List of top Questions asked in KEAM- 2025

If \(\cos\left(2\sin^{-1}\alpha\right) = \frac{47}{72}\), where \(0 < \alpha < 1\), then the value of \(\alpha\) is

If \(x \neq 11\) satisfies the inequality \(\frac{2x - 21}{x - 11} \geq 3\), then \(x\) lies in the interval

\(\frac{3\tan 15^{\circ} - \tan^3 15^{\circ}}{1 - 3\tan^2 15^{\circ}}\)

The set of all \(x\) satisfying the inequality \(|3 - 4x| \leq 11\) is

The numbers \(a_{1},a_{2},a_{3},a_{4},a_{5}\) and \(a_{6}\) are in G.P. If \(a_{1} = 2\) and the common ratio \(r = \frac{1}{2}\), then the value of \(\left| \begin{array}{lll}a_{1} & a_{2} & 1 a_{3} & a_{4} & 1 a_{5} & a_{6} & 1 \end{array} \right|\) is equal to

Let \(A = \begin{bmatrix} a & -1 & -a 0 & 1 & -1 1 & 0 & 4 \end{bmatrix}\). If \(|A| = 26\), then the value of \(a\) is equal to

If the matrix \(A = \left[ \begin{array}{cc}1 & -1 4\lambda & 8 \end{array} \right]\) is singular, then the value of \(\lambda\) is equal to

If \(\sum_{k = 0}^{n + 1} \binom{n+1}{k} = 512\) , then \(\sum_{k = 0}^{n} \binom{n}{k} =\)

\(\frac{8}{4}\left(^{7}P_{4}\right)\) equals

If \(X = A^{-1}B,\) Where \(A = \left[ \begin{array}{cc}1 & -1 2 & 1 \end{array} \right],B = \left[ \begin{array}{c}3 6 \end{array} \right]\) and \(X = \left[ \begin{array}{c}x_{1} x_{2} \end{array} \right],\) then \(x_{1} + x_{2} =\)

Let S be the set of all 5-digit numbers having only the digits 0 and 1. Then \(n(S) =\)

Let \(A = \{a,b,c,d,e,f\}\) Then the number of subsets of \(A\) with an odd number of elements is

The constant term in the binomial expansion of \(\left(2x - \frac{5}{x^2}\right)^6\) is

The positive numbers \(\alpha\) and \(\beta\) have geometric mean 6. If \(\alpha\) and \(\beta\) are roots of the equation \(2x^{2} - 25x + \lambda = 0\) then the value of \(\lambda\) is equal to

Three numbers a, b, and c are in G.P. If abc = 27 and a + c = 10, then a² + b² + c² =

In a class there are $n$ students. The mean of marks obtained by these $n$ students in an exam is 65. If the mark of one student is increased from 50 to 75 and the new mean is 66, then the value of $n$ is equal to

Let $S = \{2,\,5,\,8,\,11,\,14,\,17,\,20,\,23\}$. Two integers $m,\,n$ are chosen one by one from $S$ with replacement. Then the probability that $mn$ is odd, is

Let $X = \{a,\,b\}$ and $Y = \{1,\,3,\,4,\,5\}$. A subset of $X\times Y$ is selected at random. If $A$ is an event of selecting a subset of $X\times Y$ containing exactly three elements, then $P(A) =$

A box contains 8 red balls, 10 white balls and 17 black balls. Two balls are drawn one by one without replacement. The probability that the first ball drawn is white and the second ball drawn is black, is

If the line $\dfrac{x+1}{4} = \dfrac{y+2}{-3} = \dfrac{z-\alpha}{-2}$ passes through the point $(-1,\,-2,\,-3)$, then the value of $\alpha$ is

If the lines $\dfrac{x-4}{m} = \dfrac{y-3}{2} = \dfrac{z+2}{1}$ and $\dfrac{x-3}{1} = \dfrac{y-4}{1} = \dfrac{z+3}{m}$ are coplanar, then the values of $m$ are

If $|\vec{a}| = 3$, $|\vec{b}| = 2$, then $(2\vec{a} + 3\vec{b})\cdot(2\vec{a} - 3\vec{b})$ is equal to

The line $\dfrac{x+1}{2} = \dfrac{y-4}{4} = \dfrac{z-2}{5}$ passes through the point

The angle between the lines $\dfrac{x-3}{-4} = \dfrac{y+2}{3} = \dfrac{z-1}{5}$ and $\dfrac{x-2}{2} = \dfrac{y-4}{1} = \dfrac{z+3}{3}$ is

If $|\vec{a}| = 8$, $|\vec{b}| = 5$ and $|\vec{a} - \vec{b}| = 7$, then the angle between $\vec{a}$ and $\vec{b}$ is equal to