg left(z_{1} z_{2}right)=frac{pi}{2} ) then ( arg left(z_{2}right) ) is equals to: A ( cdot frac{3 pi}{8} ) B. ( frac{pi}{8} ) c. ( frac{5 pi}{8} ) D. ( frac{-7 pi}{8} ) |
11 |
| 547 | 23. Equation 1 + x2 + 2x sin(cos’y) = 0 is satisfied by a. exactly one value of x b. exactly two values of x a c. exactly one value of y d. exactly two values of y |
11 |
| 548 | f ( x+i y=frac{-1+sqrt{3} i}{1+i}, ) then find ( x ) and ( y ) | 11 |
| 549 | The imaginary roots of the equation ( left(x^{2}+2right)^{2}+8 x^{2}=6 xleft(x^{2}+2right) ) are A. ( 1+i ) в. ( 2 pm i ) c. ( -1 pm i ) D. noneofthese |
11 |
| 550 | Find the value of ( i^{i} ) | 11 |
| 551 | The complex number system, denoted by ( C, ) is the set of all ordered pairs of real numbers (that is, ( boldsymbol{R} times boldsymbol{R} ) ) with the operations of addition (denoted by ( +) ) which satisfy A ( cdot(a, b)+(c, d)=(a+d, b+c) ) в. ( (a, b)+(c, d)=(a c-b d, b c-a d) ) C. ( (a, b)+(c, d)=(a+c, b+d) ) D. None of these |
11 |
| 552 | Amplitude of ( frac{1+i}{1-i} ) is : A. в. ( pi ) ( c cdot frac{pi}{2} ) D. – |
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| 553 | 9. The real number x when added to its inverse gives the minimum value of the sum at x equal to [2003] (a) -2 (b) 2 (c) 1 (d) -1 |
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| 554 | If ( a, b notin R, ) then ( left|e^{a+i b}right| ) is equal to A ( cdot e^{a} ) в. ( e^{b} ) c. 1 D. None of these |
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| 555 | f ( z_{1}, z_{2}, z_{3} ) are unlmodular complex numbers then the greatest value of ( left|z_{1}-z_{2}right|^{2}+left|z_{2}-z_{3}right|^{2}+left|z_{3}-z_{1}right|^{2} ) equal to ( A cdot 3 ) B. 6 ( c cdot s ) ( D cdot frac{27}{2} ) |
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| 556 | Find the modulus and argument of the following complex numbers and hence express each of them in the polar form: ( frac{-mathbf{1 6}}{mathbf{1 + i sqrt { 3 }}} ) |
11 |
| 557 | If ( a=cos 2 alpha+i sin 2 alpha, b=cos 2 beta+ ) ( i sin 2 beta operatorname{then} sqrt{frac{a}{b}}+sqrt{frac{b}{a}}= ) A. ( 2 i sin (alpha-beta) ) B. ( 2 i sin (alpha+beta) ) ( c cdot 2 cos (alpha+beta) ) D. ( 2 cos (alpha-beta) ) |
11 |
| 558 | The value of ( (1+i)^{5} times(1-i)^{5} ) | 11 |
| 559 | 10. If iz3 + z2z+i=0, then show that|zl=1. (1995 – 5 Marks) |
11 |
| 560 | Find the multiplicative inverse of the complex numbers given the following: ( -i ) |
11 |
| 561 | Simplify ( left[i^{17}+left(frac{1}{i}right)^{25}right]^{3} ) | 11 |
| 562 | ( operatorname{Let} z=frac{(1+i)^{2}}{a-i},(a>0) ) and ( |z|=sqrt{frac{2}{5}} ) then ( bar{z} ) is equal to A. ( -frac{1}{5}-frac{3 i}{5} ) в. ( frac{1}{5}+frac{3 i}{5} ) ( c cdot frac{3}{5}-frac{1 i}{5} ) D. ( -frac{3}{5}+frac{1 i}{5} ) |
11 |
| 563 | ( operatorname{Let} z=x+i y & operatorname{amp}left(e^{z^{2}}right)=operatorname{amp} ) ( left(e^{(z+i)}right) . ) If ( y=(x) ) is a function, then ( boldsymbol{y}(mathbf{3}) ) is equal to A ( cdot frac{1}{2} ) в. ( frac{1}{3} ) ( c cdot frac{1}{4} ) D. |
11 |
| 564 | If ( boldsymbol{z}_{1}=boldsymbol{x}_{1}+boldsymbol{i} boldsymbol{y}_{1}, quad boldsymbol{z}_{2}=boldsymbol{x}_{2}+boldsymbol{i} boldsymbol{y}_{2}, ) then ( mathbf{2} ileft|begin{array}{ll}boldsymbol{x}_{2} & boldsymbol{y}_{2} \ boldsymbol{x}_{1} & boldsymbol{y}_{1}end{array}right| ) equals A ( cdot overline{z_{1}} z_{2}-z_{1} overline{z_{2}} ) B . ( z_{1} overline{z_{2}}-z_{2} overline{z_{1}} ) C . ( left|z_{1}right|^{2}-left|z_{2}right|^{2} ) D . ( left|z_{1}right|^{2}-left|z_{1}-z_{2}right|^{2} ) |
11 |
| 565 | If ( a ) and ( b ) are real numbers between 0 and 1 such that the points ( z_{1}=a+ ) ( i, z_{2}=1+b i ) and ( z_{3}=0 ) from an equilateral triangle, then find the values of ( ^{prime} a^{prime} ) and ( ^{prime} b^{prime} ) |
11 |
| 566 | Two complex numbers are represented by ordered pairs ( z_{1}:(6,4) & z_{2}:(4,-5) ) which of the following is real part of ( z_{1}+z_{2} ? ) A . -1 B. 10 ( c cdot 6 ) D. None of these |
11 |
| 567 | ( i log left(frac{x-i}{x+i}right) ) is equal to ( mathbf{A} cdot 2 i log (x-i)-i log left(x^{2}+1right) ) B. ( 2 i log (x-i)+i log left(x^{2}+1right) ) ( mathbf{c} cdot 2 i log (x+i)-3 i log left(x^{2}+1right) ) D. ( 2 i log (x-i)-i log left(x^{2}+iright) ) |
11 |
| 568 | Let ( a ) be a fixed nonzero complex number with ( |boldsymbol{a}|<mathbf{1} ) and ( boldsymbol{w}= ) ( left(frac{z-a}{1-bar{a} z}right), ) where ( z ) is a complex number. Then, A. there exists a complex number z with ( |z|1 ) B. ( |w|>1 ) for all z such that ( |z|<1 ) C ( .|w|<1 ) for all ( z ) such that ( |z|<1 ) D. there exists z such that ( |z|<1 ) and ( |w|=1 ) |
11 |
| 569 | The amplitude and modulus of the complex number ( -2+2 sqrt{3} i . ) is 4 and ( underline{boldsymbol{pi}} ) ( overline{mathbf{3}} ) A. True B. False |
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| 570 | 22. a, b, c are integers, not all simultaneously equal and ois cube root of unity (o + 1), then minimum value of + a + ba) + caº| is (2005) (a) 0 (b) 1 |
11 |
| 571 | ( left(1+x^{2}right)left(1+y^{2}right)left(1+z^{2}right) ) can be expressed as ( left(1-sum x yright)^{2}+ ) ( left(sum x-x y zright)^{2} . ) If this is true enter 1 else enter 0 |
11 |
| 572 | Find the modulus and amplitude of ( -2 i ) A ( cdot|z|=2 ; operatorname{amp}(z)=-frac{3 pi}{2} ) В ( cdot|z|=2 i ; a m p(z)=frac{pi}{2} ) c. ( |z|=2 ; a m p(z)=frac{pi}{2} ) D ( cdot|z|=2 ; operatorname{amp}(z)=-frac{pi}{2} ) |
11 |
