$\text { Let } A=\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & \alpha & \beta \\ 0 & \beta & \alpha \end{array}\right] \text { and }|2 \mathrm{~A}|^3=2^{21} \text { where } \alpha, \beta \in Z \text {, Then a value of } \alpha \text { is }$

1

In an A.P., the sixth term $a_6 = 2$. If the product $a_1 a_4 a_9$ is the greatest, then the common difference of the A…

Topic: JEE Main 2025 (29 January Morning Shift)

2

A fair die is thrown until $2$ appears. Then, the probability that $2$ appears in an even number of throws is:

Topic: JEE Main 2025 (29 January Morning Shift)

3

If in a G.P. of $64$ terms, the sum of all the terms is $7$ times the sum of the odd terms of the G.P., then the common…

Topic: JEE Main 2025 (29 January Morning Shift)

4

If $\alpha,\ -\tfrac{\pi}{2} < \alpha < \tfrac{\pi}{2}$ is the solution of $4\cos\theta + 5\sin\theta = 1$, then the va…

Topic: JEE Main 2025 (29 January Morning Shift)

5

Let $(5, \tfrac{a}{4})$ be the circumcenter of a triangle with vertices $A(a, -2)$, $B(a, 6)$ and $C\left(\tfrac{a}{4},…

Topic: JEE Main 2025 (29 January Morning Shift)

6

Let $\vec a,\vec b,\vec c$ be three non-zero vectors such that $\vec b$ and $\vec c$ are non-collinear. If $\ \vec a+5\…

Topic: JEE Main 2025 (29 January Morning Shift)

7

Suppose$ \displaystyle f(x)=\frac{(x^{2}+2-x),\tan x;\sqrt{\tan^{-1}!\left(\frac{x^{2}-x+1}{x}\right)}}{(7x^{2}+3x+1)^{…

Topic: JEE Main 2025 (29 January Morning Shift)

1

Let $R$ be a relation on $\mathbb{Z} \times \mathbb{Z}$ defined by $(a,b) R (c,d)$ if and only if $ad - bc$ is divisibl…

Topic: JEE Main 2025 (29 January Morning Shift)

2

If $f(x)=\left\{\begin{array}{cc}2+2 x, & -1 \leq x < 0 \\ 1-\frac{x}{3}, & 0 \leq x \leq 3\end{array} ; g(x)=\left\{\b…

Topic: JEE Main 2025 (29 January Morning Shift)

3

Let $O$ be the origin and the position vectors of $A$ and $B$ be $\vec{A} = 2\hat{i} + 2\hat{j} + \hat{k}$ and $\vec{B}…

Topic: JEE Main 2025 (29 January Morning Shift)

4

In a $\triangle ABC$, suppose $y = x$ is the equation of the bisector of the angle $B$ and the equation of the side $AC…

Topic: JEE Main 2025 (29 January Morning Shift)

5

For $x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, if $y(x)=\int \frac{\operatorname{cosec} x+\sin x}{\operatorname…

Topic: JEE Main 2025 (29 January Morning Shift)

6

Let $\mathrm{A}$ be a square matrix such that $\mathrm{AA}^{\mathrm{T}}=\mathrm{I}$. Then $\frac{1}{2} A\left[\left(A+A…

Topic: JEE Main 2025 (29 January Morning Shift)

7

If $z = \dfrac{1}{2} - 2i$ is such that $|z + 1| = \alpha z + \beta (1 + i)$, $i = \sqrt{-1}$ and $\alpha, \beta \in \m…

Topic: JEE Main 2025 (29 January Morning Shift)

8

Consider the function $f:\left[\frac{1}{2}, 1\right] \rightarrow \mathbb{R}$ defined by $f(x)=4 \sqrt{2} x^3-3 \sqrt{2}…

Topic: JEE Main 2025 (29 January Morning Shift)

9

Let $P Q R$ be a triangle with $R(-1,4,2)$. Suppose $M(2,1,2)$ is the mid point of $\mathrm{PQ}$. The distance of the c…

Topic: JEE Main 2025 (29 January Morning Shift)

10

$\mathop {\lim }\limits_{x \to {\pi \over 2}} \left( {{1 \over {{{\left( {x - {\pi \over 2}} \right)}^2}}}\int\limits…

Topic: JEE Main 2025 (29 January Morning Shift)