List of top Mathematics Questions asked in KEAM

The angle between the lines \[ \frac{x-1}{2} = \frac{2y+3}{4} = \frac{z+5}{-2} \quad {and} \quad \frac{x-3}{4} = \frac{y+1}{-4} = \frac{z+3}{-4} \] is equal to
The symmetric equation of the straight line passing through the points \( (-1, 4, 2) \) and \( (-3, 0, 5) \) is
The angle between the vectors \( \vec{a} \) and \( \vec{b} \) is \( \frac{\pi}{3} \). If \( | \vec{a} \cdot \vec{b} |^2 = 15 \), then \( | \vec{a} \times \vec{b} |^2 \) is equal to
The line joining the points \( (2,2,2) \) and \( (6,6,6) \) meets the line \[ \frac{x - 1}{3} = \frac{y - 2}{2} = \frac{z - 5}{-1} \] at the point
If \( \vec{a} = 5\hat{i} - 7\hat{j} + 9\hat{k} \) and \( \vec{b} = -5\hat{i} + 7\hat{j} - 9\hat{k} \), then \( \vec{a} \cdot (\vec{a} \times \vec{b}) + (\vec{a} + \vec{b}) \cdot \hat{b} \) is equal to
Let \( A(0,3,-3) \), \( B(1,1,1) \) and \( C(2,0,3) \) be three points in space. Then the projection of \( \overrightarrow{AB} \) on \( \overrightarrow{AC} \) is equal to
Let \( \alpha, \beta, \gamma \) be the direction cosines of a vector \( \vec{a} = x \hat{i} + y \hat{j} + z \hat{k} \), where \( z<0 \). If \( \alpha = \frac{-4}{\sqrt{105}} \) and \( \beta = \frac{\sqrt{5}}{\sqrt{21}} \), then \( \gamma \) is equal to
If \( (a,-6) \) lies on the perpendicular bisector of the line segment joining \( (-2,-1) \) and \( (4,-13) \), then the value of \( a \) is equal to
For an ellipse, the foci are \( F(3,0) \) and \( F'(-3,0) \). If the length of the minor axis is 8, then the length of the major axis is equal to
The eccentricity of the ellipse \[ p x^2 + 5y^2 = 80, \quad \text{where } p > 5, \] is \( \frac{\sqrt{3}}{2} \). Then the value of \( p \) is
Find the equation of the parabola with focus at \( (3,1) \) and vertex at \( (5,1) \).
The coordinates of the vertex of the parabola \[ y = 2x^2 - 12x + 26 \] are
Find the values of \( \alpha \) for which the circle \[ x^2 + y^2 + \alpha x - 8y + 56 = 0 \] has radius 3.
Find the equation of the line perpendicular to the line \[ 7x - 5y = 11 \] and passing through the point \( (7, -9) \).

The value of \( \cosec x + \cot x \) is

Evaluate \[ \frac{\cosec^2(\theta) - 1}{\cosec^2(\theta)} - \frac{\sec^2(\theta) - 1}{\sec^2(\theta)} \]

If \[ 7 \cos^2 x + 3 \sin^2 x = 6, \] then the value of \( \cos 2x \) is equal to
Let \[ \sum_{k=1}^{15} \sin (t_k) = 0 \quad {and} \quad \sum_{k=1}^{15} \sin (3t_k) = \frac{-24}{5}, \] where \( t_1, t_2, t_3, \dots \) are real numbers. Then the value of the sum \[ \sum_{k=1}^{15} \sin^3 (t_k) \] is equal to
The value of \[ \tan^{-1} \left( \frac{1}{3} \right) + \tan^{-1} \left( \frac{2}{3} \right) + \cot^{-1} \left( \frac{9}{7} \right) \] is equal to
The value of \( \sin \left( 2 \cos^{-1} \left( \frac{5}{12} \right) + \sin^{-1} \left( \frac{5}{12} \right) \right) \) is equal to
If \( \sin t + \cos t = \sqrt{2} \), then \( \tan t + \cot t \) is equal to
The value of \( \tan \left( \cos^{-1} \left( \frac{-24}{25} \right) \right) \) is equal to

Simplify: \( \tan x - \cot x + \csc x \sec x \)

Let \( A \) and \( B \) be two events. If \( P(A) = 0.49 \), \( P(B) = 0.3 \) and \( P(A | B^c) = 0.4 \), then \( P(A | B) \) is equal to
If two dice are rolled simultaneously, then the probability that the difference of the numbers on the two dice equals to zero is