ECAT Maths MCQs —Download ECAT Math Papers

Q1 A quadratic equation in x is an equation that can be written in the form ax² + bx + c = 0, where a, b and c are real and:

• A) a ≠ 0
• B) b ≠ 0
• C) c ≠ 0
• D) a = 0, b ≠ c ≠ 0

Q2 x = (–b – √(b² – 4ac)) / (–2a) is one of the roots of:

• A) ax² – bx + c = 0
• B) –ax² – bx – c = 0
• C) ax² – bx – c = 0
• D) –ax² + bx – c = 0

Q3 To reduce ax²ⁿ + bxⁿ + c = 0; a ≠ 0 to a quadratic equation, the suitable substitution is:

• A) x²ⁿ = y
• B) xⁿ = y
• C) x^–n = y
• D) ax²ⁿ = y

Q4 Equation √3 x^–6 + √7 x^–3 + 11 = 0 reduces to quadratic equation by putting:

• A) x² = y
• B) x^–6 = y
• C) x^–3 = y
• D) 1/x = y

Q5 x = 9 is a root of equation:

• A) (x–7)(x+3)(x+1)(x+5) – 1680 = 0
• B) (x–7)(x–3)(x+1)(x+5) – 1680 = 0
• C) (x+7)(x+3)(x+1)(x+5) – 1680 = 0
• D) None of these

Q6 Solution set of equation 4^1+x + 4^1–x = 10 is:

• A) {0, ½}
• B) {–½, ½}
• C) {–½, 0}
• D) {–1, 1}

Q7 Which one is NOT a reciprocal equation?

• A) x = 1
• B) x + 1/x = 1
• C) x = 0
• D) x = 1/x

Q8 (2^x – 8)(2^–x – 4) = 0 ⇒ x =

• A) (3, 2)
• B) (2, 3)
• C) (3, –2)
• D) (–3, –2)

Q9 Which one is NOT the square root of unity?

• A) 1
• B) –1
• C) 0
• D) (a) and (c)

Q10 The condition for polynomial equation ax² + bx + c = 0 to be quadratic is:

• A) a > 0
• B) a < 0
• C) a ≠ 0
• D) a ≠ 0, b ≠ 0

Q11 If α, β are non-real roots of ax² + bx + c = 0 (a, b, c ∈ Q), then:

• A) α = β
• B) αβ = 1
• C) α = β̄
• D) α = 1

Q12 The roots of (x – a)(x – b) = ab x² are always:

• A) Real
• B) Depends upon a
• C) Depends upon b
• D) Depends upon a and b

Q13 Both the roots of the equation (x – b)(x – c) + (x – c)(x – a) + (x – a)(x – b) = 0 are always:

• A) Positive
• B) Negative
• C) Real
• D) None of these

Q14 If the roots of ax² + b = 0 are real and distinct, then:

• A) ab > 0
• B) a = 0
• C) ab < 0
• D) a > 0, b > 0

Q15 If a > 0, b > 0, c > 0, then the roots of the equation ax² + bx + c = 0 are:

• A) Real and negative
• B) Non-real with negative real parts
• C) Real and positive
• D) Nothing can be said

Q16 The quadratic equation 8 sec² θ – 6 sec θ + 1 = 0 has:

• A) Infinitely many roots
• B) Exactly two roots
• C) Exactly four roots
• D) No roots

Q17 If the roots of ax² + bx + c = 0 are equal in magnitude but opposite in sign, then:

• A) a = 0
• B) b = 0
• C) c = 0
• D) None of these

Q18 If α, β are the roots of ax² + bx + c = 0, then roots of cx² + bx + a = 0 are:

• A) –α, –β
• B) α, 1/β
• C) β, 1/α
• D) 1/α, 1/β

Q19 The roots of the equation 2^2x – 10·2^x + 16 = 0 are:

• A) 2, 8
• B) 1, 3
• C) 1, 8
• D) 2, 3

Q20 The number of real roots of the equation 1 + a₁x + a₂x² + … + a_nxⁿ = 0, where |x| < 1/3 and |a_n| < 2, is:

• A) n if n is even
• B) 0 for any natural number n
• C) 1 if n is odd
• D) None of these

Q21 The set of real roots of the equation log_(3x+4)(2x+3)³ – log_(2x+3)(10x² + 23x + 12) = 1 is:

• A) {–1}
• B) {–3/5}
• C) Empty set
• D) {–1/3}

Q22 The equation x^{(3/4)(log₂x)² + log₂x – 5/4} = √2 has:

• A) Only one real solution
• B) Exactly three real solutions
• C) Exactly one rational solution
• D) Non-real roots

Q23 The equation (cos p – 1)x² + x(cos p) + sin p = 0 in the variable x has real roots, then p can take any value in the interval:

• A) (0, 2π)
• B) (–π, 0)
• C) (–π/2, π/2)
• D) (0, π)

