JEE Main 2023 (Online) 10th April Evening Shift

Let the image of the point \(\mathrm{P}(1,2,6)\) in the plane passing through the points \(\mathrm{A}(1,2,0), \mathrm{B}(1,4,1)\) and \(\mathrm{C}(0,5,1)\) be \(\mathrm{Q}(\alpha, \beta, \gamma)\). Then $$\left(\alpha^{2}+\beta^{2}+\gamma^{...

Let the line \(\frac{x}{1}=\frac{6-y}{2}=\frac{z+8}{5}\) intersect the lines \(\frac{x-5}{4}=\frac{y-7}{3}=\frac{z+2}{1}\) and \(\frac{x+3}{6}=\frac{3-y}{3}=\frac{z-6}{1}\) at the points \(\mathrm{A}\) and \(\mathrm{B}\) respectively. Then ...

Let the foot of perpendicular from the point \(\mathrm{A}(4,3,1)\) on the plane \(\mathrm{P}: x-y+2 z+3=0\) be N. If B\((5, \alpha, \beta), \alpha, \beta \in \mathbb{Z}\) is a point on plane P such that the area of the triangle ABN is $$3 \...

Let \(\mathrm{g}(x)=f(x)+f(1-x)\) and \(f^{\prime \prime}(x) > 0, x \in(0,1)\). If \(\mathrm{g}\) is decreasing in the interval \((0, a)\) and increasing in the interval \((\alpha, 1)\), then $$\tan ^{-1}(2 \alpha)+\tan ^{-1}\left(\frac{1}{...

Let the quadratic curve passing through the point \((-1,0)\) and touching the line \(y=x\) at \((1,1)\) be \(y=f(x)\). Then the \(x\)-intercept of the normal to the curve at the point \((\alpha, \alpha+1)\) in the first quadrant is ________...

If the area of the region \(\left\{(x, \mathrm{y}):\left|x^{2}-2\right| \leq y \leq x\right\}\) is \(\mathrm{A}\), then \(6 \mathrm{A}+16 \sqrt{2}\) is equal to __________.

Let the number \((22)^{2022}+(2022)^{22}\) leave the remainder \(\alpha\) when divided by 3 and \(\beta\) when divided by 7. Then \(\left(\alpha^{2}+\beta^{2}\right)\) is equal to :

If the coefficients of \(x\) and \(x^{2}\) in \((1+x)^{\mathrm{p}}(1-x)^{\mathrm{q}}\) are 4 and \(-\)5 respectively, then \(2 p+3 q\) is equal to :

Let A be the point \((1,2)\) and B be any point on the curve \(x^{2}+y^{2}=16\). If the centre of the locus of the point P, which divides the line segment \(\mathrm{AB}\) in the ratio \(3: 2\) is the point C\((\alpha, \beta)\), then the len...

Let \(S = \left\{ {z = x + iy:{{2z - 3i} \over {4z + 2i}}\,\mathrm{is\,a\,real\,number}} \right\}\). Then which of the following is NOT correct?

Let \(f\) be a continuous function satisfying \(\int_\limits{0}^{t^{2}}\left(f(x)+x^{2}\right) d x=\frac{4}{3} t^{3}, \forall t > 0\). Then

\(f\left(\frac{\pi^{2}}{4}\right)\) is equal to :

Let the tangent at any point P on a curve passing through the points (1, 1) and \(\left(\frac{1}{10}, 100\right)\), intersect positive \(x\)-axis and \(y\)-axis at the points A and B respectively. If \(\mathrm{PA}: \mathrm{PB}=1: k\) and $$...

Let a circle of radius 4 be concentric to the ellipse \(15 x^{2}+19 y^{2}=285\). Then the common tangents are inclined to the minor axis of the ellipse at the angle :

For \(\alpha, \beta, \gamma, \delta \in \mathbb{N}\), if $$\int\left(\left(\frac{x}{e}\right)^{2 x}+\left(\frac{e}{x}\right)^{2 x}\right) \log _{e} x d x=\frac{1}{\alpha}\left(\frac{x}{e}\right)^{\beta x}-\frac{1}{\gamma}\left(\frac{e}{x}\r...

