1

Complex Numbers
Topic 1 Complex Number in Iota Form
Objective Questions I (Only one correct option) 2 + 3i sin θ
6. A value of θ for which is purely imaginary, is
2z − n 1 − 2i sin θ (2016 Main)
1. Let z ∈ C with Im (z ) = 10 and it satisfies = 2i − 1
 3
2z + n π π −1  1 
(a) (b) (c) sin −1   (d) sin  
for some natural number n, then (2019 Main, 12 April II) 3 6  4   3
(a) n = 20 and Re(z ) = − 10 (b) n = 40 and Re(z ) = 10 6i –3 i 1
(c) n = 40 and Re(z ) = − 10 (d) n = 20 and Re(z ) = 10 7. If 4 3i –1 = x + iy, then (1998, 2M)
α + i  20 3 i
2. All the points in the set S =  : α ∈ R (i = −1 ) lie
α − i  (a) x = 3, y = 1 (b) x = 1, y = 1 (c) x = 0, y = 3 (d) x = 0, y = 0
on a (2019 Main, 9 April I) 13
(a) circle whose radius is 2. 8. The value of sum ∑ (i n + i n + 1 ), where i = −1, equals
(b) straight line whose slope is −1. n =1
(1998, 2M)
(c) circle whose radius is 1. (a) i (b) i − 1 (c) − i (d) 0
n
(d) straight line whose slope is 1.  1 + i
5 + 3z 9. The smallest positive integer n for which   = 1, is
3. Let z ∈ C be such that|z|< 1. If ω = , then 1 − i
5(1 − z ) (a) 8 (b) 16 (1980, 2M)
(2019 Main, 9 April II) (c) 12 (d) None of these
(a) 4 Im(ω) > 5 (b) 5 Re (ω) > 1
(c) 5 Im (ω) < 1 (d) 5 Re(ω) > 4 Objective Question II
3
x + iy (One or more than one correct option)
4. Let  −2 − i =
1
(i = −1 ), where x and y are real
 3  27 10. Let a , b, x and y be real numbers such that a − b = 1 and
numbers, then y − x equals (2019 Main, 11 Jan I) y ≠ 0. If the complex number z = x + iy satisfies
(a) 91 (b) 85 (c) – 85 (d) – 91  az + b
Im   = y, then which of the following is(are)
 π  3 + 2i sin θ   z+1
5. Let A = θ ∈  − , π : is purely imaginary 
  2  1 − 2i sin θ  possible value(s) of x? (2017 Adv.)

Then, the sum of the elements in A is (2019 Main, 9 Jan I) (a) 1 − 1 + y2 (b) − 1 − 1 − y2
3π 5π 2π
(a) (b) (c) π (d) (c) 1 + 1 + y2 (d) − 1 + 1 − y2
4 6 3

Topic 2 Conjugate and Modulus of a Complex Number

Objective Questions I (Only one correct option) (1 + i )2 2
2. If a > 0 and z = , has magnitude , then z is
1. The equation|z − i| = |z − 1|, i = −1, represents a−i 5
equal to (2019 Main, 10 April I)
1 (2019 Main, 12 April I)
(a) a circle of radius 1 3 1 3
2 (a) − i (b) − − i
5 5 5 5
(b) the line passing through the origin with slope 1 1 3 3 1
(c) a circle of radius 1 (c) − + i (d) − − i
5 5 5 5
(d) the line passing through the origin with slope − 1
2 Complex Numbers

