Class 11 Mathematics Complex Numbers and Quadratic Equation MCQs Set 06

Question: If the roots of the equation a/x – a + b/x – b = 1 are equal in magnitude and opposite in sign, then
(a) a = b
(b) a + b = 1
(c) a – b = 1
(d) a + b = 0 
Answer: d

Question: For the equation 3x² + px + 3 = 0, p > 0, if one of the root is square of the other, then p is equal to
(a) 1/3
(b) 1
(c) 3
(d) 2/3 
Answer: c

Question: The amplitude of sinπ/5 + i(1 – cosπ /5) is
(a) π/5
(b) 2π/5
(c) π/10 
(d) π/15 
Answer: c

Question: If the roots of the equation ax² + bx + c = 0 are α, β, then the value of αβ² + α²β + αβ will be
(a) c(a – b )/a²
(b) 0
(c) – bc/a²
(d) None of these 
Answer: a

Question: If the two roots of the equation, (a – 1)(x⁴ + x2 + 1) + (a + 1)(x² + x + 1)² = 0 are real and distinct, then the set of all values of ‘a’ is :
(a) (0, 1/2)
(b) ( – 1/2)∪(0, 1/2)
(c) ( – 1/2, 0)
(d) (-∞ , -2)∪(2, ∞)
Answer: b

Question: 2x² – (p + 1) x + (p – 1) = 0. If α – β = αβ, then what is the value of p?
(a) 1
(b) 2
(c) 3
(d) –2 
Answer: b

Question: If (1+i / 1-i)^x = 1 then
(a) x = 2n +1 , where n is any positive integer
(b) x = 4n , where n is any positive integer
(c) x = 2n , where n is any positive integer
(d) x = 4n +1 , where n is any positive integer.
Answer: b

Question: If z₁, z₂ and z₃, z₄ are 2 pairs of complex conjugate numbers, then arg(z₁/z₄) + arg(z₂/z₃) equals:
(a) 0
(b) π/2
(c) 3π/2
(d) π
Answer: a

Question: If z₁ and z₂ are two non-zero complex numbers such that | z₁ + z₂ | = | z₁ | + | z₂ | , then arg z₁ – arg z₂ is equal to
(a) π/2
(b) – π
(c) 0
(d) –π/2 
Answer: c

Question: Let z ≠ – i be any complex number such that z–i / z+i is a purely imaginary number. Then Z + z is :
(a) zero
(b) any non- zero real number other than 1.
(c) any non- zero real number.
(d) a purely imaginary number.
Answer: c

Question: The set of all real values of λ for which the quadratic equations, (λ² +1)x² – 4λx + 2 = 0 always have exactly one root in the interval (0, 1) is :
(a) (0, 2)
(b) (2, 4]
(c) (1, 3]
(d) (–3, –1)
Answer: c

Question: If z = x + iy, z^{1/ 3} = a – ib, then x/a – y/b = k ( a² – b² ) where k is equal to
(a) 1
(b) 2
(c) 3
(d) 4 
Answer: d

Question: If c + i/c – i = a + ib, where a, b, c are real, then a² + b² is equal to:
(a) 7
(b) 1
(c) c²
(d) – c² 
Answer: b

Question: If p and q are non- ero real numbers and α³ + β³ = – p,αβ = q, then a quadratic equation whose roots are α²/β , β²/α is :
(a) px² – qx + p² = 0
(b) qx² + px + q² = 0
(c) px² + qx + p² = 0
(d) qx² – px + q² = 0
Answer: b

Question: Let p, q, r ∈ R and r > p > 0. If the quadratic equation px² + qx + r = 0 has two complex roots α and β, then |α| + |β| is
(a) equal to1
(b) less than 2 but not equal to 1
(c) greater than 2
(d) equal to 2
Answer: c

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

Question: If one root of the equation x² + px +12 = 0 is 4, while the equation x² + px + q = 0 has equal roots , then the value of ‘q’ is
(a) 4
(b) 12
(c) 3
(d) 49/4
Answer: d

Question: The value of ‘α’ for which one root of the quadratic equation (α² – 5α + 3)x² + (3α -1)x + 2 = 0 is twice as large as the other is
(a) – 1/3
(b) 2/3
(c) – 2/3
(d) 1/3
Answer: b

