List of top Mathematics Questions asked in KEAM

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

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

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

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

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

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

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

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:

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

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 axis of the parabola \( x^2 + 6x + 4y + 5 = 0 \) 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:

If \( 3\hat{i} + 2\hat{j} - 5\hat{k} = x(2\hat{i} - \hat{j} + \hat{k}) + y(\hat{i} + 3\hat{j} - 2\hat{k}) + z(-2\hat{i} + \hat{j} - 3\hat{k}) \), then

Two dice of different colours are thrown at a time. The probability that the sum is either 7 or 11 is

If \( \sin\theta - \cos\theta = 1 \), then the value of \( \sin^3\theta - \cos^3\theta \) is

The order and degree of the differential equation \( (y'')^2 + (y''')^3 - (y')^4 + y^5 = 0 \) is

If \( |\vec{a}|=3, |\vec{b}|=1, |\vec{c}|=4 \) and \( \vec{a}+\vec{b}+\vec{c}=0 \), then the value of \( \vec{a}\cdot\vec{b} + \vec{b}\cdot\vec{c} + \vec{c}\cdot\vec{a} \) is

If the vectors \( \vec{a} = \hat{i} - \hat{j} + 2\hat{k}, \vec{b} = 2\hat{i} + 4\hat{j} + \hat{k} \) and \( \vec{c} = 2\lambda \hat{i} + 9\hat{j} + \mu \hat{k} \) are mutually orthogonal, then \( \lambda + \mu \) is equal to

Let \( \vec{a} = \hat{i}+\hat{j}+\hat{k}, \vec{b} = \hat{i}+3\hat{j}+5\hat{k}, \vec{c} = 7\hat{i}+9\hat{j}+11\hat{k} \). Then the area of parallelogram with diagonals \( \vec{a}+\vec{b} \) and \( \vec{b}+\vec{c} \) is

\( \frac{\sqrt{3}}{\sin(20^\circ) - \frac{1}{\cos(20^\circ)}} = \)