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The minimum value of (x² + (250 / x)), is

1) 75 2) 50 3) 25 4) 55 Solution: (1) 75 y = f (x) = (x2 + (250 / x)) dy / dx = f'(x) = 2x - [250 / x2] Put f (x) = 0 2x3 - 250... View Article

The function y = a (1 – cos x) is maximum when x =

1) π 2) π / 2 3) - π / 2 4) - π / 6 Solution: (1) π y = a (1 - cos x) y' = a sin x y' = 0 sinx = 0 x = 0, π y'' = a cos x... View Article

If PQ and PR are the two sides of a triangle, then the angle between them which gives maximum area of the triangle is

1) π 2) π / 3 3) π / 4 4) π / 2 Solution: (4) π / 2 Assume PQ = a and PR = b. Δ = (1 / 2) ab sinθ (- 1 ≤ sinθ ≤ 1) The area is... View Article

If ab = 2a + 3b, a > 0, b > 0 then the minimum value of ab is

1) 12 2) 24 3) 1 / 4 4) None of these Solution: (2) 24 ab = 2a + 3b (a - 3) b = 2a b = 2a / [a - 3] z = ab = [2a2] / [a - 3] dz / da = 2 [(a... View Article

The maximum value of xy = 8 is

1) 20 2) 16 3) 24 4) 8 5) 4 Solution: (4) 8 x + 2y = 8 y = [8 - x] / 2 f (x) = xy = x ([8 - x] / 2) = 4x - (x2 / 2) For... View Article

On [1, e] the greatest value of x² log x

1) e2 2) (1 / e) log (1 / √e) 3) e2 log √e 4) None of these Solution: (1) e2 f (x) = x2 log x f' (x) = (2 logx + 1)x Put f' (x)... View Article

In (- 4, 4), the function f (x) = Integral from -10 to x (t⁴ – 4) e^-4t dt has

1) No extrema 2) One extremum 3) Two extrema 4) Four extrema Solution: (3) Two extrema f (x) = ∫-10x (t4 - 4) e-4t dt f’ (x) =... View Article

If y = a log x + bx² + x has its extremum value at x = – 1 and x = 2, then (a, b) =

1) (1, 1 / 2) 2) (1 / 2, 2) 3) (2, - 1 / 2) 4) (- 2 / 3, - 1 / 6) Solution: (3) (2, - 1 / 2) y = a log x + bx2 + x has its... View Article

The perimeter of a sector is p. The area of the sector is maximum when its radius is

1) √p 2) 1 / √p 3) p / 2 4) p / 4 Solution: (4) p / 4 A = (1 / 2) r2θ (1) Length of the arc is s = rθ θ = s / r The perimeter... View Article

If f (x) = x + (1 / x), x > 0 then its greatest value is

1) –2 2) 0 3) 3 4) None of these Solution: (4) None of these f (x) = x + [1 / x] f' (x) = 1 - [1 / x2] Put f (x) = 0, x = −1, 1.... View Article

If f (x) = 1 / [4x² + 2x + 1], then its maximum value is

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

The function f (x) = 2x³ – 3x² – 12x + 4 has

1) No maxima and minima 2) One maximum and one minimum 3) Two maxima 4) Two minima Solution: (2) One maximum and one minimum f (x) = 2x3 -... View Article

Differential coefficient of √(sec√x) is

(1) (1/4√x)(sec√x)3/2sin √x (2) (1/2√x)(sec√x)3/2sin √x (3) (1/2√x)(sec√x)sin √x (4) none of these Solution: Let y = √(sec√x)... View Article

For the curve √x + √y = 1, dy/dx at (¼, ¼) is

(1) 1 (2) ½ (3) -1 (4) 2 Solution: Given √x + √y = 1 Differentiate w.r.t.x 1/2√x + (1/2√y)dy/dx = 0 dy/dx = (-1/2√x )/(1/2√y)... View Article

If y = (cos x²)², then dy/dx is equal to

(1) -4x sin 2x2 (2) -x sin x2 (3) -2x sin 2x2 (4) -x cos 2x2 Solution: Given y = (cos x2)2 Differentiate w.r.t.x dy/dx = 2 cos... View Article

If y = (x+1)(x+2)(x+3)(x+4)(x+5), then the value of dy/dx at x = 0 is equal to

(1) 742 (2) 472 (3) 374 (4) 274 Solution: Given y = (x+1)(x+2)(x+3)(x+4)(x+5) Differentiate w.r.t.x dy/dx = (x+2)(x+3)(x+4)(x+5) +... View Article

If [latex]f(x)= 3e^{x^{2}}[/latex], then f’(x) – 2xf(x) + ⅓ f(0) – f’(0) =

(1) 1 (2) 0 (3) 7/3 (4) none of these Solution: Given f’(x) - 2xf(x) = = = 0 f’(x) - 2xf(x) + ⅓ f(0) - f’(0) = 0 +... View Article

If y = xⁿ log x + x(log x)ⁿ, then dy/dx =

(1) xn-1( 1 + n log x ) + (log x)n-1 (n + log x) (2) xn-2( 1 + n log x ) + (log x)n-1 (n + log x) (3) xn-1( 1 + n log x ) + (log x)n-1 (n -... View Article

The derivative of tan x – x with respect to x is

(1) 1-tan2x (2) tan x (3) -tan2x (4) tan2x Solution: Let y = tan x - x Differentiate w.r.t.x dy/dx = sec2x - 1 = tan2 x Hence... View Article

If y = sin^-1√(1-x) + cos^-1 √x, then dy/dx =

(1) 1/√(x(1-x)) (2) -1/√(x(1-x)) (3) 1/√(x(1+x)) (4) none of these Solution: Given y = sin-1√(1-x) + cos-1 √x dy/dx =... View Article

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