List of top Questions asked in KEAM

The value of \( \tan 15^\circ + \tan 75^\circ \) is equal to

If \( x = 5 + 2\sec \theta \) and \( y = 5 + 2\tan \theta \), then \( (x-5)^2 - (y-5)^2 \) is equal to

If \( \tan \frac{\theta}{2} = \frac{1}{2} \), then the value of \( \sin \theta \) is

Let \(p, q, r\) be three statements. Then \( \sim (p \vee (q \wedge r)) \) is equal to

For any two statements \(p\) and \(q\), the statement \( \sim(p \vee q) \vee (\sim p \wedge q) \) is equivalent to

If \( |x-3| < 2x+9 \), then \(x\) lies in the interval

The area and perimeter of a rectangle are \(A\) and \(P\) respectively. Then \(P\) and \(A\) satisfy the inequality

If \( A = \begin{pmatrix} x & x-1 \\ 2x & 1 \end{pmatrix} \) and if \( \det A = -9 \), then the values of \(x\) are

The value of the determinant \[ \begin{vmatrix} \cos^2 54^\circ & \cos^2 36^\circ & \cot 135^\circ \\ \sin^2 53^\circ & \cot 135^\circ & \sin^2 37^\circ \\ \cot 135^\circ & \cos^2 25^\circ & \cos^2 65^\circ \end{vmatrix} \] is equal to

If \( A \) and \( B \) are square matrices of same order and \( A = A^T, B = B^T \), then \( (ABA)^T = \)

If \( \Delta = \begin{vmatrix}1& 2& 3 \\ 2& 3& 5 \\ 3& 6& 12\end{vmatrix} \) and \( \Delta' = \begin{vmatrix}4& 8& 15 \\ 3& 6& 12 \\ 2& 3& 5\end{vmatrix} \), then

The roots of the equation \[ \begin{vmatrix} 1+x & 3 & 5 \\ 2 & 2+x & 5 \\ 2 & 3 & x+4 \end{vmatrix} = 0 \] are

If \[ \begin{vmatrix} 0 & 3 & 2b \\ 2 & 0 & 1 \\ 4 & -1 & 6 \end{vmatrix} \] is singular, then the value of \(b\) is

If \( m_1 \) and \( m_2 \) satisfy the relation \( {}^{m+5}P_{m+1} = \frac{11}{2}(m-1)({}^{m+3}P_m) \), then \( m_1 + m_2 \) is

The number of 5-digit numbers (no digit is repeated) that can be formed by using the digits \(0,1,2,\ldots,7\) is

The value of \( x \) satisfying the relation \( 11\binom{x}{3} = 24\binom{x+1}{2} \) is

If \( a, b, c \) are in A.P. and their squares in same order form a G.P., then \( (a+c)^4 = \)

If 4th term of a G.P. is 32 whose common ratio is half of the first term, then the 15th term is

If the roots of the equation \( (x-a)(x-b) + (x-b)(x-c) + (x-c)(x-a) = 0 \) are equal, then \( a^2 + b^2 + c^2 = \)

If one root of a quadratic equation is \( \frac{1}{1+\sqrt{3}} \), then the quadratic equation is

An A.P. consists of 23 terms. If the sum of the 3 terms in the middle is 141 and the sum of the last 3 terms is 261, then the first term is

Let \( S(n) \) denote the sum of the digits of a positive integer \(n\). Then the value of \( S(1)+S(2)+\cdots+S(99) \) is

The 5th and 8th terms of a G.P. are 1458 and 54 respectively. The common ratio of the G.P. is

If the difference between the roots of \( x^2 + 2px + q = 0 \) is two times the difference between the roots of \( x^2 + qx + \frac{p}{4} = 0 \), where \( p \neq q \), then

Sum of the roots of the equation \( |x-3|^2 + |x-3| - 2 = 0 \) is equal to