JEE Main 2022 (Online) 28th July Morning Shift

The foot of the perpendicular from a point on the circle \(x^{2}+y^{2}=1, z=0\) to the plane \(2 x+3 y+z=6\) lies on which one of the following curves?

Let \(\mathrm{P}(-2,-1,1)\) and \(\mathrm{Q}\left(\frac{56}{17}, \frac{43}{17}, \frac{111}{17}\right)\) be the vertices of the rhombus PRQS. If the direction ratios of the diagonal RS are \(\alpha,-1, \beta\), where both \(\alpha\) and $$\b...

If the minimum value of \(f(x)=\frac{5 x^{2}}{2}+\frac{\alpha}{x^{5}}, x>0\), is 14 , then the value of \(\alpha\) is equal to :

The remainder when \(7^{2022}+3^{2022}\) is divided by 5 is :

For \(\mathrm{t} \in(0,2 \pi)\), if \(\mathrm{ABC}\) is an equilateral triangle with vertices \(\mathrm{A}(\sin t,-\cos \mathrm{t}), \mathrm{B}(\operatorname{cost}, \sin t)\) and \(C(a, b)\) such that its orthocentre lies on a circle with c...

Let \(C\) be the centre of the circle \(x^{2}+y^{2}-x+2 y=\frac{11}{4}\) and \(P\) be a point on the circle. A line passes through the point \(\mathrm{C}\), makes an angle of \(\frac{\pi}{4}\) with the line \(\mathrm{CP}\) and intersects th...

Let \(S_{1}=\left\{z_{1} \in \mathbf{C}:\left|z_{1}-3\right|=\frac{1}{2}\right\}\) and \(S_{2}=\left\{z_{2} \in \mathbf{C}:\left|z_{2}-\right| z_{2}+1||=\left|z_{2}+\right| z_{2}-1||\right\}\). Then, for \(z_{1} \in S_{1}\) and $$z_{2} \in ...

The minimum value of the twice differentiable function \(f(x)=\int\limits_{0}^{x} \mathrm{e}^{x-\mathrm{t}} f^{\prime}(\mathrm{t}) \mathrm{dt}-\left(x^{2}-x+1\right) \mathrm{e}^{x}\), \(x \in \mathbf{R}\), is :

If \(\int\limits_{0}^{\sqrt{3}} \frac{15 x^{3}}{\sqrt{1+x^{2}+\sqrt{\left(1+x^{2}\right)^{3}}}} \mathrm{~d} x=\alpha \sqrt{2}+\beta \sqrt{3}\), where \(\alpha, \beta\) are integers, then \(\alpha+\beta\) is equal to __________.

Let the solution curve of the differential equation \(x \mathrm{~d} y=\left(\sqrt{x^{2}+y^{2}}+y\right) \mathrm{d} x, x>0\), intersect the line \(x=1\) at \(y=0\) and the line \(x=2\) at \(y=\alpha\). Then the value of \(\alpha\) is :

If \(y=y(x), x \in(0, \pi / 2)\) be the solution curve of the differential equation \(\left(\sin ^{2} 2 x\right) \frac{d y}{d x}+\left(8 \sin ^{2} 2 x+2 \sin 4 x\right) y=2 \mathrm{e}^{-4 x}(2 \sin 2 x+\cos 2 x)\), with $$y(\pi / 4)=\mathrm...

Let \(\alpha, \beta\) and \(\gamma\) be three positive real numbers. Let \(f(x)=\alpha x^{5}+\beta x^{3}+\gamma x, x \in \mathbf{R}\) and \(g: \mathbf{R} \rightarrow \mathbf{R}\) be such that \(g(f(x))=x\) for all \(x \in \mathbf{R}\). If $...

For \(\mathrm{p}, \mathrm{q} \in \mathbf{R}\), consider the real valued function \(f(x)=(x-\mathrm{p})^{2}-\mathrm{q}, x \in \mathbf{R}\) and \(\mathrm{q}>0\). Let \(\mathrm{a}_{1}\), \(\mathrm{a}_{2^{\prime}}\) \(\mathrm{a}_{3}\) and $$\ma...

For the hyperbola \(\mathrm{H}: x^{2}-y^{2}=1\) and the ellipse \(\mathrm{E}: \frac{x^{2}}{\mathrm{a}^{2}}+\frac{y^{2}}{\mathrm{~b}^{2}}=1\), a \(>\mathrm{b}>0\), let the
(1) eccentricity of \(\mathrm{E}\) be reciprocal of the eccentricity ...

