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If k ? sin^-1 x + cos^-1 x + tan^-1 x ? k, then

1) k = 0, k = π 2) k = 0, k = π / 2 3) k = π / 4, k = 3π / 4 4) None of these Solution: (3) k = π / 4, k = 3π / 4 sin-1 x + cos-1... View Article

If a < 1 / 32, then the number of solutions of (sin^-1 x)³ + (cos^-1 x)³ = a?³ is

1) 0 2) 1 3) 2 4) Infinite Solution: (1) 0 (π3 / 32) ≤ (sin-1 x)3 + (cos-1 x)3 ≤ 7π / 8 Here a < 1 / 32, the number of... View Article

The greatest and the least values of (sin^-1 x)³ + (cos^-1 x)³ are

1) (- π / 2, π / 2) 2) (- π3 / 8, π3 / 8) 3) (7π3 / 8, π3 / 32) 4) None of these Solution: (3) (7π3 / 8, π3 / 32) f (x) = (sin-1... View Article

sin^-1 (4 / 5) + 2 tan^-1 (1 / 3) =

1) π / 2 2) π / 3 3) π / 4 4) None of these Solution: (1) π / 2 sin-1 (4 / 5) + 2 tan-1 (1 / 3) = sin-1 (4 / 5) + tan-1 [2 * (1 /... View Article

If x² + y² + z² = r², then tan^-1 (xy / zr) + tan^-1 (yz / xr) + tan^-1 (zx / yr) =

1) π 2) π / 2 3) 0 4) None of these Solution: (2) π / 2 tan-1 (xy / zr) + tan-1 (yz / xr) + tan-1 (zx / yr) = tan-1 [(xy / zr) +... View Article

If ?A = 90^o in the triangle ABC, then tan^-1 c / [a + b] + tan^-1 b / [a + c] =

1) 0 2) 1 3) π / 4 4) π / 6 5) π / 8 Solution: (3) π / 4 ABC is a right-angled triangle at A ⇒ a2 = b2 + c2 …(i) tan-1(c / (a... View Article

sin^-1 (3 / 5) + tan^-1 (1 / 7) =

1) π / 4 2) π / 2 3) cos-1 (4 / 5) 4) π Solution: (1) π / 4 sin-1 (3 / 5) + tan-1 (1 / 7) = tan-1 (3 / 4) + tan-1 (1 / 7) =... View Article

The points of extrema of[latex]f(x)=\int_{0}^{x}\frac{\sin t}{t}dt[/latex] in the domain x> 0 are

(1) nπ; n = 1,2,.. (2) (2n+1)π/2; n = 1,2,.. (3) (4n+1)π/2; n = 1,2,.. (4) (2n+1)π/2; n = 1,2,<.. Solution: Given f’(x) =... View Article

If f(x) = [x – 2], where [x] denotes the greatest integer less than or equal to x, then f’(2.5) is equal to

1) 1/2 2) 0 3) 1 4) Does not exist Solution: f(x) = [x - 2] = 0 if 0 ≤ x -2 < 1 [x] = 0, 0 ≤ x < 1 When x = 2.5 f(2.5) =... View Article

Differential function f devised for all x> 0 and satisfies f(x³) = x⁴for all x > 0, f’(27) (f’ denotes the derivative) is equal to

1) 4 2) 27 3) 8 4) 2 Solution: f(x3) = x4 Differentiate w.r.t.x f’(x3)3x2 = 4x3 f’(x3) = 4x3/3x2 = 4x/3 f’(27) = f’(33) =... View Article

If f(x) = cos x, g(x) = log x and y = (gof)(x), then dy/dx at x = 0 is

(1) 0 (2) 1 (3) -1 (4) none of these Solution: f(x) = cos x g(x) = log x y = (gof)(x) = g(cos x) = log (cos x) dy/dx = (1/cos... View Article

If log_e5 = 1.609, then the approximate value of log 5.1 is

1) 1.611 2) 1.701 3) 1.809 4) 1.629 Solution: Given loge5 = 1.609 Let x = 5 x + dx = 5.1 dx = 0.1 y = log x dy/dx = 1/x dy =... View Article

If φ(x) is the inverse of the function f(x) and f’(x) = 1/(1+x⁵), then d/dx φ(x) is

(1) 1/(1 + {φ(x)}5) (2) 1 + f(x) (3) (1 + {φ(x)}5) (4) 1/(1 + {f(x)}5) Solution: Given φ(x) is the inverse of a function f(x).... View Article

If the function f(x) = ax³ + bx² +11x – 6 satisfies the conditions of Rolle’s theorem for the interval [1,3] and f’(2 + 1/√3) = 0, then the values of a and b are respectively

(1) 1, -6 (2) -2, 1 (3) -1, 2 (4) -1, 6 Solution: Given f(x) = ax3 + bx2 +11x - 6 f(x) satisfies Rolle’s theorem for the interval... View Article

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