Question: |z₁ + z₂| = |z₁| + |z₂| is possible, if
(a) z₂ = z¯₁
(b) z₂ = 1/z₁
(c) arg(z₁) = arg(z₂)
(d) |z₁| = |z₂| 
Answer: c

Question: If z = i^–39, then simplest form of z is equal to a + i. The value of ‘a’ is 
(a) 0
(b) 1
(c) 2
(d) 3 
Answer: a

Question: The modulus of the complex number z such that |z + 3 – i | = 1 and arg(z) = π is equal to
(a) 3
(b) 2
(c) 9
(d) 4
Answer: a

Question: The square root of i is
(a) ±1/√2(–1 + i )
(b) ±1/√2(1 + i )
(c) ±1/√2(1 – i )
(d) None of these 
Answer: b

Question: If z₁ = √2 [cosπ/4 + isinπ/4] and z₂ = √3 [cosπ/3 + isinπ/3] then |z₁ z₂| is equal to m . Value of m is
(a) 6
(b) 3
(c) 2
(d) 5 
Answer: a

Question: If α, β are the roots of the equation ax² + bx + c = 0, then α/aβ + b + β/aα + b =
(a) 2/a 
(b) 2/b
(c) 2/c
(d) –2/a 
Answer: d

Question: The complex number z which satisfies the condition l i+ z/i – z l = 1 lies on
(a) circle x² + y² = 1
(b) the x-axis
(c) the y-axis
(d) the line x + y = 1
Answer: b

Question: If a + ib = x + iy, then possible value of a – ib is
(a) x² + y²
(b) √x² + y²
(c) x + iy
(d) x – iy 
Answer: d

STATEMENT TYPE QUESTIONS

Question: Consider the following statements.
I. Modulus of 1+i /1–i is 1.
II. Argument of 1+i /1–i is π/2
Choose the correct option.
(a) Only I is correct.
(b) Only II is correct.
(c) Both are correct.
(d) Both are incorrect. 
Answer: c

Question: Consider the following statements
I. Additive inverse of (1 – i) is equal to –1 + i.
II. If z₁ and z₂ are two complex numbers, then z₁ – z₂ represents a complex number which is sum of z₁ and additive inverse of z₂.
III. Simplest form of 5 + √2i/1–√2i is 1 + 2√2 i.
Choose the correct option.
(a) Only I and II are correct.
(b) Only II and III are correct.
(c) I, II and III are correct.
(d) I, II and III are incorrect. 
Answer: c

Question: Statement – I : Roots of quadratic equation x² + 3x + 5 = 0 is x = –3 ± i√11/2.
Statement – II : If x² – x + 2 = 0 is a quadratic equation, then its roots are 1 ± i√7/2.
(a) Statement I is correct
(b) Statement II is correct
(c) Both are correct
(d) Both are incorrect 
Answer: a

Question: Consider the following statements.
I. Representation of z = x + iy in terms of r and θ is called polar form of the complex number.
II. arg (z₁/z₂)= arg (z₁) – arg (z₂)
Choose the correct option.
(a) Only I is incorrect.
(b) Only II is correct.
(c) Both I and II are incorrect.
(d) Both I and II are correct. 
Answer: d

Question: Let Z₁ and Z₂ be any two complex number.
Statement 1: |Z₁ – Z₂| ≥ |Z₁| – |Z₂|
Statement 2: |Z₁ + Z₂| ≤ |Z₁| + |Z₂|
(a) Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation of Statement 1.
(b) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation of Statement 1.
(c) Statement 1 is true, Statement 2 is false.
(d) Statement 1 is false, Statement 2 is true.
Answer: b

Question: Consider the following statements.
I. If z, z₁, z₂ be three complex numbers then zz¯ = |z|²
II. The modulus of a complex number z = a + ib is defined as |z| = √a² + b² .
III. Multiplicative inverse of z = 3 – 2i is 3/13 + 2/13i
Choose the correct option.
(a) Only I and II are correct.
(b) Only II and III are correct.
(c) Only I and III are correct.
(d) All I, II and III are correct. 
Answer: d