JEE Main 2023 (Online) 24th January Evening Shift

If the foot of the perpendicular drawn from (1, 9, 7) to the line passing through the point (3, 2, 1) and parallel to the planes \(x+2y+z=0\) and \(3y-z=3\) is (\(\alpha,\beta,\gamma\)), then \(\alpha+\beta+\gamma\) is equal to :

Let the plane containing the line of intersection of the planes P1 : \(x+(\lambda+4)y+z=1\) and P2 : \(2x+y+z=2\) pass through the points (0, 1, 0) and (1, 0, 1). Then the distance of the point (2\(\lambda,\lambda,-\lambda\)) from the plane...

If the shortest between the lines \({{x + \sqrt 6 } \over 2} = {{y - \sqrt 6 } \over 3} = {{z - \sqrt 6 } \over 4}\) and \({{x - \lambda } \over 3} = {{y - 2\sqrt 6 } \over 4} = {{z + 2\sqrt 6 } \over 5}\) is 6, then the square of sum of al...

If the area of the region bounded by the curves \(y^2-2y=-x,x+y=0\) is A, then 8 A is equal to __________

If \({({}^{30}{C_1})^2} + 2{({}^{30}{C_2})^2} + 3{({}^{30}{C_3})^2}\, + \,...\, + \,30{({}^{30}{C_{30}})^2} = {{\alpha 60!} \over {{{(30!)}^2}}}\) then \(\alpha\) is equal to :

Let the sum of the coefficients of the first three terms in the expansion of \({\left( {x - {3 \over {{x^2}}}} \right)^n},x \ne 0.~n \in \mathbb{N}\), be 376. Then the coefficient of \(x^4\) is __________.

The locus of the mid points of the chords of the circle \({C_1}:{(x - 4)^2} + {(y - 5)^2} = 4\) which subtend an angle \({\theta _i}\) at the centre of the circle \(C_1\), is a circle of radius \(r_i\). If $${\theta _1} = {\pi \over 3},{\t...

The value of \({\left( {{{1 + \sin {{2\pi } \over 9} + i\cos {{2\pi } \over 9}} \over {1 + \sin {{2\pi } \over 9} - i\cos {{2\pi } \over 9}}}} \right)^3}\) is

\(\int\limits_{{{3\sqrt 2 } \over 4}}^{{{3\sqrt 3 } \over 4}} {{{48} \over {\sqrt {9 - 4{x^2}} }}dx}\) is equal to :

Let \(f\) be \(a\) differentiable function defined on \(\left[ {0,{\pi \over 2}} \right]\) such that \(f(x) > 0\) and \(f(x) + \int_0^x {f(t)\sqrt {1 - {{({{\log }_e}f(t))}^2}} dt = e,\forall x \in \left[ {0,{\pi \over 2}} \right]}\). The...

Let \(y=y(x)\) be the solution of the differential equation \((x^2-3y^2)dx+3xy~dy=0,y(1)=1\). Then \(6y^2(e)\) is equal to

If \(f(x) = {x^3} - {x^2}f'(1) + xf''(2) - f'''(3),x \in \mathbb{R}\), then

Let \(f(x)\) be a function such that \(f(x+y)=f(x).f(y)\) for all \(x,y\in \mathbb{N}\). If \(f(1)=3\) and \(\sum\limits_{k = 1}^n {f(k) = 3279}\), then the value of n is

If \(f(x) = {{{2^{2x}}} \over {{2^{2x}} + 2}},x \in \mathbb{R}\), then \(f\left( {{1 \over {2023}}} \right) + f\left( {{2 \over {2023}}} \right)\, + \,...\, + \,f\left( {{{2022} \over {2023}}} \right)\) is equal to

The set of all values of \(a\) for which \(\mathop {\lim }\limits_{x \to a} ([x - 5] - [2x + 2]) = 0\), where [\(\alpha\)] denotes the greatest integer less than or equal to \(\alpha\) is equal to

Let p and q be two statements. Then \(\sim \left( {p \wedge (p \Rightarrow \, \sim q)} \right)\) is equivalent to

Let A be a 3 \(\times\) 3 matrix such that \(\mathrm{|adj(adj(adj~A))|=12^4}\). Then \(\mathrm{|A^{-1}~adj~A|}\) is equal to

If the system of equations
\(x+2y+3z=3\)
\(4x+3y-4z=4\)
\(8x+4y-\lambda z=9+\mu\)
has infinitely many solutions, then the ordered pair (\(\lambda,\mu\)) is equal to :

The equations of the sides AB and AC of a triangle ABC are \((\lambda+1)x+\lambda y=4\) and \(\lambda x+(1-\lambda)y+\lambda=0\) respectively. Its vertex A is on the y-axis and its orthocentre is (1, 2). The length of the tangent from the p...

The number of integers, greater than 7000 that can be formed, using the digits 3, 5, 6, 7, 8 without repetition is :

The number of square matrices of order 5 with entries from the set {0, 1}, such that the sum of all the elements in each row is 1 and the sum of all the elements in each column is also 1, is :

Three urns A, B and C contain 4 red, 6 black; 5 red, 5 black; and \(\lambda\) red, 4 black balls respectively. One of the urns is selected at random and a ball is drawn. If the ball drawn is red and the probability that it is drawn from urn...

The number of real solutions of the equation \(3\left( {{x^2} + {1 \over {{x^2}}}} \right) - 2\left( {x + {1 \over x}} \right) + 5 = 0\), is

If \({{{1^3} + {2^3} + {3^3}\, + \,...\,up\,to\,n\,terms} \over {1\,.\,3 + 2\,.\,5 + 3\,.\,7\, + \,...\,up\,to\,n\,terms}} = {9 \over 5}\), then the value of \(n\) is

The minimum number of elements that must be added to the relation R = {(a, b), (b, c), (b, d)} on the set {a, b, c, d} so that it is an equivalence relation, is __________.

Let the six numbers \(\mathrm{a_1,a_2,a_3,a_4,a_5,a_6}\), be in A.P. and \(\mathrm{a_1+a_3=10}\). If the mean of these six numbers is \(\frac{19}{2}\) and their variance is \(\sigma^2\), then 8\(\sigma^2\) is equal to :

The equations of the sides AB, BC and CA of a triangle ABC are : \(2x+y=0,x+py=21a,(a\pm0)\) and \(x-y=3\) respectively. Let P(2, a) be the centroid of \(\Delta\)ABC. Then (BC)\(^2\) is equal to ___________.

Let \(\mathrm{S = \{ \theta \in [0,2\pi ):\tan (\pi \cos \theta ) + \tan (\pi \sin \theta ) = 0\}}\). Then \(\sum\limits_{\theta \in S} {{{\sin }^2}\left( {\theta + {\pi \over 4}} \right)}\) is equal to __________.

Let \(\overrightarrow \alpha = 4\widehat i + 3\widehat j + 5\widehat k\) and \(\overrightarrow \beta = \widehat i + 2\widehat j - 4\widehat k\). Let \({\overrightarrow \beta _1}\) be parallel to \(\overrightarrow \alpha\) and $${\ove...

Let $$\overrightarrow a = \widehat i + 2\widehat j + \lambda \widehat k,\overrightarrow b = 3\widehat i - 5\widehat j - \lambda \widehat k,\overrightarrow a \,.\,\overrightarrow c = 7,2\overrightarrow b \,.\,\overrightarrow c + 43 = 0,\...