Title 02. Complex Numbers(Q).pdf ✅

COMPLEX NUMBERS(Q) 2:1. Algebra of Complex Numbers 1. If the set R = {(a, b): a + 5b = 42, a, b ∈ N} has m elements and ∑n=1 m (1 − i n! ) = x + iy, where i = √−1, then the value of m + x + y is (a) 8 (b) 5 (c) 4 (d) 12 ( 8 th April 1 st Shift 2024) 2. Let z be a complex number such that the real part of z−2i z+2i is zero. Then, the maximum value of |z − (6 + 8i)| is equal to (a) 12 (b) ∞ (c) 8 (d) 10 (9 9 th April 2 nd Shift 2024) 3. If z is a complex number, then the number of common roots of the equations z 1985 + z 100 + 1 = 0 and z 3 + 2z 2 + 2z + 1 = 0, is equal to (a) 2 (b) 3 (c) 1 (d) 0 (30 th Jan 2 nd Shift 2024) 4. Let a ≠ b be two non-zero real numbers. Then the number of elements in the set X = {z ∈ C : Re(az 2 + bz) = a and Re(bz 2 + az) = b} is equal to (a) 2 (b) 0 (c) 3 (d) 1 (6 6 th April 2 nd Shift 2023) 5. Let A = {θ ∈ (0,2π): 1+2isinθ 1−isin θ is purely imaginary }. Then the sum of the elements in A is (a) 2π (b) 4π (c) π (d) 3π (8th April 2 nd Shift 2023) 6. The value of ( 1+sin2π 9 +icos 2π 9 1+sin2π 9 −icos 2π 9 ) 3 is (a) − 1 2 (1 − i√3) (b) 1 2 (√3 + i) (c) − 1 2 (√3 − i)
(d) 1 2 (1 − i√3) (24 th Jan 2 nd Shift 2023, 2 nd Sept 1 st Shift 2020) 7. For two non-zero complex numbers z1 and z2, if Re(z1z2 ) = 0 and Re(z1 + z2 ) = 0, then which of the following are possible? (A) Im(z1 ) > 0 and Im(z2 ) > 0 (B) Im(z1 ) < 0 and Im(z2 ) > 0 (C) Im(z1 ) > 0 and Im(z2 ) < 0 (D) Im(z1 ) < 0 and Im(z2 ) < 0 Choose the correct answer from the options given below: (a) A and B (b) B and C (c) B and D (d) A and C (29 th Jan 1 st Shift 2023) 8. If the real part of the complex number (1 − cos θ + 2isin θ) −1 is 1/5, for θ ∈ (0, π), then the value of the integral ∫ θ sin xdx is equal to (a) 1 (b) 0 (c) -1 (d) 2 (20 th July 2 nd Shift 2021) 9. Let u = 2z+i z−ki , z = x + iy and k > 0. If the curve represented by Re(u) + Im(u) = 1 intersects the y-axis at the points P and Q, where PQ = 5, then the value of k is (a) 1/2 (b) 3/2 (c) 4 (d) 2 (4 4 th Sept 1 st Shift 2020) 10. If a and b are real numbers such that (2 + α) 4 = a + bα, where α = −1+i√3 2 , then a + b is equal to (a) 57 (b) 9 (c) 24 (d) 33 (4 4 th Sept 2 nd Shift 2020) 11. The value of ( −1+i√3 1−i ) 30 is (a) −2 15 (b) 2 15i (c) −2 15i (d) 6 5 (5 5 th Sept 2 nd Shift 2020) 12. Let z ∈ C with Im(z) = 10 and it satisfies 2z−n 2z+n = 2i − 1 for some natural number n. Then (a) n = 20 and Re(z) = 10 (b) n = 20 and Re(z) = −10 (c) n = 40 and Re(z) = −10 (d) n = 40 and Re(z) = 10(12th April 2 nd Shift 2019)
