Summary Best questions set for jee and cat

COMPLEX NUMBER
**SINGLE OPTION CORRECT :-
Q.1 For any complex number w = a + bi, where a, b  R.
If w = cos 40° + i sin40°, then | w + 2w2 + 3w3 + ..... + 9w9 | –1 equals
1 2 1 9
(A) sin 40 (B) sin 20 (C) cos 40 (D) cosec 20
9 9 9 2

 z  
Q.2 The locus of the complex variable z when Arg   = (where  is a complex number) in the argand
z 3
plane is :
(A) a straight line (B) a circle (C) a parabola (D) a segment of a circle
Q.3 Suppose we are given a point P on the Argand plane represented by the complex number Z, moving
anticlockwise along the circle with | z | = 2 from the point of complex number 2 + 0i to 0 + 2i.
Z 1
If  = then the path described by Q which represents the complex number  is
Z2
(A) a line with gradient 1/3 passing through the origin.
(B) a circle
(C) a segment of line 4x – 3 =0 with y varying from [0, 1/4]
(D) a line 4x – 3 = 0 with y varying from [0, 1]
Q.4 Let z1, z2 & z3 be the complex numbers representing the vertices of a triangle ABC respectively . If P is a
point representing the complex number z0 satisfying ;
a (z1  z0) + b (z2  z0) + c (z3  z0) = 0 , then w.r.t. the triangle ABC, the point P is its :
(A) centroid (B) orthocentre (C) circumcentre (D) incentre
Q.5 Let z = Rei then |eiz| has the value equal to
(A) Re–R sin  (B) Re–R cos (C) e–R sin (D) e–R cos


Q.6 The roots of the cubic equation ( Z+)3 = 3 (  0) , represent the vertices of a triangle of sides of length
|| 1
(A) 3  (B) (C) 3 | | (D) ||
3 3
Q.7 Locus of z in the argand plane if z satisfies the condition
cot–1( log3 | 2z + 1 | ) > cot–1 (log3 | 2z – 1 | ) is
(A) Re | z | > 0 (B) Re (Z) < 0 (C) Im (z) > 0 (D) Im (Z) < 0

 | z  1| 4  2
Q.8 Locus of complex number z satisfying inequality :- log (1/ 2)    1 (where | z  1| ) is :
 3 | z  1| 2  3
(A) a circle (B) an interior of a circle
(C) the exterior of the circle (D) none of these

1  i 3 
x
Q.9 The solution set of the equation ,  2x = 0
(A) form an A.P. (B) form a G.P. (C) form an H.P. (D) is a empty set
Q.10 The equation z3 – (n + 1)z + (m + 2i) = 0 has three roots, where 'n' and 'm' are real constants. If the square
of the modulus of the product of the roots is 5, then the value of 'm' is
(A) ± 1 (B) ± 3 (C) ± 21 (D) none

Prepared by Gaurav Sir

, Q.11 Let p and q be the complex numbers with (q  0). If the roots of the quadratic equation x2 + px + q2 = 0
have the same absolute value then p q
(A) is purely real (B) is purely imaginary (C) is imaginary (D) nothing definite can be said

1
Q.12 If  & 2 are the roots of the equation, 8x2  10x + 3 = 0, wehre 2 > , then the equation whose roots
2
are ( + i )100 & (i )100 is :
(A) x2  x + 1 = 0 (B) x2 + x + 1 = 0 (C) x2  x  1 = 0 (D) x2 + x  1 = 0

z  
Q.13 If tangents drawn to circle | z | = 4 at points A(z1) and B(z2) intersect at P such that arg  2  = , then
 z1  2
locus of P is
(A) |z| = 2 (B) |z| = 4 2 (C) |z| = 2 2 (D) |z| = 8
Q.14 Let z1, z2 be roots of the equation z2 + 8(i – 1)z + 63 – 16i = 0, where i2 = –1 . The area of triangle formed
by O, z1 and z2 (where O being the origin) is equal to
(A) 24 (B) 26 (C) 28 (D) 30
Q.15 If z1 and z2 are the roots of the equation az2 + bz + c = 0, where a, b, c  R and 4ac > b2, then which one
of the following quantities is real?
(A) z1 – z2 (B) z1 z 2 (C) (z1 – z2)i (D) (z1 + z2)i

2 2 2 2
Q.16 If z  C then area enclosed by the curve 2 z  | z |  3 z  | z | = 6 | z | , z  0 is
(A) 3 sq. units (B) 6 sq. units (C) 3 sq. units (D) 6 sq. unit
Q.17 Let  be a complex cube root of unity with 0 < arg() < 2. A fair die is thrown three times.
If a, b, c are number obtained on the die, probability that (a + b + c2) (a + b2 + c) = 1, is equal to
1 1 5 1
(A) (B) (C) (D)
18 9 36 6

z1  1 z2  4
Q.18 If = 2 and = 2, then the value of z1  z 2  z1  z 2 is
z1  4 z2  1 max min

(A) 8 (B) 9 (C) 10 (D) 11
Q.19 A function f (x) is defined on the complex number by f (z) = (a + bi)z, where a and b are positive numbers.
This function has the property that the image of each point in the complex plane is equidistant from that point
m
and the origin. Given that | a + bi | = 8 and that b2 = , where m and n are relatively prime positive integers.
n
Then the value of (m + n) is
(A) 259 (B) 229 (C) 139 (D) 299
Q.20 Triangle ABC is inscribed in the circle | z | = a, (a  R+). If z1, z2, z3 are the complex numbers corresponding
to the vertices A, B, C of triangle ABC respectively and the internal bisector of angle A meets the circle at D
with complex number z4 , then
(A) z1z4 = z2z3 (B) z1z3 = z2z4
(C) z1z2 = z3z4 (D) z2z3 = z4



Prepared by Gaurav Sir