# 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 NoQuestionsClass
1If ( z_{1}, z_{2} ) are roots of equation ( z^{2}-a z+ )
( a^{2}=0, ) then ( left|frac{z_{1}}{z_{2}}right|= )
11
2ff ( z=sqrt{20 i-21}+sqrt{21+20 i} ) then
principal value of arg z can not be
11
3If ( 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
4Number 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
5Let ( 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
6Find 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
7Evalaute ( :left[i^{18}+left(frac{1}{i}right)^{25}right]^{3} )11
821. 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
9If ( 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
10Find the real and imaginary parts of the complex number ( z=frac{3 i^{20}-i^{19}}{2 i-1} )11
1124. 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
12For a positive integer ( n ) ( (1-i)^{n}left(1-frac{1}{i}right)^{n}=k^{n}, ) find the value
of ( k )
11
13State 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
14If ( 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
15The smallest integer n such that ( left(frac{1+i}{1-i}right)^{n}=1 ) is
A . 16
B. 12
( c cdot 8 )
D.
11
16f ( 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
17Represent follow complex no. in polar
form.
( boldsymbol{z}=-mathbf{1}+sqrt{mathbf{3}} boldsymbol{i} )
11
18Solve: ( left(frac{1}{1-2 i}+frac{3}{1+i}right)left(frac{3+4 i}{2-4 i}right) )11
19Amplitude of ( frac{1+sqrt{3} i}{sqrt{3}+i} ) is
A ( cdot frac{pi}{3} )
в. ( frac{pi}{2} )
( c cdot 0 )
D.
11
20If ( 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
21If ( 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
22If ( (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
23If ( bar{z} ) lies in the third quadrant then ( z ) lies
in the
11
242.
[2002]
If|z-410
(b) Re(z)3
(d) Re(z)>2
11
25If ( |z-2+i| leq 2 ), then find the greatest
value of ( |z| )
11
26For ( boldsymbol{a}<mathbf{0}, ) arg ( boldsymbol{a}= )
A ( cdot frac{pi}{2} )
в. ( frac{-pi}{2} )
( c . pi )
D. – ( pi )
11
27Find 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
28For 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
29Complex 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
30A 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
31The 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
32Simplify 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
33Which 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
34If ( 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
35Find ( arg (1+sqrt{2}+i) )
A ( . pi / 16 . )
в. ( pi / 8 )
c. ( pi / 12 )
D. ( pi / 10 )
11
36Evaluate: ( 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
38The argument of every complex number
is
A. Double valued
B. single valued
c. Many valued
D. Triple valued
11
39Calculate ( sqrt[3]{-1} )11
40If ( i z^{3}+z^{2}-z+i=0, ) then ( |z|=? )
B . |z| = 2
c. ( |z|=3 )
D. ( |z|=4 )
11
41Locate the complex numbers ( z=x+ )
iy such that
( |z-i|=1, arg frac{z}{z+i}=frac{pi}{2} )
11
42Let ( 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
43If ( 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
44Express 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
46Show that if ( left|frac{z-3 i}{z+3 i}right|=1, ) then ( z ) is a
real number.
11
47Find 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
48If ( 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
49Let ( n ) be a positive integer. Then
( (i)^{4 n+1}+(-i)^{4 n+5}= )
( A )
в. 2
( c )
D. ( -i )
11
50If ( 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
51If ( 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
53Solve for ( z:(i-z)(1+i)=2 i )11
54The 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
55Let ( 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
56If ( 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
57Express 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
59Express the following complex numbers
in the form ( r(cos theta+i sin theta) )
( 1+i tan alpha )
11
60Find the modulus and the argument of the complex number ( z=-sqrt{3}+i )11
61Represent the following complex number in trigonometric form:
( 3-4 i )
11
62If ‘ ( 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
6316. 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
64If ( 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
65The 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
66Suppose ( 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
67What 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
68Express ( frac{-1+i}{sqrt{2}} ) in the polar form11
69The 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
70If ( 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
71If ( 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
7236. 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
(d) 4
11
73Represent the following complex number in trigonometric form:
( (-sqrt{3}+i)^{3} )
11
74If ( (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
75If ( 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
76Find 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
77Let ( 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
78Express 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
79Find the real numbers ( x ) and ( y, ) if ( (x- ) ( i y)(1+i) ) is the conjugate of ( -3-2 i )11
802.
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
81Given ( : 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
82If ( z=frac{sqrt{3}+i}{2}, ) then the value of ( z^{69} ) is
A . ( -i )
B.
( c )
D.
