List of top Questions asked in KEAM- 2025

$\tan(315^{\circ}) \cot(-405^{\circ}) = $ ________.

If $\alpha+\beta+\gamma=2\pi$, then $\tan\frac{\alpha}{2}+\tan\frac{\beta}{2}+\tan\frac{\gamma}{2} = $ ________.

$\cos 75^{\circ} \cos 45^{\circ} \cos 15^{\circ} = $ ________.

Let $x$ be a real number such that $x+\frac{x}{4}+\frac{x}{3}<13$. Then the solution set is ________.

Let $A$ be a square matrix of order 3 and $|A|=9$. Then $|adj(adj A)|=$ ________.

Let $A$ be $3 \times 3$, $B$ be $3 \times 2$ and $C$ be $3 \times 1$. Which one of the following products is not defined? ________.

If $n$ is a positive integer and the coefficient of $x$ in the expansion of $(x^{2}+\frac{1}{x^{3}})^{n}$ is $^{n}C_{2}$, then $n$ is equal to ________.

The constant term in the expansion of $(2x^{2}-\frac{1}{x^{2}})^{6}$ is ________.

Five digit number is formed using the digits 0, 1, 2, 3, 4 and 5 without repetitions. Number of five digit numbers which are divisible by 10 is ________.

The coefficient of $x^{9}$ in the expansion of $(4-\frac{x^{2}}{4})^{12}$ is ________.

The number of integers greater than 7000 using 2, 4, 6, 7, 8 without repetition is ________.

Let $\alpha$ and $\beta$ be the roots. If $A$ is the A.M. between $\alpha$ and $\beta$ and also $G$ is the G.M. between $\alpha$ and $\beta$, then $\alpha^{2}+\beta^{2}=$ ________.

Let $a_{1},a_{2},....,a_{n}$ be positive non-zero real numbers. If $a_{1},a_{2},....,a_{n}=k$, then the minimum value of $a_{1}+a_{2}+....+a_{n}$ is ________.

Let $a, \frac{3}{4}, ar^{2}, ar^{3}, \dots$ be in G.P. where $r>0$. If the product of the first four terms is $\frac{3^{6}}{4^{5}}$, then $a$ is equal to ________.

If $a, b, c$ are real numbers such that $(a-2)^{2}+(b-2)^{2}+(c-2)^{2}=0$, then ________.

Let $z=x+iy,$ where $x,y \in \mathbb{R}$ and $i^{2}=-1$. If $|z-i|=|z-1|$, then $y=$ ________.

If $x,y \in \mathbb{R}$ and $x+iy=-(6+i)^{3}, i^{2}=-1$, then $x-y$ is equal to ________.

$\sum_{n=1}^{2025}i^{n}(1+i), i^{2}=-1$ is equal to ________.

All the points in $A=\{\frac{\lambda+i}{\lambda-i}; \lambda \in \mathbb{R}\}$ lie on ________.

The domain of the function $f(x)=\sqrt{x^{2}+2x-15}$ is ________.

The range of the function $f(x)=\log_{e}(4x^{2}-4x+1)$ where $x \ne \frac{1}{2}$ is ________.

Let $A=\{1,2,3,4\}$ and $B=\{7,8,3,4\}$. Then the number of elements common to both $A \times B$ and $B \times A$ is ________.

The relation $R = \{(4,4), (4,5), (5, 7), (4, 8), (5, 5), (7, 8), (7, 7), (7, 5), (8, 8), (8, 7), (8, 5), (9, 9)\}$ on the set $A=\{4,5,7,8,9\}$ is ________.

If $f(x)=x|x|$, then $f^{\prime}(-10)=$ ________.

The solution set for $-12x > 38$, where $x$ is a natural number, is ________.