The Rolle’s theorem is applicable in the interval – 1 ≤ x ≤ 1 for the function 1) f (x) = x 2) f (x) = x2 3) f (x) = 2x3 + 3 4) f (x) = |x| Solution: (2) f (x) = x2 f (x) = x2 f (1) = f (-1) = 1 f (x) is... View Article
If sin(x + y) = log(x + y), then dy/dx is equal to 1) 2 2) –2 3) 1 4) –1 Solution: Given sin(x + y) = log(x + y) Differentiate w.r.t.x, we get cos (x + y) (1 + dy/dx) = (1/(x +... View Article
If f(x) = x/(1 + x) and g(x) = f[f(x)], then g'(x) is equal to If f(x) = x/(1 + x) and g(x) = f[f(x)], then g'(x) is equal to (1) 1/(2x+3)2 (2) 1/(x+1)2 (3) 1/x2 (4) 1/(2x+1)2 Solution: Given... View Article
If f(x) = x2 + bx + 7. If f ‘(5) = 2f ‘(7/2), then the value of b is 1) 4 2) 3 3) –4 4) –3 Solution: Given f(x) = x2 + bx + 7 f’(x) = 2x + b f '(5) = 10 + b f '(7/2) = 7 + b Given f '(5) = 2f... View Article
If y2 = P(x), a polynomial of degree 3, then 2(d/dx)(y3 d2y/dx2) equals (1) P’’’(x) + P’(x) (2) P’’(x) P’’’(x) (3) P(x) P’’’(x) (4) a constant Solution: Given y2 = P(x) Differentiate w.r.t.x, we get... View Article
If g is the inverse of a function f and f’(x) = 1/(1+x5), then g’(x) is equal to 1) 1 + x5 2) 5x4 3) 1/(1 + {g(x)}5) 4) 1 + {g(x)}5 Solution: Given g is the inverse of a function f. g(x) = f-1x So fog(x) = x... View Article
If z = sec^-1 (x^4 + y^4-8x^2y^2)/(x^2+y^2), then x ∂z/∂x + y ∂z/∂y is equal to (1) cot z (2) 2 cot z (3) 2 tan z (4) 2 sec z Solution: Given z = sec-1 (x4 + y4-8x2y2)/(x2+y2) sec z = (x4 + y4-8x2y2)/(x2+y2)... View Article
If f(x,y) = cos (x-4y)/cos (x+4y), then ∂f/∂x at (0, π/4) is equal to (1) -1 (2) 0 (3) 2 (4) 1 Solution: f(x,y) = cos (x-4y)/cos (x+4y) ∂f/∂x = [ - cos (x+4y) sin (x-4y) + cos (x-4y) sin (x+4y)]/... View Article
If u = sin-1 (x/y) + tan-1(y/x), then the value of x ∂u/∂x + y ∂u/∂y is (1) 0 (2) 1 (3) 2 (4) none of these Solution: u = sin-1 (x/y) + tan-1(y/x) = sin-1 (1/(y/x)) + tan-1(y/x) = x0f(y/x) Here u is a... View Article
If [latex]u = \tan^{-1}\left ( \frac{x^{3}+y^{3}}{x-y} \right )[/latex], then [latex]x\frac{\partial u}{\partial x}+y\frac{\partial u}{\partial y}[/latex] is equal to (1) -sin 2u (2) sin 2u (3) cos 2u (4) -cos 2u Solution: u = tan-1 (x3+y3)/(x-y) tan u = (x3+y3)/(x-y) Let tan u = z …(i) So z =... View Article
If y = 1 – x + x2/2! – x3/3! + x4/4! -…., then d2y/dx2 is equal to (1) -x (2) x (3) y (4) -y Solution: y = 1 - x + x2/2! - x3/3! + x4/4! -.... dy/dx = -1 + 2x/2! - 3x2/3! + 4x3/4! - … = -1 + x -... View Article
The value of [latex]\frac{d}{dx}\left [ \tan^{-1}\left \{ \frac{\sqrt{x}(3-x)}{1-3x} \right \} \right ][/latex] is (1) 1/2(1+x)√x (2) 3/(1+x)√x (3) 2/(1+x)√x (4) 3/2(1+x)√x Solution: Let y = [ tan-1 {√x(3-x)/(1-3x)}] Put √x = tan θ So y = [... View Article
If x sin (a + y) + sin a cos (a + y) = 0, then dy/dx is equal to (1) sin2(a+ y)/sin a (2) cos2(a+ y)/cos a (3) sin2(a+ y)/cos a (4) cos2(a+ y)/sin a Solution: Given x sin (a + y) + sin a cos (a +... View Article
If f (x) = 10 cos x + (13 + 2x) sin x, then f ” (x) + f (x) is equal to 1) cos x 2) 4 cos x 3) sin x 4) 4 sinx Solution: Given f (x) = 10 cos x + (13 + 2x) sin x f’(x) = -10 sin x + 13 cos x + 2 sin x... View Article
If y = x2emx, where m is a constant, then d3y/dy3 is equal to (1) memx (6mx + 6 + m2x2) (2) 2m3x emx (3) memx (2mx + 2 + m2x2) (4) none of these Solution: Given y = x2emx y’ = 2xemx +... View Article
nth derivative of (x + 1)n is equal to 1) (n – 1)! 2) (n + 1)! 3) n! 4) n[(n + 1)]n–1 Solution: Let y = (x + 1)n y’ = n(x+1)n-1 y’’ = n(n-1)(x+1)n-2 y’’’ =... View Article
If sin (x + y) + cos (x + y) = log (x + y), then d2y/dx2 is equal to 1) –y/x 2) 0 3) –1 4) 1 Solution: Given sin (x + y) + cos (x + y) = log (x + y) Differentiate w.r.t.x cos (x+y) (1+ dy/dx) - sin... View Article
If y = t10 + 1 and x = t8 + 1, then d2y/dx2 is equal to 1) 5/2 t 2) 20 t8 3) 5/16 t6 4) None of these Solution: Given y = t10 + 1 dy/dt = 10t9 x = t8 + 1 dx/dt = 8t7 dy/dx =... View Article
If x = sin t, y = cos pt, then If x = sin t, y = cos pt, then 1) (1 – x2) y2 + xy1 + p2y = 0 2) (1 – x2) y2 + xy1 - p2y = 0 3) (1 + x2) y2 - xy1 + p2y = 0 4) (1 –x2)... View Article
[latex]\frac{d^{\, n}}{dx^{n}}(\log x)[/latex] is equal to is equal to (1) (n-1)!/xn (2) n!/xn (3) (n-2)!/xn (4) (-1)n-1(n-1)!/xn Solution: Let y = log x y’ = 1/x y’’ = -1/x2 y’’’ =... View Article