HSC Arts (English Medium) 12th Standard Board Exam - Maharashtra State Board Question Bank Solutions for Mathematics and Statistics

If 2y = `sqrt(x + 1) + sqrt(x - 1)`, show that 4(x² – 1)y₂ + 4xy₁ – y = 0.

If y = sin (m cos^–1x), then show that `(1 - x^2)(d^2y)/(dx^2) - x"dy"/"dx" + m^2y` = 0.

If x² + 6xy + y² = 10, show that `(d^2y)/(dx^2) = (80)/(3x + y)^3`.

If x = a sin t – b cos t, y = a cos t + b sin t, show that `(d^2y)/(dx^2) = -(x^2 + y^2)/(y^3)`.

Find the n^th derivative of the following : (ax + b)^m 

Find the n^th derivative of the following:

`(1)/x`

Find the n^th derivative of the following : e^ax+b 

Find the nth derivative of the following : a^px+q 

Find the n^th derivative of the following : cos x

Find the n^th derivative of the following : sin (ax + b)

Find the n^th derivative of the following : cos (3 – 2x)

Find the n^th derivative of the following : `(1)/(3x - 5)`

Find the n^th derivative of the following : y = e^ax . cos (bx + c)

Find the n^th derivative of the following:

y = e^8x . cos (6x + 7)

Choose the correct option from the given alternatives : 

Let `f(1) = 3, f'(1) = -(1)/(3), g(1) = -4 and g'(1) =-(8)/(3).` The derivative of `sqrt([f(x)]^2 + [g(x)]^2` w.r.t. x at x = 1 is 

Choose the correct option from the given alternatives :

If y = sec (tan ^–1x), then `"dy"/"dx"` at x = 1, is equal to

Choose the correct option from the given alternatives :

If f(x) = `sin^-1((4^(x + 1/2))/(1 + 2^(4x)))`, which of the following is not the derivative of f(x)?

Choose the correct option from the given alternatives :

If y = sin (2sin^–1 x), then _dx = ........

Choose the correct option from the given alternatives :

If y = `tan^-1(x/(1 + sqrt(1 - x^2))) + sin[2tan^-1(sqrt((1 - x)/(1 + x)))] "then" "dy"/"dx"` = ...........

Choose the correct option from the given alternatives :

If `xsqrt(y + 1) + ysqrt(x + 1) = 0 and x ≠ y, "then" "dy"/"dx"` = ........