11
83Let ( 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
84The 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
85What is the value of ( i^{i} )
Where ( i=sqrt{-1} )
11
86If ( 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
87If ( z ) be a complex number satisfying
( z^{4}+z^{3}+2 z^{2}+z+1=0, ) then find the
value of ( |vec{z}| )
11
88If the square of ( (a+i b) ) is real, then
( boldsymbol{a} boldsymbol{b}= )
( mathbf{A} cdot mathbf{0} )
B.
c. -1
D. 2
11
89Represent the following complex number in trigonometric form:
( 1+i tan alpha )
11
90If ( frac{(1+i)^{2}}{2-i}=x-i y, ) then find the value of ( boldsymbol{x}+boldsymbol{y} )11
91The 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
92Find 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
93Evaluate ( 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
94The 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
95Find the value of ( (-1+sqrt{-3})^{2}+ ) ( (-1-sqrt{-3})^{2} )11
96If ( (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
97V3+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
98Verify 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
9927. 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
100If ( (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
101f ( 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
10231. 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
1036.
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
104The greatest value of ( |z+1| ) if ( |z+4| leq )
3 is
A . 4
B. 5
( c cdot 6 )
D. None of these
11
10518)
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
106If ( 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
107Solve 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
108Find 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
110Find the conjugate of ( frac{(3-2 i)(2+3 i)}{(1+2 i)(2-i)} )11
111locate the point representing the
complex numbers ( z ) on the Argand diagram for which
( |z-1|=|z-3|=|z-i| )
11
112In 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
113Prove 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
114If ( 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
115What 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
117If ( 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
118What is the square of the modulus of
the complex number ( 2+3 i )
11
119Find 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
120If 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
121When ( 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
122Represent the following complex number in trigonometric form:
( frac{3}{2}-i frac{sqrt{3}}{2} )
11
123Represent 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
124If ( 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
125For ( 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
126Let ( 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
12729. 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
128Simplify the following ( : i^{457} )11
129If ( 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
130Consider ( 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
131If ( z=frac{1+i sqrt{3}}{sqrt{3}+i}, ) then ( (bar{z})^{100} ) lies in
c. ( I I I ) quadrant
D. ( I V ) quadrant
11
132Evaluate:
( 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
133The number of solutions of ( z^{2}+|z|=0 )
is
( mathbf{A} cdot mathbf{1} )
B . 2
( c .3 )
D. Infinite
11
134In 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
135Let ( 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
13618. 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
137The 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
138If ( 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
139Evaluate :
( (-sqrt{-1})^{4 n+3}, n in N )
A . ( -i )
B.
( c cdot 1 )
D. –
11
140State whether the given statement is true or false ( overline{left(z^{-1}right)}=(bar{z})^{-1} )11
141If ( 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
143If ( frac{z+1}{z+i} ) is purely imaginary, then z lies
on a
A. straight lone
B. circle
D. circle passing through (1,1)
11
144Find the multiplicative of ( 2-3 i a=2 )
( boldsymbol{b}=-mathbf{3} )
11
145In 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
146Find 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
14727. 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
150Find 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
151The 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
152Solve ( frac{1}{1+i} )11
153If ( 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
154The 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
155If 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
156The 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
157Find 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
15820. 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
159What is value of ( (-i)^{12} )11
160State 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
161If ( 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
162If 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
163Assertion (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
164The 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
166Indicate the point of the complex plane
( z ) which satisfy the following equation ( boldsymbol{operatorname { R e }} boldsymbol{z}^{2}=mathbf{0} )
11
167The 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
16826. 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
169Let ( 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
170Find 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
172Find 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
17428. 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
17525. 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
176Find the value of ( sum_{n=0}^{100} i^{n !}(text {where}, i= )
( sqrt{-1}) )
11
17737. 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
179The 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
180Find the modulus and argument of ( z= ) ( frac{3+2 i}{-2+i} )11
181Find 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
182Find the value of the complex number
( left(i^{25}right)^{3} )
11
183Given 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
184If ( 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
185Find 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
186Find the multiplicative inverse of the
complex number ( 4-3 i )
11
187If ( 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
188If ( 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
189In 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
190For 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
191Represent the complex number ( 2+3 i )
in argand plane
11
192The 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
19322. If z is a complex number of unit modulus and
argument e, then arg (1 ) equals: JJEE M 2013]
Itz
1+2
(a) –
(b)

(d) – 0
11
1949.
(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
1952.
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
196If ( z_{1}=2-i, z_{2}=-2+i ),find
[
operatorname{Im}left(frac{1}{z_{1} overline{z_{2}}}right)
]
11
197Find the value of ( 2 i^{2}+6 i^{3}+3 i^{16}- )
( mathbf{6} i^{19}+4 i^{25} )
11
1989.
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
199If ( 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
200If ( 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
201If ( 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
202Find the number of integral solution of
( (1-i)^{x}=2^{x} )
11
203The number of complex numbers ( z )
satisfies ( boldsymbol{R e}left(boldsymbol{z}^{2}right)=mathbf{0},|boldsymbol{z}|=sqrt{mathbf{3}} )
11
204If ( 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
206If ( 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
207If ( 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
20825.
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
20925. A value of for which
2+ Zi sine
one is purely imaginary, is:
JEEM 2016
Sin
Blo
11
210If ( 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
211If 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
212If ( A r gleft(frac{z+1}{z-1}right)=frac{pi}{6}, ) then find the locus of
( mathbf{z} )
11
213The 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
214Find 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
215If ( 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
216If ( 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
217The 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
21815.
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
219If ( |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
2208.
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
221f ( 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
222If ( 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
223If ( 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
224If ( 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
225Inequality ( 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
226The 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
2277.
Let z and w be complex numbers such that z +iū= 0 and
arg zw=1. Then arg z equals
[2004]
11
228If ( (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
229Represent the following complex number in trigonometric form:
-1
11
230The 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
23111.
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
232Find the smallest natural number such
that, ( left(frac{1+i}{1-i}right)^{n}=1 )
11
233If ( 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
235If ( 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
23632. 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
237If ( 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
238Complex 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
239If ( 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
240The 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
241For 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
242If ( 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
243Express 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
244If ( 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
245Find 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
246For 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
247The 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
248If ( 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
249If ( 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
250Find real values of ( theta ) for which
( left(frac{4+3 i sin theta}{1-2 i sin theta}right) ) is purely real
11
251Write 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
253ff ( z_{1}=1+i=sqrt{3}+i, ) then the principle ( arg left(frac{z_{1}}{z_{2}}right) )11
254Number 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
255If ( frac{a+3 l}{2+i b}=1-1, ) show that ( (5 a- ) ( mathbf{7 b})=mathbf{0} )11
256If ( 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
257The simplified value of ( frac{1+i}{1-i} ) is
( mathbf{A} cdot mathbf{1} )
в.
( c cdot-i )
D. ( 2 i )
11
258The principal argument of ( 1+sqrt{2}+i ) is
A ( cdot frac{pi}{3} )
в.
c.
D.
11
259The 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
260If ( 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
26116. 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
(d) i<m<4
11
262( sqrt{-3} sqrt{-75}= )
A . 15
в. 15
c. -15
D. – -15
11
263A 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
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 to11
266( frac{1+i}{1-i}-frac{1-i}{1+i} )
( mathbf{A} cdot-2 i )
B.
( c cdot 2 i )
( D )
11
267The 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
269Let ( 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
270The 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
271Given: ( 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
272Solve:
( left(frac{2 i}{1+i}right)^{2} )
( mathbf{A} cdot-i )
B. ( i )
( mathbf{c} cdot 2 i )
D. ( 1-i )
11
273The value of ( i^{2}+i^{4}+i^{6}+ldots i^{2(2 n+1)}=? )
A . -1
B.
( c cdot-i )
( D )
11
274What is the smallest positive integer for which ( (1+i)^{2 n}=(1-i)^{2 n} ? )11
275Let ( z=x+ ) iy and ( v=frac{1-i z}{z-i}, ) show
that if ( |boldsymbol{v}|=1, ) then ( boldsymbol{z} ) is purely real
11
276If ( 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
277For 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
278Simplify 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
280The 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
2814. 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
28233. 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
284Find 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
285What 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
286Evaluate (i) ( boldsymbol{i}^{998} )11
287The amplitude of ( 1+cos x-i sin x ) is
A ( cdot frac{x}{2} )
B.
( c cdot-frac{x}{2} )
D. ( 2 x )
11
28830. 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
289Two 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
29023. 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
291Find 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
292Find the multiplicative inverse of the
complex numbers given the following:
( 4-3 i )
11
293Let ( 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
294Find 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
295The 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
D. none of these
11
296Simplify 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
298If ( (x+i y)^{3}=u+i v, ) then prove that ( frac{u}{x}+frac{v}{y}=4left(x^{2}-y^{2}right) )11
299If ( 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
300The 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
301If ( 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
30215. If I z + 4 ls 3, then the maximum value of
12+1| is
[20071
(a) 6 (b) 0 C) 4 (d) 10
11
303If ( 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
304Let ( 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
305Conjugate surd of ( a+sqrt{6} ) is
A ( .6-sqrt{a} )
B. ( 6+sqrt{a} )
c. ( sqrt{6}-a )
D. ( a-sqrt{6} )
11
306The 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
307If ( 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
308The 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
309If ( 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
310The 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
311The 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
313Find the multiplicative inverse of each of the complex numbers given in the Exercises 11 to 13
( 4-3 i )
11
3144.
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
316If ( |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
317if ( 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
318Find the greatest and the least values of
the moduli of complex numbers ( z )
satisfying the equation ( |z-4 / z|=2 )
11
319If ( 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
320If ( 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
321Find the amplitude of ( 1+i sqrt{3} )11
322Prove 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
324If 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
325If ( 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
326Find the value of ( (1+i)^{3}+(1-i)^{6} )11
327The 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
328Simplify: ( (-sqrt{3}+sqrt{-2})(2 sqrt{3}-i) )11
329If ( 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
330If ( 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
331Evaluate the following
( left(i^{77}+i^{70}+i^{87}+i^{414}right)^{3} )
11
332sum
6. The value of the sum 2 (+*+ **), where i = -1, equals
(1998 – 2 Marks)
(a) i 6 i-1 © -i (d) 0
11
3331.
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
334The 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
335The 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
338Simplify 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
340If ( alpha ) and ( beta ) are different complex number with ( |beta|=1 ) then find ( left|frac{beta-alpha}{1-bar{alpha} beta}right| )11
341If ( 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
343f ( 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
344The 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
345Two 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
346The 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
347If ( a=frac{(3 i+1)}{10} ) and ( b=(2 i+3) times 10 ), then
the value of ( (2 a b) ) is
11
3483.
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
34910. 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
350The 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
351Two 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
353Prove 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
354Let ( 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
355A 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
356Represent ( frac{1}{1-cos theta+2 i sin theta} ) in the
form ( boldsymbol{A}+boldsymbol{i} boldsymbol{B} )
11
357If ( (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
358locate 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
360Find 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
361The 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
363In 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
364Find the value of
( (1+i)^{6}+(1-i)^{6} )
11
365Find the principal argument of the complex number ( sin frac{6 pi}{5}+i(1+ )
( left.cos frac{6 pi}{5}right) )
11
3669.
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
368For ( z=x+i y ) find the real and
imaginary part of ( e^{z} )
11
369Simplify: ( (-sqrt{3}+sqrt{-2})(2 sqrt{3}-i)=(a+ )
ib) ( (-sqrt{3}+sqrt{-2}) . ) Find value of a and ( b )
11
370The complex number ( frac{1+2 i}{1-i} ) lies in the
( A )
B. I
c. ॥॥
D. IV
11
371If ( 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
372f ( left|x^{2}-7right| leq 9 ) then find the values of ( x )11
373Two 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
374Assertion
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
375Find the multiplicative inverse of the complex numbers given the following:
( sqrt{5}+3 i )
11
376Dividing ( 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
377Write ( (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
378Express 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
379Find 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
380Find the additive inverse of ( frac{9}{2+i sqrt{5}} )11
381Find the modulus and the argument of the complex number ( z=-1-i sqrt{3} )11
382If ( 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
383Write the conjugate of complex number ( frac{5 i}{7+i} )11
384Which 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
386If ( 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
387The 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
388Find the modulus and argument of the complex numbers. ( frac{5-i}{2-3 i} )11
389Find the Modulas and argument of ( frac{1+i}{1-i} )11
39023. 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
391Let 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
(d) Arg (2-2)= Arg (22-24)
11
392Find the value of ( sum_{n=1}^{13} i^{n}+i^{n+1} )11
393Express ( left(frac{1-i}{1+i}right)^{1000} ) in the form of ( a+ )
ib. Find ( a+b )
11
394The 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
395Indicate the point of the complex plane
( z ) which satisfy the following equation. ( z^{2}+|bar{z}|=0 )
11
396If ( 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
397If 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
398If ( 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
399The value of ( 1+i^{2}+i^{4}+i^{6}+i^{8}+ )
( +i^{20} ) is :
11
400If ( 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
401The 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
402If ( 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
403If ( 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
404Solve ( sin 2 x+cos 4 x=2 )11
405If ( left|z+frac{2}{z}right|=2, ) then the maximum value of ( |z| ) is ( sqrt{m}+1 . ) Find ( m )11
406A 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
407If ( 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
40920. 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)
11
410In 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
411The 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
412If ( |z|=3, ) then ( frac{9+z}{1+bar{z}} ) equals
( A )
B.
