JEE Main 2023 (Online) 24th January Morning Shift

The distance of the point (7, \(-\)3, \(-\)4) from the plane passing through the points (2, \(-\)3, 1), (\(-\)1, 1, \(-\)2) and (3, \(-\)4, 2) is :

The distance of the point (\(-1,9,-16\)) from the plane \(2x+3y-z=5\) measured parallel to the line \({{x + 4} \over 3} = {{2 - y} \over 4} = {{z - 3} \over {12}}\) is :

The shortest distance between the lines \({{x - 2} \over 3} = {{y + 1} \over 2} = {{z - 6} \over 2}\) and \({{x - 6} \over 3} = {{1 - y} \over 2} = {{z + 8} \over 0}\) is equal to ________

The area enclosed by the curves \({y^2} + 4x = 4\) and \(y - 2x = 2\) is :

The value of \(\sum\limits_{r = 0}^{22} {{}^{22}{C_r}{}^{23}{C_r}}\) is

Suppose \(\sum\limits_{r = 0}^{2023} {{r^2}{}~^{2023}{C_r} = 2023 \times \alpha \times {2^{2022}}}\). Then the value of \(\alpha\) is ___________

Let \(\mathrm{p,q\in\mathbb{R}}\) and \({\left( {1 - \sqrt 3 i} \right)^{200}} = {2^{199}}(p + iq),i = \sqrt { - 1}\) then \(\mathrm{p+q+q^2}\) and \(\mathrm{p-q+q^2}\) are roots of the equation.

The value of \(12\int\limits_0^3 {\left| {{x^2} - 3x + 2} \right|dx}\) is ____________

The value of \({8 \over \pi }\int\limits_0^{{\pi \over 2}} {{{{{(\cos x)}^{2023}}} \over {{{(\sin x)}^{2023}} + {{(\cos x)}^{2023}}}}dx}\) is ___________

Let \(y = y(x)\) be the solution of the differential equation \({x^3}dy + (xy - 1)dx = 0,x > 0,y\left( {{1 \over 2}} \right) = 3 - \mathrm{e}\). Then y (1) is equal to

Let C be the largest circle centred at (2, 0) and inscribed in the ellipse \({{{x^2}} \over {36}} + {{{y^2}} \over {16}} = 1\). If (1, \(\alpha\)) lies on C, then 10 \(\alpha^2\) is equal to ____________

Let a tangent to the curve \(9{x^2} + 16{y^2} = 144\) intersect the coordinate axes at the points A and B. Then, the minimum length of the line segment AB is ________

\({\tan ^{ - 1}}\left( {{{1 + \sqrt 3 } \over {3 + \sqrt 3 }}} \right) + {\sec ^{ - 1}}\left( {\sqrt {{{8 + 4\sqrt 3 } \over {6 + 3\sqrt 3 }}} } \right)\) is equal to :

\(\mathop {\lim }\limits_{t \to 0} {\left( {{1^{{1 \over {{{\sin }^2}t}}}} + {2^{{1 \over {{{\sin }^2}t}}}}\, + \,...\, + \,{n^{{1 \over {{{\sin }^2}t}}}}} \right)^{{{\sin }^2}t}}\) is equal to

Let \(f(x) = \left\{ {\matrix{ {{x^2}\sin \left( {{1 \over x}} \right)} & {,\,x \ne 0} \cr 0 & {,\,x = 0} \cr } } \right.\)
Then at \(x=0\)

The compound statement \(\left( { \sim (P \wedge Q)} \right) \vee \left( {( \sim P) \wedge Q} \right) \Rightarrow \left( {( \sim P) \wedge ( \sim Q)} \right)\) is equivalent to

If A and B are two non-zero n \(\times\) n matrices such that \(\mathrm{A^2+B=A^2B}\), then :

Let \(\alpha\) be a root of the equation \((a - c){x^2} + (b - a)x + (c - b) = 0\) where a, b, c are distinct real numbers such that the matrix $$\left[ {\matrix{
{{\alpha ^2}} & \alpha & 1 \cr
1 & 1 & 1 \cr
a & b & c \cr

...

Let a tangent to the curve \(\mathrm{y^2=24x}\) meet the curve \(xy = 2\) at the points A and B. Then the mid points of such line segments AB lie on a parabola with the :

A boy needs to select five courses from 12 available courses, out of which 5 courses are language courses. If he can choose at most two language courses, then the number of ways he can choose five courses is __________

The number of 9 digit numbers, that can be formed using all the digits of the number 123412341 so that the even digits occupy only even places, is ______________.

Let N denote the number that turns up when a fair die is rolled. If the probability that the system of equations
\(x + y + z = 1\)
\(2x + \mathrm{N}y + 2z = 2\)
\(3x + 3y + \mathrm{N}z = 3\)
has unique solution is \({k \over 6}\), then the ...

Let \(\Omega\) be the sample space and \(\mathrm{A \subseteq \Omega}\) be an event.
Given below are two statements :
(S1) : If P(A) = 0, then A = \(\phi\)
(S2) : If P(A) = 1, then A = \(\Omega\)
Then :

The equation \({x^2} - 4x + [x] + 3 = x[x]\), where \([x]\) denotes the greatest integer function, has :

Let \(\lambda \in \mathbb{R}\) and let the equation E be \(|x{|^2} - 2|x| + |\lambda - 3| = 0\). Then the largest element in the set S = {\(x+\lambda:x\) is an integer solution of E} is ______

For three positive integers p, q, r, \({x^{p{q^2}}} = {y^{qr}} = {z^{{p^2}r}}\) and r = pq + 1 such that 3, 3 log\(_yx\), 3 log\(_zy\), 7 log\(_xz\) are in A.P. with common difference \(\frac{1}{2}\). Then r-p-q is equal to

The 4\(^\mathrm{th}\) term of GP is 500 and its common ratio is \(\frac{1}{m},m\in\mathbb{N}\). Let \(\mathrm{S_n}\) denote the sum of the first n terms of this GP. If \(\mathrm{S_6 > S_5 + 1}\) and \(\mathrm{S_7 < S_6 + \frac{1}{2}}\), the...

The relation \(\mathrm{R = \{ (a,b):\gcd (a,b) = 1,2a \ne b,a,b \in \mathbb{Z}\}}\) is :

Let PQR be a triangle. The points A, B and C are on the sides QR, RP and PQ respectively such that \({{QA} \over {AR}} = {{RB} \over {BP}} = {{PC} \over {CQ}} = {1 \over 2}\). Then \({{Area(\Delta PQR)} \over {Area(\Delta ABC)}}\) is equal ...

Let \(\overrightarrow u = \widehat i - \widehat j - 2\widehat k,\overrightarrow v = 2\widehat i + \widehat j - \widehat k,\overrightarrow v .\,\overrightarrow w = 2\) and $$\overrightarrow v \times \overrightarrow w = \overrightarrow u...