Class 11 Mathematics Complex Numbers and Quadratic Equation MCQs Set 05

Question: The polar form of the complex number (i²⁵)³ is
(a) cosπ/2 + isinπ/2
(b) cosπ/2 – isinπ/2
(c) cosπ/3 + isinπ/3
(d) cosπ/3 – isinπ/3 
Answer: b

Question: (x – iy) (3 + 5i) is the conjugate of (–6 – 24i), then x and y are
(a) x = 3, y = –3
(b) x = –3, y = 3
(c) x = –3, y = –3
(d) x = 3, y = 3 
Answer: a

Question: A value of k for which the quadratic equation x² – 2x(1 + 3k) + 7(2k + 3) = 0 has equal roots is
(a) 1
(b) 2
(c) 3
(d) 4 
Answer: b

Question: If the equations k (6x² + 3) + rx + 2x² – 1 = 0 and 6k (2x² – 1) + px + 4x² + 2 = 0 have both roots common, then the value of (2r – p) is :
(a) 0
(b) 1/2
(c) 1
(d) None of these
Answer: a

Question: If the difference between the roots of the equation x² + ax + 1 = 0 is less than √5 , then the set of possible values of a is
(a) (3,∞)
(b) (-∞,-3)
(c) (– 3, 3)
(d) (-3, ∞)
Answer: c

Question: If ω(≠1) is a cube root of unity, and (1+ω)⁷ = A+ Bω. Then (A, B) equals
(a) (1, 1)
(b) (1, 0)
(c) (–1, 1)
(d) (0, 1)
Answer: a

Question: If | z + 4 | ≤ 3, then the maximum value of | z + 1 | is
(a) 6
(b) 0
(c) 4
(d) 10
Answer: a

Question: For all complex numbers of the form 1 + iα, α ∈ R , if z² = x + iy, then
(a) y² – 4x + 2 = 0
(b) y² + 4x – 4 = 0
(c) y² – 4x – 4 = 0
(d) y² + 4x + 2 = 0
Answer: b

Question: The number of complex numbers such that |z – 1| = |z + 1| = |z – i| equals
(a) 1
(b) 2
(c) ∝
(d) 0
Answer: a

Question: If z = 1 + i, then the multiplicative inverse of z² is (where, i = √–1 )
(a) 2i
(b) 1– i
(c) –i/2
(d) i/2 
Answer: c

Question: The roots of the equation 4x – 3 . 2^{x + 3} + 128 = 0 are
(a) 4 and 5
(b) 3 and 4
(c) 2 and 3
(d) 1 and 2 
Answer: b

Question: If z = 7 – i/3 – 4i then |z|¹⁴ =
(a) 2⁷
(b) 2⁷ i
(c) –2⁷
(d) –2⁷ i 
Answer: a

Question: If (1 – i)ⁿ = 2ⁿ, then the value of n is 
(a) 1
(b) 2
(c) 0
(d) None of these 
Answer: c

Question: If z₁ = 6 + 3i and z₂ = 2 – i, then z₁/z₂ is equal to
(a) 1/5(9 + 12i ) 
(b) 9 + 12i
(c) 3 + 2i
(d) 1/5(12 + 9i)
Answer: a

Question: If the equation (m – n)x²+ (n – l)x + l – m = 0 has equal roots, then l, m and n satisfy
(a) 2l = m + n
(b) 2m = n + l
(c) m = n + l
(d) l = m + n 
Answer: b

Question: If z = 2 –3i, then value of z² – 4z + 13 is
(a) 0
(b) 1
(c) 2
(d) 3 
Answer: a

Question: If | z – 4 | < | z – 2 |, its solution is given by
(a) Re(z) > 0
(b) Re(z) < 0
(c) Re(z) > 3
(d) Re(z) > 2 
Answer: c

Question: The sum of the roots of the equation, x² + |2x – 3| – 4 = 0, is:
(a) 2
(b) – 2
(c) √2
(d) – √2
Answer: c

Question: If 5, 5r, 5r² are the lengths of the sides of a triangle, then r cannot be equal to:
(a) 3/4
(b) 5/4
(c) 7/4
(d) 3/2
Answer: c

Question: If z = 5i(–3/5i) , then z is equal to 3 + bi. The value of ‘b’ is
(a) 1
(b) 2
(c) 0
(d) 3 
Answer: c

Question: If α, β are the roots of the equation (x – a) (x – b) = 5, then the roots of the equation (x – α) (x – β) + 5 = 0 are
(a) a, 5
(b) b, 5
(c) a, α
(d) a, b 
Answer: d

