JEE Main 2023 (Online) 11th April Evening Shift

Let the line passing through the points \(\mathrm{P}(2,-1,2)\) and \(\mathrm{Q}(5,3,4)\) meet the plane \(x-y+z=4\) at the point \(\mathrm{R}\). Then the distance of the point \(\mathrm{R}\) from the plane \(x+2 y+3 z+2=0\) measured paralle...

Let P be the plane passing through the points \((5,3,0),(13,3,-2)\) and \((1,6,2)\).

For \(\alpha \in \mathbb{N}\), if the distances of the points \(\mathrm{A}(3,4, \alpha)\) and \(\mathrm{B}(2, \alpha, a)\) from the plane P are 2 and 3 re...

Let the line \(l: x=\frac{1-y}{-2}=\frac{z-3}{\lambda}, \lambda \in \mathbb{R}\) meet the plane \(P: x+2 y+3 z=4\) at the point \((\alpha, \beta, \gamma)\). If the angle between the line \(l\) and the plane \(P\) is $$\cos ^{-1}\left(\sqrt{...

If A is the area in the first quadrant enclosed by the curve \(\mathrm{C: 2 x^{2}-y+1=0}\), the tangent to \(\mathrm{C}\) at the point \((1,3)\) and the line \(\mathrm{x}+\mathrm{y}=1\), then the value of \(60 \mathrm{~A}\) is _________.

The sum of the coefficients of three consecutive terms in the binomial expansion of \((1+\mathrm{x})^{\mathrm{n}+2}\), which are in the ratio \(1: 3: 5\), is equal to :

If the \(1011^{\text {th }}\) term from the end in the binominal expansion of \(\left(\frac{4 x}{5}-\frac{5}{2 x}\right)^{2022}\) is 1024 times \(1011^{\text {th }}\)R term from the beginning, then \(|x|\) is equal to

For \(a \in \mathbb{C}\), let \(\mathrm{A}=\{z \in \mathbb{C}: \operatorname{Re}(a+\bar{z}) > \operatorname{Im}(\bar{a}+z)\}\) and \(\mathrm{B}=\{z \in \mathbb{C}: \operatorname{Re}(a+\bar{z})<\operatorname{Im}(\bar{a}+z)\}\). Then among th...

Let \(\mathrm{S}=\left\{z \in \mathbb{C}-\{i, 2 i\}: \frac{z^{2}+8 i z-15}{z^{2}-3 i z-2} \in \mathbb{R}\right\}\). If \(\alpha-\frac{13}{11} i \in \mathrm{S}, \alpha \in \mathbb{R}-\{0\}\), then

\(242 \alpha^{2}\) is equal to _________.

If \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a continuous function satisfying \(\int_\limits{0}^{\frac{\pi}{2}} f(\sin 2 x) \sin x d x+\alpha \int_\limits{0}^{\frac{\pi}{4}} f(\cos 2 x) \cos x d x=0\), then the value of \(\alpha\) is :

Let the function \(f:[0,2] \rightarrow \mathbb{R}\) be defined as
$$f(x)= \begin{cases}e^{\min \left\{x^{2}, x-[x]\right\},} & x \in[0,1) \\ e^{\left[x-\log _{e} x\right]}, & x \in[1,2]\end{cases}$$
where \([t]\) denotes the greatest intege...

Let \(y=y(x)\) be the solution of the differential equation \(\frac{d y}{d x}+\frac{5}{x\left(x^{5}+1\right)} y=\frac{\left(x^{5}+1\right)^{2}}{x^{7}}, x > 0\). If \(y(1)=2\), then \(y(2)\) is equal to :

If the radius of the largest circle with centre (2,0) inscribed in the ellipse \(x^2+4y^2=36\) is r, then 12r\(^2\) is equal to :

The domain of the function \(f(x)=\frac{1}{\sqrt{[x]^{2}-3[x]-10}}\) is : ( where \([\mathrm{x}]\) denotes the greatest integer less than or equal to \(x\) )

Let \(\mathrm{A}=\{1,2,3,4,5\}\) and \(\mathrm{B}=\{1,2,3,4,5,6\}\). Then the number of functions \(f: \mathrm{A} \rightarrow \mathrm{B}\) satisfying \(f(1)+f(2)=f(4)-1\) is equal to __________.