| 573 | If ( a ) and ( b ) are real, then show that the principal value of arg ( a ) is 0 or ( pi ) according to ( a ) is positive or negative and that of arg ( b ) is ( pi / 2 ) or ( -pi / 2 ) according to ( b ) is positive or negative. |
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| 574 | If ( z_{1}=a+i b ) and ( z_{2}=c+i d ) are complex numbers such that ( left|z_{1}right|= ) ( left|z_{2}right|=1 ) and ( operatorname{Re}left(z_{1} bar{z}_{2}right)=0, ) then the pair of complex numbers ( omega_{1}=a+i c ) and ( omega_{2}=b+i d ) satisfies This question has multiple correct options ( mathbf{A} cdotleft|omega_{1}right|=1 ) В ( cdotleft|omega_{2}right|=1 ) c. ( operatorname{Re}left(omega_{1} overline{omega_{2}}right)=0 ) D. ( omega_{1} bar{omega}^{2}=0 ) |
11 |
| 575 | The greatest positive argument of complex number satisfying ( |z-4|= ) ( operatorname{Re}(z) ) is A. в. ( frac{2 pi}{3} ) c. ( frac{pi}{2} ) D. |
11 |
| 576 | Put in the form ( A+i B ) ( frac{(cos x+i sin x)(cos y+i sin y)}{[cot u+i](1+i tan v)} ) ( mathbf{A} cdot sin u cos v[cos (x+y-u-v)-i sin (x+y-u-v)] ) B ( cdot sin u cos v[cos (x+y-u-v)+i sin (x+y-u-v)] ) ( mathbf{C} cdot sin v cos u[cos (x+y-u-v)+i sin (x+y-u-v)] ) ( mathbf{D} cdot sin v cos u[cos (x+y-u-v)-i sin (x+y-u-v)] ) |
11 |
| 577 | ff ( i^{2}=-1 ) then the value of ( sum_{n=1}^{200} i^{2 n} ) is: A . 50 B . 10 ( c cdot 0 ) D. 100 |
11 |
| 578 | Find the real values of ( x ) and ( y, ) if ( (x+i y)(2-3 i)=4+i ) |
11 |
| 579 | Find the modulus and argumrent of the following complex numbers and hence express each of them in the polar form: ( frac{1+2 i}{1-3 i} ) |
11 |
| 580 | ( P ) represents the variable complex number ( z . ) Find the locus of ( P, ) if ( operatorname{lm} ) ( left[frac{2 z+i}{i z-1}right]=-1 ) |
11 |
| 581 | If ( z=(sqrt{3}+i) ) then find ( operatorname{Re}(z) ) and ( operatorname{lm}(z) ) |
11 |
| 582 | When simplified the value of ( left[boldsymbol{i}^{57}-right. ) ( left.left(1 / i^{25}right)right] ) is? ( mathbf{A} cdot mathbf{0} ) в. ( 2 i ) c. ( -2 i ) D. 2 |
11 |
| 583 | If ( Z=cos theta+i sin theta ) find the complex representation of ( frac{Z}{1-2 Z} ) |
11 |
| 584 | Show that ( (-1+sqrt{3} i)^{3} ) is a real number. |
11 |
| 585 | If ( boldsymbol{A}=(mathbf{3}-mathbf{4} boldsymbol{i}) ) and ( boldsymbol{B}=(mathbf{9}+boldsymbol{k} boldsymbol{i}), ) where ( k ) is a constant. If ( A B-15=60, ) then the value of ( k ) is ( mathbf{A} cdot mathbf{6} ) B . 24 c. 12 D. 3 |
11 |
| 586 | What is the multiplicative inverse of ( -1 times frac{-2}{5} ) | 11 |
| 587 | ff ( left(frac{1+i sqrt{3}}{1-i sqrt{3}}right)^{n} ) is an integer, then ( n ) is ( A ) B. 2 ( c .3 ) D. |
11 |
| 588 | Write the correct letter from column I against the entry number in column lin your answer book, ( z neq 0 ) is a complex number |
11 |
| 589 | Locate the points representing the complex number z for which ( frac{pi}{3}<arg z leq frac{pi}{2} ) represents portion of the first quadrant located between rays emerging from origin at angles of ( frac{pi}{3} ) and ( frac{pi}{2} . ) If this is true enter 1 , else enter 0 |
11 |
| 590 | Convert the complex number ( frac{-16}{1+i sqrt{3}} ) into polar form. | 11 |
| 591 | If ( i^{2}=-1, ) then ( i^{162} ) is equal to A. ( -i ) B. – ( c cdot 0 ) D. ( E ) |
11 |
| 592 | If ( z=frac{-1}{2}+i frac{sqrt{3}}{2}, ) then ( 8+10 z+7 z^{2} ) is equal to: A ( cdot-frac{1}{2}-i frac{sqrt{3}}{2} ) B ( cdot frac{1}{2}+i frac{sqrt{3}}{2} ) ( ^{mathrm{C}}-frac{1}{2}+i frac{3 sqrt{3}}{2} ) D. ( frac{sqrt{3}}{2} ) E ( -frac{sqrt{3}}{2} i ) |
11 |
| 593 | If ( z_{1}=2-i, z_{2}=1+i, ) find ( left|frac{z_{1}+z_{2}+1}{z_{1}+z_{2}+i}right| ) |
11 |
| 594 | Find the conjugate of the following complex number. ( (15+3 i)-(4-20 i) ) |
11 |
| 595 | State true or false: The region of the z-plane for which ( left|frac{boldsymbol{z}-boldsymbol{a}}{boldsymbol{z}+overline{boldsymbol{a}}}right|=1(boldsymbol{R} e boldsymbol{a} neq 0) ) is ( X ) -axis. |
11 |
| 596 | State whether the following statement is true or false. If ( Z_{r}=cos frac{pi}{3^{r}}+i sin frac{pi}{3^{r}}, r= ) ( mathbf{1}, mathbf{2}, mathbf{3}, dots dots operatorname{then} z_{1} z_{2} z_{3} dots dots dots=i ) A. True B. False |
11 |
| 597 | 20. The quadritic equations x2 – 6x +a=0 and x2 – cx+6=0 have one root in common. The other roots of the first and second equations are integers in the ratio 4:3. Then the common root is [2009] (a) 1 (6) 4 (c) 3 (d) 2 |
11 |
| 598 | If ( z neq 0 ), then ( int_{0}^{50} arg (-|z|) d x ) equals A. 50 B. not defined ( c cdot 0 ) D. ( 50 pi ) |
11 |
| 599 | If we plot ( left|Z_{1}right|=2 ) and ( left|Z_{2}-6-8 iright|=4 ) on the argand plane, the locus of ( Z_{1} ) and ( Z_{2} ) are A. two circle touching each other B. two circles neither touching nor intersecting c. two circles intersecting D. none of these |
11 |
| 600 | The conjugate of a complex number is ( frac{1}{i-1} . ) Then, that complex number is A ( cdot frac{-1}{i+1} ) B. ( frac{1}{i-1} ) c. ( frac{-1}{i-1} ) D. ( frac{1}{i+1} ) |
11 |
| 601 | Argument and modulus of ( frac{1+i^{2013}}{1-i} ) are respectively A ( cdot frac{-pi}{2} ) and 1 в. ( frac{pi}{2} ) and ( sqrt{2} ) c. 0 and ( sqrt{2} ) D ( cdot frac{pi}{2} ) and 1 |
11 |
| 602 | ( [(cos theta+i sin theta)(cos theta-i sin theta)]^{-1} ) ( mathbf{A} cdot i ) B. ( c cdot-i ) D. – |
11 |
| 603 | If ( left|frac{z_{1}+z_{2}}{z_{1}-z_{2}}right|=1 ) then ( frac{z_{1}}{z_{2}} ) is A . positive real B. negative real c. purely imaginary D. |
11 |
| 604 | Which of the following is true about ( boldsymbol{f}(boldsymbol{x}) ? ) A. ( f(x) ) decreases for ( x epsilon[2 n pi,(2 n+1) pi], n epsilon Z ) в. ( f(x) ) decreases for ( x epsilonleft[(2 n-1) frac{pi}{2},(2 n+1) frac{pi}{2}right], n epsilon Z ) c. ( f(x) ) is non-monotonic function. D. ( f(x) ) increases for ( x in R ) |