Considering only the principal values of the inverse trigonometric functions, the domain of the function \(f(x)=\cos ^{-1}\left(\frac{x^{2}-4 x+2}{x^{2}+3}\right)\) is :

Considering the principal values of the inverse trigonometric functions, the sum of all the solutions of the equation \(\cos ^{-1}(x)-2 \sin ^{-1}(x)=\cos ^{-1}(2 x)\) is equal to :

Let \(f:[0,1] \rightarrow \mathbf{R}\) be a twice differentiable function in \((0,1)\) such that \(f(0)=3\) and \(f(1)=5\). If the line \(y=2 x+3\) intersects the graph of \(f\) at only two distinct points in \((0,1)\), then the least numbe...

\(\lim\limits_{x \rightarrow 0}\left(\frac{(x+2 \cos x)^{3}+2(x+2 \cos x)^{2}+3 \sin (x+2 \cos x)}{(x+2)^{3}+2(x+2)^{2}+3 \sin (x+2)}\right)^{\frac{100}{x}}\) is equal to ___________.

Let the operations \(*, \odot \in\{\wedge, \vee\}\). If \((\mathrm{p} * \mathrm{q}) \odot(\mathrm{p}\, \odot \sim \mathrm{q})\) is a tautology, then the ordered pair \((*, \odot)\) is :

Let the matrix $$A=\left[\begin{array}{lll}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0\end{array}\right]$$ and the matrix \(B_{0}=A^{49}+2 A^{98}\). If \(B_{n}=A d j\left(B_{n-1}\right)\) for all \(n \geq 1\), then $$\operatorname{det}\left(B_{4}\r...

Let $$A=\left[\begin{array}{cc}1 & -1 \\ 2 & \alpha\end{array}\right]$$ and $$B=\left[\begin{array}{cc}\beta & 1 \\ 1 & 0\end{array}\right], \alpha, \beta \in \mathbf{R}$$. Let \(\alpha_{1}\) be the value of \(\alpha\) which satisfies $$(\m...

If the tangents drawn at the points \(\mathrm{P}\) and \(\mathrm{Q}\) on the parabola \(y^{2}=2 x-3\) intersect at the point \(R(0,1)\), then the orthocentre of the triangle \(P Q R\) is :

Let \(S\) be the set of all passwords which are six to eight characters long, where each character is either an alphabet from \(\{A, B, C, D, E\}\) or a number from \(\{1,2,3,4,5\}\) with the repetition of characters allowed. If the number ...

Out of \(60 \%\) female and \(40 \%\) male candidates appearing in an exam, \(60 \%\) candidates qualify it. The number of females qualifying the exam is twice the number of males qualifying it. A candidate is randomly chosen from the quali...

The sum of all real values of \(x\) for which \(\frac{3 x^{2}-9 x+17}{x^{2}+3 x+10}=\frac{5 x^{2}-7 x+19}{3 x^{2}+5 x+12}\) is equal to __________.

Consider the sequence \(a_{1}, a_{2}, a_{3}, \ldots\) such that \(a_{1}=1, a_{2}=2\) and \(a_{n+2}=\frac{2}{a_{n+1}}+a_{n}\) for \(\mathrm{n}=1,2,3, \ldots .\) If $$\left(\frac{\mathrm{a}_{1}+\frac{1}{\mathrm{a}_{2}}}{\mathrm{a}_{3}}\right)...

For \(\alpha \in \mathbf{N}\), consider a relation \(\mathrm{R}\) on \(\mathbf{N}\) given by \(\mathrm{R}=\{(x, y): 3 x+\alpha y\) is a multiple of 7\(\}\). The relation \(R\) is an equivalence relation if and only if :

Let \(x_{1}, x_{2}, x_{3}, \ldots, x_{20}\) be in geometric progression with \(x_{1}=3\) and the common ratio \(\frac{1}{2}\). A new data is constructed replacing each \(x_{i}\) by \(\left(x_{i}-i\right)^{2}\). If \(\bar{x}\) is the mean of...

Let the vectors \(\vec{a}=(1+t) \hat{i}+(1-t) \hat{j}+\hat{k}, \vec{b}=(1-t) \hat{i}+(1+t) \hat{j}+2 \hat{k}\) and \(\vec{c}=t \hat{i}-t \hat{j}+\hat{k}, t \in \mathbf{R}\) be such that for $$\alpha, \beta, \gamma \in \mathbf{R}, \alpha \ve...

Let a vector \(\vec{a}\) has magnitude 9. Let a vector \(\vec{b}\) be such that for every \((x, y) \in \mathbf{R} \times \mathbf{R}-\{(0,0)\}\), the vector \((x \vec{a}+y \vec{b})\) is perpendicular to the vector $$(6 y \vec{a}-18 x \vec{b}...