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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 of the equation, x2 - ax + b = 0, so each of them must satisfy the equation.... 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 - 16 = 0 4) x2 + 18x + 16 = 0 Solution: (2) x2 - 18x + 16 = 0 Let α and β be two... 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 - 12 = 0 4) x2 + 2x - 8 = 0 Solution: (4) x2 + 2x - 8 = 0 (α + β)3 = α3 + β3 + 3 αβ (α +... 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 + β2 + γ2 = 6 α3 + β3 + γ3 = 8 (α + β + γ)2 = 4 α2 + β2 + γ2 + 2 ∑αβ = 4 2 ∑αβ = 4... 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| - 1) (|x| - 2) = 0 |x| = 1, 2 x = 1, -1, 2, -2 Hence, four real solutions exist.... 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 - 1 = 0 4) x3 + 4x + 1 = 0 Solution: (3) x3 + 4x - 1 = 0 Given α, β and γ are the roots of... 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 + β2 = (α + β)2 - 2αβ = 4 - 8 = - 4 (α2 + β2)3 = α6 + β6 + 3 α2 β2 (α2 + β2)... 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 - β ∑x1 = sin 2β ∑x1x2 = cos 2β ∑x1x2x3 = cos β x1x2x3x4 = sin β tan-1 x1 + tan-1 x2 +... 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 root of the equation. x2 + ax + 12 = 0 42 + 4a + 12 = 0 4a = -... 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 + c = 0 4) ax2 + bx + 3c = 0 Solution: (1) ax2 + 3bx + 3c = 0 If α, β are the roots... 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 Since a and b are the roots of the given equation, ɑ2 - a (ɑ - 1) + b = 0 β2 - a... 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 + n = 0. m2 - 4ln = 0 It has real roots. [x - (9 / 2)]2 = 0... 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 pairs. The other root is 2 + i. 2 + i + 2 - i = - 12... 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 / (x + q) = 1 / r ⇒ r (2x + p + q) = [x2 + (p + q) x + pq] ⇒ x2 + (p + q -... 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; ɑβ = j / ƛ (ɑ / β) + (β / ɑ) = 4 / 5 [ɑ2 + β2] / ɑβ = 4 / 5 [(ɑβ)2... View Article

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