HSC Science (Computer Science) 12th Standard Board Exam - Maharashtra State Board Question Bank Solutions

If x = 2cos⁴(t + 3), y = 3sin⁴⁽t + 3), show that `"dy"/"dx" = -sqrt((3y)/(2x)`.

If x = log(1 + t²), y = t – tan^–1t,show that `"dy"/"dx" = sqrt(e^x - 1)/(2)`.

If x = sin^–1(e^t), y = `sqrt(1 - e^(2t)), "show that"  sin x + dy/dx` = 0

If x = `(2bt)/(1 + t^2), y = a((1 - t^2)/(1 + t^2)), "show that" "dx"/"dy" = -(b^2y)/(a^2x)`.

Differentiate 3^x w.r.t. log_x3.

Find the second order derivatives of the following : x³.logx

Find the second order derivatives of the following : log(logx)

If y = `log(x + sqrt(x^2 + a^2))^m`, show that `(x^2 + a^2)(d^2y)/(dx^2) + x "d"/"dx"` = 0.

If y = log (log 2x), show that xy₂ + y₁ (1 + xy₁) = 0.

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

Find the n^th derivative of the following : log (2x + 3)

Choose the correct option from the given alternatives :

If x^y = y^x, then `"dy"/"dx"` = ..........

If y = A cos (log x) + B sin (log x), show that x²y² + xy₁ + y = 0.

Verify Lagrange’s mean value theorem for the following function:

f(x) = log x, on [1, e]

Verify Lagrange’s mean value theorem for the following functions : f(x) = (x – 1)(x – 2)(x – 3) on [0, 4].

Verify Lagrange’s mean value theorem for the following function:

`f(x) = x^2 - 3x - 1, x ∈ [(-11)/7 , 13/7]`.

Verify Lagrange’s mean value theorem for the following functions : f(x) = 2x – x², x ∈ [0, 1].

Verify Lagrange’s mean value theorem for the following functions : f(x) = `(x - 1)/(x - 3)` on [4, 5].

A die is thrown 6 times. If ‘getting an odd number’ is a success, find the probability of 5 successes. 

A die is thrown 6 times. If ‘getting an odd number’ is a success, find the probability of at least 5 successes.