For some $a, b,$ let $f(x)=\left|\begin{array}{ccc}\mathrm{a}+\frac{\sin x}{x} & 1 & \mathrm{~b} \\ \mathrm{a} & 1+\frac{\sin x}{x} & \mathrm{~b} \\ \mathrm{a} & 1 & \mathrm{~b}+\frac{\sin x}{x}\end{array}\right|, x \neq 0, \lim \limits_{x \rightarrow 0} f(x)=\lambda+\mu \mathrm{a}+\nu \mathrm{b}.$ Then $(\lambda+\mu+v)^2$ is equal to :
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Let $A={,x\in(0,\pi)-{\tfrac{\pi}{2}}: \log_{(2/\pi)}|\sin x|+\log_{(2/\pi)}|\cos x|=2,}$ and $B={,x\ge 0:\sqrt{x}(\sqr…
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Let $(2,3)$ be the largest open interval in which the function $f(x)=2\log_e(x-2)-x^2+ax+1$ is strictly increasing and …
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Let $A = [a_{ij}]$ be a square matrix of order $2$ with entries either $0$ or $1$. Let $E$ be the event that $A$ is an …
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In an arithmetic progression, if $S_{40} = 1030$ and $S_{12} = 57$, then $S_{30} - S_{10}$ is equal to
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The area of the region enclosed by the curves $y=\mathrm{e}^x, y=\left|\mathrm{e}^x-1\right|$ and $y$-axis is :
Topic: JEE Main 2025 (24 January Evening Shift)
If $\alpha>\beta>\gamma>0$, then the expression $\cot ^{-1}\left\{\beta+\frac{\left(1+\beta^2\right)}{(\alpha-\beta)}\r…
Topic: JEE Main 2025 (24 January Evening Shift)
If the system of equations
$x+2y-3z=2$,
$2x+\lambda y+5z=5$,
$14x+3y+\mu z=33$
has infinitely many solutions, then $\la…
Topic: JEE Main 2025 (24 January Evening Shift)
Suppose $A$ and $B$ are the coefficients of $30^{\text{th}}$ and $12^{\text{th}}$ terms respectively in the binomial ex…
Topic: JEE Main 2025 (24 January Evening Shift)
Let the position vectors of three vertices of a triangle be $4\vec p+\vec q-3\vec r$, $-5\vec p+\vec q+2\vec r$ and $2\…
Topic: JEE Main 2025 (24 January Evening Shift)
If $7 = 5 + \frac{1}{7}(5+\alpha) + \frac{1}{7^2}(5+2\alpha) + \frac{1}{7^3}(5+3\alpha) + \cdots$, then the value of $\…
Topic: JEE Main 2025 (24 January Evening Shift)
If the equation of the parabola with vertex $V!\left(\frac{3}{2},,3\right)$ and directrix $x+2y=0$ is $\alpha x^2+\beta…
Topic: JEE Main 2025 (24 January Evening Shift)
Let $\vec a = 3\hat{i}-\hat{j}+2\hat{k}$, $\vec b=\vec a \times (\hat{i}-2\hat{k})$ and $\vec c=\vec b \times \hat{k}$.…
Topic: JEE Main 2025 (24 January Evening Shift)
The equation of the chord of the ellipse $\dfrac{x^2}{25}+\dfrac{y^2}{16}=1$, whose mid-point is $(3,1)$, is:
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Let the points $\left(\dfrac{11}{2},,\alpha\right)$ lie on or inside the triangle with sides $x+y=11$, $x+2y=16$ and $2…
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