11 |
| 605 | locate the point representing the complex numbers ( z ) on the Argand diagram for which ( |z|-4=|z-i|-|z+5 i|=0 ) |
11 |
| 606 | If ( z ) is a complex number such that ( z+ ) ( |z|=8+12 i, ) then the value of ( left|z^{2}right| ) is A . 228 в. 144 ( c cdot 121 ) D. 169 E. 189 |
11 |
| 607 | Let ( z, omega ) be complex numbers such that ( vec{z}+i vec{omega}=0 ) and ( A r g(z omega)=pi ) then ( A r g(z)= ) A ( cdot frac{pi}{4} ) в. ( frac{5 pi}{4} ) c. ( frac{3 pi}{4} ) D. |
11 |
| 608 | If ( frac{x+3 i}{2+i y}=1-i, ) then the value of ( (5 x-7 y)^{2} ) is A . 1 B. 0 ( c cdot 2 ) D. |
11 |
| 609 | If ( frac{pi}{5} ) and ( frac{pi}{3} ) are respectively the arguments of ( bar{z}_{1} ) and ( z_{2}, ) what is the value of ( left(a r g z_{1}+a r g z_{2}right) ? ) |
11 |
| 610 | Solve: ( (i+3 i)^{2}(3+1) ) |
11 |
| 611 | Evaluate and write in standard form: ( (-3+2 i)^{2}-3(3-i)(-2+2 i), ) where ( i^{2}=-1 ) A ( .17+36 ) B. ( 17-36 i ) c. ( -17-36 i ) D. ( -17+36 i ) |
11 |
| 612 | Interpret the following equations geometrically on the Argand plane. ( mathbf{1}<|boldsymbol{z}-mathbf{2}-mathbf{3} boldsymbol{i}|<mathbf{4} ) A. Annular B. Straight line c. A point D. Ringg |
11 |
| 613 | 1. If the cube roots of unity are 1, o, then the roots of the equation (-1)-8=0 are (1979) (2) -1.1+2 1+202 (b) -1,1-20 1-20- c) – 1,-1,-1 (d) None of these |
11 |
| 614 | If ( boldsymbol{a}=frac{-1+sqrt{3 i}}{2}, boldsymbol{b}=frac{-1-sqrt{3 i}}{2} ) then show that ( a^{2}=b ) and ( b^{2}=a ) |
11 |
| 615 | 33. Let a, ß be the roots of the equation x2 – px + r = 0 and 9, 2B be the roots of the equation x2 -qx+r=0. Then the value ofr is (2007-3 marks) (a) (p=9)(24-p) © £(q-2p/24-p) (b) (9-p}(2p-9) (a) (20-9(29- p) |
11 |
| 616 | 19. If one the vertices of the square circumscribing the circle 2-11 = 2 is 2+ 3 i. Find the other vertices of the square. (2005- 4 Marks) |
11 |
| 617 | ff ( i^{2}=-1 ), then the value of ( sum_{n=1}^{200} i^{n} ) is A . 50 B. – -50 ( c cdot 0 ) D. 100 |
11 |
| 618 | 15. Let a complex number a, a #1, be a root of the equation zpty – – 24+1=0, where p,q are distinct primes. Show that either 1 +a+a?+ … + QP-1=0 or 1+a+ a2 + … + a9-1=0, but not both together. (2002 – 5 Marks) either 1 to 1 0, where p, q are be a root of the e |
11 |
| 619 | If ( A ) and ( B ) be two complex numbers satisfying ( frac{boldsymbol{A}}{boldsymbol{B}}+frac{boldsymbol{B}}{boldsymbol{A}}=1 . ) Then the two points represented by A and B and the origin form the vertices of A. An equilateral triangle B. An isosceles triangle which is not equilateral c. An isosceles triangle which is not right angled D. A right angled triangle |
11 |
| 620 | If ( arg left(frac{z_{1}}{z_{2}}right)=frac{pi}{2}, ) then find the value of ( left|frac{z_{1}+z_{2}}{z_{1}-z_{2}}right| ) |
11 |
| 621 | if ( z=frac{1+3 i}{1+i} ) then This question has multiple correct options ( mathbf{A} cdot operatorname{Re}(z)=2 operatorname{Im}(z) ) B. ( operatorname{Re}(z)+2 operatorname{Im}(z)=0 ) ( mathbf{c} .|z|=sqrt{5} ) D. ( a m p z=tan ^{-1} 2 ) |
11 |
| 622 | If ( frac{left(a^{2}+1right)^{2}}{2 a-i}=x+i y, ) then ( x^{2}+y^{2} ) is equal to ( ^{mathrm{A}} cdot frac{left(a^{2}+1right)^{4}}{4 a^{2}+1} ) B. ( frac{(a+1)^{2}}{4 a^{2}+1} ) c. ( frac{left(a^{2}-1^{2}right)}{left(4 a^{2}-1right)^{2}} ) D. None of these |
11 |
| 623 | Let ( z ) and ( w ) be two nonzero complex numbers such that ( |z|=|w| ) and ( arg (z)+arg (w)=pi ) Then prove that ( z=-bar{w} ) |
11 |
| 624 | If ( alpha ) and ( beta ) are different complex numbers with ( |boldsymbol{alpha}|=1, ) then what is ( left|frac{boldsymbol{alpha}-boldsymbol{beta}}{mathbf{1}-boldsymbol{alpha} overline{boldsymbol{beta}}}right| ) equal to? ( mathbf{A} cdot|beta| ) B. 2 ( c . ) D. |
11 |
| 625 | If ( boldsymbol{z}=mathbf{1}+boldsymbol{i}, ) then the multiplicative inverse of ( left.z^{2} text { is (where } i=sqrt{-1}right) ) ( mathbf{A} cdot 2 i ) в. ( 1-i ) c. ( -frac{i}{2} ) D. |
11 |
| 626 | If ( z ) is a complex number ( z=9-12 i ) find ( |z| ) A . 15 B. 16 c. 17 D. 8 |
11 |
| 627 | 18. Find the centre and radius of circle given by k.kz1 |z-B, where, z=x+iy, a=a, +id, B=B, +iß, (2004 – 2 Marks) ibing the circle |
11 |
| 628 | Solve: ( left(i^{25}right)^{3} times i ) | 11 |
| 629 | Write ( left[sqrt{2}left(cos 30^{circ}+i sin 30^{circ}right)right]^{2} ) in the form ( a+b i ) A ( .2+i sqrt{3} ) B . ( frac{3}{2}+frac{1}{2} i ) ( mathrm{c} cdot 1-i sqrt{3} ) D. ( frac{3}{2}-frac{1}{2} i ) ( E cdot 1+i sqrt{3} ) |
11 |
| 630 | If ( z ) is uni modular complex number ( frac{1+z}{1+bar{z}} ) is equal to? A. ( bar{z} ) в. ( y+i x ) c. ( y-i x ) D. |
11 |
| 631 | Find the multiplicative inverse of the complex numbers given. ( sqrt{5}+3 i ) | 11 |
| 632 | If ( z(neq-1) ) is a complex number such that ( frac{z-1}{z+1} ) is purely imaginary, then find ( |z| ) |
11 |
| 633 | Show that: ( left|begin{array}{ccc}mathbf{1} & -mathbf{2} boldsymbol{i} & mathbf{- 1} \ mathbf{3} boldsymbol{i} & boldsymbol{i}^{mathbf{3}} & -mathbf{2} \ mathbf{1} & mathbf{- 3} & -boldsymbol{i}end{array}right|=-mathbf{7}+ ) ( mathbf{1 8 i}, ) where ( boldsymbol{i}=sqrt{-mathbf{1}} ) |
11 |
| 634 | Find the square root of following: (i) ( 3+4 i ) (ii) ( -5+12 i ) |
11 |
| 635 | ( P ) represents the variable complex number ( z . ) Find the locus of ( boldsymbol{P}, ) if ( mid boldsymbol{z}- ) ( mathbf{5 i}|=| boldsymbol{z}+mathbf{5 i} mid ) |
11 |
| 636 | The simplest form of the expression ( frac{10-sqrt{-12}}{1-sqrt{-27}} ) is ( A cdot-frac{2}{7} ) в. ( frac{28}{3} ) c. ( -frac{2}{7}+i sqrt{3} ) D. ( 1+i sqrt{3} ) |
11 |
| 637 | ( frac{sqrt{3}-1}{sqrt{3}+1}=a+b sqrt{3} ; ) then what is the value of ( boldsymbol{a}+boldsymbol{b} ? ) A . -3 B. ( c cdot 3 ) D. – |
11 |
| 638 | Find the value of ( left(frac{1-i}{1+i}right)^{40} ) | 11 |
| 639 | Write the real and imaginary part of ( (i-sqrt{3})^{3} ) | 11 |
| 640 | If ( z_{1}, z_{2}, varepsilon C ) are such that ( left|z_{1}+z_{2}right|^{2}= ) ( left|z_{1}right|^{2}+left|z_{2}right|^{2} ) then ( frac{z_{1}}{z_{2}} ) is A . zero B. purely real c. purely imaginary D. complex |
11 |
| 641 | Let ( z_{k}(k=0,1,2, dots, 6) ) be the roots of the equation ( (z+1)^{7}+z^{7}=0, ) then ( sum_{k=0}^{6} R eleft(z_{k}right) ) is equal to ( mathbf{A} cdot mathbf{0} ) B. ( frac{3}{2} ) c. ( -frac{7}{2} ) D. ( frac{7}{2} ) |
11 |
| 642 | Solve the equation ( z^{2}=bar{z} ) | 11 |
| 643 | ( boldsymbol{n} in boldsymbol{N},left(frac{1+i}{sqrt{2}}right)^{8 n}+left(frac{1-i}{sqrt{2}}right)^{8 n}= ) ( A ) B. ( c ) D. – |
11 |
| 644 | If ( z neq 1 ) and ( frac{z^{2}}{z-1} ) is real, then the point represented by the complex number z lies: A. either on the real axis or on a circle passing through the origin. B. on a circle with centre at the origin c. either on the real axis or on a circle not passing through the origin. D. on the imaginary axis |
11 |
| 645 | Prove that ( (1+i)^{4}left(1+frac{1}{i}right)^{4}=16 ) | 11 |
| 646 | If ( z_{1}=frac{1}{a+i}, a neq 0 ) and ( z_{2}= ) ( frac{1}{1+b i}, b neq 0 ) are such that ( z_{1}=bar{z}_{2} ) then В. ( a=1, b=-1 ) c. ( a=2, b=1 ) D. ( a=1, b=2 ) 2 |
11 |
| 647 | Prove that ( left|frac{1-z_{1} bar{z}_{2}}{z_{1}-z_{2}}right|<1left|z_{1}right|<1< ) ( left|z_{2}right| ) |
11 |
| 648 | Find ( sqrt{boldsymbol{i}}+sqrt{-boldsymbol{i}} ) |
11 |
| 649 | Let ( z ) be a complex number of constant modulus such that ( z^{2} ) is purely imaginary then the number of possible values of z is A . 2 в. 1 ( c . ) D. infinite |
11 |
| 650 | Evaluate: ( left[i^{18}+left(frac{1}{i}right)^{25}right]^{3} ) | 11 |
| 651 | If the conjugate of ( (x+i y)(1-2 i) ) is ( (1+i), ) then A. ( x+i y=1-i ) в. ( x+i y=frac{1-i}{1-2 i} ) c. ( x-i y=frac{1-i}{1+2 i} ) D. ( _{x-i y}=frac{1-i}{1+i} ) |
11 |
| 652 | ( frac{3+2 i}{2-5 i}+frac{3-2 i}{2+5 i} ) | 11 |
| 653 | if ( z_{1}=3+7 i ) then ( left|z_{1}right| ) is A ( cdot sqrt{28} ) B. ( sqrt{58} ) ( c cdot sqrt{68} ) D. none of these |
11 |
| 654 | If ( z ) is a non-real complex number, then the minimum value of ( frac{operatorname{Im} z^{5}}{(operatorname{Im} z)^{5}} ) is ( A cdot-2 ) B . – – ( c cdot-5 ) D. – |
11 |
| 655 | 11. Let z and a be two complex numbers such that|z1 s 1, 10l s 1 and z+io |=|z-iāl=2 then z equals (1995) (a) 1 ori (b) i or-i (c) 1 or -1 (d) ior-1 lufth ynrosion |
11 |
| 656 | If ( a+i b=frac{(x+i)^{2}}{2 x^{2}+1}, ) prove that ( a^{2}+ ) ( b^{2}=frac{left(x^{2}+1right)^{2}}{left(2 x^{2}+1right)^{2}} ) |
11 |
| 657 | Assertion The greatest value of the moduli of complex numbers ( z ) satisfying the equation ( left|z-frac{4}{z}right|=2 ) is ( sqrt{5}+1 ) Reason For any two complex number ( z_{1} ) and ( z_{2} ) ( left|z_{1}-z_{2}right| geqleft|z_{1}right|-left|z_{2}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 |
11 |
| 658 | The complex number ( frac{2^{n}}{(1+i)^{2 n}}+ ) ( frac{(1+i)^{2 n}}{2^{n}}, quad n in ) ( boldsymbol{Z}, quad boldsymbol{i s} quad boldsymbol{e q u a l} quad boldsymbol{t o} ) |
11 |
| 659 | The given figure represents a multiplication operation, where each alphabet represents a different number, then what is the value of ( A ) ( A cdot O ) B. 3 ( c cdot 2 ) D. 4 |
11 |
| 660 | f ( z_{1}, z_{2} ) and ( z_{3} ) are complex numbers such that ( left|z_{1}right|=left|z_{2}right|=left|z_{3}right|= ) ( left|frac{1}{z_{1}}+frac{1}{z_{2}}+frac{1}{z_{3}}right|=1, ) then find the value of ( left|z_{1}+z_{2}+z_{3}right| ) |
11 |
| 661 | If ( frac{z-1}{z+1} ) is purely imaginary then в. ( |z|>1 ) c. ( |z|<1 ) D. |z|<2 |
11 |
| 662 | Two complex numbers are represented by ordered pairs ( z_{1}:(3,4) & z_{2}:(4,5) ) which of the following is true for ( z_{1}+z_{2} ) ( ? ) This question has multiple correct options ( mathbf{A} cdot z_{1}+z_{2}=(7,9) ) B . ( z_{1}+z_{2}=(7+9 i) ) ( mathbf{c} cdot z_{1}+z_{2}=(1,9) ) D. None of these |
11 |
| 663 | If ( (x+i y)(2-3 i)=4+ileft(frac{1}{2}right) ) then ( boldsymbol{x}+boldsymbol{y}= ) ( A cdot frac{3}{2} ) B. c. D. |
11 |
| 664 | Find the condition on the complex constants ( alpha, beta ) if ( z^{2}+alpha z+beta=0 ) has two distinct roots on the line ( operatorname{Re}(z)=1 ) |
11 |
| 665 | Locus of z if ( left|z-z_{1}right|=left|z-z_{2}right|, ) where ( z_{1} ) and ( z_{2} ) are complex numbers with the greatest and the least moduli, is A. Line parallel to the real axis B. Line parallel to the imaginary axis c. Line having a positive slope D. Line having a negative slope |
11 |
| 666 | If ( |z|=5 ) and ( w=frac{z-5}{z+5}, ) then the ( boldsymbol{R} e(boldsymbol{w}) ) is equal to ( mathbf{A} cdot mathbf{0} ) в. ( frac{1}{25} ) c. 25 D. |
11 |
| 667 | 31. Let complex numbers a and lie on circles (x-xo)? +(y-yo)2 = r2 and (x – Xo)2+(y – y)2 = 4r2. respectively. If zo = xo + iy, satisfies the equation 2/zol =r?+2, then cl= (JEE Adv. 2013) |
11 |
| 668 | Find the value of ( (4+2 i)(4-2 i) ) given that ( i^{2}=-1 ) ( mathbf{A} cdot 12 ) B . 20 ( c cdot 16-4 i ) D. ( 4+16 i ) E . ( 12-16 i ) |
11 |
| 669 | The value of ( (a+2 i)(b-i) ) is ( mathbf{A} cdot a+b-i ) B. ( a b+2 ) c. ( a b+(2 b-a) i+2 ) D. ( a b-2 ) E ( . a b+(2 b-a) i-2 ) |
11 |
| 670 | The conjugate of ( (2+i) /(3+i) ) in the form of ( a+i b ) is ( mathbf{A} cdot 13 / 2+i(15 / 2) ) B . ( 7 / 10+i(-1 / 10) ) c. ( 13 / 10+i(-15 / 2) ) D. ( 13 / 10+i(9 / 10) ) |
11 |
| 671 | Express the complex number ( 1+i sqrt{3} ) in modulus amplitude form. |
11 |
| 672 | Evaluate ( :(1+i)^{6}+(1-i)^{3} ) | 11 |
| 673 | Find multiplicative inverse of : ( frac{3+4 i}{4-5 i} ) |
11 |
| 674 | 38. The quadratic equation px)=0 with real coefficients has purely imaginary roots. Then the equation p(p(x)) = 0 has (JEE Adv. 2014) (a) one purely imaginary root (b) all real roots c) two real and two purely imaginary roots (d) neither real nor purely imaginary roots |
11 |
| 675 | 5. Let the complex number 2, 2, and zz be the vertices of a equilateral triangle. Letz, be the circumcentre of the triangle Then prove that z 2 + z 2 + z 2 = 322. (1981 – 4 Marks |
11 |
| 676 | If the number ( frac{z-2}{z+2} ) is purely imaginary number, then modulus value of z satisfies A. less than 2 B. greater than 2 c. lies between 2 and 2 D cdot ( |z|=2 ) |
11 |
| 677 | If ( boldsymbol{x}+boldsymbol{i} boldsymbol{y}=frac{boldsymbol{3}}{boldsymbol{2}+cos boldsymbol{theta}+boldsymbol{i} sin boldsymbol{theta}} ) then the value of ( (x-3)(x-1)+y^{2}= ) ( mathbf{A} cdot mathbf{0} ) B. ( c .-1 ) D. |
11 |
| 678 | Find the principal argument of the complex number ( sin frac{6 pi}{5}+ ) ( ileft(1+cos frac{6 pi}{5}right) ) A ( cdot arg (z)=frac{9 pi}{10},|z|=-2 cos frac{3 pi}{5} ) B. ( arg (z)=frac{pi}{10},|z|=-2 cos frac{3 pi}{5} ) c. ( arg (z)=frac{9 pi}{10},|z|=2 cos frac{3 pi}{5} ) D. ( arg (z)=frac{9 pi}{10},|z|=-2 cos frac{2 pi}{5} ) |
11 |
| 679 | Find the value of ( x^{3}+7 x^{2}-x+16 ) when ( boldsymbol{x}=mathbf{1}+mathbf{2} boldsymbol{i} ) ( mathbf{A} cdot-11+24 i ) B . ( -17+24 i ) c. ( -17-24 i ) D. ( -1+24 i ) |
11 |
| 680 | Perform the indicated operations: ( (8-2 i)-(-2-6 i) ) ( mathbf{A} cdot 6+4 i ) B. ( 10+4 i ) ( c cdot 10+8 i ) D. ( 10-8 i ) |
11 |
| 681 | ( boldsymbol{i}^{3}=frac{mathbf{1}}{boldsymbol{i}}=frac{mathbf{1}}{boldsymbol{i}} times frac{boldsymbol{i}}{boldsymbol{i}}=frac{boldsymbol{u}}{boldsymbol{1}}=-boldsymbol{i} ) | 11 |
| 682 | 10. Let a, beR and a² + b + 0. Suppose s={zeC:Zatibe+ER,t+0), where i=1-1. Ifz=x+iy and z e S, then (x, y) lies on (JEE Adv. 2016) (a) the circle with radius za and centre (20) for a>0, be 0 (b) the circle with radius-za and centre 2.0 for a<0,b=0 ©) the x-axis for a 60,b=0 (d) the y-axis for a = 0,0 1. L ai numbers auch that a b= 1 and |
11 |
| 683 | IF ( z_{1}=1+i, z_{2}=1-i ) find ( z_{1} z_{2} ) A. ( z_{1}+z_{2} ) В. ( z_{1}-z_{2} ) c. ( z_{1} / z_{2} ) in D. None |
11 |
| 684 | The simplified value of ( frac{1-i}{1+i} ) is: ( mathbf{A} cdot i ) B. ( -i ) ( c .1 ) D. ( -2 i ) |
11 |
| 685 | [2003] 6. If(4+) = 1 then (a) x=2n+1, where n is any positive integer (b) x= 4n , where n is any positive integer c) x=2n, where n is any positive integer (d) x = 4n+1, where n is any positive integer. |
11 |
| 686 | Find the least value of ( n ) for which ( left(frac{1+i}{1-i}right)^{n}=1 ) This question has multiple correct options A .4 B. 3 ( c .-4 ) D. |
11 |
| 687 | Express the following in the form of ( a+ ) ( boldsymbol{b} boldsymbol{i} ) (i) ( (-i)(2 i)left(-frac{1}{8} iright)^{3} ) |
11 |
| 688 | 1. I eta, bx and y be real numbers such that a b= 1 and y* 0. If the complex number z = x + iy satisfies az +b) Im ry, then which of the following is(are) possible value(s) of x? (JEE Adv. 2017) (6) -1-1-y? (d) 1-v1+y2 |
11 |
| 689 | Let tangents at ( Aleft(z_{1}right) ) and ( Bleft(z_{1}right) ) are drawn to the circle ( |z|=2 . ) Then which of the following is/are CORRECT ? A ( cdot ) The equation of tangent at ( A ) is given by ( frac{z}{z_{1}}+frac{bar{z}}{overline{z_{1}}}=2 ) B. If tangents at ( Aleft(z_{1}right) ) and ( Bleft(z_{2}right) ) intersect at ( Pleft(z_{p}right) ), then ( z_{p}=frac{2 z_{1} z_{2}}{z_{1}+z_{2}} ) c. slope of tangent at ( Aleft(z_{1}right) ) is ( frac{1}{i}left(frac{z_{1}+bar{z}_{1}}{z_{1}-bar{z}_{1}}right) ) D. If points ( Aleft(z_{1}right) ) and ( Bleft(z_{2}right) ) on the circle ( |z|=2 ) are such that ( z_{1}+z_{2}=0, ) then tangents intersect at ( frac{pi}{2} ) |
11 |
| 690 | The points z, z, zaz, in the complex plane are the vertices of a parallelogram taken in order if and only if (1983 – 1 Mark) (a) 2, + 24 = 22 + Zz (b) z, + Zg=Zz + ZA I z +22= 23 + ZA (d) None of these |
11 |
| 691 | Evaluate and write in standard form ( (4-2 i)(-3+3 i), ) where ( i^{2}=-1 ) A ( .6+18 i ) B. ( -6+18 i ) c. ( 12+18 i ) D. ( 6-18 i ) |
11 |
| 692 | 14. Ifarg(z) <0, then arg (-2) – arg(z)= (2000) |
11 |
| 693 | Calculate, ( sqrt[4]{-1 frac{1}{2}-i frac{sqrt{3}}{2}} ) | 11 |
| 694 | If ( z_{1} ) and ( z_{2} ) are two non zero complex numbers such that ( left|z_{1}+z_{2}right|=left|z_{1}right|+ ) ( left|z_{2}right| ) then ( arg z_{1}-arg z_{2} ) is equal to A. ( -pi ) в. ( frac{pi}{2} ) ( c cdot-frac{pi}{2} ) D. |
11 |
| 695 | Simplify: ( left(frac{2 i}{1+i}right)^{2} ) |
11 |
| 696 | The additive inverse of ( z ) is ( A cdot 0 ) B. ( z ) ( c .-z ) D. |
11 |
| 697 | If ( r ) is non-real and ( r=sqrt[5]{1}, ) then the value of ( 2left|1+r+r^{2}+r^{-2}-r^{-1}right| ) is equal to |
11 |
| 698 | If ( boldsymbol{x}=mathbf{9}^{frac{1}{3}} mathbf{9}^{frac{1}{9}} mathbf{9}^{frac{1}{27}} dots dots infty, boldsymbol{y}= ) ( mathbf{4}^{frac{1}{3}} mathbf{4}^{frac{-1}{9}} mathbf{4}^{frac{1}{27}} ldots . infty, ) and ( boldsymbol{z}=sum_{r=1}^{infty}(mathbf{1}+boldsymbol{i})^{-boldsymbol{r}} ) then ( arg (x+y z) ) is equal to A. B・tan” ( ^{-1}left(frac{sqrt{2}}{3}right) ) c. ( -tan ^{-1}left(frac{sqrt{2}}{3}right) ) ( D cdot-tan ^{-1}left(frac{2}{sqrt{3}}right) ) |
