The slope of the tangent to a curve $C: y=y(x)$ at any point $(x, y)$ on it is $\dfrac{2e^{2x}-6e^{-x}+9}{2+9e^{-2x}}$. If $C$ passes through the points $\left(0, \tfrac{1}{2}+\tfrac{\pi}{2\sqrt{2}}\right)$ and $\left(\alpha, \tfrac{1}{2}e^{2\alpha}\right)$, then $e^{\alpha}$ is equal to :
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For any real number $x$, let $[x]$ denote the largest integer less than equal to $x$. Let $f$ be a real valued function…
Topic: JEE Main 2022 (25 July Morning Shift)
The area of the region given by$A=\left\{(x, y): x^{2} \leq y \leq \min \{x+2,4-3 x\}\right\}$ is :
Topic: JEE Main 2022 (25 July Morning Shift)
The curve $y(x)=a x^{3}+b x^{2}+c x+5$ touches the $x$-axis at the point $\mathrm{P}(-2,0)$ and cuts the $y$-axis at th…
Topic: JEE Main 2022 (25 July Morning Shift)
If the absolute maximum value of the function $f(x)=\left(x^{2}-2 x+7\right) \mathrm{e}^{\left(4 x^{3}-12 x^{2}-180 x+3…
Topic: JEE Main 2022 (25 July Morning Shift)
If $\mathop {\lim }\limits_{n \to \infty } \left( {\sqrt {{n^2} - n - 1} + n\alpha + \beta } \right) = 0$, then $8(\a…
Topic: JEE Main 2022 (25 July Morning Shift)
The number of $\theta \in(0,4 \pi)$ for which the system of linear equations$\begin{aligned}&3(\sin 3 \theta) x-y+z=2 \…
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The general solution of the differential equation
$(x - y^2),dx + y(5x + y^2),dy = 0$ is:
Topic: JEE Main 2022 (25 July Morning Shift)
Let the locus of the centre $(\alpha,\beta)$, $\beta>0$, of the circle which touches the circle $x^2+(y-1)^2=1$ externa…
Topic: JEE Main 2022 (25 July Morning Shift)
Let $ABC$ be a triangle such that $\overrightarrow{BC}=\vec a,\ \overrightarrow{CA}=\vec b,\ \overrightarrow{AB}=\vec c…
Topic: JEE Main 2022 (25 July Morning Shift)
If the numbers appeared on the two throws of a fair six faced die are $\alpha$ and $\beta$, then the probability that $…
Topic: JEE Main 2022 (25 July Morning Shift)