3. Let z1 and z2 be two complex numbers satisfying| z1 | = 9 12. If w = α + iβ, where β ≠ 0 and z ≠ 1, satisfies the
and | z2 − 3 − 4i | = 4. Then, the minimum value of  w − wz 
condition that   is purely real, then the set of
| z1 − z2|is (2019 Main, 12 Jan II)  1−z 
(a) 1 (b) 2 (c) 2 (d) 0 values of z is (2006, 3M)
z −α (a)| z | = 1, z ≠ 2 (b)| z | = 1 and z ≠ 1
4. If (α ∈ R) is a purely imaginary number and
z+α (c) z = z (d) None of these
|z| = 2, then a value of α is (2019 Main, 12 Jan I) z −1
13. If|z| = 1 and w = (where, z ≠ − 1), then Re (w) is
(a) 2 (b)
1
(c) 1 (d) 2 z+1 (2003, 1M)
2 1  1  1 2
(a) 0 (b) (c) ⋅ (d)
5. Let z be a complex number such that | z | + z = 3 + i |z + 1|2
z +  |z + 1|
1 2
|z + 1|2
(where i = − 1).
14. For all complex numbers z1 , z2 satisfying |z1| = 12 and
Then,| z |is equal to (2019 Main, 11 Jan II)
|z2 − 3 − 4i| = 5, the minimum value of|z1 − z2|is
34 5 41 5
(a) (b) (c) (d) (a) 0 (b) 2 (2002, 1M)
3 3 4 4
(c) 7 (d) 17
6. A complex number z is said to be unimodular, if z ≠ 1. 15. If z1 , z2 and z3 are complex numbers such that
z – 2 z2
If z1 and z2 are complex numbers such that 1 is 1 1 1
2 – z1z2 |z1| = |z2| = |z3| =  + + = 1, then |z1 + z2 + z3|is
z1 z2 z3
unimodular and z2 is not unimodular.
(a) equal to 1 (b) less than 1 (2000, 2M)
Then, the point z 1 lies on a (2015 Main)
(c) greater than 3 (d) equal to 3
(a) straight line parallel to X-axis
16. For positive integers n1 , n2 the value of expression
(b) straight line parallel toY -axis
(1 + i )n 1 + (1 + i3 )n1 + (1 + i5 )n 2 + (1 + i7 )n 2 , here
(c) circle of radius 2
i = −1 is a real number, if and only if (1996, 2M)
(d) circle of radius 2
(a) n1 = n2 + 1 (b) n1 = n2 − 1
7. If z is a complex number such that |z| ≥ 2, then the (c) n1 = n2 (d) n1 > 0, n2 > 0
1 17. The sin x + i cos 2x
minimum value of z + complex numbers and
2 (2014 Main) cos x − i sin 2x are conjugate to each other, for
(a) is equal to 5/2 (a) x = nπ (b) x = 0 (1988, 2M)
(b) lies in the interval (1, 2) (c) x = (n + 1/2) π (d) no value of x
(c) is strictly greater than 5/2 18. The points z1 , z2, z3 and z4 in the complex plane are the
(d) is strictly greater than 3/2 but less than 5/2 vertices of a parallelogram taken in order, if and only if
8. Let complex numbers α and 1 /α lies on circles (a) z1 + z4 = z2 + z3 (b) z1 + z3 = z2 + z4 (1983, 1M)
(x − x0 )2 + ( y − y0 )2 = r 2 and (x − x0 )2 + ( y − y0 )2 = 4r 2, (c) z1 + z2 = z3 + z4 (d) None of these
respectively. 19. If z = x + iy and w = (1 − iz ) / (z − i ), then |w| = 1 implies
If z0 = x0 + iy0 satisfies the equation 2|z0|2 = r 2 + 2, then
that, in the complex plane (1983, 1M)
|α |is equal to (2013 Adv.)
1 1 1 1 (a) z lies on the imaginary axis (b) z lies on the real axis
(a) (b) (c) (d) (c) z lies on the unit circle (d) None of these
2 2 7 3
9. Let z be a complex number such that the imaginary part 20. The inequality |z − 4| < |z − 2| represents the region
of z is non-zero and a = z + z + 1 is real. Then, a cannot
2 given by (1982, 2M)
take the value (2012) (a) Re (z ) ≥ 0 (b) Re (z ) < 0
1 1 3 (c) Re (z ) > 0 (d) None of these
(a) − 1 (b) (c) (d)
5 5
3 2 4  3 i  3 i
10. Let z = x + iy be a complex number where, x and y are 21. If z =  +  + −  , then
 2 2   2 2 (1982, 2M)
integers. Then, the area of the rectangle whose vertices
are the root of the equation zz3 + zz3 = 350, is (2009)
(a) Re (z ) = 0 (b) Im (z ) = 0
(c) Re (z ) > 0, Im (z ) > 0 (d) Re (z ) > 0, Im (z ) < 0
(a) 48 (b) 32
(c) 40 (d) 80 22. The complex numbers z = x + iy which satisfy the
z z − 5i 
11. If|z|= 1 and z ≠ ± 1, then all the values of lie on equation  = 1, lie on
1 − z2 z + 5i  (1981, 2M)
(a) a line not passing through the origin (2007, 3M) (a) the X-axis
(b)|z|= 2 (b) the straight line y = 5
(c) the X-axis (c) a circle passing through the origin
(d) the Y-axis (d) None of the above
 Complex Numbers 3

