Complex Numbers And Quadratic Equations Questions

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|= )
11
2 ff ( z=sqrt{20 i-21}+sqrt{21+20 i} ) then
principal value of arg z can not be
11
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.
11
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
11
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} )
11
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.
11
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
11
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
11
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
11
12 For a positive integer ( n ) ( (1-i)^{n}left(1-frac{1}{i}right)^{n}=k^{n}, ) find the value
of ( k )
11
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} )
11
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.
11
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.
11
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
11
17 Represent follow complex no. in polar
form.
( boldsymbol{z}=-mathbf{1}+sqrt{mathbf{3}} boldsymbol{i} )
11
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.
11
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 )
11
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 )
11
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
11
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
11
24 2.
[2002]
If|z-410
(b) Re(z)3
(d) Re(z)>2
11
25 If ( |z-2+i| leq 2 ), then find the greatest
value of ( |z| )
11
26 For ( boldsymbol{a}<mathbf{0}, ) arg ( boldsymbol{a}= )
A ( cdot frac{pi}{2} )
в. ( frac{-pi}{2} )
( c . pi )
D. – ( pi )
11
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 )
11
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} )
11
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
11
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 )
11
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} )
11
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
11
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) )
11
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
11
35 Find ( arg (1+sqrt{2}+i) )
A ( . pi / 16 . )
в. ( pi / 8 )
c. ( pi / 12 )
D. ( pi / 10 )
11
36 Evaluate: ( i^{24}+left(frac{1}{i}right)^{26} )
( A cdot 0 )
B.
( c cdot-1 )
D.
11
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
11
38 The argument of every complex number
is
A. Double valued
B. single valued
c. Many valued
D. Triple valued
11
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 )
11
41 Locate the complex numbers ( z=x+ )
iy such that
( |z-i|=1, arg frac{z}{z+i}=frac{pi}{2} )
11
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) )
11
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.
11
44 Express the following in the form of ( a+ )
( boldsymbol{b} boldsymbol{i} )
(i) ( (-i)(2 i)left(-frac{1}{8} iright)^{3} )
11
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.
11
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.
11
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} )
11
49 Let ( n ) be a positive integer. Then
( (i)^{4 n+1}+(-i)^{4 n+5}= )
( A )
в. 2
( c )
D. ( -i )
11
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
11
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.
11
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 )
11
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.
11
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
11
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)
11
57 Express the following in the form ( A+i B )
( frac{1}{1-cos theta+2 i sin theta} )
11
58 ( i^{n} . i^{n+1} i^{n+3}= )
( mathbf{A} cdot(i)^{n} )
B. ( -i )
( mathbf{C} cdot(-i)^{n} )
D. 1
11
59 Express the following complex numbers
in the form ( r(cos theta+i sin theta) )
( 1+i tan alpha )
11
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 )
11
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.
11
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
11
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} )
11
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 )
11
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
11
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 )
11
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 )
11
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
11
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
11
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
11
73 Represent the following complex number in trigonometric form:
( (-sqrt{3}+i)^{3} )
11
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)} )
11
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.
11
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}} )
11
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}} )
11
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})} )
11
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.
11
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
11
82 If ( z=frac{sqrt{3}+i}{2}, ) then the value of ( z^{69} ) is
A . ( -i )
B.
( c )
D.
11
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) )
11
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} )
11
85 What is the value of ( i^{i} )
Where ( i=sqrt{-1} )
11
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} )
11
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}| )
11
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
11
89 Represent the following complex number in trigonometric form:
( 1+i tan alpha )
11
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)
11
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 )
11
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.
11
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.
11
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) )
11
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)
11
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) )
11
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
11
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
11
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) )
11
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
11
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)
11
104 The greatest value of ( |z+1| ) if ( |z+4| leq )
3 is
A . 4
B. 5
( c cdot 6 )
D. None of these
11
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
11
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
11
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 )
11
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.
11
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| )
11
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 )
11
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
11
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 )
11
116 ( operatorname{Let} z=left(a-frac{i}{2}right) ; in mathrm{R} . ) Then ( |i+z|^{2} )
( |i-z|^{2} ) is equal to
11
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 )
11
118 What is the square of the modulus of
the complex number ( 2+3 i )
11
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 )
11
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} )
11
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} )
11
122 Represent the following complex number in trigonometric form:
( frac{3}{2}-i frac{sqrt{3}}{2} )
11
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.
11
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} )
11
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
11
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 )
11
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
11
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
11
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
11
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
11
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.
11
133 The number of solutions of ( z^{2}+|z|=0 )
is
( mathbf{A} cdot mathbf{1} )
B . 2
( c .3 )
D. Infinite
11
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
11
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
11
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).
