JEE Main 2023 (Online) 6th April Evening Shift

A plane P contains the line of intersection of the plane \(\vec{r} \cdot(\hat{i}+\hat{j}+\hat{k})=6\) and \(\vec{r} \cdot(2 \hat{i}+3 \hat{j}+4 \hat{k})=-5\). If \(\mathrm{P}\) passes through the point \((0,2,-2)\), then the square of dista...

Let the line \(\mathrm{L}\) pass through the point \((0,1,2)\), intersect the line \(\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}\) and be parallel to the plane \(2 x+y-3 z=4\). Then the distance of the point \(\mathrm{P}(1,-9,2)\) from the li...

If the lines \(\frac{x-1}{2}=\frac{2-y}{-3}=\frac{z-3}{\alpha}\) and \(\frac{x-4}{5}=\frac{y-1}{2}=\frac{z}{\beta}\) intersect, then the magnitude of the minimum value of \(8 \alpha \beta\) is _____________.

Let a curve \(y=f(x), x \in(0, \infty)\) pass through the points \(P\left(1, \frac{3}{2}\right)\) and \(Q\left(a, \frac{1}{2}\right)\). If the tangent at any point \(R(b, f(b))\) to the given curve cuts the \(\mathrm{y}\)-axis at the point ...

The number of points, where the curve \(y=x^{5}-20 x^{3}+50 x+2\) crosses the \(\mathrm{x}\)-axis, is ____________.

The area bounded by the curves \(y=|x-1|+|x-2|\) and \(y=3\) is equal to :

If the coefficient of \({x^7}\) in \({\left( {a{x^2} + {1 \over {2bx}}} \right)^{11}}\) and \({x^{ - 7}}\) in \({\left( {ax - {1 \over {3b{x^2}}}} \right)^{11}}\) are equal, then :

Among the statements :
(S1) : \(2023^{2022}-1999^{2022}\) is divisible by 8
(S2) : \(13(13)^{n}-12 n-13\) is divisible by 144 for infinitely many \(n \in \mathbb{N}\)

If the tangents at the points \(\mathrm{P}\) and \(\mathrm{Q}\) on the circle \(x^{2}+y^{2}-2 x+y=5\) meet at the point \(R\left(\frac{9}{4}, 2\right)\), then the area of the triangle \(\mathrm{PQR}\) is :

Let \(a \neq b\) be two non-zero real numbers. Then the number of elements in the set \(X=\left\{z \in \mathbb{C}: \operatorname{Re}\left(a z^{2}+b z\right)=a\right.\) and \(\left.\operatorname{Re}\left(b z^{2}+a z\right)=b\right\}\) is equ...

For \(\alpha, \beta, z \in \mathbb{C}\) and \(\lambda > 1\), if \(\sqrt{\lambda-1}\) is the radius of the circle \(|z-\alpha|^{2}+|z-\beta|^{2}=2 \lambda\), then \(|\alpha-\beta|\) is equal to __________.

Let \(f(x)\) be a function satisfying \(f(x)+f(\pi-x)=\pi^{2}, \forall x \in \mathbb{R}\). Then \(\int_\limits{0}^{\pi} f(x) \sin x d x\) is equal to :

\(\lim _\limits{n \rightarrow \infty}\left\{\left(2^{\frac{1}{2}}-2^{\frac{1}{3}}\right)\left(2^{\frac{1}{2}}-2^{\frac{1}{5}}\right) \ldots . .\left(2^{\frac{1}{2}}-2^{\frac{1}{2 n+1}}\right)\right\}\) is equal to :

Let \(f(x)=\frac{x}{\left(1+x^{n}\right)^{\frac{1}{n}}}, x \in \mathbb{R}-\{-1\}, n \in \mathbb{N}, n > 2\).
If \(f^{n}(x)=\left(f \circ f \circ f \ldots .\right.\). upto \(n\) times) \((x)\), then
$$\lim _\limits{n \rightarrow \infty} \in...

If the solution curve \(f(x, y)=0\) of the differential equation
\(\left(1+\log _{e} x\right) \frac{d x}{d y}-x \log _{e} x=e^{y}, x > 0\),
passes through the points \((1,0)\) and \((\alpha, 2)\), then \(\alpha^{\alpha}\) is equal to :

In a group of 100 persons 75 speak English and 40 speak Hindi. Each person speaks at least one of the two languages. If the number of persons, who speak only English is \(\alpha\) and the number of persons who speak only Hindi is \(\beta\),...

Let the sets A and B denote the domain and range respectively of the function \(f(x)=\frac{1}{\sqrt{\lceil x\rceil-x}}\), where \(\lceil x\rceil\) denotes the smallest integer greater than or equal to \(x\). Then among the statements
(S1) :...

Let the eccentricity of an ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) is reciprocal to that of the hyperbola \(2 x^{2}-2 y^{2}=1\). If the ellipse intersects the hyperbola at right angles, then square of length of the latus-rectu...

Among the statements
(S1) : \((p \Rightarrow q) \vee((\sim p) \wedge q)\) is a tautology
(S2) : \((q \Rightarrow p) \Rightarrow((\sim p) \wedge q)\) is a contradiction

Let \(P\) be a square matrix such that \(P^{2}=I-P\). For \(\alpha, \beta, \gamma, \delta \in \mathbb{N}\), if \(P^{\alpha}+P^{\beta}=\gamma I-29 P\) and \(P^{\alpha}-P^{\beta}=\delta I-13 P\), then \(\alpha+\beta+\gamma-\delta\) is equal t...

For the system of equations
\(x+y+z=6\)
\(x+2 y+\alpha z=10\)
\(x+3 y+5 z=\beta\), which one of the following is NOT true?

All the letters of the word PUBLIC are written in all possible orders and these words are written as in a dictionary with serial numbers. Then the serial number of the word PUBLIC is :

The number of 4-letter words, with or without meaning, each consisting of 2 vowels and 2 consonants, which can be formed from the letters of the word UNIVERSE without repetition is __________.

Three dice are rolled. If the probability of getting different numbers on the three dice is \(\frac{p}{q}\), where \(p\) and \(q\) are co-prime, then \(q-p\) is equal to :

If \(\operatorname{gcd}~(\mathrm{m}, \mathrm{n})=1\) and \(1^{2}-2^{2}+3^{2}-4^{2}+\ldots . .+(2021)^{2}-(2022)^{2}+(2023)^{2}=1012 ~m^{2} n\) then \(m^{2}-n^{2}\) is equal to :

If
\((20)^{19}+2(21)(20)^{18}+3(21)^{2}(20)^{17}+\ldots+20(21)^{19}=k(20)^{19}\),
then \(k\) is equal to ___________.

If the mean and variance of the frequency distribution

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The value of \(\tan 9^{\circ}-\tan 27^{\circ}-\tan 63^{\circ}+\tan 81^{\circ}\) is __________.

Let the vectors \(\vec{a}, \vec{b}, \vec{c}\) represent three coterminous edges of a parallelopiped of volume V. Then the volume of the parallelopiped, whose coterminous edges are represented by \(\vec{a}, \vec{b}+\vec{c}\) and $$\vec{a}+2 ...

The sum of all values of \(\alpha\), for which the points whose position vectors are \(\hat{i}-2 \hat{j}+3 \hat{k}, 2 \hat{i}-3 \hat{j}+4 \hat{k},(\alpha+1) \hat{i}+2 \hat{k}\) and \(9 \hat{i}+(\alpha-8) \hat{j}+6 \hat{k}\) are coplanar, is...