If the system of linear equations :

$\begin{aligned} & x+y+2 z=6 \\ & 2 x+3 y+\mathrm{az}=\mathrm{a}+1 \\ & -x-3 y+\mathrm{b} z=2 \mathrm{~b} \end{aligned}$

where $a, b \in \mathbf{R}$, has infinitely many solutions, then $7 a+3 b$ is equal to :

1

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Topic: JEE Main 2025 (22 January Evening Shift)

2

Let \alpha_\theta and \beta_\theta be the distinct roots of $2x^2+(\cos\theta)x-1=0$, $\theta\in(0,2\pi)$. If $m$ and $…

Topic: JEE Main 2025 (22 January Evening Shift)

1

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Topic: JEE Main 2025 (22 January Evening Shift)

2

Let $\alpha, \beta, \gamma$ and $\delta$ be the coefficients of $x^7, x^5, x^3$ and $x$ respectively in the expansion o…

Topic: JEE Main 2025 (22 January Evening Shift)

3

If $A$ and $B$ are two events such that $P(A \cap B)=0.1$, and $P(A \mid B)$ and $P(B \mid A)$ are the roots of the equ…

Topic: JEE Main 2025 (22 January Evening Shift)

4

The perpendicular distance of the line $\dfrac{x-1}{2}=\dfrac{y+2}{-1}=\dfrac{z+3}{2}$ from the point $P(2,-10,1)$ is:

Topic: JEE Main 2025 (22 January Evening Shift)

5

If $x=f(y)$ is the solution of the differential equation $(1+y^{2})+\big(x-2e^{\tan^{-1}y}\big)\dfrac{dy}{dx}=0,\ y\in\…

Topic: JEE Main 2025 (22 January Evening Shift)

6

For a $3\times3$ matrix $M$, let $\operatorname{trace}(M)$ denote the sum of all the diagonal elements of $M$. Let $A$ …

Topic: JEE Main 2025 (22 January Evening Shift)

7

If $\displaystyle \int e^{x}\!\left(\frac{x\sin^{-1}x}{\sqrt{1-x^{2}}}+\frac{\sin^{-1}x}{(1-x^{2})^{3/2}}+\frac{x}{1-x^…

Topic: JEE Main 2025 (22 January Evening Shift)

8

Let $\vec{a}$ and $\vec{b}$ be two unit vectors such that the angle between them is $\frac{\pi}{3}$. If $\lambda \vec{a…

Topic: JEE Main 2025 (22 January Evening Shift)

9

The area of the region enclosed by the curves $y=x^2-4 x+4$ and $y^2=16-8 x$ is :

Topic: JEE Main 2025 (22 January Evening Shift)

10

In a group of 3 girls and 4 boys, there are two boys $B_1$ and $B_2$. The number of ways in which these girls and boys …

Topic: JEE Main 2025 (22 January Evening Shift)