List of top Mathematics Questions asked in KEAM

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:
Let \( \vec{a}, \vec{b}, \vec{c} \) be three vectors with magnitudes 4, 4, and 2, respectively. If \( \vec{a} \) is perpendicular to \( (\vec{b} + \vec{c}) \), \( \vec{b} \) is perpendicular to \( (\vec{c} + \vec{a}) \), and \( \vec{c} \) is perpendicular to \( (\vec{a} + \vec{b}) \), then the value of \( |\vec{a} + \vec{b} + \vec{c}| \) is:
Let \( \vec{a} = \hat{i} - \hat{j} \), \( \vec{b} = \hat{j} - \hat{k} \), and \( \vec{c} = \hat{k} - \hat{i} \), then the value of \( \vec{b} \cdot (\vec{a} + \vec{c}) \) is:
Let \( \mathbf{u}, \mathbf{v}, \mathbf{w} \) be vectors such that \( \mathbf{u} + \mathbf{v} + \mathbf{w} = \mathbf{0} \). If \( |\mathbf{u}| = 3 \), \( |\mathbf{v}| = 4 \), and \( |\mathbf{w}| = 5 \), then \( \mathbf{u} \cdot \mathbf{v} + \mathbf{w} \cdot \mathbf{u} \) is:
If a vector makes angles \( \frac{\pi}{3}, \frac{\pi}{4} \) and \( \gamma \) with \( \hat{i}, \hat{j} \), and \( \hat{k} \), respectively, where \( \gamma \in \left( \frac{\pi}{2}, \pi \right) \), then the angle \( \gamma \) is:
If \( \cos \theta = \frac{2 \cos \alpha + 1}{2 + \cos \alpha} \), then \( \tan^2 \left( \frac{\theta}{2} \right) \) is equal to:
Evaluate \( \cos \left( \cot^{-1} \left( \frac{7}{24} \right) \right) \):
If \( 3 \sin \theta + 5 \cos \theta = 5 \), then the value of \( 5 \sin \theta - 3 \cos \theta \) is:
The value of \( \tan \left[ \tan^{-1} \left( \frac{3}{4} \right) + \tan^{-1} \left( \frac{2}{3} \right) \right] \) is:
If \( \alpha = \tan^2 x + \cot^2 x \), where \( x \in \left( 0, \frac{\pi}{2} \right) \), then \( \alpha \) lies in the interval:
The domain of the function \( f(x) = \frac{\sin^{-1} \left( x-3 \right)}{\sqrt{9 - x^2}} \) is:
The period of \( 2 \sin 4x \cos 4x \) is: