Let $\alpha, \beta, \gamma$ and $\delta$ be the coefficients of $x^7, x^5, x^3$ and $x$ respectively in the expansion of $\begin{aligned} & \left(x+\sqrt{x^3-1}\right)^5+\left(x-\sqrt{x^3-1}\right)^5, x>1 \text {. If } u \text { and } v \text { satisfy the equations } \\\\ & \alpha u+\beta v=18, \\\\ & \gamma u+\delta v=20, \end{aligned}$ then $\mathrm{u+v}$ equals :
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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)
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)
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)
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)
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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)
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)
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)
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)
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)
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)
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)
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)
If $\displaystyle \lim_{x\to\infty}\left(\frac{e}{1-e}\left(\frac{1}{e}-\frac{x}{1+x}\right)\right)^{x}=\alpha$, then t…
Topic: JEE Main 2025 (22 January Evening Shift)
Let $f(x)=\displaystyle \int_{0}^{e^{x^{2}}}\frac{t^{2}-8t+15}{e^{t}}\,dt,\ x\in\mathbb{R}$.
Then the numbers of loca…
Topic: JEE Main 2025 (22 January Evening Shift)