( c cdot 3 z )
D. ( z+bar{z} )
11
413The 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
41412.
If w
and
1-1, then z lies on 120051
(a)
(c)
an ellipse
a straight line
(b) a circle
(d) a parabola
11
41512. Find all non-zero complex numbers Z satisfying Z = iZ.
(1996 – 2 Marks)
11
416locate the point representing the
complex numbers ( z ) on the Argand diagram for which
( |z| geq 3 )
11
417For ( boldsymbol{a}>mathbf{0}, ) arg ( (boldsymbol{i} boldsymbol{a})= )
A ( cdot frac{pi}{2} )
B. ( -frac{pi}{2} )
( c . pi )
D. – ( pi )
11
418The 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
420The value of ( (i+sqrt{3})^{100}+ ) ( (i-sqrt{3})^{100}+2^{100}= )
A .
B. – –
( c cdot 0 )
D.
11
421If ( 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
422If ( 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
423If ( 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
424If ( 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
42517. Prove that there exists no complex number z such that
<
and
a,z' = 1 where la <2. (2003 – 2 Marks)
r=1
11
426In 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
427Find ( (-5 i)left(frac{3}{5} iright) )11
428If ( z ) is purely real and ( R e(z)<0, ) then
( arg (x) ) is
A . 0
в. ( pi )
c. ( -pi )
D.
11
42917. Ifx is real, the maximum value of –
3×2 +9x+17
3×2 + 9x +7
is
[2006]
(b) 41
(d) 17
1
11
43038. 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
431If 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
432Represent ( (-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
434If ( 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
435If ( 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
436uw
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
437Show that:
( left(frac{1+i}{sqrt{2}}right)^{8}+left(frac{1-i}{sqrt{2}}right)^{8}=2 )
11
438Find 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
440Find ‘x’ and ‘y’ if ( x^{2}-y^{2}-i(2 x+y)= )
( mathbf{2} i )
11
441If ( 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
442Modulus of nonzero complex number z
satisfying ( bar{z}+z=0 ) and ( |z|^{2}-4 z i= )
( z^{2} ) is
11
443The 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
444Evaluate:
( 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
445Find 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
446If ( left|frac{z-5 i}{z+5 i}right|=1, ) prove that ( z ) is real.11
44721.
(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
448The 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
449If 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
450ff ( 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
451Find the multiplicative inverse of
( frac{sqrt{3}}{2}-frac{1}{2} i )
11
452The 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
4531. 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 to11
455Which 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
456If ( 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
457Ifa+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
459Find the multiplicative inverse of the complex number ( sqrt{5}+3 i )11
460Find 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
461Evaluate 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
463Let ( 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
464Prove that ( x^{2}+y^{2}=9 ) where ( z=x+ )
( i y ) and ( |z+6|=|2 z+3| )
11
465f ( z_{1}=6+i z_{2}=3-4 i ) then find ( z_{1} z_{2} )11
466Difference 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
(d) a-5+4=0
11
467If ( 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
468Find 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
470The 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
471The value of ( sqrt{-36} ) is
( A cdot 6 )
в. -6
( c cdot 6 i )
D. None of these
11
47224. 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
474Solve the problem:( left(frac{1}{5}+i frac{2}{5}right)-left(4+i frac{5}{2}right) )11
4758.
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
47632. 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
477Find 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
478Complex 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
479Consider 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
480The 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
481If ( 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
483If 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
484If ( 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
485If ( 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
48723. 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
488Write the argument of ( (1+sqrt{3})(1+ )
( i)(cos theta+i sin theta) )
11
489If ( 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
490Find 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
491Letf(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
492If ( 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
493Find 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
494If ( arg z<0 ) then ( arg (-z)-arg z ) is
equal to
A . ( pi )
в. ( -pi )
( c cdot-frac{pi}{2} )
D.
11
495Write principal argument of ( frac{-sqrt{11} i}{17} )11
496If ( 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
497Simplify: ( (14+2 i)(7+12 i) ) where ( i= )
( sqrt{-1} )
11
498Find 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
499The conjugate surd of ( 2-sqrt{3} ) is11
500The 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
501If ( 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
502C
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
503If ( 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
504Solve ( :-frac{i^{592}+i^{590}+i^{588}+i^{586}+i^{584}}{i^{582}+i^{580}+i^{578}+i^{576}+i^{574}} )11
50517. 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
507If ( 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
5088.
