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If one root of the equation x² + px + q = 0 is 2 + √3, then the values of p and q are respectively

1) - 4, 1 2) 4, -1 3) 2,√3 4) - 2, -√3 Solution: (1) - 4, 1 If 2 + √3 is a root and 2 - √3 is its conjugate root. (2 + √3 + 2 - √3) = - p p... View Article

If α and β are the roots of the equation ax² + bx + c = 0 (c ≠ 0), then the equation, whose roots are 1 / [aα + b] and 1 / [aβ + b], is

1) acx2 - bx + 1 = 0 2) x2 - acx + bc + 1 = 0 3) acx2 + bx - 1 = 0 4) x2 + acx + 11 = 0 Solution: (1) acx2 - bx + 1 = 0 ɑ, β are roots of... View Article

If α and β are the roots of the equation x² – x + 1 = 0, then α²⁰⁰⁹ + β²⁰⁰⁹ is equal to

1) - 2 2) - 1 3) 1 4) 2 Solution: (3) 1 x2 - x + 1 = 0 x3 + 1 = (x + 1) (x2 - x + 1) = 0 x3 = - 1 If ɑ, β are roots then ɑ + β = 1, ɑβ = 1... View Article

Let p and q be real numbers such that p ≠ 0, p³ ≠ q and p³ ≠ – q. If α and β are non – zero complex numbers satisfying α + β = – p and α³ + β³ = q, then a quadratic equation having α / β and α / β as its roots is

1) (p3 + q)x2 - (p3 + 2q)x + (p3 + q ) = 0 2) (p3 + q)x2 - (p3 - 2q)x + (p3 + q ) = 0 3) (p3 - q)x2 - (5p3 - 2q)x + (p3 - q ) = 0 4) (p3 -... View Article

If α, β and γ are the roots of x³ – 2x + 1 = 0 then the value of ∑ [1] / [α + β – γ] is

1) -1 / 2 2) -1 3) 0 4) 1/2 Solution: (2) -1 x3 - x - x + 1 = 0 x2 (x2 - 1) - (x - 1) = 0 x = 1, x2 + x - 1 = 0 α = 1, β = [- 1 + √5] / 2, γ... View Article

If α and β are the roots of the equation x² – 2x + 4 = 0, then the value of αⁿ + βⁿ will be

1) i2n+1 sin (nπ/3 ) 2) 2n+1 cos (nπ/3 ) 3) i 2n-1 sin (nπ/3 ) 4) 2n-1 cos (nπ/3 ) Solution: (2) 2n+1 cos (nπ / 3) Since α and β are the... View Article

If the product of the roots of the equation x² – 2 √2kx + 2e^2logk – 1 = 0 is 31, then the roots of the equation are real for k, is equal to

1) - 4 2) 1 3) 4 4) 0 Solution: (3) 4 Product = 31 2e2logk - 1 = 31 2e2logk = 32 k2 = 16 k = ± 4 But log k > 0, k ≠ - 4, k = 4... View Article

If (α + √β ) and (α -√β ) are the roots of the equation where x² + px + q = 0, where α, β p and q are real, then the roots of the equation (p² – 4q) (p²x² + 4px) – 16q = 0 are

1) [(1 / ɑ) + (1 / √β)] and [(1 / ɑ) - (1 / √β)] 2) [(1 / √ɑ) + (1 / β)] and [(1 / √ɑ) - (1 / β)] 3) [(1 / √ɑ) + (1 / √β)] and [(1 / √ɑ) - (1 /... View Article

If α , β and γ are the roots of the equation x³ + 4x + 2 = 0, then α³ + β³ + γ³ is equal to

1) 2 2) 6 3) - 2 4) - 6 Solution: (4) - 6 x3 + 4x + 2 = 0 ∑ɑ = 0, ∑ɑβ = 4 / 1 = 4, ɑβγ = - 2 / 1 = - 2 α3 + β3 + γ3 = 3ɑβγ = 3 * - 2 = - 6... View Article

If (x² + px + 1) is a factor of (ax³ + bx + c), then

1) a2 + c2 = - ab 2) a2 - c2 = - ab 3) a2 - c2 = ab 4) none of these Solution: (3) a2 - c2 = ab Since (x2 + px + 1) is a factor of (ax3 +... View Article

If the roots of the equation ax² + bx + c = 0 are of the form [k + 1] / k and [k + 2] / [k + 1] then (a + b + c)² =

1) b2 - 4ac 2) b2 - 2 ac 3) 2b2 - ac 4) ∑a2 Solution: (1) b2 - 4ac [k + 1] / k + [k + 2] / [k + 1] = - b / a, [k + 2] / k = c / a 1 + (1 /... View Article

If the arithmetic mean of the roots of a quadratic equation is 8 and the geometric mean is 5, then the equation is

1) x2 - 16 x - 25 = 0 2) x2 + 16 x - 25 = 0 3) x2 - 16 x + 25 = 0 4) x2 - 8 x - 5 = 0 Solution: (3) x2 - 16 x + 25 = 0 [α + β] / 2 = 8, √αβ... View Article

If the roots of ax² + bx + c = 0 are sin α and cos β for some α, then which one of the following is correct?

1) a2 + b2 = 2ac 2) b2 - c2 = 2ab 3) b2 - a2 = 2ac 4) b2 + c2 = 2ab Solution: (3) b2 - a2 = 2ac sin α + cos α = - b / a and sin α cos α = c... View Article

If α and β are the roots of the equation ax² + bx + c = 0 and S_n = αⁿ + βⁿ and a S_n+1 + b S_n + c S_n-1 is equal to

1) 0 2) abc 3) a + b + c 4) None of these Solution: (1) 0 α + β = - b / a, αβ = c / a Sn = αn + βn Sn+1 = αn+1 + βn+1 αn+1 + βn+1 + αnβ +... View Article

Let f (x) = 2x² + 5x + 1. If we write f (x ) as f (x) = [a (x + 1) (x – 2) + b (x – 2) (x – 1) + c(x – 1)(x + 1)] for real numbers a,b and c, then

1) there are infinite numbers of choices for a, b and c 2) only one choice for a but infinite numbers of choices for b and c 3) exactly one... View Article

If α, β are the roots of ax² + bx + c = 0 (a ≠ 0) and α + h, β + h are the roots of px² + qx + r = 0 (p ≠ 0), then the ratio of the squares of their discriminants is

1) a2 : p2 2) a : p2 3) a2 : p 4) a : 2p Solution: (1) a2 : p2 Let α, β are the roots of the equation. ax2 + bx + c = 0 and α + h, β + h are... View Article

If a, b and c are positive numbers in a GP, then the roots of the quadratic equation (log_e a) x² – 2 (log_e b) x + (log_e c) = 0 are

1) - 1 and [loge c] / [loge a] 2) 1 and - [loge c] / [loge a] 3) 1 and loga c 4) - 1 and logc a Solution: (3) 1 and loga c If a + b + c =... View Article

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