The angle of elevation of the top \(\mathrm{P}\) of a tower from the feet of one person standing due South of the tower is \(45^{\circ}\) and from the feet of another person standing due west of the tower is \(30^{\circ}\). If the height of...

Let \(f\) and \(g\) be two functions defined by
$$f(x)=\left\{\begin{array}{cc}x+1, & x < 0 \\ |x-1|, & x \geq 0\end{array}\right.$$ and $$\mathrm{g}(x)=\left\{\begin{array}{cc}x+1, & x < 0 \\ 1, & x \geq 0\end{array}\right.$$
Then $$(g \ci...

The converse of \(((\sim p) \wedge q) \Rightarrow r\) is

If the system of linear equations
$$ \begin{aligned} & 7 x+11 y+\alpha z=13 \\\\ & 5 x+4 y+7 z=\beta \\\\ & 175 x+194 y+57 z=361 \end{aligned} $$
has infinitely many solutions, then \(\alpha+\beta+2\) is equal to :

$$\left|\begin{array}{ccc}x+1 & x & x \\ x & x+\lambda & x \\ x & x & x+\lambda^{2}\end{array}\right|=\frac{9}{8}(103 x+81)$$, then \(\lambda, \frac{\lambda}{3}\) are the roots of the equation :

Let the tangent to the parabola \(\mathrm{y}^{2}=12 \mathrm{x}\) at the point \((3, \alpha)\) be perpendicular to the line \(2 x+2 y=3\). Then the square of distance of the point \((6,-4)\) from the normal to the hyperbola $$\alpha^{2} x^{2...

If the letters of the word MATHS are permuted and all possible words so formed are arranged as in a dictionary with serial numbers, then the serial number of the word THAMS is :

Let the probability of getting head for a biased coin be \(\frac{1}{4}\). It is tossed repeatedly until a head appears. Let \(\mathrm{N}\) be the number of tosses required. If the probability that the equation $$64 \mathrm{x}^{2}+5 \mathrm{...

The number of points, where the curve \(f(x)=\mathrm{e}^{8 x}-\mathrm{e}^{6 x}-3 \mathrm{e}^{4 x}-\mathrm{e}^{2 x}+1, x \in \mathbb{R}\) cuts \(x\)-axis, is equal to _________.

Let \(a, b, c\) and \(d\) be positive real numbers such that \(a+b+c+d=11\). If the maximum value of \(a^{5} b^{3} c^{2} d\) is \(3750 \beta\), then the value of \(\beta\) is

For \(k \in \mathbb{N}\), if the sum of the series \(1+\frac{4}{k}+\frac{8}{k^{2}}+\frac{13}{k^{3}}+\frac{19}{k^{4}}+\ldots\) is 10 , then the value of \(k\) is _________.

Let \(\mathrm{A}=\{1,3,4,6,9\}\) and \(\mathrm{B}=\{2,4,5,8,10\}\). Let \(\mathrm{R}\) be a relation defined on \(\mathrm{A} \times \mathrm{B}\) such that $$\mathrm{R}=\left\{\left(\left(a_{1}, b_{1}\right),\left(a_{2}, b_{2}\right)\right):...

Let the mean of 6 observations \(1,2,4,5, \mathrm{x}\) and \(\mathrm{y}\) be 5 and their variance be 10 .

Then their mean deviation about the mean is equal to :

If the line \(l_{1}: 3 y-2 x=3\) is the angular bisector of the lines \(l_{2}: x-y+1=0\) and \(l_{3}: \alpha x+\beta y+17=0\), then \(\alpha^{2}+\beta^{2}-\alpha-\beta\) is equal to _________.

If four distinct points with position vectors \(\vec{a}, \vec{b}, \vec{c}\) and \(\vec{d}\) are coplanar, then \([\vec{a} \,\,\vec{b} \,\,\vec{c}]\) is equal to :

Let \(\vec{a}=\hat{i}+2 \hat{j}+3 \hat{k}\) and \(\vec{b}=\hat{i}+\hat{j}-\hat{k}\). If \(\vec{c}\) is a vector such that \(\vec{a} \cdot \vec{c}=11, \vec{b} \cdot(\vec{a} \times \vec{c})=27\) and $$\vec{b} \cdot \vec{c}=-\sqrt{3}|\vec{b}|$...