Assignments Class 11 Mathematics Complex Numbers and Quadratic Equations

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Complex Numbers and Quadratic Equations Assignments Class 11 Mathematics

Question. If z = a + ib, then real and imaginary part of z are
(a) Re(z) = a, Im(z) = b
(b) Re(z) = b, Im(z) = a
(c) Re(z) = a, Im(z) = ib
(d) None of these 

Question. The conjugate of the complex number 2+5i/4–3i is equal to :
(a) 7– 26i/25
(b) –7– 26i/25
(c) – 7+26i/25
(d) 7+26i/25

Question. The solution of √3x² – 2 = 2x –1 are :
(a) (2, 4)
(b) (1, 4)
(c) (3, 4)
(d) (1, 3) 

Question. If z = x –i y and z³ = p + iq, then (x/p + y/q) / (p² + q²) is equal to
(a) –2
(b) –1
(c) 2
(d) 1 

Question. Simplify the complex numbers given in column-I and match with column-II. 

Codes:
     A B C
(a) 1 2 3
(b) 2 1 3
(c) 1 3 2
(d) 2 3 1 

Question. If one root of ax² + bx + c = 0 be square of the other, then the value of b³ + ac² + a² c is
(a) 3abc
(b) –3abc
(c) 0
(d) None of these

Question. The value of 

(a) 1 – √2
(b) 1 + √2
(c) 1 ± √2
(d) None of these 

Question. If z is a complex number such that z – 1/z + 1 is purely imaginary, then
(a) |z | = 0
(b) |z | = 1
(c) |z | > 1
(d) |z | < 1 

Question. Find the value of a such that the sum of the squares of the roots of the equation x² – (a – 2)x – (a + 1) = 0 is least.
(a) 4
(b) 2
(c) 1
(d) 3

Question. If z₁ = 2 + 3i and z₂ = 3 – 2i, then z₁ – z₂ equals to –1 + bi. The value of ‘b’ is
(a) 1
(b) 2
(c) 3
(d) 5 

Question. If z is a complex number such that z₂ = (z̄)² , then
(a) z is purely real
(b) z is purely imaginary
(c) either z is purely real or purely imaginary
(d) None of these 

Question. Simplify the complex numbers given in column-I and match with column-II 

Codes:
     A B C D
(a) 1 2 3 4
(b) 2 1 3 4
(c) 2 3 4 1
(d) 1 4 3 2 

Question. The roots of the given equation (p – q) x² + (q – r) x + (r – p) = 0 are :
(a) p–q/r–p,1
(b) q–r/p–q,1
(c) r–p/p–q,1
(d) None of these 

Question. If x = √–16, then
(a) x = 4i
(b) x = 4
(c) x = – 4
(d) All of these 

Question. If (x + iy)^1/3 = a + ib, where x, y, a, b ∈ R, then x/a – y/b =
(a) a² – b²
(b) –2(a² + b²)
(c) 2(a² – b²)
(d) a² + b² 

Question. Product of real roots of the equation t²x²+|x|+9 = 0
(a) is always positive
(b) is always negative
(c) does not exist
(d) None of these 

Question. The value of (1 + i)⁴ (1 + 1/1)⁴ is
(a) 12
(b) 2
(c) 8
(d) 16 

Question. If α, β be the roots of the equation 2x² – 35x + 2 = 0, then the value of (2α – 35)3 . (2β – 35)3 is equal to
(a) 1
(b) 64
(c) 8
(d) None of these 

Question. The roots of equation x–2/x–1 = 1– 2/x–1 is
(a) one
(b) two
(c) infinite
(d) None of these

Question. The solutions of the quadratic equation ax² + bx + c = 0, where a, b, c ∈ R, a ≠ 0, b² – 4ac < 0, are given by x = ? 