11
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
11
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
11
139 Evaluate :
( (-sqrt{-1})^{4 n+3}, n in N )
A . ( -i )
B.
( c cdot 1 )
D. –
11
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) )
11
142 ( (i)^{457} )
( A cdot-1 )
B . ( -i )
( mathbf{c} cdot i )
D. 1
11
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)
11
144 Find the multiplicative of ( 2-3 i a=2 )
( boldsymbol{b}=-mathbf{3} )
11
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
11
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 )
11
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
11
148 ( z ) is a complex number. If ( a=|x|+|y| )
and ( b=sqrt{2}|x+i y| )
11
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} )
11
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}) )
11
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 )
11
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 )
11
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
11
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
11
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} )
11
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} )
11
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
11
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 )
11
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
11
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
11
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} )
11
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
11
166 Indicate the point of the complex plane
( z ) which satisfy the following equation ( boldsymbol{operatorname { R e }} boldsymbol{z}^{2}=mathbf{0} )
11
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} )
11
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
11
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} )
11
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 )
11
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| )
11
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)
11
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
11
176 Find the value of ( sum_{n=0}^{100} i^{n !}(text {where}, i= )
( sqrt{-1}) )
11
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
11
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} )
11
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 )
11
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} )
11
182 Find the value of the complex number
( left(i^{25}right)^{3} )
11
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 )
11
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
11
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 )
11
186 Find the multiplicative inverse of the
complex number ( 4-3 i )
11
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} )
11
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}} )
11
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.
11
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) )
11
191 Represent the complex number ( 2+3 i )
in argand plane
11
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} )
11
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
11
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
11
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
11
196 If ( z_{1}=2-i, z_{2}=-2+i ),find
[
operatorname{Im}left(frac{1}{z_{1} overline{z_{2}}}right)
]
11
197 Find the value of ( 2 i^{2}+6 i^{3}+3 i^{16}- )
( mathbf{6} i^{19}+4 i^{25} )
11
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
11
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
11
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
11
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 )
11
202 Find the number of integral solution of
( (1-i)^{x}=2^{x} )
11
203 The number of complex numbers ( z )
satisfies ( boldsymbol{R e}left(boldsymbol{z}^{2}right)=mathbf{0},|boldsymbol{z}|=sqrt{mathbf{3}} )
11
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
11
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 )
11
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 )
11
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
11
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)
11
209 25. A value of for which
2+ Zi sine
one is purely imaginary, is:
JEEM 2016
Sin
Blo
11
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 )
11
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
11
212 If ( A r gleft(frac{z+1}{z-1}right)=frac{pi}{6}, ) then find the locus of
( mathbf{z} )
11
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) )
11
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} )
11
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 )
11
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
11
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
11
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}} )
11
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
11
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) )
11
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
11
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} )
11
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}} )
11
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 )
11
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} )
11
227 7.
Let z and w be complex numbers such that z +iū= 0 and
arg zw=1. Then arg z equals
[2004]
11
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)
11
229 Represent the following complex number in trigonometric form:
-1
11
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 )
11
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) *
11
232 Find the smallest natural number such
that, ( left(frac{1+i}{1-i}right)^{n}=1 )
11
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 )
11
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 )
11
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
11
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)
11
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| )
11
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
11
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}} )
11
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} )
11
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 )
11
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 )
11
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} )
11
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
11
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.
11
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
11
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 )
11
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.
11
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} )
11
250 Find real values of ( theta ) for which
( left(frac{4+3 i sin theta}{1-2 i sin theta}right) ) is purely real
11
251 Write the additive inverse of the
complex number ( 4-3 i )
11
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
11
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
11
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 )
11
257 The simplified value of ( frac{1+i}{1-i} ) is
( mathbf{A} cdot mathbf{1} )
в.
( c cdot-i )
D. ( 2 i )
11
258 The principal argument of ( 1+sqrt{2}+i ) is
A ( cdot frac{pi}{3} )
в.
c.
D.
11
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 )
11
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
11
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
11
262 ( sqrt{-3} sqrt{-75}= )
A . 15
в. 15
c. -15
D. – -15
11
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} )
11
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
11
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 )
11
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} )
11
268 ( sqrt{i}+sqrt{-i}=? )
A ( cdot sqrt{2} )
B. ( -sqrt{2} )
( c cdot pm frac{1}{sqrt{2}} )
D. ( pm sqrt{2} )
11
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} )
11
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.
11
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) )
11
272 Solve:
( left(frac{2 i}{1+i}right)^{2} )
( mathbf{A} cdot-i )
B. ( i )
( mathbf{c} cdot 2 i )
D. ( 1-i )
11
273 The value of ( i^{2}+i^{4}+i^{6}+ldots i^{2(2 n+1)}=? )
A . -1
B.
( c cdot-i )
( D )
11
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
11
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
11
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} )
11
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
11
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}} )
11
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
11
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
11
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
11
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
11
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 )
11
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 )
11
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
11
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
11
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)
11
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.
11
292 Find the multiplicative inverse of the
complex numbers given the following:
( 4-3 i )
11
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. –
11
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 )
11
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
11
296 Simplify the following:
( frac{3-i}{2+i}+frac{3+i}{2-i} )
11
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 )
11
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.
11
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.
11
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
11
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
11
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
11
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
11
305 Conjugate surd of ( a+sqrt{6} ) is
A ( .6-sqrt{a} )
B. ( 6+sqrt{a} )
c. ( sqrt{6}-a )
D. ( a-sqrt{6} )
11
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
11
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} )
11
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} )
11
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 )
11
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) )
11
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 )
11
312 ( left(frac{1+i}{1-i}right)^{4}+left(frac{1-i}{1+i}right)^{4}= )
A.
B.
( c cdot 2 )
( D )
11
313 Find the multiplicative inverse of each of the complex numbers given in the Exercises 11 to 13
( 4-3 i )
11
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)<
11
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
11
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
11
318 Find the greatest and the least values of
the moduli of complex numbers ( z )
satisfying the equation ( |z-4 / z|=2 )
11
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 )
11
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} )
11
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
11
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 )
11
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
11
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}} )
11
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} )
11
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
11
331 Evaluate the following
( left(i^{77}+i^{70}+i^{87}+i^{414}right)^{3} )
11
332 sum
6. The value of the sum 2 (+*+ **), where i = -1, equals
(1998 – 2 Marks)
(a) i 6 i-1 © -i (d) 0
11
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
11
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
11
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}} ) )
11
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) )
11
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
11
339 ( (1+i)^{-1} )
A ( cdot frac{1+i}{2} )
B. ( frac{1-i}{2} )
( mathbf{C} cdot 1+i )
D. ( 1-i )
11
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} )
11
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| )
11
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
11
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.
11
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
11
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
11
347 If ( a=frac{(3 i+1)}{10} ) and ( b=(2 i+3) times 10 ), then
the value of ( (2 a b) ) is
11
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
11
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) –
11
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} )
11
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
11
352 ( boldsymbol{i}^{242}= )
( mathbf{A} cdot i )
B. ( -i )
c. 1
D. –
11
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} )
11
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
11
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
11
356 Represent ( frac{1}{1-cos theta+2 i sin theta} ) in the
form ( boldsymbol{A}+boldsymbol{i} boldsymbol{B} )
11
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} )
11
358 locate the point representing the
complex numbers ( z ) on the Argand diagram for which
( |z+i|=|z-2| )
11
359 ( mathbf{f}|boldsymbol{z}-mathbf{1}+boldsymbol{i}|+|boldsymbol{z}+boldsymbol{i}|=mathbf{1} ) then find
range of principle argument of z.
11
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 )
11
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.
11
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 )
11
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.
11
364 Find the value of
( (1+i)^{6}+(1-i)^{6} )
11
365 Find the principal argument of the complex number ( sin frac{6 pi}{5}+i(1+ )
( left.cos frac{6 pi}{5}right) )
11
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
11
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} )
11
369 Simplify: ( (-sqrt{3}+sqrt{-2})(2 sqrt{3}-i)=(a+ )
ib) ( (-sqrt{3}+sqrt{-2}) . ) Find value of a and ( b )
11
370 The complex number ( frac{1+2 i}{1-i} ) lies in the
quadrant :
( A )
B. I
c. ॥॥
D. IV
11
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} )
11
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
11
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
11
375 Find the multiplicative inverse of the complex numbers given the following:
( sqrt{5}+3 i )
11
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 )
11
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) )
11
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) )
11
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) )
11
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} )
11
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 )
11
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.
11
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 )
11
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 )
11
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
11
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)
11
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 )
11
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
11
395 Indicate the point of the complex plane
( z ) which satisfy the following equation. ( z^{2}+|bar{z}|=0 )
11
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.
11
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} )
11
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} )
11
399 The value of ( 1+i^{2}+i^{4}+i^{6}+i^{8}+ )
( +i^{20} ) is :
11
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
11
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
11
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 )
11
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 )
11
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
11
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
11
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 )
11
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)
11
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,
11
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 )
11
412 If ( |z|=3, ) then ( frac{9+z}{1+bar{z}} ) equals
( A )
B.
( c cdot 3 z )
D. ( z+bar{z} )
11
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
11
415 12. Find all non-zero complex numbers Z satisfying Z = iZ.
(1996 – 2 Marks)
11
416 locate the point representing the
complex numbers ( z ) on the Argand diagram for which
( |z| geq 3 )
11
417 For ( boldsymbol{a}>mathbf{0}, ) arg ( (boldsymbol{i} boldsymbol{a})= )
A ( cdot frac{pi}{2} )
B. ( -frac{pi}{2} )
( c . pi )
D. – ( pi )
11
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.
11
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} )
11
420 The value of ( (i+sqrt{3})^{100}+ ) ( (i-sqrt{3})^{100}+2^{100}= )
A .
B. – –
( c cdot 0 )
D.
11
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}} )
11
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
11
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 )
11
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
11
425 17. Prove that there exists no complex number z such that
<
and
a,z' = 1 where la <2. (2003 – 2 Marks)
r=1
11
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
11
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.
11
429 17. Ifx is real, the maximum value of –
3×2 +9x+17
3×2 + 9x +7
is
[2006]
(b) 41
(d) 17
©
1
11
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
11
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} )
11
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 )
11
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
11
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.
11
437 Show that:
( left(frac{1+i}{sqrt{2}}right)^{8}+left(frac{1-i}{sqrt{2}}right)^{8}=2 )
11
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} )
11
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 )
11
440 Find ‘x’ and ‘y’ if ( x^{2}-y^{2}-i(2 x+y)= )
( mathbf{2} i )
11
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) )
11
442 Modulus of nonzero complex number z
satisfying ( bar{z}+z=0 ) and ( |z|^{2}-4 z i= )
( z^{2} ) is
11
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} )
11
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} )
11
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} )
11
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
11
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} )
11
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| )
11
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
11
451 Find the multiplicative inverse of
( frac{sqrt{3}}{2}-frac{1}{2} i )
11
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
11
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
11
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
11
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
11
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
11
458 ( (1+i)^{8}+(1-i)^{8}= )
A . 16
B. -16
( c .32 )
D. -32
11
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) )
11
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} )
11
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 )
11
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} )
11
464 Prove that ( x^{2}+y^{2}=9 ) where ( z=x+ )
( i y ) and ( |z+6|=|2 z+3| )
11
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
11
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.
11
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 )
11
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 )
11
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} )
11
471 The value of ( sqrt{-36} ) is
( A cdot 6 )
в. -6
( c cdot 6 i )
D. None of these
11
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}
11
473 ( i+frac{1}{i}= )
A.
B. –
c.
D. 2
11
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
11
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
11
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
11
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
11
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 )
11
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} )
11
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.
11
482 ( frac{(1+i)^{2011}}{(1-i)^{2009}}= )
A . -1
B.
( c cdot 2 )
D. –
11
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} )
11
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 )
11
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 )
11
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)
11
488 Write the argument of ( (1+sqrt{3})(1+ )
( i)(cos theta+i sin theta) )
11
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
11
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
11
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
11
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)= )
11
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}} )
11
494 If ( arg z<0 ) then ( arg (-z)-arg z ) is
equal to
A . ( pi )
в. ( -pi )
( c cdot-frac{pi}{2} )
D.
11
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.
11
497 Simplify: ( (14+2 i)(7+12 i) ) where ( i= )
( sqrt{-1} )
11
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 )
11
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 )
11
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 )
11
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
11
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
11
508 8.
The number of real solutions of the equation
*2-3x/+2 = 0 is
(a) 3
(b) 2
(c) 4
(d) 1
11
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
11
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?
11
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 )
11
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
11
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
11
516 Evaluate ( : sqrt{-mathbf{1 6}}+mathbf{3} sqrt{-mathbf{2 5}}+sqrt{-mathbf{3 6}}- )
( sqrt{-mathbf{6 2 5}} )
11
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} )
11
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
11
519 Find the square root of ( 4 a b-2left(a^{2}-right. )
( left.b^{2}right) i )
11
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)
11
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 )
11
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| )
11
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. –
11
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) )
11
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
11
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)
11
527 Represent the following complex number in trigonometric form:
( sqrt{3} i )
11
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.
11
529 Find real values of ( x ) and ( y ) if
( frac{x-1}{3+i}+frac{y-1}{3-i}=i )
11
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
11
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
11
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
11
534 If ( n=4 m+3, m ) is an integer, then ( i^{n} )
is equal to:
A . ( -i )
B.
( c . i )
D. –
11
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 )
11
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 )
11
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
11
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. –
11
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
11
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
11
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} )
11
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
11
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.
11
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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