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Given, P (x) = x⁴ + bx² + cx + d such that x = 0 is the only real root of P’(x) = 0. If P(–1) < P(1), then in the interval [–1, 1]

1) P(–1) is not minimum and P(1) is the maximum of P 2) P(–1) is not minimum but P(1) is the maximum of P 3) P(–1) is the minimum but P(1)... View Article

If a²x⁴ + b²y⁴ = c⁶, then maximum value of xy is

1) c2 / √ab 2) c3 / ab 3) c3 / √2ab 4) c3 / 2ab Solution: (3) c3 / √2ab... View Article

If P = (1, 1), Q = (3, 2) and R is a point on x – axis then the value of PR + RQ will be minimum at

1) (5 / 3, 0) 2) (1 / 3, 0) 3) (3, 0) 4) (1, 0) Solution: (1) (5 / 3, 0) Let coordinate of R (x, 0). It is given that P (1, 1)... View Article

The function f (x) = ax + (b / x), a, b, x > 0 takes on the least value at x =

1) b 2) √a 3) √b 4) √b / a Solution: (4) √b / a f (x) = ax + (b / x) f’ (x) = a - (b / x2) f’ (x) = 0 x = √b / a f’’ (x) = 2b... View Article

The function f (x) = x + sin x has

1) A minimum but no maximum 2) A maximum but no minimum 3) Neither maximum nor minimum 4) Both maximum and minimum Solution: (3)... View Article

Let f (x) = 1 + 2x² + 2² x⁴ + …. 2¹⁰x²⁰, then f (x) has

1) More than one minimum 2) Exactly one minimum 3) At least one maximum 4) None of the above Solution: (2) Exactly one minimum f... View Article

A one of maximum value is inscribed in a given sphere, then ration of the height of the cone to diameter of the sphere is

1) 2 / 3 2) 3 / 4 3) 1 / 3 4) 1 / 4 Solution: (1) 2 / 3 Let the diameter of the sphere be AE - 2r, the radius of cone be x... View Article

The real number which most exceeds its cube is

1) 1 / 2 2) 1 / √3 3) 1 / √2 4) None of these Solution: (2) 1 / √3 Let the number be x, then the cube of the number is x3. f (x)... View Article

The denominator of a fraction number is greater than 16 of the square of the numerator, then the least value of the number is

1) - 1 / 4 2) - 1 / 8 3) 1 / 2 4) 1 / 16 Solution: (2) - 1 / 8 f (x) = x / [x2 + 16] f’ (x) = [[x2 + 16] 1 - x (2x)] / (x2 + 16)2... View Article

The real number x when added to its inverse gives the minimum value of the sum at x equal to

1) –2 2) 2 3) 1 4) –1 Solution: (3) 1 f (x) = x + (1 / x) f’ (x) = 1 - (1 / x2) Put f’ (x) = 0 1 - (1 / x2) = 0 x = ± 1 f’’... View Article

If A + B = π / 2, the maximum values of cos A cos B is

1) 1 / 2 2) 3 / 4 3) 1 4) 4 / 3 Solution: (1) 1 / 2 f (A) = cos A cos B = cos A cos (π / 2 - A) = cos A sin A f’ (A) = cos2 A -... View Article

Suppose the cubic x³ – px + q has 3 distinct real roots where p > 0 and q > 0. Then which one of the following holds

1) The cubic has minima at - √p / 3 and maxima at √p / 3 2) The cubic has minima at both √p / 3 and - √p / 3 3) The cubic has maxima at... View Article

Area of the greatest rectangle that can be inscribed in the ellipse x² / a² + y² / b² = 1 is

1) √ab 2) a / b 3) 2ab 4) ab Solution: (3) 2ab Area of rectangle ABCD = (2a cos θ). (2b sin θ) = 2ab sin 2θ The area of... View Article

If x is real, then greatest and least values of [x² – x + 1] / [x² + x + 1] are

1) 3, - 1 / 2 2) 3, 1 / 3 3) - 3, 1 / 3 4) None of these Solution: (2) 3, 1 / 3 Let y = [x2 - x + 1] / [x2 + x + 1] dy / dx =... View Article

x / [1 + x tan x] is maxima at

1) x = sin x 2) x = cos x 3) x = π / 3 4) x = tan x Solution: (2) x = cos x [x / (1 + x tanx)] is maximum then [(1 + x tanx) / x]... View Article

The maximum value of sin x (1 + cos x) will be at the

1) x = π / 2 2) x = π / 6 3) x = π / 3 4) x = π Solution: (3) x = π / 3 y = sin x (1 + cos x) --- (i) dy / dx = cos x (1 + cos x)... View Article

The maximum and minimum values of x³ – 18x² + 96x in the interval (0, 9) are

1) 160, 0 2) 60, 0 3) 160, 128 4) 120, 28 Solution: (3) 160, 128 y = x3 – 18x2 + 96x y’ = 3x2 - 36x + 96 = 0 x2 - 12x + 32 = 0... View Article

The minimum value of 2x + 3y, where xy = 6 is

1) 12 2) 9 3) 8 4) 6 Solution: (1) 12 f (x) = 2x + 3y f (x) = 2x + 18x (xy = 6) f′ (x) = 2 − 18x2 f′ (x) = 0 for maximum or... View Article

What are the minimum and maximum values of the function x⁵ – 5x⁴ + 5x³ – 10

1) –37, –9 2) 10, 0 3) It has 2 min. and 1 max.values 4) It has 2 max. and 1 min. values Solution: (1) –37, –9 y = x5 – 5x4 + 5x3... View Article

Let x and y be two variables such that x > 0 and xy = 1. Then the minimum value of x + y is

1) 2 2) 3 3) 4 4) 0 Solution: (1) 2 Let f (x) = x + y, where xy = 1. f (x) = x + (1 / x) f’ (x) = 1 - (1 / x2) = [x2 - 1] / x2... View Article

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