The number of real solutions of the equation
*2-3x/+2 = 0 is
(a) 3
(b) 2
(c) 4
(d) 1
11
509For ( 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
511If ( 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
512Solve the system of equations ( boldsymbol{operatorname { R e }}left(z^{2}right)=mathbf{0},|z|=mathbf{2} )11
513Real part of ( frac{(1+i)^{2}}{3-i}= )
A. ( -1 / 5 )
в. ( 1 / 5 )
c. ( 1 / 10 )
D. ( -1 / 10 )
11
514Simplify 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
5153.
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
516Evaluate ( : sqrt{-mathbf{1 6}}+mathbf{3} sqrt{-mathbf{2 5}}+sqrt{-mathbf{3 6}}- )
( sqrt{-mathbf{6 2 5}} )
11
517Let ( 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
518ff ( 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
519Find the square root of ( 4 a b-2left(a^{2}-right. )
( left.b^{2}right) i )
11
52026.
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
522Prove 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
523If ( i=sqrt{-1}, ) then ( 1+i^{2}+i^{3}-i^{6}+i^{8} ) is
equal to –
A. 2-
в.
( c cdot-3 )
D. –
11
524The 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
525Topic-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
526M-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
527Represent the following complex number in trigonometric form:
( sqrt{3} i )
11
528If ( 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
529Find real values of ( x ) and ( y ) if
( frac{x-1}{3+i}+frac{y-1}{3-i}=i )
11
53018. 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
53115
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
5325.
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
533If ( 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
534If ( n=4 m+3, m ) is an integer, then ( i^{n} )
is equal to:
A . ( -i )
B.
( c . i )
D. –
11
535If ( 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
536Let 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
537Solve: ( boldsymbol{x}^{2}-(boldsymbol{3} sqrt{mathbf{2}}-mathbf{2} boldsymbol{i}) boldsymbol{x}-sqrt{mathbf{2}} boldsymbol{i}=mathbf{0} )11
538Reduce ( left(frac{1}{1-4 i}-frac{2}{1+i}right)left(frac{3-4 i}{5+i}right) )
the standard form.
11
539Find ( 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
540The 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
541The 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
542Two 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
544Find conjugate: ( -i(9+6 i)(2-i)^{-1} )11
545Among 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
546Let ( 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
54723. 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
548f ( x+i y=frac{-1+sqrt{3} i}{1+i}, ) then find ( x ) and ( y )11
549The 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
550Find the value of ( i^{i} )11
551The 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
552Amplitude of ( frac{1+i}{1-i} ) is :
A.
в. ( pi )
( c cdot frac{pi}{2} )
D. –
11
5539.
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
554If ( 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
555f ( 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
556Find 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
557If ( 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
558The value of ( (1+i)^{5} times(1-i)^{5} )11
55910.
If iz3 + z2z+i=0, then show that|zl=1.
(1995 – 5 Marks)
11
560Find the multiplicative inverse of the complex numbers given the following:
( -i )
11
561Simplify ( 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
564If ( 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
565If ( 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
566Two 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
568Let ( 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
569The 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
57022. 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
572Find 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
573If ( 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
574If ( 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
575The 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
576Put 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
577ff ( 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
578Find the real values of ( x ) and ( y, ) if
( (x+i y)(2-3 i)=4+i )
11
579Find 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
581If ( z=(sqrt{3}+i) ) then find ( operatorname{Re}(z) ) and
( operatorname{lm}(z) )
11
582When 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
583If ( Z=cos theta+i sin theta ) find the complex
representation of ( frac{Z}{1-2 Z} )
11
584Show that ( (-1+sqrt{3} i)^{3} ) is a real
number.
11
585If ( 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
586What is the multiplicative inverse of ( -1 times frac{-2}{5} )11
587ff ( 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
588Write the correct letter from column I
against the entry number in column lin
your answer book, ( z neq 0 ) is a complex number
11
589Locate 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
590Convert the complex number ( frac{-16}{1+i sqrt{3}} ) into polar form.11
591If ( i^{2}=-1, ) then ( i^{162} ) is equal to
A. ( -i )
B. –
( c cdot 0 )
D.
