The area enclosed by the closed curve $\mathcal{C}$ given by the differential equation $\dfrac{dy}{dx}+\dfrac{x+a}{\,y-2\,}=0,\quad y(1)=0$ is $4\pi$. Let $P$ and $Q$ be the points of intersection of the curve $\mathcal{C}$ with the $y$-axis. If the normals at $P$ and $Q$ on $\mathcal{C}$ intersect the $x$-axis at points $R$ and $S$ respectively, then the length of the line segment $RS$ is:
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Let $f(x)=\begin{vmatrix}
1+\sin^{2}x & \cos^{2}x & \sin 2x\\
\sin^{2}x & 1+\cos^{2}x & \sin 2x\\
\sin^{2}x & \cos^{2}x…
Topic: JEE Main 2023 (1 February Morning Shift)
Let $f(x)=2x+\tan^{-1}x$ and $g(x)=\log_{e}\!\big(\sqrt{1+x^{2}}+x\big),\ x\in[0,3]$. Then
Topic: JEE Main 2023 (1 February Morning Shift)
The mean and variance of 5 observations are 5 and 8 respectively. If 3 observations are 1, 3, 5, then the sum of cubes …
Topic: JEE Main 2023 (1 February Morning Shift)
If the center and radius of the circle $\left|\dfrac{z-2}{z-3}\right|=2$ are respectively $(\alpha,\beta)$ and $\gamma$…
Topic: JEE Main 2023 (1 February Morning Shift)
Let $S=\left\{x:\ x\in\mathbb{R}\ \text{and}\ (\sqrt{3}+\sqrt{2})^{\,x^{2}-4}+(\sqrt{3}-\sqrt{2})^{\,x^{2}-4}=10\right\…
Topic: JEE Main 2023 (1 February Morning Shift)
Let $S$ denote the set of all real values of $\lambda$ such that the system of equations
$\lambda x + y + z = 1$
$x…
Topic: JEE Main 2023 (1 February Morning Shift)
The shortest distance between the lines $\dfrac{x-5}{1}=\dfrac{y-2}{2}=\dfrac{z-4}{-3}$ and $\dfrac{x+3}{1}=\dfrac{y+5}…
Topic: JEE Main 2023 (1 February Morning Shift)
Let $R$ be a relation on $\mathbb{R}$, given by $R=\{(a,b):\,3a-3b+\sqrt{7}\text{ is an irrational number}\}$. Then $R$…
Topic: JEE Main 2023 (1 February Morning Shift)
Let $S$ be the set of all solutions of the equation $\cos^{-1}(2x)-2\cos^{-1}\!\big(\sqrt{1-x^{2}}\big)=\pi,\ x\in\left…
Topic: JEE Main 2023 (1 February Morning Shift)
The value of $\displaystyle \frac{1}{1\cdot 50!}+\frac{1}{3\cdot 48!}+\frac{1}{5\cdot 46!}+\cdots+\frac{1}{49\cdot 2!}+…
Topic: JEE Main 2023 (1 February Morning Shift)
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If the orthocentre of the triangle whose vertices are $(1,2)$, $(2,3)$ and $(3,1)$ is $(\alpha,\beta)$, then the quadra…
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If $y=y(x)$ is the solution curve of the differential equation $\dfrac{dy}{dx}+y\tan x=x\sec x,\ 0\le x\le \dfrac{\pi}{…
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