List of top Mathematics Questions asked in KEAM

If \( \int_{-\sqrt{3}}^{1} (-6x^2 + 18)\,dx = \alpha + \beta\sqrt{3} \), then the value of \( \alpha + \beta \) is equal to

\( \int x^7 (x^8 + 1)^{-3/4} \, dx = \)

\( \int \frac{\sin^{-1}x}{\sqrt{1-x^2}} \, dx = \)

\( \int e^x(x^2-2)\cos(e^x(x^2-2x)) dx = \)

\( \int e^x \sec x (1+\tan x) dx = \)

\( \int \frac{2x^2 + 4x + 3}{x^2 + x + 1} \, dx = \)

Let \( f(x) = x^2 + ax + \beta \). If \( f \) has a local minimum at \( (2,6) \), then \( f(0) \) is equal to

The function \( f(x) = 2x^3 - 3x^2 - 36x + 28 \) is increasing in

Let \( f(x) = (\cos^2 x)(a + \cos x) \). If \( f'\left(\frac{\pi}{3}\right) = 0 \), then the value of \( a \) is equal to

If \( y = \tan^{-1}(x^2 - x) \), then \( \frac{dy}{dx} = \)

The function \( f(x) = x^2(x-2) \) is strictly decreasing in

The surface area of a solid hemisphere is increasing at the rate of \( 8 \, \text{cm}^2/\text{sec} \) (retaining its shape). Then the rate of change of its volume (in \( \text{cm}^3/\text{sec} \)), when the radius is \( 5 \,\text{cm} \), is

If \( f(x) = \frac{3^x}{3^x + \sqrt{3}} \), then \( f(x) + f(1-x) \) is equal to

Let \( f(x) = |\sin 3x| - |\cos 3x| \), where \( \frac{\pi}{6} \le x \le \frac{\pi}{3} \). Then the value of \( f\left(\frac{\pi}{4}\right) \) is

\( \lim_{x \to 0} \frac{\sqrt{\cos 2x + 3} - \sqrt{\cos^2 x + \sin x + 3}}{x} \) is equal to

If \( f(x) = |x^2 + x - 6| \) is not differentiable at \( x = a \) and \( x = b \), then \( a^2 + b^2 = \)

Let \( h(x) = f(\sqrt{g(x)}) \). If \( f'(3) = 6 \), \( g'(3) = 3 \) and \( g(3) = 9 \), then the value of \( h'(3) \) is equal to

The domain of the function \( f(x) = \sqrt{x - 3 + 4\sqrt{5 - x}} \) is

An unbiased die is tossed until 5 appears. If \( X \) denotes the number of tosses required, \( \frac{25}{P(X=5)} \) is equal to

\( \lim_{x \to 0} \frac{x^2}{\sqrt{2} - \sqrt{1 + \cos x}} \) is equal to

The standard deviation of the numbers \( -3, 0, 3, 8 \) is

A box contains 4 red and 6 white marbles. Two successive draws of 3 balls are made without replacement. The probability that in first draw all the 3 balls are white and in second draw all the 3 balls are red, is

Let \( A \) and \( B \) be two events. If \( P(A|B)=0.4 \), \( P(A|B')=0.7 \) and \( P(B)=0.7 \), then \( P(A) \) is

Let \( \alpha, \beta \) and \( \gamma \) be the angles made by a straight line with the x-axis, y-axis and z-axis respectively. If \( \cos\alpha + \cos\beta + \cos\gamma = \frac{5}{3} \), then the value of \( \cos\alpha \cos\beta + \cos\beta \cos\gamma + \cos\gamma \cos\alpha \) is equal to