List of top Questions asked in KEAM- 2026

If $\cos^{-1}(x - 2) = \sin^{-1}(y + 1)$, then the variables $x$ and $y$ satisfy the equation}

If $\alpha$ and $\beta$ are real constants such that $\alpha - \beta = \frac{\pi}{4}$, then the value of $(\sin \alpha + \sin \beta)^2 + (\cos \alpha + \cos \beta)^2$ is equal to

If $(3 \cos x - 2 \sec x)^2 = 9 \cos^2 x + 4 \tan^2 x + k$, where $k$ is a constant, then the value of $k$ is equal to

If $\tan \alpha = \frac{1}{2}$, then the value of $\tan^2(2\alpha) \sec^2(2\alpha)$ is equal to

The value of the product $\cot 10^\circ \cot 20^\circ \cot 30^\circ \cot 45^\circ \cot 60^\circ \cot 70^\circ \cot 80^\circ$ is equal to

In a right-angled trapezium $ABCD$, $\angle A = 90^\circ$, $\angle D = 90^\circ$, $AB = 7a + 1$, $CD = 3a + 1$ and $AD = 3a$. If the perimeter of the trapezium is greater than 56 but less than 92, then the range of possible values of $a$ is

The solution set of the inequality $6(2x + 3) + x > 53 - 2x$ is

Let $A$ be a non-singular square matrix of order 3. If $A^2 - A = 20I$, where $I$ is the unit matrix of order 3, then $A^{-1} =$}

If the coefficient of $x^3$ in the binomial expansion of $(2 + x)^n$ is 160, then the coefficient of $x^6$ in the binomial expansion of $(2 - x^2)^n$ is

The number of arrangements of the letters of the word BANANA so that the arrangement starts and ends with the same letter, is

Let $(2 - x)^9 = a_0 + a_1x + a_2x^2 + \dots + a_9x^9$. Then the value of $a_1 + a_2 + a_3 + \dots + a_8$ is equal to

There are 3 boys and 4 girls in a group. The number of ways they can sit in a row so that between any two boys there is a girl and between any two girls there is a boy, is

Let $S$ be the set of all 3-digit numbers containing the digits 3, 5 and 7 without repetition. The sum of all numbers in $S$ is

The A.M. and G.M. of two positive real numbers $a$ and $b$ ($a > b$) are $A$ and $G$ respectively. If $A:G = 5:3$, then $(a^2 + b^2) : ab = $

If $\cos \theta, \sqrt{2}\sin \theta$ and $\sqrt{3}\tan \theta$, where $0 < \theta < \frac{\pi}{2}$, are the second, third and fourth terms of a geometric series, respectively, then the first term of the geometric series is

The sum of four consecutive terms in a geometric progression is 960. If the fourth term is 8 times as large as the first term, then the smallest number in the geometric progression is

The sum of the first two terms of a geometric series is 12 and the third term is 16. Then the common ratio $r > 0$ of the geometric progression is

If $\alpha$ is a real number satisfying $\alpha^2 - \frac{1}{\alpha^2} = 2$, then the value of $(\alpha + \frac{i}{\alpha})^{16}$ is equal to

The complex number $z$ satisfying the equation $\frac{Re(z)}{2+i} + \frac{Im(z)}{1+2i} = \frac{3}{1-2i}$, is

Let $z = 1 + i$, where $i = \sqrt{-1}$. If $z - \frac{24\bar{z}}{z^2} = \lambda z$, then the value of $\lambda$ is equal to

If the complex number $z = x + iy$ satisfies the equation $5z - 2\bar{z} = \frac{7 - 7i}{1 + i}$, then the value of $x + y$ is equal to

Let $f(x) = x^2 - 10x$ and $g(x) = e^x + 5$ for $x \in \mathbb{R}$. Then, for all $x$, $g(2x) - (f \circ g)(x) = $}

Let $f(x) = \frac{10}{7 + 4\sin x + 3\cos x}, x \in \mathbb{R}$. Then the range of the function $f$ is

Let $f(x) = \sin^{-1}x$ and $g(x) = x - 2$. To define the composite function $f \circ g$, the largest domain of $g(x)$ has to be}

Let $A$ and $B$ be two subsets of a universal set $U$. If $n(U) = 116$, $n(A \cup B) = 99$, $n(B) = 61$ and $n(A \cap B) = 28$, then $n(A')$ is equal to