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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.
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)
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)
Let the complex number be z = x + iy. So,
|x|+|y| = 4 (1)
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)
Since, z is purely imaginary, then
Now, the sum of elements in A =
Q.5. Let z0 be a root of the quadratic equation, then arg z is equal (2019)
∵ z0 is a root of quadratic equation
Q.6. Let z1 and z2 be any two non-zero complex numbers such that ,then: (2019)
(a) Re(z) = 0
(d) Im(z) = 0
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)
Q.8. Let where x and y are real numbers then y - x equals: (2019)
Q.9. Let z be a complex number such that |z| + z = 3 + i
Then |z| is equal to: (2019)
Since, |z| + z = 3 + i
Let z = a + ib, then
Compare real and imaginary coefficients on both sides
Q.10. is a purely imaginary number and |z| = 2, then a value of α is: (2019)
∵ t is purely imaginary number.
Q.11. Let z1 and z2 be two complex numbers satisfying |z1| = 9 and |z2- 3 - 4i|=4. Then the minimum value of |z1 -z2| is: (2019)
|Z1| = 9, |z2 - 3 - 4i| = 4
z1 lies on a circle with centre C1(0, 0) and radius r1 = 9
z2 lies on a circle with centre C2(3, 4) and radius r2 = 4
So, minimum value of |z1 -z2| is zero at point of contact (i.e. A)
Q.12. If then (1 + iz + z5 + iz8)9 is equal to:
(c) (-1 + 2i)9
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.
Let z∈S then
Since, z is a complex number and let z = x + iy
Then, (by rationalisation)
Then compare both sides
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
Q.15. If a > 0 and z = has magnitude then is equal to: (2019)
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)
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.
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
On comparing real and imaginary parts,
Hence, Re(z) = -10
Q.19. If α ,β ∈ c are the distinct roots, of the equation x2 - x + 1 =0 , then α101 + β107 is equal to: (2018)
x2 - 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
(c) equal to R
As ω is purely imaginary
If Re(z) ≠ 1
then, α = 0
Q.21. The least positive integer n for which is: (2018)
∴ Least positive integer n is 3.
Q.22. Let z ∈ C, the set of complex numbers. Then the equation, 2|z + 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
Q.23. The equation represents a part of a circle having radius equal to: (2017)
Let = x + y
Q.24. A value of θ for which is purely imaginary, is: (2016)
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
Let P(2 + i)
By rotation theorem
Q.26. Let z = 1 + ai be a complex number, a > 0 such that z3 is a real number. Then the sum 1 + z + z2 +....+ z11 is equal to (2016)
(a) -1250 √3 i
(b) 1250 √3 i
(c) -1365 √3 i
(d) 1365 √3 i