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If [latex]y=\log \left ( \frac{1+x}{1-x} \right )^{\frac{1}{4}}-\frac{1}{2}\tan^{-1}x[/latex], then dy/dx =

(1) x2/(1-x4) (2) 2x2/(1-x4) (3) x2/2(1-x4) (4) none of these Solution: y = log ((1+x)/(1-x))1/4 - ½ tan-1 x = (1/4) log... View Article

If f(x) = log_x (log x), then f’(x) at x = e is

(1) e (2) 1/e (3) 1 (4) none of these Solution: f(x) = logx (log x) = (log log x)/ log x Use quotient rule of differentiation... View Article

d/dx [ sinⁿ x cos nx] =

(1) n sinn-1x cos (n+1)x (2) n sinn-1x cos nx (3) n sinn-1x cos (n-1)x (4) n sinn-1x sin (n+1)x Solution: Let y = sinn x cos nx... View Article

If y = b cos log (x/n)ⁿ, then dy/dx =

(1) -nb sin log (x/n)n (2) nb sin log (x/n)n (3) (-nb/x) sin log (x/n)n (4) none of these Solution: Given y = b cos log (x/n)n... View Article

If y = log₁₀x + log_x 10 + log_xx + log₁₀10, then dy/dx =

(1) 1/x log 10 - log 10/x(log x)2 (2) 1/x log210 - 1/x log10 e (3) 1/x log10 + log10/x(log x)2 (4) none of these Solution: y =... View Article

[latex]\frac{d}{dx}\left ( \frac{\cot ^{2}x-1}{\cot ^{2}x+1} \right )[/latex]

(1) - sin 2x (2) 2 sin 2x (3) 2 cos 2x (4) -2 sin 2x Solution: (d/dx) (cot2x -1)/(cot2 x+1) = (d/dx)(cos2x- sin2x)/(cos2x+ sin2x)... View Article

If f(x) = x tan^-1 x, then f’(1) =

(1) 1 + π/4 (2) ½ + π/4 (3) ½ - π/4 (4) 2 Solution: Given f(x) = x tan-1 x Differentiate w.r.t.x f’(x) = tan-1x + x/(1+x2)... View Article

If y = √((1-x)(1+x)), then

(1) (1-x2)dy/dx - xy = 0 (2) (1-x2)dy/dx + xy = 0 (3) (1-x2)dy/dx - 2xy = 0 (4) (1-x2)dy/dx + 2xy = 0 Solution: Given y =... View Article

d/dx (tan^-1 √((1+cos x/2)/(1-cos x/2)) is equal to

(1) -¼ (2) ¼ (3) ½ (4) -½ Solution: d/dx (tan-1 √((1+cos x/2)/(1-cos x/2)) = d/dx (tan-1 √(2 cos2 x/4)/(2 sin2 x/4) = d/dx (tan-1... View Article

If y = x[(cos (x/2) + sin (x/2)) (cos (x/2) – sin (x/2)) + sin x] + 1/2√x, then dy/dx

(1) (1+x) cos x + (1-x) sin x - 1/4x√x (2) (1-x) cos x + (1-x) sin x + 1/4x√x (3) (1+x) cos x + (1+x) sin x - 1/4x√x (4) none of these... View Article

(d/dx) tan^-1 (ax-b)/(bx+a) =

(1) 1/(1+x2) - a2/(a2+b2) (2) -1/(1+x2) - a2/(a2+b2) (3) 1/(1+x2) + a2/(a2+b2) (4) none of these Solution: Let y = tan-1... View Article

The differential coefficient of a^x + log x sin x is

(1) ax log a + (sin x)/x + log x cos x (2) ax + (sin x)/x + log x cos x (3) axlog a + (cos x)/x + sin x log x (4) none of these... View Article

If y = 3x⁵ + 4x⁴ + 2x + 3, then

(1) y4 = 0 (2) y5 = 0 (3) y6 = 0 (4) none of these Solution: y = 3x5 + 4x4 + 2x + 3 y1 = 15x4 + 16 x3 + 2 y2 = 60x3 + 48x2 y3... View Article

If f(x) = mx+c, f(0) = f’(0) = 1, then f(2) =

(1) 1 (2) 2 (3) 3 (4) -3 Solution: Given f(x) = mx+c Given f(0) = 1 f(0) = c So c = 1 Given f’(0) = 1 f’(x) = m f’(0) = 1... View Article

If f(x) = x² -3x, then the points at which f(x) = f’(x) are

(1) 1,3 (2) 1,-3 (3) -1,3 (4) none of these Solution: Given f(x) = x2-3x f’(x) = 2x-3 f(x) = f’(x) => x2-3x = 2x-3 x2- 3x -... View Article

If y = a sin x + b cos x, then y² + (dy/dx)² is a

(1) function of x (2) function of y (3) function of x and y (4) constant Solution: y = a sin x + b cos x dy/dx = a cos x - b sin... View Article

(d/dx) log |x| = ….(x≠ 0)

(1) 1/x (2) -1/x (3) x (4) -x Solution: Let y = log |x| If x >0, y = log x Then dy/dx = 1/x If x < 0, y = log -x Then... View Article

If y = sin^-1( x√(1-x) + √x√(1-x²), then dy/dx =

(1) -2x/√(1-x2) + 1/2√(x-x2) (2) -1/√(1-x2) - 1/2√(x-x2) (3) 1/√(1-x2) + 1/2√(x-x2) (4) none of these Solution: Given y = sin-1(... View Article

02 Feb 2021: UPSC Exam Comprehensive News Analysis

02 Feb 2021 CNA:- Download PDF Here TABLE OF CONTENTS A. GS 1 Related B. GS 2 Related INTERNATIONAL RELATIONS 1. Military seizes power in Myanmar... View Article

Jainism [History Notes for UPSC & Govt. Exams]

Jainism is a very important topic for the UPSC history, and art and culture segments. It is an important ancient religion that originated in... View Article

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