( E )
11
592If ( 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
593If ( z_{1}=2-i, z_{2}=1+i, ) find
( left|frac{z_{1}+z_{2}+1}{z_{1}+z_{2}+i}right| )
11
594Find the conjugate of the following
complex number. ( (15+3 i)-(4-20 i) )
11
595State 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
596State 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
59720. 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
598If ( z neq 0 ), then ( int_{0}^{50} arg (-|z|) d x ) equals
A. 50
B. not defined
( c cdot 0 )
D. ( 50 pi )
11
599If 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
600The 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
601Argument 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
603If ( 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
604Which 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
605locate the point representing the
complex numbers ( z ) on the Argand diagram for which
( |z|-4=|z-i|-|z+5 i|=0 )
11
606If ( 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
607Let ( 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
608If ( 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
609If ( 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
610Solve:
( (i+3 i)^{2}(3+1) )
11
611Evaluate 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
612Interpret 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
6131.
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
614If ( 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
61533. 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)
(b) (9-p}(2p-9)
(a) (20-9(29- p)
11
61619. 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
617ff ( i^{2}=-1 ), then the value of ( sum_{n=1}^{200} i^{n} ) is
A . 50
B. – -50
( c cdot 0 )
D. 100
11
61815. 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
619If ( 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
620If ( 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
621if ( 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
622If ( 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
623Let ( 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
624If ( 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
625If ( 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
626If ( z ) is a complex number ( z=9-12 i )
find ( |z| )
A . 15
B. 16
c. 17
D. 8
11
62718. 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
628Solve: ( left(i^{25}right)^{3} times i )11
629Write ( 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
630If ( 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
631Find the multiplicative inverse of the complex numbers given. ( sqrt{5}+3 i )11
632If ( z(neq-1) ) is a complex number such
that ( frac{z-1}{z+1} ) is purely imaginary, then find
( |z| )
11
633Show 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
634Find 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
636The 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
638Find the value of ( left(frac{1-i}{1+i}right)^{40} )11
639Write the real and imaginary part of ( (i-sqrt{3})^{3} )11
640If ( 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
641Let ( 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
642Solve 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
644If ( 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
645Prove that ( (1+i)^{4}left(1+frac{1}{i}right)^{4}=16 )11
646If ( 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
647Prove that ( left|frac{1-z_{1} bar{z}_{2}}{z_{1}-z_{2}}right|<1left|z_{1}right|<1< )
( left|z_{2}right| )
11
648Find
( sqrt{boldsymbol{i}}+sqrt{-boldsymbol{i}} )
11
649Let ( 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
650Evaluate: ( left[i^{18}+left(frac{1}{i}right)^{25}right]^{3} )11
651If 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
653if ( 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
654If ( 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
65511. 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
656If ( 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
657Assertion
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
658The complex number ( frac{2^{n}}{(1+i)^{2 n}}+ ) ( frac{(1+i)^{2 n}}{2^{n}}, quad n in )
11
659The 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
660f ( 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
661If ( frac{z-1}{z+1} ) is purely imaginary then
в. ( |z|>1 )
c. ( |z|<1 )
D. |z|<2
11
662Two 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
663If ( (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
664Find 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
665Locus 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
666If ( |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
66731. 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=
11
668Find 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
669The 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
670The 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
671Express the complex number ( 1+i sqrt{3} )
in modulus amplitude form.
11
672Evaluate ( :(1+i)^{6}+(1-i)^{3} )11
673Find multiplicative inverse of :
( frac{3+4 i}{4-5 i} )
11
67438. The quadratic equation px)=0 with real coefficients has
purely imaginary roots. Then the equation p(p(x)) = 0 has
(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
6755.
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
676If 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
677If ( 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
678Find 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
679Find 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
680Perform 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
68210. 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
(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
683IF ( 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
684The 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
686Find 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
687Express the following in the form of ( a+ )
( boldsymbol{b} boldsymbol{i} )
(i) ( (-i)(2 i)left(-frac{1}{8} iright)^{3} )
11
6881.
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?
(6) -1-1-y?
(d) 1-v1+y2
11
689Let 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
690The 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
691Evaluate 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
69214.
Ifarg(z) <0, then arg (-2) – arg(z)=
(2000)
11
693Calculate, ( sqrt[4]{-1 frac{1}{2}-i frac{sqrt{3}}{2}} )11
694If ( 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
695Simplify:
( left(frac{2 i}{1+i}right)^{2} )
11
696The additive inverse of ( z ) is
( A cdot 0 )
B. ( z )
( c .-z )
D.
11
697If ( 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
698If ( 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
69916. Ifz, and z, are two complex numbers such tahtz,l<l<z2
1-2,22
(2003 – 2 Marks)
then prove that
| 21 – 22 |
11
700Find 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
702If ( frac{a+3 i}{2+i b}=1-i, ) show that ( (5 a- ) ( 7 b)=0 )11
703If ( 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
704Find 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
705If ( 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
70622. f – -4 = 2, then then
= 2, then the maximum value of|Z is equal to :
[2009]
(a) J5+1
(b) 2
(d) 73+1
11
707Lets, 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?
(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
708Simplify the following:
( frac{3}{1+i}-frac{2}{2-i}+frac{2}{1-i} )
11
709Find the multiplicative inverse of each
of the complex numbers given in the
Exercises 11 to 13
( -i )
11
710If ( 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
711For 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
712If ( 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
713Let ( 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
714Find the modulus of the complex number ( sqrt{mathbf{2}} boldsymbol{i}-sqrt{-mathbf{2}} boldsymbol{i} )11
715Perform 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
717Put the following in the form ( A+i B: ) ( frac{(1+i)^{2}}{3-i} )11
718If ( |z-2+i| leq 2 ),then find the least
value of ( |z| )
11
719Find the value of ( frac{i^{4 n+1}-i^{4 n-1}}{2} )
A . -1
B.
( c .-i )
( D )
11
720The 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
721The inequality ( |z-4|0 )
B ( cdot operatorname{Re}(z)2 )
D. None of these
11
722A 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
724f ( |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
725The 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
72613. 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
727The 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
728Find the real and imaginary parts of the complex number ( frac{a+i b}{a-i b} )11
729The 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
730Show 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
731If ( 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
732Simplify 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
733If ( 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
734Simplify: ( 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
736If ( 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
737show 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
73914. 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
740If ( 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
741Simplify 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
742The 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
743State true or false
if ( z ) is a complex number then ( z bar{z} ) is
purely real
A. True
B. False
11
744The 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
745For 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
74627. 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
748If ( 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
749Express the complex number given in the form ( a+i b )
( boldsymbol{i}^{-mathbf{3 9}} )
11
750For ( |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
751Find the value of :
( frac{i^{6}+i^{7}+i^{9}}{i^{2}+i^{3}} ? )
11
752All the values/s of ( (1+i)^{frac{1}{2}} ) are11
753The inequality ( |z-4|0 )
в. ( operatorname{Re}(z)2 )
D. none of these
11
754Two 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
755If ( z=i-1, ) then ( bar{z}= )
A . ( i+1 )
B. ( -i-1 )
( c cdot-1 )
D. none of these
11
756If ( 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
757Find the amplitude of -411
758Z-1
nd @=
(where z
-1), then Rew is
19
z +1
JP 2005)
(2003)
11
759Find the value of the principal argument of the complex number ( z=frac{(1+i sqrt{3})^{2}}{(1-i)^{3}} )11
76012. Ifo is an imaginary cube root of unity then the value of
(1994)
11
761if ( 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
762If ( |z|=1, ) then ( |z-1| ) is
( mathbf{A} cdot|a r g(z)| )
( mathbf{C} cdot=|arg (z)| )
D. None of these
11
76314.
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
764Express ( (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
766The value of ( 2 sqrt{-49} ) is equal to
A . -14
B. None of these
c. 14
D. 14
11
767Let ( 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
76812.
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
(d) n,>0,n,> 0
11
769If ( 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
770The 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
7725.
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
773If ( 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
774If ( 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
776If ( 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
777For a complex number z, the minimum
value of ( |z|+|z-2| ) is
A. 1
в. 2
( c .3 )
D. None of these
11
77816. The conjugate of a complex number is
complex number is
, then that
i-1
[2008]
@
#
(a
11
779Find 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
780Express the given complex number in
the form ( a+i b: )
( (1-i)^{4} )
11
781There 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
78215. Ifz, z, and z, are complex numbers such that
(20005)
lal= (22=1231=
=1, then 1a + 2a + zs is.
(a) equal to 1
(b) less than 1
(d) equal to 3
11th
11
78327. 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
784If ( 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
7854.
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
786What is ( operatorname{cis} 0 ? )11
7874. 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
788The value of ( sqrt{-1} ) is
( A cdot 1 )
B. –
c. ( i ) (iota)
D. none of these
11
789Simplify the following expressions:
(A) ( 7 i^{2} )
(B) ( -6 i^{8} )
( (C) 8 i^{7} )
11
79012.
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
791If ( (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
792Let 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
793If ( left(frac{1+i}{1-i}right)^{n} ) is 1 find the least value of ( n ) where ( boldsymbol{n} in boldsymbol{N} )11
794Express ( frac{mathbf{5}+i sqrt{2}}{2 i} ) in the form of ( x+i y )11
795If ( 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
796Let ( 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
797Express the complex number ( frac{2+i}{3-4 i} ) in
( a+i b ) form.
11

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