List of top Questions asked in KEAM

The area of the region bounded by the curves \( y = |x - 2| \), \( x = 1 \), \( x = 3 \) and \( y = 0 \) is:

For two data sets, each of size 5, the variances are given to be 4 and 5 and the corresponding means are given to be 2 and 4, respectively. The variance of the combined data set is

A bag contains 5 red balls and some blue balls. If the probability of drawing a blue ball is double that of red ball, the number of blue balls must be

The maximum value of \( y = \left(\frac{1}{x}\right)^x \), \( x > 0 \) is

The value of the integral \( \int_{0}^{\pi} \frac{\cos x}{1+\sin^2 x} \, dx \) is

The solution of the differential equation \( 5y\,dx = 2x\,dy \) passing through the point \( (1,1) \) is:

The equation of the tangent to the curve given by \( x^2 + 2x - 3y + 3 = 0 \) at the point \( (1,2) \) is

The area enclosed between the curves \( y = 2x^2 + 1 \) and \( y = x^2 + 5 \) is:

If \( \sin^{-1} x + \cos^{-1} 2x = \frac{\pi}{6} \), then the value of \( x \) is

The value of \( \tan\left[ \sin^{-1}\left(\frac{-1}{\sqrt{2}}\right) \right] \) is

If \( x = 2\cos t - \cos 2t \) and \( y = 2\sin t - \sin 2t \), then \( \frac{dy}{dx} \) at \( t = \frac{\pi}{2} \) is

The complex number \( \sqrt{2}\left[ \sin \frac{\pi}{8} + i \cos \frac{\pi}{8} \right]^6 \) represents

Let \( z_1 = 1 + i\sqrt{3} \) and \( z_2 = 1 + i \), then \( \arg\left( \frac{z_1}{z_2} \right) \) is

If \( z^2 + z + 1 = 0 \), where \( z \) is a complex number, then the value of \[ \left( z + \frac{1}{z} \right)^2 + \left( z^2 + \frac{1}{z^2} \right)^2 + \left( z^3 + \frac{1}{z^3} \right)^2 + \cdots + \left( z^6 + \frac{1}{z^6} \right)^2 \] is

If \( f : \mathbb{R} \to \mathbb{R} \) is a function defined by \( f(x) = \sin x \), then which of the following is true?

If \[ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \] is the inverse of the matrix \[ \begin{pmatrix} 1 & 5 \\ 7 & -3 \end{pmatrix}, \] then \( d \) equals

Consider the set \( M = \{1,2,3\} \) along with the relation \( R = \{(1,2), (1,1), (3,1), (3,4), (3,3), (4,3)\ \). Which of the following statements is true?

If \( a, b \) and \( c \) are distinct reals and the determinant \[ \begin{vmatrix} a^3+1 & a^2 & a \\ b^3+1 & b^2 & b \\ c^3+1 & c^2 & c \end{vmatrix} = 0, \] then the product \( abc \) is

If \( (x,y,z) \) is the solution of the equations \[ x - y - 2z = 3, 2x + y + 4z = 5, 4x - y - 2z = 11, \] then the value of \( y \) equals

A particle is displaced from the point \( (2,1,-1) \) to \( (4,3,-4) \) by the force \( 2i + 4j - 5k \). Then the work done by the force is

The value of \( m \) if the vectors \( 4i - 3j + 5k \) and \( mi - 4j + k \) are perpendicular, is

The circle passing through \( (1, -2) \) and touching the \(x\)-axis at \( (3, 0) \) also passes through the point:

If \( \alpha \) and \( \beta \) are the roots of the equation \( x^2 + \alpha x + \beta = 0 \), then

If \( A \) and \( B \) are two matrices such that \[ 3A + B = \begin{pmatrix} 9 & 11 & 3 \\ 12 & 14 & 19 \end{pmatrix} \] and \[ 2A - 3B = \begin{pmatrix} -16 & 11 & 2 \\ -3 & -22 & 9 \end{pmatrix}, \] then the matrix \( B \) is

The value of \( k \), if the circles \( 2x^2 + 2y^2 - 4x + 6y = 3 \) and \( x^2 + y^2 + kx + y = 0 \) cut orthogonally is: