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 $M$ are the minimum and the maximum values of $\alpha_\theta^{4}+\beta_\theta^{4}$, then $16(M+m)$ equals:

1

Let a line pass through two distinct points $P(-2,-1,3)$ and $Q$, and be parallel to the vector $3\hat i+2\hat j+2\hat …

Topic: JEE Main 2025 (22 January Evening Shift)

2

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+\mat…

Topic: JEE Main 2025 (22 January Evening Shift)

3

The sum of all values of $\theta \in [0,2\pi]$ satisfying $2\sin^2\theta=\cos 2\theta$ and $2\cos^2\theta=3\sin\theta$ …

Topic: JEE Main 2025 (22 January Evening Shift)

4

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)

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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)

6

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)

7

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)

8

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)

9

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)

10

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)