1 Crore+ students have signed up on EduRev. Have you? 
Q.1. If where z = x + iy, then the point (x, y) lies on a (2020)
(a) Circle whose centre is at
(b) Straight line whose slope is 2/3.
(c) Straight line whose slope is 3/2.
(d) Circle whose diameter is √5/2.
Ans. d
We have
...(1)
Therefore,
Center and radius
Hence, the diameter of circle is √5/2.
Q.2. If is a real number, then an argument of sinθ + i cosθ is (2020)
(a) π  tan^{1}(4/3)
(b) π  tan^{1}(3/4)
(c)  tan^{1}(3/4)
(d)  tan^{1}(4/3)
Ans. a
We have
Given that the number is real, then
4 sinθ + 3 cosθ = 0 ⇒ tan θ = 3/4
Hence, the argument of sinθ + i cosθ is
Q.3. If z be a complex number satisfying Re(z) + Im(z) = 4, then z cannot be (2020)
(a)
(b) √10
(c) √7
(d) √8
Ans. c
Let the complex number be z = x + iy. So,
x+y = 4 (1)
Now,
Minimum value of
Maximum value of z = 4
So, z ∈ (2√2, 4).Hence, the value of z cannot be √7.
Q.4. Let Then the sum of the elements in A is: (2019)
(a) 5π/6
(b) π
(c) 3π/4
(d) 2π/3
Ans. d
Since, z is purely imaginary, then
Now, the sum of elements in A =
Q.5. Let z_{0} be a root of the quadratic equation, then arg z is equal (2019)
(a) π/4
(b) π/6
(c) π/3
(d) 0
Ans. a
∵ z_{0} is a root of quadratic equation
Q.6. Let z_{1} and z_{2} be any two nonzero complex numbers such that ,then: (2019)
(a) Re(z) = 0
(b)
(c)
(d) Im(z) = 0
Ans. Bonus
Q.7. Let If R(z) and I(z) respectively denote the real and imaginary parts of z, then: (2019)
(a) I(z) = 0
(b) R(z) > 0 and I(z) > 0
(c) R(z) < 0 and I(z) > 0
(d) R(z) = (3)
Ans. a
Q.8. Let where x and y are real numbers then y  x equals: (2019)
(a) 91
(b) 85
(c) 85
(d) 91
Ans. a
Q.9. Let z be a complex number such that z + z = 3 + i
Then z is equal to: (2019)
(a)
(b) 5/3
(c)
(d) 5/4
Ans. b
Since, z + z = 3 + i
Let z = a + ib, then
Compare real and imaginary coefficients on both sides
Then,
Q.10. is a purely imaginary number and z = 2, then a value of α is: (2019)
(a) 2
(b) 1
(c) 1/2
(d)
Ans. a
∵ t is purely imaginary number.
Q.11. Let z_{1} and z_{2} be two complex numbers satisfying z_{1} = 9 and z_{2} 3  4i=4. Then the minimum value of z_{1} z_{2} is: (2019)
(a) 0
(b) √2
(c) 1
(d) 2
Ans. a
Z_{1} = 9, z_{2}  3  4i = 4
z_{1} lies on a circle with centre C_{1}(0, 0) and radius r_{1} = 9
z_{2} lies on a circle with centre C_{2}(3, 4) and radius r_{2} = 4
So, minimum value of z_{1} z_{2} is zero at point of contact (i.e. A)
Q.12. If then (1 + iz + z^{5} + iz^{8})^{9} is equal to:
(a) 0
(b) 1
(c) (1 + 2i)^{9}
(d) 1
Ans. d
where ω is imaginary cube root of unity.
Q.13. All the points in the set lie on a: (2019)
(a) straight line whose slope is 1.
(b) circle whose radius is 1.
(c) circle whose radius is √2 .
(d) straight line whose slope is 1.
Ans. b
Let z∈S then
Since, z is a complex number and let z = x + iy
Then, (by rationalisation)
Then compare both sides
...(1)
...(2)
Now squaring and adding equations (1) and (2)
Q.14. Let z ∈ C be such that z < 1. If ω = then: (2019)
(a) 5 Re (ω) > 4
(b) 4 Im (ω) > 5
(c) 5 Re (ω) >1
(d) 5 Im (ω) < 1
Ans. c
Q.15. If a > 0 and z = has magnitude then is equal to: (2019)
(a)
(b)
(c)
(d)
Ans. a
Then, from equation (1),
Now, square on both side; we get
Q.16. If z and ω are two complex numbers such that zω = 1 and arg(z)  arg(ω) = π/2, then: (2019)
(a)
(b)
(c)
(d)
Ans. c
Q.17. The equation represents: (2019)
(a) a circle of radius 1/2.
(b) the line through the origin with slope 1.
(c) a circle of radius 1.
(d) the line through the origin with slope 1.
Ans. b
Given equation is, z  1 = z  i
Hence, locus is straight line with slope 1.
Q.18. Let z ∈ C with Im(z) = 10 and it satisfies for some natural number n. Then: (2019)
(a) n = 20 and Re(z) = 10
(b) n = 40 and Re(z) = 10
(c) n = 40 and Re(z) = 10
(d) n = 20 and Re(z) = 10
Ans. c
On comparing real and imaginary parts,
Hence, Re(z) = 10
Q.19. If α ,β ∈ c are the distinct roots, of the equation x^{2}  x + 1 =0 , then α^{101} + β^{107} is equal to: (2018)
(a) 1
(b) 0
(c) 1
(d) 2
Ans. c
x^{2}  x+ 1 = 0
Q.20. The set of all α ∈ R, for which ω =is a purely imaginary number, for all z ∈ C Satifying z = 1 and Re z ≠ 1, is: (2018)
(a) An empty set
(b)
(c) equal to R
(d) {0}
Ans. d
As ω is purely imaginary
If Re(z) ≠ 1
then, α = 0
Q.21. The least positive integer n for which is: (2018)
(a) 2
(b) 5
(c) 6
(d) 3
Ans. d
Also,
∴ Least positive integer n is 3.
Q.22. Let z ∈ C, the set of complex numbers. Then the equation, 2z + 3i – z – i = 0 represents a circle with radius (2017)
(a) a cirlce with radius 8/3
(b) an ellipse with length of minor axis 16/9
(c) an ellipse with length of major axis 16/3
(d) a circle with diameter 10/3
Ans. a
Q.23. The equation represents a part of a circle having radius equal to: (2017)
(a) 1
(b) 2
(c) 3/4
(d) 1/2
Ans. c
Let = x + y
Q.24. A value of θ for which is purely imaginary, is: (2016)
(a) π/3
(b) π/6
(c) sin^{1}
(d) sin^{1}
Ans. d
To be purely imaginary if
Q.25. The point represented by 2 + i in the Argand plane moves 1 unit eastwards, then 2 units northwards and finally from there 2√2 units in the south westwards direction. Then its new position in the Argand plane is at the point represented by: (2016)
(a) 2 + 2i
(b) – 2 – 2i
(c) 1 + i
(d) – 1 – i
Ans. c
Let P(2 + i)
By rotation theorem
Q.26. Let z = 1 + ai be a complex number, a > 0 such that z^{3} is a real number. Then the sum 1 + z + z^{2} +....+ z^{11} is equal to (2016)
(a) 1250 √3 i
(b) 1250 √3 i
(c) 1365 √3 i
(d) 1365 √3 i
Ans. c
130 videos359 docs306 tests
