List of top Questions asked in KEAM

Evaluate \( \int x \cos x \, dx \):
The integral \( \int \frac{dx}{1 + e^x} \) is:
The limit \( \lim_{x \to 10} \frac{x - 10}{\sqrt{x + 6} - 4} \) is equal to:
The maximum value of \( y = 12 - |x - 12| \) in the range \( -11 \leq x \leq 11 \) is:
The function \( f(x) = 2x^3 + 9x^2 + 12x - 1 \) is decreasing in the interval:
If \( y = \frac{x^2}{x - 1} \), then \( \frac{dy}{dx} \) at \( x = -1 \) is:
If \( y = \tan^{-1} \left( \frac{\cos x - \sin x}{\cos x + \sin x} \right) \), \( \frac{-\pi}{2}<x<\frac{\pi}{2} \), then \( \frac{dy}{dx} \) is:
If \( f(x) = \sin^{-1}(\cos x) \), then \( \frac{d^2 y}{dx^2} \) at \( x = \frac{\pi}{4} \) is:
If \( f(x) = \cos x - \sin x \), and \( x \in \left( \frac{\pi}{4}, \frac{\pi}{2} \right) \), then \( f' \left( \frac{\pi}{3} \right) \) is equal to:
Let \( f(x) = x - \lfloor x \rfloor \), where \( \lfloor \cdot \rfloor \) denotes the greatest integer function and \( x \in (-1, 2) \). The number of points at which the function is not continuous is:
If \( f(x) = \left\{ \begin{array}{ll} mx + 1, & \text{when } x \leq \frac{\pi}{2} \\ \sin x + n, & \text{when } x > \frac{\pi}{2} \end{array} \right. \) is continuous at \( x = \frac{\pi}{2} \), then the values of \( m \) and \( n \) are:
Evaluate \( \lim_{x \to 0} \frac{\sin 2x + \sin 5x}{\sin 4x + \sin 6x} \):
If \( g(x) = -\sqrt{25 - x^2} \), then \( g'(1) \) is:
The mean deviation of the numbers 3, 10, 10, 4, 7, 10 and 5 from the mean is:
If \( \frac{1 + 3p}{4}, \frac{1 - p}{3}, \frac{1 - 3p}{2} \) are the probabilities of three mutually exclusive and exhaustive events, then the value of \( p \) is:
If three distinct numbers are chosen randomly from the first 50 natural numbers, then the probability that all of them are divisible by 2 and 3 is:
The angle between the lines \( \frac{x}{1} = \frac{y}{1} = \frac{z}{1} \) and \( \frac{x}{0} = \frac{y}{1} = \frac{z}{-1} \) is:
If the straight line \( \frac{x - a}{1} = \frac{y - b}{2} = \frac{z - 3}{-1} \) passes through \( (-1, 3, 2) \), then the values of \( a \) and \( b \) are, respectively:
A ray of light passing through the point \( (1, 2) \) is reflected on the \( x \)-axis at a point \( P \) and passes through the point \( (5, 6) \). Then the abscissa of the point \( P \) is:
The line \( \frac{x}{5} + \frac{y}{b} = 1 \) passes through the point \( (13, 32) \) and is parallel to the line \( \frac{x}{c} + \frac{y}{3} = 1 \). Then the values of \( b \) and \( c \) are, respectively:
The equation of the line passing through the point \( (1, 2) \) and perpendicular to the line \( x + y + 1 = 0 \) is:
The length of the minor axis of the ellipse with foci \( (\pm 2, 0) \) and eccentricity \( \frac{1}{3} \) is:
If the focus of a parabola is \( (0, -3) \) and its directrix is \( y = 3 \), then its equation is:
If one end of a diameter of the circle \( x^2 + y^2 - 4x - 6y + 11 = 0 \) is \( (3, 4) \), then the coordinate of the other end of the diameter is:
If two vectors \( \vec{a} = \cos \alpha \hat{i} + \sin \alpha \hat{j} + \sin \frac{\alpha}{2} \hat{k} \) and \( \vec{b} = \sin \alpha \hat{i} - \cos \alpha \hat{j} + \cos \frac{\alpha}{2} \hat{k} \) are perpendicular, then the values of \( \alpha \) are: