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

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

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

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

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

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

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 \(x_{1} + x_{2} =\)

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

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

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

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 =

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

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

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

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

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

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

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

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

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