11 |
| 699 | 16. Ifz, and z, are two complex numbers such tahtz,l<l<z2 1-2,22 (2003 – 2 Marks) then prove that | 21 – 22 | |
11 |
| 700 | Find the value of : ( (mathbf{5} i)left(-frac{mathbf{3}}{mathbf{5}} iright) ) |
11 |
| 701 | ( |mathbf{f}| z-i R e(z)|=| z-operatorname{Im}(z) mid ) then prove that ( z ) lies on the bisectors of the quadrants. |
11 |
| 702 | If ( frac{a+3 i}{2+i b}=1-i, ) show that ( (5 a- ) ( 7 b)=0 ) | 11 |
| 703 | If ( boldsymbol{x}+boldsymbol{i} boldsymbol{y}=frac{boldsymbol{3}}{boldsymbol{2}+cos boldsymbol{theta}+boldsymbol{i} sin boldsymbol{theta}}, ) then ( x^{2}+y^{2} ) is equal to A ( .3 x-4 ) B. ( 4 x-3 ) c. ( 4 x+3 ) D. None of these |
11 |
| 704 | Find modulus of following ( (mathrm{i}) pm(4+3 i) ) ( (mathrm{ii}) pm sqrt{2}+0 i ) (iii) ( mathbf{0} pm sqrt{mathbf{2}} boldsymbol{i} ) |
11 |
| 705 | If ( left|z_{1}-z_{2}right|=left|z_{1}right|+left|z_{2}right|, ) then This question has multiple correct options A ( cdot arg left(frac{z_{1}}{z_{2}}right)=frac{pi}{2} ) B ( cdot arg left(frac{z_{1}}{z_{2}}right)=(2 n+1) pi, n in I ) ( mathbf{c} cdot z_{1} overline{z_{2}}+overline{z_{1}} z_{2} leq 0 ) D. ( z_{1}=l z_{2}, l in R ) |
11 |
| 706 | 22. f – -4 = 2, then then = 2, then the maximum value of|Z is equal to : [2009] (a) J5+1 © 2+52 (b) 2 (d) 73+1 |
11 |
| 707 | Lets, ir be non-zero complex numbers and I be the set -1) of the equation of solutions z=x+iy (x, y, ER.i= SZ + Iz + r = 0, where 3 = x – iy. Then, which of the following statement(s) is (are) TRUE? (JEE Adv. 2018) (a) IfL has exactly one element, then s (b) If si=1t|, then L has infinitely many elements c) The number of elements in Ln {z: 12-1+i)=5} is at most 2 (d) If L has more than one element, then L has infinitely many elements |
11 |
| 708 | Simplify the following: ( frac{3}{1+i}-frac{2}{2-i}+frac{2}{1-i} ) |
11 |
| 709 | Find the multiplicative inverse of each of the complex numbers given in the Exercises 11 to 13 ( -i ) |
11 |
| 710 | If ( z in C, ) the minimum value of ( |z|+ ) ( |z-5| ) is attained by This question has multiple correct options ( mathbf{A} cdot z=0 ) B . ( z=5 ) c. ( z=5 / 2 ) D. For all ( z in[0,5] ) |
11 |
| 711 | For any two complex numbers ( z_{1}, z_{2} ) we have ( left|z_{1}+z_{2}right|^{2}=left|z_{1}right|^{2}+left|z_{2}right|^{2}, ) then A ( cdot operatorname{Re}left(frac{z_{1}}{z_{2}}right)=0 ) в. ( operatorname{Im}left(frac{z_{1}}{z_{2}}right)=0 ) c. ( operatorname{Re}left(z_{1} z_{2}right)=0 ) D・Im ( left(z_{1} z_{2}right)=0 ) |
11 |
| 712 | If ( cos alpha+2 cos beta+3 cos gamma=sin alpha+ ) ( 2 sin beta+3 sin gamma=0, ) then the value of ( sin 3 alpha+8 sin 3 beta+27 sin 3 gamma ) is ( A cdot sin (a+b+gamma) ) B. ( 3 sin (alpha+beta+gamma) ) c. ( 18 sin (alpha+beta+gamma) ) D. ( sin (alpha+2 beta+3) ) |
11 |
| 713 | Let ( z_{1} ) and ( z_{2} ) be two roots of the equation ( z^{2}+a z+b=0, z ) being complex, Further, assume that the origin ( z_{1} ) and ( z_{2} ) form an equilateral triangle. Then, A ( cdot a^{2}=b ) в. ( a^{2}=2 b ) ( mathbf{c} cdot a^{2}=3 b ) D. ( a^{2}=4 b ) |
11 |
| 714 | Find the modulus of the complex number ( sqrt{mathbf{2}} boldsymbol{i}-sqrt{-mathbf{2}} boldsymbol{i} ) | 11 |
| 715 | Perform the indicated operations: ( (5+3 i)(3-2 i) ) A . ( 21-2 i ) В. ( 19-3 i ) ( c cdot 11-2 i ) D. ( 21-i ) |
11 |
| 716 | ( left(1+i+i^{2}+i^{3}+i^{4}+i^{5}right)(1+i)= ) ( mathbf{A} cdot i ) B . ( 2 i ) ( c .3 i ) D. 4 |
11 |
| 717 | Put the following in the form ( A+i B: ) ( frac{(1+i)^{2}}{3-i} ) | 11 |
| 718 | If ( |z-2+i| leq 2 ),then find the least value of ( |z| ) |
11 |
| 719 | Find the value of ( frac{i^{4 n+1}-i^{4 n-1}}{2} ) A . -1 B. ( c .-i ) ( D ) |
11 |
| 720 | The complex number ( e^{i theta} ) can be expressed in vector form by ( A cdot sin theta+i cos theta ) ( mathbf{B} cdot cos theta+i sin theta ) c. both ( (a) ) and ( (b) ) D. none of these |
11 |
| 721 | The inequality ( |z-4|0 ) B ( cdot operatorname{Re}(z)2 ) D. None of these |
11 |
| 722 | A complex number is represented by an ordered pair ( z:(3,4), ) which of the following is true for ( z ? ) A. ( z=3+4 i ) В. ( z=4+3 i ) ( mathbf{c} cdot z=3+4 ) D. None of these |
11 |
| 723 | ( arg (mathrm{bi}),(b>0) ) is A . ( pi ) в. ( frac{pi}{2} ) ( c cdot-frac{pi}{2} ) D. |
11 |
| 724 | f ( |z|=1, z neq i, ) then ( z ) can be written in the form A ( cdot frac{1+x}{1-x}(x in R) ) B. ( frac{1+i x}{1-i x}(x in R) ) c. ( frac{i+x}{1-x}(x in R) ) D. None of these |
11 |
| 725 | The principal argument of ( sqrt{2}left[cos frac{5 pi}{3}+i sin frac{5 pi}{3}right] ) is A ( cdot frac{5 pi}{3} ) B. ( frac{pi}{3} ) ( c cdot-frac{pi}{3} ) D. ( -frac{pi}{2} ) |
11 |
| 726 | 13. In a triangle PQR, ZR=. Iftan (9) and -tan () are an аге [2005] the roots of ax? + bx+c=0, a 0 then (a) a=b+c (b) c= a + b (c) b=0 (d) b=a+c |
11 |
| 727 | The value of the sum ( sum_{n=1}^{13}left(i^{n}+i^{n+1}right) ) where ( boldsymbol{i}=sqrt{-mathbf{1}} ) ( mathbf{A} cdot i ) B . ( i-1 ) ( c cdot-i ) D. |
11 |
| 728 | Find the real and imaginary parts of the complex number ( frac{a+i b}{a-i b} ) | 11 |
| 729 | The resultant complex number when ( (4+6 i) ) is divided by ( (10-5 i) ) is A ( cdot frac{2}{25}+frac{16}{25} i ) B ( cdot frac{2}{25}-frac{16}{25} i ) c. ( frac{2}{5}+frac{6}{5} ) D. ( frac{2}{5}-frac{6}{5} ) |
11 |
| 730 | Show that ( frac{sqrt{8}+i sqrt{2}}{sqrt{8}-i sqrt{2}}+frac{sqrt{8}-i sqrt{2}}{sqrt{8}+i sqrt{2}} ) is real. |
11 |
