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

A random variable X has the following has the following:

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

Negation of the Boolean expression `p Leftrightarrow (q \implies p)` is ______. 

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 ______.

∼ ((∼ p) ∧ q) is equal to ______.

If α and β are roots of the equation x² + 5|x| – 6 = 0, then the value of |tan^–1α – tan^–1 β| is ______.

If M is the arithmetic mean of two distinct real number l and n (I, n > 1) and G₁, G₂ and G₃ are three geometric means between l and n, then `G_1^4 + 2G_2^4 + G_3^4` is equal to ______.

If log₂ x + log₂ y ≥ 6, then the least value of x + y is ______.

If θ lies in the first quadrant and 5 tan θ = 4, then `(5 sin θ - 3 cos θ)/(sin θ + 2 cos θ)` is equal to ______.

The means of two samples of sizes 60 and 120 respectively are 35.4 and 30.9 and the standard deviations are 4 and 5. Obtain the standard deviation of the sample of size 180 obtained by combining the two samples.

If x = e^2y, then `("d"^2y)/"dx"^2*("d"^2x)/"dy"^2` is equal to

If x = a sin t - b cos t, y = a cos t+ b sin t, then `"y"^3 ("d"^2"y")/"dx"^2 + x^2 + y^2` = ?

If f(x) = log (sin x), x ∈ `[π/6, (5π)/6]`, then value of 'c' by applying LMVT is ____________.

The differential equation obtained from the function y = a(x − a)² is ____________.

lf f(x) = cos^-1 `[(1 - (log x)^2)/(1 + (log x)^2)]`, then f' (e) = _______.

A stone is dropped into a pond. Waves in the form of circles are generated and radius of outermost ripple increases at the rate of 5 cm/sec. Then area increased after 2 seconds is ______.

If x = sin θ, y = sin³ θ then `("d"^2"y")/"dx"^2  "at" theta = pi/2` is ______.

If for a function f(x), f'(a) = 0, f"(a) = 0, f"'(a) > 0, then at x = a, f(x) is ______

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