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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 =... 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!... 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... 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... 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... View Article

If y = x²e^mx, 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... 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 +... 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) =... 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... View Article

If x = sin t, y = cos pt, then

If x = sin t, y = cos pt, then 1) (1 – x2) y2 + xy1 + p2y =... View Article

[latex]\frac{d^{\, n}}{dx^{n}}(\log x)[/latex] is equal to

\(\begin{array}{l}\frac{d^{\, n}}{dx^{n}}(\log x)\end{array} \) 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... View Article

If y = a^x b^2x-1, then d²y/dx²is equal to

If y = ax b2x-1, then d2y/dx2 is equal to (1) y2 log ab2 (2) y log ab2 (3) y2... View Article

If y²= ax² + bx + c, where a, b and c are constant, then y³ d²y/dx² is

1) a constant 2) a function of x 3) a function of y 4) a function of x and y... View Article

If f(x) = e^x sin x, then f’’(x) is equal to

(1) e6x sin 6x (2) 2ex cos x (3) 8ex sin x (4) 8ex cos x Solution: f(x) = ex... View Article

If y = e^ax sin bx, then (d²y/dx²) – 2a (dy/dx) + a²y is equal to

1) 0 2) 1 3) -b2y 4) -by Solution: y = eax sin bx….(i) dy/dx = a eax sin bx... View Article

If x = e^t sin t, y = e^t cos t, then d²y/dx² at t = π is equal to

1) 2 eπ 2) (1/2) eπ 3) 1/2eπ 4) 2/eπ Solution: Given x = et sin t dx/dt = et... View Article

If f (x) = sin x, g (x) = x² and h (x) = log_ex. If F (x) = (hogof) (x), then F ” (x) is equal to

1) a cosec3 x 2) 2 cot x2 – 4x2 cosec2 x2 3) 2 cot x2 4) – 2 cosec2... View Article

The second-order derivative of a sin³ t with respect to a cos³ t at t = π/4 is

1) 2 2) 1/12a 3) 4√2/3a 4) 3a/4√2 Solution: Let u = a sin3 t du/dt = 3 a sin2t... View Article

The derivative of sin^-1 (2x√(1-x²) with respect to sin^-1 (3x- 4x³)

(1) 2/3 (2) 3/2 (3) 1/2 (4) 1 Solution: Let u = sin-1 (2x√(1-x2) Put x = sin θ Then... View Article

If x = a(1 + cos θ), y = a(θ + sin θ) , then d²y/ dx² at θ = π/2 is equal to

1) – (1/a) 2) 1/a 3) –1 4) –2 Solution: Given x = a(1 + cos θ) dx/dθ = -a... View Article

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