| 731 | If ( a, b, c, d epsilon R ) are such that ( a^{2}+b^{2}=4 ) and ( c^{2}+d^{2}=2 ) and if ( |a+i b|^{2}=mid c+ ) ( left.i dright|^{2}|x+i y| operatorname{then} x^{2}+y^{2}= ) ( A cdot 4 ) B. 3 ( c cdot 2 ) ( D ) |
11 |
| 732 | Simplify the multiplication of complex numbers: ( (boldsymbol{x}, boldsymbol{y}) times(mathbf{0}, mathbf{0}) ) A. ( (-x, 0) ) ) в. ( (0,-y) ) D. None of these |
11 |
| 733 | If ( sin x+i cos 2 x, cos x-i sin 2 x ) are conjugate to each other, then ( mathbf{x}= ) ( mathbf{A} cdot n pi ) B. ( (n+1) frac{pi}{2} ) ( c cdot phi ) D. ( (n+1) pi ) |
11 |
| 734 | Simplify: ( i^{2}+i^{3}+i^{4}+i^{5} ) | 11 |
| 735 | ( z_{1} z_{2} in C, z_{1}^{2}+z_{2}^{2} in ) ( boldsymbol{R}, boldsymbol{z}_{1}left(boldsymbol{z}_{1}^{2}-boldsymbol{3} boldsymbol{z}_{2}^{2}right)=boldsymbol{2} ) and ( z_{2}left(3 z_{1}^{2}-z_{2}^{2}right)=11, ) then the value of ( z_{1}^{2}+z_{2}^{2} ) is A . 2 B. 3 ( c cdot 4 ) D. 5 |
11 |
| 736 | If ( z=frac{sqrt{3}}{2}+frac{i}{2}(i=sqrt{-1}), ) then ( (1+ ) ( left.i z+z^{5}+i z^{8}right)^{9} ) is equal to A . -1 B. 1 c. 0 D ( cdot(-1+2 i)^{9} ) |
11 |
| 737 | show that: The modulus and argument of the complex number ( z_{1}=z^{2}-z, ) if ( z=cos phi+i sin phi . ) is ( mathbf{2}|sin phi / mathbf{2}|,left(frac{mathbf{3} boldsymbol{pi}+mathbf{3} boldsymbol{phi}}{mathbf{2}}right) ) |
11 |
| 738 | -TUPIL WIVU U 18. For all complex numbers 21, 22 satisfying 12, =12 and 2,-3-4i=5, the minimum value of|z,-22lis (2002) (a) 0 (6) 2 C) 7 (d) 17 |
11 |
| 739 | 14. For complex numbers z and w, prove that (zla w-lw z=z-w if and only if z=wor z w = 1. (1999 – 10 Marks) |
11 |
| 740 | If ( alpha=cos theta+i sin theta, ) then ( frac{1+alpha}{1-alpha} ) is equal to A ( cdot cot frac{theta}{2} ) B. ( cot theta ) ( mathrm{c} cdot_{i cot frac{theta}{2}} ) D. ( i tan frac{theta}{2} ) |
11 |
| 741 | Simplify the multiplication of complex numbers: ( (mathbf{0}, mathbf{1}) times(mathbf{0}, mathbf{1}) ) A ( cdot(-1,-1) ) в. (-1,0) c. (0,1) () D. None of these |
11 |
| 742 | The complex number system, denoted by ( C, ) is the set of all ordered pairs of real numbers (that is, ( boldsymbol{R} times boldsymbol{R} ) ) with the operation (denoted by ( times ) ) which satisfy multiplication ( mathbf{A} cdot(a, b) times(c, d)=(a c+b d, b c-a d) ) B. ( (a, b) times(c, d)=(a c-b d, b c+a d) ) C ( .(a, b) times(c, d)=(a+c, b+d) ) D. None of these |
11 |
| 743 | State true or false if ( z ) is a complex number then ( z bar{z} ) is purely real A. True B. False |
11 |
| 744 | The solution of the equation ( |z| z=1+2 i ) is A ( cdot frac{-2}{5^{0.25}}+frac{4}{5^{0.25}} ) B. ( frac{1}{5^{0.25}}+frac{2}{5^{0.25}} ) c. ( frac{0.5}{5^{0.25}}-frac{1}{5^{0.25}} ) D. ( frac{0.25}{5^{0.25}}-frac{0.5}{5^{0.25}} ) |
11 |
| 745 | For any two complex numbers ( z_{1} ) and ( z_{2} ) with ( left|z_{1}right| neqleft|z_{2}right|,left|sqrt{2} z_{1}+i sqrt{3} overline{z_{2}}right|^{2}+ ) ( left|sqrt{3} overline{z_{1}}+i sqrt{2} z_{2}right|^{2} ) is A ( cdot ) Less than ( 5left(left|z_{1}right|^{2}+left|z_{2}right|^{2}right) ) B cdot Greater than ( 10 mid z_{1} z_{2} ) c. Equal to ( left(2left|z_{1}right|^{2}+3left|z_{2}right|^{2}right) ) D. zero |
11 |
| 746 | 27. Let a and ß be two roots of the equation x2 + 2x +2=0. then a 15 + B15 is equal to: [JEEM 2019-9 Jan (M) (a) -256 (b) 512 (c) -512 (d) 256 |
11 |
| 747 | ( (-sqrt{-1})^{4 n+3}(n,+i v e text { integer }) ) ( mathbf{A} cdot-i ) B. ( c cdot 1 ) D. – |
11 |
| 748 | If ( z_{1} ) and ( z_{2} ) are two complex numbers such that ( left|z_{1}right|=left|z_{2}right| ) and ( arg left(z_{1}right)+ ) ( operatorname{rag}left(z_{2}right)=pi, ) then show that ( z_{1}=-bar{z}_{2} ) |
11 |
| 749 | Express the complex number given in the form ( a+i b ) ( boldsymbol{i}^{-mathbf{3 9}} ) |
11 |
| 750 | For ( |z-1|=1, ) find ( tan left[arg frac{((z-1)}{left.left.left(2-2 frac{i}{z}right)right)right]}right. ) ( mathbf{A} cdot i ) B. ( c cdot-i ) D. – |
11 |
| 751 | Find the value of : ( frac{i^{6}+i^{7}+i^{9}}{i^{2}+i^{3}} ? ) |
11 |
| 752 | All the values/s of ( (1+i)^{frac{1}{2}} ) are | 11 |
| 753 | The inequality ( |z-4|0 ) в. ( operatorname{Re}(z)2 ) D. none of these |
11 |
| 754 | Two complex numbers are represented by ordered pairs ( z_{1}:(2,4) & z_{2}:(-4,5) ) which of the following is real part for ( boldsymbol{z}_{1} times boldsymbol{z}_{2}=? ) A . -6 B. -28 ( c cdot 6 ) D. None of these |
11 |
| 755 | If ( z=i-1, ) then ( bar{z}= ) A . ( i+1 ) B. ( -i-1 ) ( c cdot-1 ) D. none of these |
11 |
| 756 | If ( z_{1}=a+i b ) and ( z_{2}=c+i d ) are complex numbers such tat ( left|z_{1}right|= ) ( left|z_{2}right|=1 & R eleft(z_{1} overline{z_{2}}right)=0, ) then the pair of complex numbers ( boldsymbol{w}_{1}=boldsymbol{a}+boldsymbol{i} boldsymbol{c} ) and ( boldsymbol{w}_{2}=boldsymbol{b}+boldsymbol{i} boldsymbol{d} ) satisfies – This question has multiple correct options ( mathbf{A} cdotleft|w_{1}right|=1 ) B ( cdotleft|w_{2}right|=1 ) ( mathbf{c} cdot operatorname{Re}left(w_{1} overline{w_{2}}right)=0 ) D. None of these |
11 |
| 757 | Find the amplitude of -4 | 11 |
| 758 | Z-1 nd @= (where z -1), then Rew is 19 z +1 JP 2005) (2003) |
11 |
| 759 | Find the value of the principal argument of the complex number ( z=frac{(1+i sqrt{3})^{2}}{(1-i)^{3}} ) | 11 |
| 760 | 12. Ifo is an imaginary cube root of unity then the value of (1994) |
11 |
| 761 | if ( boldsymbol{alpha} ) and ( beta ) are complex cube root of unity then find the value of ( boldsymbol{alpha}^{2}+boldsymbol{beta}^{2}+ ) ( boldsymbol{alpha} boldsymbol{beta} ) |
11 |
| 762 | If ( |z|=1, ) then ( |z-1| ) is ( mathbf{A} cdot|a r g(z)| ) ( mathbf{C} cdot=|arg (z)| ) D. None of these |
