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d/dx tan^-1 4√x/(1-4x) =

(1) 1/√x(1+4x) (2) 2/√x(1+4x) (3) 4/√x(1+4x) (4) none of these Solution: Let y = tan-1 4√x/(1-4x) We know tan 2A = 2 tan... View Article

If y = sin [ cos (sin x)], then dy/dx =

(1) -cos [cos (sin x)] sin (cos x) cos x (2) -cos [cos (sin x)] sin (sin x) cos x (3) cos [cos (sin x)] sin (cos x) cos x (4) cos [cos... View Article

If x = φ(t), y = ψ(t), then d²y/dx² is equal to

(1) (φ’ψ’’- ψ’φ’’)/(φ’)2 (2) (φ’ψ’’- ψ’φ’’)/(φ’)3 (3) φ’’/ψ’’ (4) ψ’’/φ’’ Solution: Given x = φ(t) and y = ψ(t). dx/dt = φ’(t)... View Article

Maximum slope of the curve y = – x³ + 3x² + 9x – 27 is

1) 0 2) 12 3) 16 4) 32 Correct option: (2) 12 Solution: Equation of the given curve is: y = - x3 + 3x2 + 9x - 27 Differentiating on both... View Article

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

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Archives

d/dx tan^-1 4√x/(1-4x) =

If y = sin [ cos (sin x)], then dy/dx =

If x = φ(t), y = ψ(t), then d²y/dx² is equal to

Maximum slope of the curve y = – x³ + 3x² + 9x – 27 is

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]

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

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

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

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

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

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

The real number which most exceeds its cube is

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

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

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

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

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

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

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

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