MAH-MHT CET (PCM/PCB) entrance exam Question Bank Solutions

Let f(x) = tan^-1x. Then f'(x) + f"(x) is equal to 0, when x is equal to ______ 

If y = cos nx, then `(d^2y)/dx^2 + n^2y` = ______ 

If y =sin (log x), then `x^2 ("d"^2"y")/"dx"^2 + x "dy"/"dx" + "y"` is equal to ______.

The order and degree of the differential equation `("d"^2y)/("d"x^2) = {y + (("d"^3y)/("d"x^3))^3}^(1/5)` are respectively ______.

Let `overlinep` and `overlineq` be the position vectors of P and Q respectively, with respect to O and `|overlinep|` = P, `|overlineq|` = q`. The points R and S divide PQ internally and externally in the ratio 2 : 3 respectively. If OR and OS are perpendicular, then ______

The differential equation representing the family of curves `y^2 = 2d(x + sqrtd)`, where d is a positive parameter, is of ______ 

If f(x) =log_x (log x), then f'(x) at x = e is ______.

If the p.d.f. of a continuous random variable X is

f(x) = k (x - x²),    0 < x < 1

= 0,             otherwise

then the value of k is ______.

if y = axⁿ⁺² + bx^-n-1, then `x^2(d^2y)/(dx^2)` is ______.

If y = `sqrt("e"^x + sqrt("e"^x + sqrt("e"^x + .... oo)`, then `("d"y)/("d"x)` is equal to ______.

`"d"^2/("d"x^2) (2 sin 3x cosx)` = ______.

If f'(x) = `2 - 5/x^4` and f(1) = `14/3`, then f(-1) = ______

A random variable X has the following has the following:

Then, the value of P(–1 < X < 3) is ______.

If f(0) = 3, f'(0) = 2, then`"d"/("d"x) { log "f"(sin x + 3x^2)}` at x  0 is ______.

If the p.d.f of a c.r.v. X is
f(x) = `{:("K.e"^(-thetax)";", theta > 0","  0 ≤ x < oo),(0";", "otherwise"):}`
Then, K = ______.

The vector equation of the line passing through the points (3, 2, 1) and (1, 3, 1) is ______.

If y = `x^3 + 1/(x^3 + 1/(x^3 + 1/(x^3 + .....oo)`, ten `("d"y)/("d"x)` = ______.

Given a family of lines a (2x + y + 4) + b(x – 2y – 3) = 0, the number of lines belonging to the family at a distance `sqrt(10)` from P(2, –3) is ______.

Suppose f(x) = e^ax + e^bx, where a ≠ b, and that f"(x) – 2f'(x) – 15f(x) = 0 for all x. Then the product ab is ______.

If x = a(cos t + t sin t) and y = a(sin t – t cos t), then `(d^2y)/(dx^2)` is ______.