Objective Questions II 29. Let z be any point in A ∩ B ∩ C and let w be any point
(One or more than one correct option) satisfying |w − 2 − i| < 3. Then, |z | − |w| + 3 lies
between
23. Let S be the set of all complex numbers z satisfying (a) − 6 and 3 (b) − 3 and 6
| z 2 + z + 1| = 1. Then which of the following statements (c) − 6 and 6 (d) − 3 and 9
is/are TRUE? (2020 Adv.)
1 1 Passage II
(a) z + ≤ for all z ∈S (b)|z|≤ 2 for all z ∈S
2 2 Let S = S1 ∩ S 2 ∩ S3 , where
1 1  z − 1 + 3 i  
(c) z + ≥ for all z ∈S S1 = { z ∈ C :|z | < 4}, S 2 = z ∈ C : lm   > 0
2 2   1− 3i  
(d) The set S has exactly four elements and S3 : { z ∈ C : Re z > 0} (2008)
24. Let s, t, r be non-zero complex numbers and L be the set of 30. Let z be any point in A ∩ B ∩ C.
solutions z = x + iy (x, y ∈ R, i = − 1 ) of the equation The|z + 1 − i|2 + |z − 5 − i|2 lies between
sz + tz + r = 0, where z = x − iy. Then, which of the
(a) 25 and 29 (b) 30 and 34
following statement(s) is (are) TRUE? (2018 Adv.)
(c) 35 and 39 (d) 40 and 44
(a) If L has exactly one element, then| s|≠ |t |
31. The number of elements in the set A ∩ B ∩ C is
(b) If|s|=|t |, then L has infinitely many elements
(a) 0 (b) 1
(c) The number of elements in L ∩ {z :| z − 1 + i| = 5} is at most 2
(c) 2 (d) ∞
(d) If L has more than one element, then L has infinitely many
elements Match the Columns
25. Let z1 and z2 be complex numbers such that z1 ≠ z2 and 32. Match the statements of Column I with those of
|z1| = |z2|. If z1 has positive real part and z2 has negative Column II.
z + z2
imaginary part, then 1 may be (1986, 2M) Here, z takes values in the complex plane and Im (z )
z1 − z2 and Re (z ) denote respectively, the imaginary part and
(a) zero (b) real and positive the real part of z (2010)
(c) real and negative (d) purely imaginary
Column I Column II
26. If z1 = a + ib and z2 = c + id are complex numbers such
A. The set of points z satisfying p. an ellipse with
that |z1| = |z2| = 1 and Re (z1z2) = 0, then the pair of
| z − i| z|| = | z + i | z|| is eccentricity 4/5
complex numbers w1 = a + ic and w2 = b + id satisfies
contained in or equal to
(a)|w1| = 1 (b)|w2| = 1 (1985, 2M)
B. The set of points z satisfying q. the set of points z
(c) Re (w1 w2 ) = 0 (d) None of these
| z + 4| + | z − 4| = 0 is satisfying Im ( z) = 0
contained in or equal to
Passage Based Problems
C. If| w| = 2 , then the set of r. the set of points z
Read the following passages and answer the questions 1
points z = w − is contained satisfying|Im( z) |≤ 1
that follow. w
Passage I in or equal to

Let A, B, C be three sets of complex number as defined D. If| w| = 1, then the set of points s. the set of points
below
1
z = w + is contained in or satisfying|Re( z)|≤ 2
t. the set of points z
A = { z : lm (z ) ≥ 1} w
equal to satisfying| z| ≤ 3
B = { z :|z − 2 − i| = 3}
C = { z : Re((1 − i )z ) = 2 } (2008, 12M)
Fill in the Blanks
27. min|1 − 3i − z|is equal to
z ∈s 33. If α , β, γ are the cube roots of p, p < 0, then for any x, y
2− 3 2+ 3 3− 3 3+ 3 xα + yβ + zγ
(a) (b) (c) (d) and z then = ... .
2 2 2 2 xβ + yγ + zα (1990, 2M)
28. Area of S is equal to
10 π 20 π 16 π 32 π 34. For any two complex numbers z1 , z2 and any real
(a) (b) (c) (d)
3 3 3 3 numbers a and b,|az1 − bz2|2+ |bz1 + az2|2 = K .
(1988, 2M)