The function f(x) = x/2 + 2/x has a local minimum at (1) x = 2 (2) x = -2 (3) x = 0 (4) x = -1 Solution: f(x) = x/2 + 2/x Differentiate w.r.t.x f’(x) = ½ - 2/x2 For minimum or... View Article
If u = tan-1 (y/x), then by Euler’s theorem the value of x ∂u/∂x + y ∂u/∂y = (1) sin u (2) tan u (3) 0 (4) cos u Solution: Given u = tan-1 (y/x) = x0f(y/x) Here u is a homogeneous function of the form xn... View Article
[latex]\frac{d}{dx}\left ( e^{\sqrt{1-x^{2}}}\tan x \right )=[/latex] (1) (2) (3) (4) none of these Solution: Let dy/dx = = = = Hence option (2) is the answer.... View Article
If u = (x+y)/(x-y), then ?u/?x + ?u/?y = (1) 1/(x-y) (2) 2/(x-y) (3) 1/(x-y)2 (4) 2/(x-y)2 Solution: u = (x+y)/(x-y) ∂u/∂x = [(x-y) - (x+y)]/(x-y)2 = -2y/(x-y)2... View Article
If y = x + ex, then d2y/dx2 is (1) ex (2) -ex/(1 + ex)3 (3) -ex/(1 + ex)2 (4) -1/(1 + ex)3 Solution: Given y = x + ex Differentiate w.r.t.x dy/dx = 1 + ex... View Article
Let f(x) = tan-1 x, then f’(x) + f’’(x) is equal to 0, where x is equal to (1) 0 (2) 1 (3) i (4) -i Solution: Given f(x) = tan–1 x Differentiate w.r.t.x f’(x) = 1/(1+x2) Again differentiate w.r.t.x... View Article
nth derivative of cos2 x = (1) 2n-1 cos (nπ/2 + 2x) (2) 2n cos (nπ/2 - 2x) (3) 2n+1 cos (nπ/2 + 2x) (4) 2n+2 cos (nπ/2 + 2x) Solution: Let y = cos2 x... View Article
sec2 (tan-1 2) + cosec2 (cot-1 3) = 1) 5 2) 13 3) 15 4) 6 Solution: (3) 15 sec2 (tan-1 2) + cosec2 (cot-1 3) = [sec (tan-1 2)]2 + [cosec (cosec-1 √10)]2 = (√5)2 +... View Article
If y = (sin-1x)2, then (1 – x2)d2y/dx2 – xdy/dx is equal to (1) 1 (2) 0 (3) -1 (4) 2 Solution: Given y = (sin-1x)2 Differentiate w.r.t.x dy/dx = 2 sin-1x (1/√(1 - x2)) Squaring both... View Article
[latex]f(x) =\begin{vmatrix} x^{3} & x^{2}&3x^{2} \\ 1 & -6 & 4\\ p&p^{2} & p^{3} \end{vmatrix}[/latex], where p is a constant, then [latex]\frac{d^{3}f(x)}{dx^{3}}[/latex] is (1) proportional to x (2) proportional to x2 (3) proportional to x3 (4) a constant Solution: Given = x3(-6p3 - 4p2) - x2(p3 -... View Article
d2/dx2 (2 cos x cos 3x) = (1) 22 (22 cos 4x + cos 2x) (2) 22 (-22 cos 4x + cos 2x) (3) 22 (22 cos 4x - cos 2x) (4) -22 (22 cos 4x + cos 2x) Solution: Let y... View Article
The nth derivative of xex vanishes when (1) x = 0 (2) x = -1 (3) x = -n (4) x = n Solution: Let y = xex dy/dx = xex + ex d2y/dx2 = xex + ex + ex = xex + 2ex... View Article
If u = x2 + y2 and x = s + 3t, y = 2s – t , then d2u/ds2 = (1) 10 (2) 12 (3) 32 (4) 36 Solution: Given x = s + 3t y = 2s - t Differentiate w.r.t.s dx/ds = 1 dy/ds = 2 Differentiate... View Article
If y = a cos (log x) + b sin (log x) where a, b are parameters then x2y’’ + xy’ = (1) y (2) -y (3) -2y (4) 2y Solution: Given y = a cos (log x) + b sin (log x) Differentiate w.r.t.x y’ = -a sin (log x) ×(1/x) +... View Article
If y = aex + be-x where a, b are parameters then y’’ = (1) y (2) y’ (3) -y’ (4) 0 Solution: Given y = aex + be-x Differentiate w.r.t.x y’ = aex - be-x Again differentiate w.r.t.x... View Article
If y = (x + √(1 + x2))n, then (1 + x2)d2y/dx2 + x dy/dx is (1) n2y (2) -n2y (3) -y (4) 2x2y Solution: y = (x + √(1 + x2))n Differentiate w.r.t.x dy/dx = n(x + √(1 + x2))n-1 [1 + 1/2√(1... View Article
If y = x3 log loge (1+x), then y’’(0) equals (1) 0 (2) -1 (3) 6 loge 2 (4) 6 Solution: Given y = x3 log loge (1+x) Differentiate w.r.t.x using product rule dy/dx = 3x2 log... View Article
If y = sin x + ex , then d2x/dy2 (1) (sin x - ex)/(cos x + ex)3 (2) (-sin x + ex)-1 (3) (sin x - ex)/(cos x + ex)2 (4) none of these Solution: Given y = sin x + ex... View Article
If f be a polynomial, then the second derivative of f(ex) is (1) f’(ex) (2) f’’(ex)ex + f’(ex) (3) f’’(ex) e2x + f’’(ex) (4) f’’(ex) e2x + f’(ex) ex Solution: Let y = f(ex) dy/dx = f’(ex)... View Article
If ey + xy = e, then the value of d2y/dx2 for x = 0 is (1) 1/e (2) 1/e2 (3) 1/e3 (4) none of these Solution: Given ey + xy = e When x = 0, y = 1 Differentiate w.r.t.x eydy/dx + x... View Article