Q24 If α and β are the roots of x² + px + q = 0 and α⁴, β⁴ are the roots of x² – rx + s = 0, then the equation x² – 4qx + 2q² – r = 0 has always:

• A) Two real roots
• B) Two positive roots
• C) Two negative roots
• D) One positive and one negative root

Q25 If the roots of ax² – bx – c = 0 change by the same quantity, then the expression in a, b, c that does not change is:

• A) (b² – 4ac) / a²
• B) (b – 4c) / a
• C) (b² + 4ac) / a²
• D) None of these

Q26 If y = tan x · cot 3x, x ∈ R, then:

• A) 1/3 < y < 1
• B) 1/3 ≤ y ≤ 1
• C) 1/3 ≤ y ≤ 3
• D) None of these

Q27 If the roots of ax² + bx + c = 0 (a > 0) be greater than unity, then:

• A) a + b + c = 0
• B) a + b + c > 0
• C) a + b + c < 0
• D) None of these

Q28 If 2 – 3x – 2x² ≥ 9, then:

• A) x ≤ –2
• B) –2 ≤ x ≤ ½
• C) x ≥ –2
• D) x ≤ ½

Q29 For the equation |x²| + |x| – 6 = 0, the roots are:

• A) One and only one real number
• B) Real with sum one
• C) Real with sum zero
• D) Real with product zero

Q30 Roots of the equation 3^x–1 + 3^1–x = 2 are:

• A) 2
• B) 1
• C) 0
• D) –1

Q31 If a(p + q)² + bpq + c = 0 and a(p + r)² + 2bpr + c = 0, then qr equals:

• A) p² + c/a
• B) p² + a/c
• C) p² – c/a
• D) p² – a/c

Q32 The number of solutions of (log 5 + log(x² + 1)) / log(x – 2) = 2 is:

• A) 2
• B) 3
• C) 1
• D) None of these

Q33 The ratio of the roots of ax² + bx + c = 0 is the same as the ratio of the roots of px² + qx + r = 0. If D₁ and D₂ are the discriminants of the two equations, then D₁ : D₂ =

• A) a²/p²
• B) b²/q²
• C) c²/r²
• D) None of these

Q34 The solution of the quadratic equation x² – 7x + 10 = 0 is:

• A) 2
• B) 5
• C) 2, 5
• D) 7

Q35 The graph of the quadratic equation is:

• A) Straight line
• B) Circle
• C) Parabola
• D) Ellipse

Q36 In the quadratic equation f(x) = ax², if a > 0, then the graph of the parabola:

• A) Opens up
• B) Opens down
• C) Closes up
• D) Symmetric w.r.t x-axis

Q37 If a parabola opens down, then its vertex is at the:

• A) Right of the parabola
• B) Left of the parabola
• C) Lowest point on the parabola
• D) Highest point on the parabola

Q38 If f(x) = ax² and a > 0, then the lowest point on the parabola is called:

• A) Vertex of parabola
• B) Co-ordinates of parabola
• C) Roots of the equation
• D) Coefficient of the equation

Q39 The maximum value of the quadratic function f(x) = –x² + 6x + 2 is:

• A) 11
• B) 6
• C) –11
• D) 13

Q40 The minimum value of the quadratic function f(x) = 5x² – 11 is:

• A) 7
• B) –7
• C) 6
• D) –11

Q41 The maximum value of the quadratic function f(x) = –2x² + 20x is:

• A) 4
• B) 3
• C) 5
• D) 7

Q42 ω¹⁵ =

• A) 0
• B) 1
• C) ω
• D) ω²

Q43 ω^–1 =

• A) 0
• B) 1
• C) ω
• D) ω²

Q44 ω⁴ =

• A) 0
• B) 1
• C) ω
• D) ω²

Q45 ω^–12 =

• A) 0
• B) 1
• C) ω
• D) ω²

Q46 ω¹¹ =

• A) 0
• B) 1
• C) ω
• D) ω²

Q47 If a polynomial P(x) is divided by x – a, then the remainder is:

• A) P(0)
• B) P(–a)
• C) P(a)
• D) None of these

Q48 The sum of roots of the equation ax² + bx + c = 0, a ≠ 0 is:

• A) b/a
• B) –b/a
• C) c/a
• D) a/b

Q49 The product of the roots of the equation ax² + bx + c = 0, a ≠ 0 is:

• A) c/a
• B) –c/a
• C) b/a
• D) –b/a

Q50 If S and P are the sum and the product of roots of a quadratic equation, then the quadratic equation is:

• A) x² + Sx – P = 0
• B) x² – Sx + P = 0
• C) x² – Sx – P = 0
• D) x² + Sx + P = 0

Q51 For any integer k, ωⁿ = _____ when n = 3k:

