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If α and β are the roots of the equations x² – (1 + n²) x+ 1 / 2 (1 + n² + n⁴) = 0, then α² + β² is equal to

1) n2 2) – n2 3) n4 4) – n4 Solution: (1) n2 (α + β) = 1 + n2;... View Article

If α and β are the roots of the equation x² – ax + b = 0 and A_n = αⁿ + βⁿ. Then A_n+1 – aA_n + bA_n-1 is equal to

1) – a 2) b 3) 0 4) a – b Solution: (3) 0 If α, β are the roots... View Article

Let the two numbers have arithmetic mean 9 and geometric mean 4. Then, these numbers are the roots of the quadratic equation

1) x2 – 18x – 16 = 0 2) x2 – 18x +16 = 0 3) x2 + 18x –... View Article

If α + β = – 2 and α³ + β³ = – 56, then the quadratic equation, whose roots are α and β, is

1) x2 + 2x – 16 = 0 2) x2 + 2x – 15 = 0 3) x2 + 2x... View Article

If α, β and γ are such that α + β + γ = 2, α² + β² + γ² = 6, α³ + β³ + γ³ = 8 then α⁴ + β⁴ + γ⁴ is equal to

1) 7 2) 12 3) 18 4) 36 Solution: (3) 18 α + β + γ = 2 α2 +... View Article

The number of real solutions of the equation |x|² – 3 | x | + 2 = 0 is

1) 4 2) 1 3) 3 4) 2 Solution: (1) 4 |x|2 – 3 |x| + 2 = 0 (|x|... View Article

If α, β and γ are the roots of x³ + 4x + 1 = 0, then the equation, whose roots are α² / (β + γ), β² / (α + γ), γ² / (α + β), is

1) x3 – 4x – 1 = 0 2) x3 – 4x + 1 = 0 3) x3 + 4x... View Article

If α and β are the roots of x² – 2x + 4 = 0, then the value of α⁶ + β⁶ is

1) 32 2) 64 3) 128 4) 256 Solution: (3) 128 α + β = 2 αβ = 4 α2... View Article

If x₁, x₂, x₃ and x₄ are roots of the equation x⁴ – x³ sin 2β + x² cos 2β – x cos β – sin β = 0. Then tan^-1 x₁ + tan^-1 x₂ + tan^-1 x₃ + tan^-1 x₄ =

1) β 2) π / 2 – β 3) π – β 4) – β Solution: (2) π / 2... View Article

If one root of equation x² + ax + 12 = 0 is 4 while the equation x² + ax + b = 0 has equal roots, then the value of b is

1) 4/49 2) 49/4 3) 7/4 4) 4/7 Solution: (2) 49 / 4 It is given that 4 is the... View Article

The quadratic equation, whose roots are three times the roots of 3ax² + 3bx + c = 0, is

1) ax2 + 3bx + 3c = 0 2) ax2 + 3bx + c = 0 3) 9ax2 + 9bx... View Article

If α and β are the roots of x² – a (x – 1) + b = 0, then the value of 1 / [ɑ² – aɑ] + 1 / [β² – aβ] + 2 / [a + b] is

1) 4 / (a + b) 2) 1 / (a + b) 3) 0 4) – 1 Solution: (3) 0... View Article

If one root of the equation lx² + mx + n = 0 is 9/2 (where, l,m and n are positive integers and m/4n = l/m, then l + n is equal to

1) 80 2) 85 3) 90 4) 95 5) 100 Solution: (2) 85 The given equation is lx2 + mx... View Article

If 2 – i is a root of the equation ax² + 12x² + b = 0(where,a and b are real), then the value of ab is

1) 45 2) 15 3) – 15 4) – 45 5) 25 Solution: (1) 45 Imaginary roots exist in conjugate... View Article

If the roots of the equation 1 / (x + p) + 1 / (x + q) = 1 / r (x ≠ – p, x ≠ – q, r ≠ 0) are equal in magnitude but opposite in sign, then p + q =

1) r 2) 2r 3) r 4) 1/r 5) 2/r Solution: (2) 2r 1 / (x + p) + 1... View Article

If α and β are the roots of the quadratic equation ƛ (x² – x) + x + 5 = 0 and ƛ₁, ƛ₂ are two values ƛ obtained from (ɑ / β) + (β / ɑ) = 4 / 5, then ƛ₁ / ƛ₁² + ƛ₂ / ƛ₂² are two values of ƛ

1) 4192 2) 4144 3) 4096 4) 4048 Solution: (4) 4048 (ɑ + β) = [ƛ – 1] / ƛ;... View Article

If the sum of the roots of the equation ax² + 2x + 3a = 0 is equal to their product, then value of a is

1) -2/3 2) -3 3) 4 4) -1/2 Solution: (1) -2 / 3 Let α and β be the roots... View Article

If α and β are the roots of the quadratic equation x² + x + 1 = 0, then the equation, whose roots are α¹⁹ and β⁷, is

1) x2 – x + 1 = 0 2) x2 – x – 1 = 0 3) x2 + x... View Article

If the roots of the equation x²2

1) – 1 2) 0 3) 1 4) 2 5) 3 Solution: (3) 1 Let α, α + 1 are... View Article

If a and b are the roots of the equation x² + ax + b = 0, a ≠ 0, b ≠ 0, the values of a and b are respectively

1) 2 and -2 2) 2 and -1 3) 1 and – 2 4) 1 and 2 5) – 1... View Article

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