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
If [latex]y = x \log \left ( \frac{x}{a+bx}\right )[/latex], then [latex]x^{3}\frac{d^{2}y}{dx^{2}}=[/latex] (1) x dy/dx - y (2) (x dy/dx - y)2 (3) y dy/dx - x (4) (y dy/dx - x)2 Solution: Given y = x log (x/(a+bx)) y/x = log x/(a+bx)... View Article
If y = sin2 α + cos2 (α+β) + 2 sin α sin β cos (α+β), then d3y/dα3 is, (keeping β as constant) (1) 1 (2) 0 (3) cos (α + 3β) (4) none of these Solution: Given y = sin2 α + cos2 (α+β) + 2 sin α sin β cos (α+β) We know 2 sin A... View Article
d20/dx20 (2 cos x cos 3x) = (1) 220 (cos 2x - 220 cos 4x) (2) 220 (cos 2x + 220 cos 4x) (3) 220 (sin 2x + 220 sin 4x) (4) 220 (sin 2x - 220 sin 4x) Solution:... View Article
If y = a + bx2: a, b are arbitrary constants, then (1) d2y/dx2 = 2xy (2) x d2y/dx2 = dy/dx (3) x d2y/dx2 - dy/dx + y = 0 (4) x d2y/dx2 = 2xy Solution: Given y = a + bx2... View Article
If y = axn+1+ bx-n, then x2d2y/dx2 = (1) n(n-1)y (2) n(n+1)y (3) ny (4) n2y Solution: Given y = axn+1+ bx-n Differentiate w.r.t.x dy/dx = (n+1)axn - bnx-n-1... View Article
