List of top Questions asked in KEAM- 2025

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

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

A straight line passing through \( (6,1,3) \) meets the line \( \frac{x-1}{2} = \frac{y}{1} = \frac{z-2}{3} \) at \( Q \). If the lines are perpendicular to each other, then the coordinates of \( Q \) are

A straight line passes through the points \( (10,8,6) \) and \( (13,9,4) \). A unit vector parallel to this line 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

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

The angle between the lines \( \frac{x-3}{1} = \frac{y+1}{-1} = \frac{z-2}{-1} \) and \( \frac{x+1}{2} = \frac{y-2}{2} = \frac{z+3}{-2} \) is

The eccentricity of the hyperbola \( \frac{(2x-6)^2}{2} - \frac{(4y+7)^2}{16} = 1 \) is

Let \( \vec{a} + \vec{b} = \lambda \hat{i} + 16\hat{j} - 18\hat{k} \) and \( \vec{a} - \vec{b} = 2\hat{i} + 8\hat{j} + \lambda \hat{k} \). If \( \vec{a} + \vec{b} \) is perpendicular to \( \vec{a} - \vec{b} \), then \( |\vec{a}| = \)

If \( |\vec{a}| = 12 \) and the projection of \( \vec{a} \) on \( \vec{b} \) is \( 6\sqrt{3} \), then the angle between \( \vec{a} \) and \( \vec{b} \) is

Let \( \vec{a} = 6\hat{i} + 2\hat{j} + 3\hat{k} \). If \( \vec{b} \) is parallel to \( \vec{a} \) and \( \vec{a} \cdot \vec{b} = \frac{49}{2} \), then \( |\vec{b}| = \)

If \( |\vec{a} + \vec{b}| = \frac{\sqrt{14}}{2} \) where \( \vec{a} \) and \( \vec{b} \) are unit vectors, then the value of \( |\vec{a} + \vec{b}|^2 - |\vec{a} - \vec{b}|^2 \) is equal to

The vertex of the parabola \( 4y = x^2 - 6x + 17 \) is

The length of major axis and minor axis of an ellipse are, respectively, \( m \) and \( n \). If \( m^2 - n^2 = 45 \) and the eccentricity of the ellipse is \( \frac{\sqrt{5}}{3} \), then the length of the major axis is

A circle touches the \( x \)-axis at \( (9,0) \). If it also touches the straight line \( y = 14 \), then the equation of the circle is

The focus of the parabola \( x^2 - 4x + 8y + 4 = 0 \) is

Let \( ABC \) be an equilateral triangle. If the coordinates of \( A \) are \( (-2,2) \) and the side \( BC \) is along the line \( x + y = 6 \), then the length of the side of the triangle is

The straight line \( ax + by + c = 0 \) passes through the point \( (-10,7) \). If the line is perpendicular to \( 11x - 7y = 13 \), then the value of \( c \) is equal to

If \( \theta = \cot^{-1}\sqrt{\frac{1-x}{1+x}} \), then \( \sec^2 \theta \) is equal to

If \( 5\sin^{-1}\alpha + 3\cos^{-1}\alpha = \pi \), then \( \alpha \) is equal to

If \( f(x) = \tan^{-1}\left(\frac{2x}{1 - x^2}\right) \), then \( f\left(\frac{1}{\sqrt{3}}\right) \) is equal to

If \( \sin \alpha = \frac{12}{13} \), where \( \frac{\pi}{2}<\alpha<\frac{3\pi}{2} \), then the value of \( \tan \alpha \) is equal to