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If ? and ? are the roots of the equations 6x² – 5x + 1 = 0, then the value of tan^-1 ? + tan^-1 ? is

1) 0 2) π/4 3) 1 4) π/2 Solution: (2) π/4 6x2 - 5x + 1 = 0 (α + β) = - (- 5) / 6 = 5 / 6 αβ = 1 / 6 tan-1 α + tan-1 β = tan-1... View Article

The value of cos [2 tan^-1 (-7)] is

1) 49/50 2) - (49/50) 3) 24/25 4) - (24/25) 5) 48/49 Solution: (4) - (24/25) cos [2 tan-1 (-7)] 2 tan-1 x = cos-1 [1 - x2] /... View Article

If a > b > 0, then the value of tan^-1 (a / b) + tan^-1 ((a + b) / (a – b))

1) both a and b 2) b but not a 3) a but not b 4) neither a nor b Solution: (4) neither a nor b tan-1 (a / b) + tan-1 ((a + b) / (a... View Article

The value of sin [sin^-1 (1 / 3) + sec^-1 (3)] + cos [tan^-1 (1 / 2) + tan^-1 (2)] is

1) 1 2) 2 3) 3 4) 4 Solution: (1) 1 y = sin [sin-1 (1 / 3) + sec-1 (3)] + cos [tan-1 (1 / 2) + tan-1 (2)] sec-1 x = cos-1 (1 /... View Article

The number of real solutions of tan^-1 ?x (x – 1) + sin^-1 ?x² + x + 1 = ? / 2, is

1) 0 2) 1 3) 2 4) ∞ Solution: (1) 0 The given function is tan-1 √x (x - 1) + sin-1 √x2 + x + 1 = π / 2. The function is defined... View Article

d/dx tan^-1 x/√(a² – x²) =

(1) a/(a2 + x2) (2) 1/√(a2 - x2) (3) -1/(a2 + x2) (4) none of these Solution: Let y = tan-1 x/√(a2 - x2) Put x = a sin θ θ =... View Article

If [latex]y=\sin^{-1}\frac{2x}{1+x^{2}}+\sec^{-1}\frac{1+x^{2}}{1-x^{2}}[/latex], then dy/dx =

(1) 4/(1 - x2) (2) 4/(1 + x2) (3) 1/(1 + x2) (4) -4/(1 + x2) Solution: Given y = sin-1 (2x/(1 + x2) + sec-1 (1 + x2)/(1 - x2) Put x = tan θ... View Article

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

(1) -1/(1+x2) (2) 1/(1+x2) (3) -2/(1+x2) (4) 2/(1+x2) Solution: Let y = cos-1 ((x - x-1)/(x + x-1)) = cos-1 ((x - 1/x)/(x + 1/x))... View Article

If y = cos^-1 [(3 cos x – 4 sin x)/5], then dy/dx =

(1) 0 (2) 1 (3) -1 (4) 1/2 Solution: Given y = cos-1 [(3 cos x - 4 sin x)/5] Let 3/5 = r cos θ and 4/5 = r sin θ Squaring and... View Article

If y = sin (2 sin^-1 x), then dy/dx =

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

If y = x^{sin x}, then dy/dx =

(1) xsin x [x cos x log x + sin x]/x (2) xsin x [x cos x log x + cos x]/x (3) y [x sin x log x + cos x]/x (4) none of these Solution:... View Article

If x^y = y^x, then dy/dx =

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

If x^py^q = (x+y)^p+q, then dy/dx =

(1) y/x (2) -y/x (3) x/y (4) -x/y Solution: Given xpyq = (x+y)p+q Take log on both sides p log x + q log y = (p+q) log (x+y)... View Article

If y = (tan x)^{cot x}, then dy/dx =

(1) y cosec2 x (1 - log (tan x)) (2) y cosec2 x (1 + log (tan x)) (3) y cosec2 x log (tan x) (4) none of these Solution: Given y =... View Article

The derivative of tan^-1 {(1 + x)/(1 – x)} with respect to x is

(1) 1 + x2 (2) 1/(1 + x2) (3) -1/(1 + x2) (4) none of these Solution: Let y = tan-1 {(1 + x)/(1 - x)} Put x = tan θ So y = tan-1 {(1 + tan... View Article

If y = x² + x^{log x}, then dy/dx =

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

If y = x^x, then dy/dx is

(1) xx log ex (2) xx(1 + 1/x) (3) xx(1 + log x) (4) xx log x Solution: Given y = xx Take log on both sides log y = x log x... View Article

If x = sin t cos 2t and y = cos t sin 2t, then at t = ?/4, the value of dy/dx is equal to

(1) -1/2 (2) 1/2 (3) -2 (4) 2 Solution: Given x = sin t cos 2t y = cos t sin 2t Using product rule, differentiate w.r.t.t dx/dt... View Article

If x = a cos⁴ θ, y = a sin⁴ θ, then dy/dx at θ = 3π/4 is

(1) -1 (2) 1 (3) -a2 (4) a2 Solution: Given x = a cos4 θ y = a sin4 θ Differentiate w.r.t.θ dx/dθ = -4a cos3 θ sin θ dy/dθ =... View Article

If x = a(t cos t – sin t), y = a(t sin t + cos t). Then dy/dx is equal to

(1) -tan t (2) tan t (3) cot t (4) -cot t Solution: Given x = a(t cos t - sin t) y = a(t sin t + cos t) Differentiate w.r.t.t... View Article

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