Question. Solve √5x² + x + √5 = 0.
(a) ± √19/5 i
(b) ± √19i/2
(c) – 1± √19i/2√5
(d)  – 1± √19i/√5

Question. If the roots of the equation x² – 5x + 16 = 0 are α, β and the roots of equation x² + px + q = 0 are α² + β², αβ/2 ,then
(a) p = 1, q = –56
(b) p = –1, q = –56
(c) p = 1, q = 56
(d) p = –1, q = 56 

Question. Evaluate: (1 + i)⁶ + (1 – i)³.
(a) –2 – 10i
(b) 2 – 10i
(c) –2 + 10i
(d) 2 + 10i 

Question. If z₁ = √3 + i√3 and z₂ = √3 + i , then in which quadrant z₁/z₂ lies?
(a) I
(b) II
(c) III
(d) IV 

Question. Value of (2i/1+i)² is
(a) i
(b) 2i
(c) 1 – i
(d) 1 – 2i 

Question. If z(2 – i) = (3 + i), then z²⁰ is equal to
(a) 2¹⁰
(b) –2¹⁰
(c) 2²⁰
(d) –2²⁰ 

Question. If z₁ = 3 + i and z₂ = i – 1, then
(a) | z₁ + z₂| > |z₁| + |z₂|
(b) | z₁+ z₂ | < |z₁| – |z₂|
(c) |z₁ + z₂| ≤ |z₁| + |z₂|
(d) |z₁ + z₂| < |z₁| + |z₂|

Question. Simplify the complex numbers given in column-I and match with column-II

Codes:
     A B C D E
(a) 5 4 3 2 1
(b) 4 1 5 2 3
(c) 4 2 5 1 3
(d) 3 1 2 5 4 

Question. Let z be any complex number such that | z | = 4 and arg (z) = 5π/6 , then value of z is
(a) –2√3 – 2i
(b) 2√3 – i
(c) √2 + 3i
(d) –2√3 + 2i 

Question. If α, β are the roots of the equation ax² + bx + c = 0, then the value of 1/aα + b + 1/aβ + b equals
(a) ac/b
(b) 1
(c) ab/c
(d) b/ac 

Question. If x² + y² = 25, xy = 12, then x =
(a) {3, 4}
(b) {3, –3}
(c) {3, 4, –3, –4}
(d) {–3, –3} 

Question. If x² + ax + 10 = 0 and x² + bx – 10 = 0 have a common root, then a² – b² is equal to
(a) 10
(b) 20
(c) 30
(d) 40 

Question. i⁵⁷ + 1 /i²⁵  when simplified has the value
(a) 0
(b) 2i
(c) – 2i
(d) 2 

Question. The modulus of (1 + i√3)(2 +2i)/(√3–i) is
(a) 2
(b) 4
(c) 3√2
(d) 2√2

Question. If the product of the roots of the equation (a + 1)x² + (2a + 3)x + (3a + 4) = 0 be 2, then the sum of roots is
(a) 1
(b) –1
(c) 2
(d) –2 

Question. Modulus of z = (1 + i√3)(cosθ+ isinθ)/2(1 – i)(cosθ– isinθ)  is
(a) 1/√3
(b) – 1/√2
(c) 1/√2
(d) 1 

Question. The argument of the complex number (i/2 – 2/i) is equal to
(a) π/4
(b) 3π/4
(c) π/12
(d) π/2 

Question. Value of k such that equations 2x² + kx – 5 = 0 and x² – 3x – 4 = 0 have one common root, is
(a) –1, –2
(b) –3, –27/4
(c) 3, 4/27
(d) –2, – 3

Question. √–3 √–6 is equal to
(a) 3√2
(b) –3√2
(c) 3√2 i
(d) –3√2i 

Question. If the complex numbers z₁, z₂,z₃ represents the vertices of an equilateral triangle such that | z₁ | = | z₂ | = | z₃ |, then value of z₁ + z₂ + z₃ is
(a) 0
(b) 1
(c) 2
(d) 3/2 

Question. If α and β are the roots of the equation x² + 2x + 4 = 0, then 1/α³ + 1/β³ is equal to
(a) – 1/2
(b) 1/2
(c) 32
(d) 1/4 