13. Let A = {θ ∈ (− π 2 , π) : 3+2isinθ 1−2isinθ is purely imaginary }. Then the sum of the elements in A is (a) 3π/4 (b) 2π/3 (c) π (d) 5π/6 14. Let z = ( √3 2 + i 2 ) 5 + ( √3 2 − i 2 ) 5 . If R(z) and I(z) respectively denote the real and imaginary parts of z, then (a) R(z) > 0 and I(z) > 0 (b) I(z) = 0 (c) R(z) < 0 and I(z) > 0 (d) R(z) = −3 (10 th Jan nd Shift 2019) 15. Let (−2 − 1 3 i) 3 = x+iy 27 (i = √−1), where x and y are real numbers, then y − x equals (a) -91 (b) -85 (c) 85 (d) 91 (11 1 th Jan 1 st Shift 2019) 16. The least positive integer n for which ( 1+i√3 1−i√3 ) n = 1, is (a) 3 (b) 5 (c) 2 (d) 6 (Online 2018) 17. Let ω be a complex number such that 2ω + 1 = z where z = √−3. If | 1 1 1 1 −ω 2 − 1 ω 2 1 ω 2 ω 7 | = 3k, then k is equal to (a) z (b) -1 (c) 1 (d) −z(2017) 18. A value of θ for which 2+3isinθ 1−2isinθ is purely imaginary is (a) π/3 (b) π/6 (c) sin−1 ( √3 4 ) (d) sin−1 ( 1 √3 ) (2016) 19. If ω(≠ 1) is a cube root of unity, and (1 + ω) 7 = A + Bω. Then (A, B) equals (a) (1,0) (b) (−1,1) (c) (0,1) (d) (1,1) (2011)
20. If z 2 + z + 1 = 0, where z is a complex number, then the value of (z + 1 z ) 2 + (z 2 + 1 z 2 ) 2 + (z 3 + 1 z 3 ) 2 + ⋯ . + (z 6 + 1 z 6 ) 2 is (a) 18 (b) 54 (c) 6 (d) 12 (2006) 21. If the cube roots of unity are 1, ω, ω 2 then the roots of the equation (x − 1) 3 + 8 = 0, are (a) −1, −1, −1 (b) −1, −1 + 2ω, −1 − 2ω 2 (c) −1,1 + 2ω, 1 + 2ω 2 (d) −1,1 − 2ω, 1 − 2ω 2 (2005) 22. If z = x − iy and z 1/3 = p + iq, then ( x p + y q ) (p2+q2) is equal to (a) 2 (b) -1 (c) 1 (d) -2 (2004) 23. If ( 1+i 1−i ) x = 1, then (a) x = 2n, where n is any positive integer (b) x = 4n + 1, where n is any positive integer (c) x = 2n + 1, where n is any positive integer (d) x = 4n, where n is any positive integer (2003) Numerical Value Type 24. Let S = {z ∈ C − {i, 2i}: z 2+8iz−15 z 2−3iz−2 ∈ R}. If α − 13 11 i ∈ S, α ∈ R − {0}, then 242α 2 is equal to . (11 th April 2 nd Shift 2023) 25. The least positive integer n such that (2i) n (1−i) n−2 , i = √−1, is a positive integer, is . (26 th Aug 2 nd Shift 2021) 26. If the real part of the complex number z = 3+2icosθ 1−3icosθ , θ ∈ (0, π 2 ) is zero, then the value of sin2 3θ + cos2 θ is equal to . (27 th July 2 nd Shift 2021) 27. Let i = √−1. If (−1+i√3) 21 (1−i) 24 + (1+i√3) 21 (1+i) 24 = k, and n = [|k|] be the greatest integral part of |k|. Then ∑j=0 n+5 (j + 5) 2 − ∑j=0 n+5 (j + 5) is equal to . (24 th Feb 2 nd Shift 2021) 28. The sum of 162th power of the roots of the equation x 3 − 2x 2 + 2x − 1 = 0 is . (26 th Febst Shift 2021)