11 |
| 763 | 14. If both the roots of the quadratic equation x -2kx + k + k-5= 0 are less than 5, then k lies in the interval [2005] (a) (5,6] (b) (6, ) (c) (- 004) (d) [4,5] |
11 |
| 764 | Express ( (1-i)-(1+i 6) ) as ( a+i b ) | 11 |
| 765 | ( sin left(log i^{i}right)=a+i b cdot ) Find ( a ) and ( b ) | 11 |
| 766 | The value of ( 2 sqrt{-49} ) is equal to A . -14 B. None of these c. 14 D. 14 |
11 |
| 767 | Let ( left|z_{i}right|=i, i=1,2,3,4 ) and ( mid 16 z_{1} z_{2} z_{3}+9 z_{1} z_{2} z_{4}+4 z_{1} z_{3} z_{4}+z_{2} z_{3} z_{4} ) ( 48, ) then the value of ( left|frac{1}{z_{1}}+frac{4}{bar{z}_{2}}+frac{9}{bar{z}_{3}}+frac{16}{bar{z}_{4}}right| ) A . 1 B. 2 ( c cdot 4 ) ( D ) |
11 |
| 768 | 12. For positive integers n,, n, the value of the expression (1+i)”1 +(1+1°)” +(1+i%)^2 + (1+i?)^2 , where i= -1 is a real number if and only if (1996 – 1 Marks) (a) n=n2 +1 (b) n=n2-1 © n=n, (d) n,>0,n,> 0 |
11 |
| 769 | If ( z_{1}, z_{2}, z_{3}, z_{4} ) be the vertices of rhombus in argand palne and ( angle C B A=pi / 3, ) then prove that ( mathbf{2} z_{2}=z_{1}(mathbf{1}+i sqrt{mathbf{3}})+z_{3}(1-i sqrt{mathbf{3}}) ) and ( 2 z_{4}=z_{1}(1-i sqrt{3})+z_{3}(1+ ) ( boldsymbol{i} sqrt{boldsymbol{3}}) ) |
11 |
| 770 | The derivative of ( f(x)=sin ^{2} 2 x ) | 11 |
| 771 | ( mathbf{f} boldsymbol{y}=log left(frac{sqrt{(boldsymbol{x}+mathbf{1})}-mathbf{1}}{sqrt{(boldsymbol{x}+mathbf{1})}+mathbf{1}}right)+ ) ( frac{sqrt{boldsymbol{x}}}{sqrt{(boldsymbol{x}+mathbf{1})}} ) the by using substitution ( x=tan ^{2} theta, y ) reduces to |
11 |
| 772 | 5. The inequality Iz-4<z- 2 represents the region given by (1982 – 2 Marks) (a) Re(z) 20 (b) Re(z)0 (d) none of these |
11 |
| 773 | If ( i^{2}=-1, ) calculate the value of ( 3 i^{2}+ ) ( boldsymbol{i}^{3}-boldsymbol{i}^{4} ) A ( .-4-i ) B. ( -2-i ) ( c cdot 2+i ) D. ( 4+i ) E ( .6+2 i ) |
11 |
| 774 | If ( z_{1}=2 sqrt{2}(1+i) ) and ( z=1+i sqrt{3} ) then ( z_{1}^{2} z_{2}^{3} ) is equal to ( mathbf{A} cdot 128 ) в. ( 64 i ) ( c .-64 ) D. ( -128 i ) E . 256 |
11 |
| 775 | ( f arg (z)<0 ) then find ( arg (-z)-arg (z) ) | 11 |
| 776 | If ( arg (z)<0, ) then ( arg (-z)- ) ( boldsymbol{a r} boldsymbol{g}(boldsymbol{z})= ) A . ( pi ) в. ( -pi ) ( c cdot frac{pi}{2} ) D. ( -frac{pi}{2} ) |
11 |
| 777 | For a complex number z, the minimum value of ( |z|+|z-2| ) is A. 1 в. 2 ( c .3 ) D. None of these |
11 |
| 778 | 16. The conjugate of a complex number is complex number is , then that i-1 [2008] @ # © (a |
11 |
| 779 | Find the real values of x and y for which the following + (1+i)x – 2i (2 – 3i) y +i equation is satisfied 3+i 3-i – =i (1980 |
11 |
| 780 | Express the given complex number in the form ( a+i b: ) ( (1-i)^{4} ) |
11 |
| 781 | There is a complex number ( z ) with imaginary part 164 and a positive integer ( n ) such that ( frac{z}{z+n}=4 i . ) The value of ( n ) is |
11 |
| 782 | 15. Ifz, z, and z, are complex numbers such that (20005) lal= (22=1231= =1, then 1a + 2a + zs is. (a) equal to 1 © greater than 3 (b) less than 1 (d) equal to 3 11th |
11 |
| 783 | 27. let and be the rest of the equation et de la 27. Let a and b be the roots of the equation x2 – 10cx -11d=0 and those of x2 -10ax – 11b =0 are c, d then the value of a+b+c+d, when a +b+c+d, is. (2006 – 6M) |
11 |
| 784 | If ( z=-3+2 i, ) then ( frac{1}{z} ) is equal to A ( cdot frac{1}{13}(3+2 i) ) B. ( -frac{1}{13}(3+2 i) ) c. ( frac{1}{sqrt{13}}(3+2 i) ) D. ( -frac{1}{sqrt{13}}(3+2 i) ) |
11 |
| 785 | 4. Ifp and q are the roots of the equation x2+px+q=0, then (a) p=1,9=-2 (b) p=0,q=1 [2002] c) p=-2,q=0 (d) p=-2,9=1 |
11 |
| 786 | What is ( operatorname{cis} 0 ? ) | 11 |
| 787 | 4. The value of (sin 24k_icos 2ck) is (1987-2 Marks) ka a -1 6 0 6 -1 (d) i k=1 (e) None |
11 |
| 788 | The value of ( sqrt{-1} ) is ( A cdot 1 ) B. – c. ( i ) (iota) D. none of these |
11 |
| 789 | Simplify the following expressions: (A) ( 7 i^{2} ) (B) ( -6 i^{8} ) ( (C) 8 i^{7} ) |
11 |
| 790 | 12. If one root of the equation x² + px +12 = 0) is 4, while the equation x2 + px+q = 0 has equal roots , then the value of ‘q’ is [2004] (a) 4 (b) 12 (C) 3 |
11 |
| 791 | If ( (boldsymbol{w}-overline{boldsymbol{w}} boldsymbol{z}) /(1-boldsymbol{z}) ) is purely real where ( boldsymbol{w}=boldsymbol{alpha}+boldsymbol{i} boldsymbol{beta}, boldsymbol{beta} neq mathbf{0} ) and ( boldsymbol{z} neq mathbf{1}, ) then set of the values of ( z ) is ( mathbf{A} cdot z:|z|=1 ) B . ( z: z=bar{z} ) c. ( z: z neq 1 ) D ( cdot z:|z|=1, z neq 1 ) |
11 |
| 792 | Let z=x +iy be a complex number where x and y are integers. Then the area of the rectangle whose vertices are the roots of the equation : zzº + z 2 = 350 is (2009) (a) 48 (b) 32 (C) 40 000 (d) 80 |
11 |
| 793 | If ( left(frac{1+i}{1-i}right)^{n} ) is 1 find the least value of ( n ) where ( boldsymbol{n} in boldsymbol{N} ) | 11 |
| 794 | Express ( frac{mathbf{5}+i sqrt{2}}{2 i} ) in the form of ( x+i y ) | 11 |
| 795 | If ( left(i^{413}right)left(i^{x}right)=1, ) then determine the one possible value of ( x ) ( mathbf{A} cdot mathbf{0} ) B. ( c cdot 2 ) D. 3 |
11 |
| 796 | Let ( z_{1}, z_{2} in C ) and ( x=left|z_{1} z_{2}right|- ) ( boldsymbol{operatorname { R e }}left(boldsymbol{z}_{1} boldsymbol{z}_{2}right)-frac{mathbf{1}}{mathbf{2}}left|overline{z_{2}}-boldsymbol{z}_{1}right|^{2}+frac{mathbf{1}}{mathbf{2}}left(left|boldsymbol{z}_{2}right|-right. ) ( left.left|z_{1}right|right)^{2} ) then ( mathbf{A} cdot x<0 ) B. ( x=0 ) c. ( x geq 1 ) D. ( 0<x<1 ) |
11 |
| 797 | Express the complex number ( frac{2+i}{3-4 i} ) in ( a+i b ) form. |
11 |
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