Question. Amplitude of 1+√3i/√3+1 is :
(a) π/6
(b) π/4
(c) π/3
(d) π/2 

Question. If the sum of the roots of the equation x² + px + q = 0 is three times their difference, then which one of the following is true?
(a) 9p² = 2q
(b) 2q² = 9p
(c) 2p² = 9q
(d) 9q² = 2p 

Question. If a < b < c < d, then the nature of roots of ( x – a) (x – c) + 2 (x – b) (x – d) = 0 is
(a) real and equal
(b) complex
(c) real and unequal
(d) None of these

Question. Which of the following is correct for any two complex numbers z₁ and z₂?
(a) |z₁ z₂| = |z₁| |z₂|
(b) arg(z₁ z₂) = arg(z₁) . arg(z₂)
(c) |z₁ + z₂| = |z₁| + |z₂|
(d) |z₁ + z₂| ≥ |z₁| – |z₂| 

Question. The value of (1 + i)⁵ (1 – i)⁵ is 2ⁿ. ‘n’ is equal to
(a) 2
(b) 3
(c) 4
(d) 5 

Question. The complex number 1+ 2i/1–i lies in:
(a) I quadrant
(b) II quadrant
(c) III quadrant
(d) IV quadrant 

Question. For the equation 1/x + a – 1/x + b = 1/x + c , if the product of roots is zero, then sum of roots is
(a)– 2bc/b+c
(b) 2ca/c + a
(c) bc/c + a
(d) –bc/b+c 

Question. Represent z = 1 + i √3 in the polar form.
(a) cosπ/3 + i sinπ/3
(b) cosπ/3 – i sinπ/3
(c) 2 (cosπ/3 + i sinπ/3)
(d) 4 cosπ/3 + i sinπ/3) 

Question. If z₁ = 2 – i and z₂ = 1 + i, then value of |z₁ + z₂ + 1/z₁ – z₂ + 1| is
(a) 2
(b) 2i
(c) 2
(d) 2i 

Question. If z₁ = 2 + 3i and z₂ = 3 + 2i, then z₁ + z₂ equals to a + ai. Value of ‘a’ is equal to
(a) 3
(b) 4
(c) 5
(d) 2 

Question. If the sum of the roots of the equation x² + px + q = 0 is equal to the sum of their squares, then
(a) p² – q² = 0
(b) p² + q² = 2q
(c) p² + p = 2q
(d) None of these 

Question. The value of (1 + i)⁸ + (1 – i)⁸ is 2n. Value of n is
(a) 2
(b) 3
(c) 4
(d) 5 

Question. If ax² + bx + c = 0 is a quadratic equation, then equation has no real roots, if
(a) D > 0
(b) D = 0
(c) D < 0
(d) None of these 

Question. If z be the conjugate of the complex number z, then which of the following relations is false?
(a) |z| =| z̄|
(b) z.z̄ = | z̄|²
(c)  z̄₁ + z̄₂ = z̄₁ + z̄₂
(d) arg z = arg z̄ 

Question. Additive inverse of 1 – i is
(a) 0 + 0i
(b) –1 – i
(c) –1 + i
(d) None of these

Question. The roots of the equation 3^2x – 10.3^x + 9 = 0 are
(a) 1, 2
(b) 0, 2
(c) 0, 1
(d) 1, 3 

Question. If 2 + i√3 is a root of the equation x² + px + q = 0, where p and q are real, then (p, q) =
(a) (–4, 7)
(b) (4, –7)
(c) (4, 7)
(d) (–4, –7) 

Question. If x² + x + 1 = 0, then what is the value of x ?
(a) 1 + √3i/2
(b) –1 ± √3i/2
(c) –1 ± √3i/3
(d) –1 ± √2i/2 

Question. (1/1–2i + 3/1+i)(3+4i/2–4i)is equal to :
(a) 1/2 + 9/2i
(b) 1/2 – 9/2i
(c) 1/4 – 9/4i
(d) 1/4 + 9/4i