If the domain of the function \(f(x)=\sec ^{-1}\left(\frac{2 x}{5 x+3}\right)\) is \([\alpha, \beta) \mathrm{U}(\gamma, \delta]\), then \(|3 \alpha+10(\beta+\gamma)+21 \delta|\) is equal to _________.

The statement \(\sim[p \vee(\sim(p \wedge q))]\) is equivalent to :

If $$\mathrm{A}=\frac{1}{5 ! 6 ! 7 !}\left[\begin{array}{ccc}5 ! & 6 ! & 7 ! \\ 6 ! & 7 ! & 8 ! \\ 7 ! & 8 ! & 9 !\end{array}\right]$$, then \(|\operatorname{adj}(\operatorname{adj}(2 \mathrm{~A}))|\) is equal to :

Let \(\mathrm{S}\) be the set of values of \(\lambda\), for which the system of equations \(6 \lambda x-3 y+3 z=4 \lambda^{2}\), \(2 x+6 \lambda y+4 z=1\), \(3 x+2 y+3 \lambda z=\lambda\) has no solution. Then $$12 \sum_\limits{i \in S}|\la...

Eight persons are to be transported from city A to city B in three cars of different makes. If each car can accommodate at most three persons, then the number of ways, in which they can be transported, is :

The sum of all the four-digit numbers that can be formed using all the digits 2, 1, 2, 3 is equal to __________.

Let a die be rolled \(n\) times. Let the probability of getting odd numbers seven times be equal to the probability of getting odd numbers nine times. If the probability of getting even numbers twice is \(\frac{k}{2^{15}}\), then $$\mathrm...

In the figure, \(\theta_{1}+\theta_{2}=\frac{\pi}{2}\) and \(\sqrt{3}(\mathrm{BE})=4(\mathrm{AB})\). If the area of \(\triangle \mathrm{CAB}\) is \(2 \sqrt{3}-3\) unit \({ }^{2}\), when \(\frac{\theta_{2}}{\theta_{1}}\) is the largest, then...

If \(\mathrm{S}_{n}=4+11+21+34+50+\ldots\) to \(n\) terms, then \(\frac{1}{60}\left(\mathrm{~S}_{29}-\mathrm{S}_{9}\right)\) is equal to :

Suppose \(a_{1}, a_{2}, 2, a_{3}, a_{4}\) be in an arithmetico-geometric progression. If the common ratio of the corresponding geometric progression is 2 and the sum of all 5 terms of the arithmetico-geometric progression is $$\frac{49}{2}$...

Let \(\mathrm{A}=\{2,3,4\}\) and \(\mathrm{B}=\{8,9,12\}\). Then the number of elements in the relation

$$\mathrm{R}=\left\{\left(\left(a_{1}, \mathrm{~b}_{1}\right),\left(a_{2}, \mathrm{~b}_{2}\right)\right) \in(A \times B, A \times B): a...

Let \(\mu\) be the mean and \(\sigma\) be the standard deviation of the distribution

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Let the equations of two adjacent sides of a parallelogram \(\mathrm{ABCD}\) be \(2 x-3 y=-23\) and \(5 x+4 y=23\). If the equation of its one diagonal \(\mathrm{AC}\) is \(3 x+7 y=23\) and the distance of A from the other diagonal is $$\ma...

Let \(S=\left\{x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right): 9^{1-\tan ^{2} x}+9^{\tan ^{2} x}=10\right\}\) and

\(\beta=\sum_\limits{x \in S} \tan ^{2}\left(\frac{x}{3}\right)\), then \(\frac{1}{6}(\beta-14)^{2}\) is equal to :

Let \(\vec{a}=2 \hat{i}+7 \hat{j}-\hat{k}, \vec{b}=3 \hat{i}+5 \hat{k}\) and \(\vec{c}=\hat{i}-\hat{j}+2 \hat{k}\). Let \(\vec{d}\) be a vector which is perpendicular to both \(\vec{a}\) and \(\vec{b}\), and \(\vec{c} \cdot \vec{d}=12\). Th...

If the points \(\mathrm{P}\) and \(\mathrm{Q}\) are respectively the circumcenter and the orthocentre of a \(\triangle \mathrm{ABC}\), then \(\overrightarrow{\mathrm{PA}}+\overrightarrow{\mathrm{PB}}+\overrightarrow{\mathrm{PC}}\) is
equal ...