• A) 1
• B) 2
• C) 0
• D) –4

Q52 ω²⁹ =

• A) 0
• B) 1
• C) ω^–1
• D) ω

Q53 ω⁷³ =

• A) 0
• B) 1
• C) ω
• D) ω²

Q54 ω²⁸ + ω³⁸ =

• A) 0
• B) 1
• C) ω
• D) –1

Q55 (2 + ω)(2 + ω²) =

• A) 1
• B) 2
• C) 3
• D) 0

Q56 If x³ + ax² – a²x – a³ is divided by x + a, then the remainder is:

• A) –a³
• B) a³
• C) 2a³
• D) 0

Q57 If ω is a complex cube root of unity, then ω =

• A) 0
• B) 1
• C) ω²
• D) ω^–2

Q58 If ω is a complex cube root of unity, then ω² =

• A) 0
• B) 1
• C) ω³
• D) ω^–1

Q59 If α, β are roots of 2x² – 4x + 5 = 0, then 1/α + 1/β =

• A) 5/4
• B) –5/4
• C) 4/5
• D) –4/5

Q60 If α, β are roots of 2x² – 4x + 5 = 0, then (α+1)(β+1) =

• A) 11/2
• B) –11/2
• C) 2/11
• D) –2/11

Q61 If α, β are roots of 2x² – 4x + 5 = 0, then α² + β² =

• A) –1
• B) 0
• C) 2
• D) 1

Q62 The cube roots of 8 are:

• A) 1, ω, ω²
• B) 2, 2ω, 2ω²
• C) –2, –2ω, –2ω²
• D) 3, 3ω, 3ω²

Q63 If ω is a complex cube root of unity, then ω²⁹ + ω²⁸ + 1 =

• A) 0
• B) 1
• C) 2
• D) 3

Q64 The quadratic equation with roots 2 – √3 and 2 + √3 is:

• A) x² – 4x + 1 = 0
• B) x² – 3x + 3 = 0
• C) x² + 4x + 1 = 0
• D) x² – 4x – 1 = 0

Q65 Which one is NOT the imaginary cube root of unity?

• A) 1
• B) (–1 + √3 i)/2
• C) (–1 – √3 i)/2
• D) None of these

Q66 ((–1 + √3 i)/2)² =

• A) (–1 – √3 i)/2
• B) (–1 + √3 i)/2
• C) (1 + √3 i)/2
• D) (1 – √3 i)/2

Q67 ((–1 – √3 i)/2)³ · ((–1 + √3 i)/2)⁴ =

• A) (–1 – √3 i)/2
• B) (–1 + √3 i)/2
• C) (–1 + √3 i + 9√3 i)/8
• D) (1 – √3 i)/2

Q68 ((–1 + √3 i)/2)⁴ + ((–1 – √3 i)/2)⁵ =

• A) (–1 – √3 i)/2
• B) (–1 + √3 i)/2
• C) –1
• D) Both (a) and (b)

Q69 Sum of all the three cube roots of unity is:

• A) 1
• B) –1
• C) 0
• D) ω

Q70 Which one CANNOT be roots of –1?

• A) 1
• B) –1
• C) (1 + i√3)/2
• D) (1 – i√3)/2

Q71 –2 / (–1 – i√3) is one of the cube roots of:

• A) –1
• B) 1
• C) 0
• D) i

Q72 ω^3n–1 where n ∈ N is equal to:

• A) 1
• B) ω
• C) ω²
• D) 0

Q73 ω^2–6n where n ∈ N is equal to:

• A) 1
• B) ω
• C) ω²
• D) 0

Q74 If ω³ⁿ⁺⁷ – a/ω⁴ = –1 where n ∈ N, then a =

• A) 1
• B) 2
• C) –1
• D) 0

Q75 (–1 + √(–3))⁴ + (–1 – √(–3))⁴ =

• A) –16
• B) 16
• C) 8
• D) –8

Q76 Which one is NOT the fourth root of unity?

• A) 1
• B) –1
• C) 0
• D) i

Q77 Sum of four fourth roots of –1 is:

• A) 1
• B) –1
• C) 0
• D) i

Q78 Which one is NOT the fourth root of –1?

• A) i√i
• B) –i√i
• C) √i
• D) –1

Q79 Product of four fourth roots of –1 is:

• A) 1
• B) –1
• C) i
• D) –i

Q80 If a is a fourth root of unity and a¹¹ = i, then a =

• A) 1
• B) –1
• C) i
• D) –i

Q81 If n is a multiple of 3, then ω^–11n + ω⁴¹ⁿ – 2ω^–7n + ω²ⁿ =

• A) 1
• B) 0
• C) –1
• D) 2

Q82 1 + ω + ω² + ω³ + ω⁴ + … + ω¹⁰⁰ =

• A) –ω²
• B) ω
• C) 0
• D) –ω

Q83 1 · ω · ω² · ω³ · … · ω¹⁰⁰ =

• A) 1
• B) ω
• C) ω²
• D) 0

Q84 If n/3 is an integer, then ω^n³ + ω³ⁿ =

• A) 1
• B) ω
• C) ω²
• D) 2