Question: Let α, b ∈ R, α ≠ 0 be such that the equation, αx² – 2bx + 5 = 0 has a repeated root α, which is also a root of the equation, x² – 2bx – 10 = 0. If β is the other root of this equation, then α² + β² is equal to :
(a) 25
(b) 26
(c) 28
(d) 24
Answer: a

Question: If α, β are the roots of (x – a) (x – b) = c, c ≠ 0, then the roots of (x – α) (x – β) + c = 0 shall be
(a) a, c
(b) b, c
(c) a, b
(d) a + c, b + c 
Answer: c

 Question: If p and q are the roots of the equation x²+px+q = 0, then
(a) p = 1, q = –2
(b) p = 0, q = 1
(c) p = –2, q = 0
(d) p = – 2, q =1 
Answer: a

Question: The modulus and amplitude of 1 + 2i/1 –(1 –i) are
(a) √2 and π/6
(b) 1 and 0
(c) 1 and π/3
(d) 1 and π/4 
Answer: b

Question: The equation whose roots are twice the roots of the equation, x² – 3x + 3 = 0 is:
(a) 4x² + 6x + 3 = 0
(b) 2x² – 3x + 3 = 0
(c) x² – 3x + 6 = 0
(d) x² – 6x + 12 = 0 
Answer: d

Question: If α,β are the roots of ax² + bx + c = 0, then αβ² + α²β + αβ equals
(a) c(a–b)/a²
(b) 0
(c) –bc/a²
(d) abc 
Answer: a

Question: If ƒ(x) is a quadratic expression such that ƒ(a) + ƒ(b) = 0, and – 1 is a root of ƒ(x) = 0, then the other root of ƒ(x) = 0 is
(a) – 5/8
(b) – 8/5
(c) 5/8
(d) 8/5
Answer: d

Question: If α and β are the roots of the equation x² + px + 2 = 0 and 1/α and 1/β are the roots of the equation 2x² + 2qx + 1 = 0, then (α – 1/α)(β – 1/β)(α + 1/β)(β + 1/α) is equal to :
(a) 9/4 (9 + q²)
(b) 9/4 (9 – q²)
(c) 9/4 (9 + p²)
(d) 9/4 (9 – p²)
Answer: d

Question: The value of i⁴ⁿ⁺¹ – i^{4n – 1} /2 is
(a) i
(b) 2i
(c) –i
(d) –2i 
Answer: a

Question: Value of i⁵⁹² + i⁵⁹⁰ + i⁵⁸⁸ + i⁵⁸⁶ + i⁵⁸⁴/i⁵⁸² + i⁵⁸⁰ + i⁵⁷⁸ + i⁵⁷⁶ + i⁵⁷⁴ – 1 is
(a) –2
(b) 0
(c) –1
(d) 1 
Answer: a

Question: If |z | = 1, (z ≠ –1) and z = x + iy, then (z –1/z + 1) is
(a) purely real
(b) purely imaginary
(c) zero
(d) undefined 
Answer: b

Question: If the sum of the square of the roots of the equation x² – (sin α – 2)x – (1 + sin α) = 0 is least, then a is equal to
(a) π/6
(b) π/4
(c) π/3
(d) π/2
Answer: d

Question: Sachin and Rahul attempted to solve a quadratic equation. Sachin made a mistake in writing down the constant term and ended up in roots (4,3). Rahul made a mistake in writing down coefficient of x to get roots (3,2). The correct roots of equation are :
(a) 6, 1
(b) 4, 3
(c) – 6, – 1
(d) – 4, – 3
Answer: a

Question: If 2 + 3i is one of the roots of the equation 2x³ – 9x² + kx – 13 = 0, k ∈ R, then the real root of this equation :
(a) exists and is equal to – 1/2.
(b) exists and is equal to 1/2.
(c) exists and is equal to 1.
(d) does not exist.
Answer: b

Question: If z and ω are two non- zero complex numbers such that |zω| =1 and Arg(z) – Arg(ω) = π/2 then z̅ω, is equal to
(a) – 1
(b) 1
(c) – i
(d) i
Answer: a

Question: If α and β are roots of the equation x² + px + 3p / 4 = 0 such that | α -β |= 10, then p belongs to the set :
(a) {2, – 5}
(b) {– 3, 2}
(c) {– 2, 5}
(d) {3, – 5}
Answer: c

Question: If a complex number z statisfies the equation z + √2 | z +1| +i = 0 , then | z | is equal to :
(a) 2
(b) √3
(c) √5
(d) 1
Answer: c

Question: If a and b are the odd integers, then the roots of the equation 2ax² + (2a + b)x + b = 0, a ≠ 0, will be
(a) rational
(b) irrational
(c) non-real
(d) equal 
Answer: a