JEE Main 2023 (Online) 25th January Morning Shift

The distance of the point P(4, 6, \(-\)2) from the line passing through the point (\(-\)3, 2, 3) and parallel to a line with direction ratios 3, 3, \(-\)1 is equal to :

Consider the lines \(L_1\) and \(L_2\) given by
\({L_1}:{{x - 1} \over 2} = {{y - 3} \over 1} = {{z - 2} \over 2}\)
\({L_2}:{{x - 2} \over 1} = {{y - 2} \over 2} = {{z - 3} \over 3}\).
A line \(L_3\) having direction ratios 1, \(-\)1, \(-\)...

Let the equation of the plane passing through the line \(x - 2y - z - 5 = 0 = x + y + 3z - 5\) and parallel to the line \(x + y + 2z - 7 = 0 = 2x + 3y + z - 2\) be \(ax + by + cz = 65\). Then the distance of the point (a, b, c) from the pla...

Let \(x=2\) be a local minima of the function \(f(x)=2x^4-18x^2+8x+12,x\in(-4,4)\). If M is local maximum value of the function \(f\) in (\(-4,4)\), then M =

Let \(f:(0,1)\to\mathbb{R}\) be a function defined \(f(x) = {1 \over {1 - {e^{ - x}}}}\), and \(g(x) = \left( {f( - x) - f(x)} \right)\). Consider two statements
(I) g is an increasing function in (0, 1)
(II) g is one-one in (0, 1)
Then,

If the area enclosed by the parabolas \(\mathrm{P_1:2y=5x^2}\) and \(\mathrm{P_2:x^2-y+6=0}\) is equal to the area enclosed by \(\mathrm{P_1}\) and \(\mathrm{y=\alpha x,\alpha > 0}\), then \(\alpha^3\) is equal to ____________.

If \(a_r\) is the coefficient of \(x^{10-r}\) in the Binomial expansion of \((1 + x)^{10}\), then \(\sum\limits_{r = 1}^{10} {{r^3}{{\left( {{{{a_r}} \over {{a_{r - 1}}}}} \right)}^2}}\) is equal to

The constant term in the expansion of \({\left( {2x + {1 \over {{x^7}}} + 3{x^2}} \right)^5}\) is ___________.

The points of intersection of the line \(ax + by = 0,(a \ne b)\) and the circle \({x^2} + {y^2} - 2x = 0\) are \(A(\alpha ,0)\) and \(B(1,\beta )\). The image of the circle with AB as a diameter in the line \(x + y + 2 = 0\) is :

Let \(\mathrm{z_1=2+3i}\) and \(\mathrm{z_2=3+4i}\). The set \(\mathrm{S = \left\{ {z \in \mathbb{C}:{{\left| {z - {z_1}} \right|}^2} - {{\left| {z - {z_2}} \right|}^2} = {{\left| {{z_1} - {z_2}} \right|}^2}} \right\}}\) represents a

The minimum value of the function \(f(x) = \int\limits_0^2 {{e^{|x - t|}}dt}\) is :

Let \(y = y(x)\) be the solution curve of the differential equation \({{dy} \over {dx}} = {y \over x}\left( {1 + x{y^2}(1 + {{\log }_e}x)} \right),x > 0,y(1) = 3\). Then \({{{y^2}(x)} \over 9}\) is equal to :

Let \(y(x) = (1 + x)(1 + {x^2})(1 + {x^4})(1 + {x^8})(1 + {x^{16}})\). Then \(y' - y''\) at \(x = - 1\) is equal to

For some a, b, c \(\in\mathbb{N}\), let \(f(x) = ax - 3\) and \(\mathrm{g(x)=x^b+c,x\in\mathbb{R}}\). If \({(fog)^{ - 1}}(x) = {\left( {{{x - 7} \over 2}} \right)^{1/3}}\), then \((fog)(ac) + (gof)(b)\) is equal to ____________.

The vertices of a hyperbola H are (\(\pm\) 6, 0) and its eccentricity is \({{\sqrt 5 } \over 2}\). Let N be the normal to H at a point in the first quadrant and parallel to the line \(\sqrt 2 x + y = 2\sqrt 2\). If d is the length of the l...

Let \(f(x) = \int {{{2x} \over {({x^2} + 1)({x^2} + 3)}}dx}\). If \(f(3) = {1 \over 2}({\log _e}5 - {\log _e}6)\), then \(f(4)\) is equal to

If the sum of all the solutions of \({\tan ^{ - 1}}\left( {{{2x} \over {1 - {x^2}}}} \right) + {\cot ^{ - 1}}\left( {{{1 - {x^2}} \over {2x}}} \right) = {\pi \over 3}, - 1 < x < 1,x \ne 0\), is \(\alpha - {4 \over {\sqrt 3 }}\), then $$\a...

The value of \(\mathop {\lim }\limits_{n \to \infty } {{1 + 2 - 3 + 4 + 5 - 6\, + \,.....\, + \,(3n - 2) + (3n - 1) - 3n} \over {\sqrt {2{n^4} + 4n + 3} - \sqrt {{n^4} + 5n + 4} }}\) is :

Let \(S = \left\{ {\alpha :{{\log }_2}({9^{2\alpha - 4}} + 13) - {{\log }_2}\left( {{5 \over 2}.\,{3^{2\alpha - 4}} + 1} \right) = 2} \right\}\). Then the maximum value of \(\beta\) for which the equation $${x^2} - 2{\left( {\sum\limits_{...

The statement \(\left( {p \wedge \left( { \sim q} \right)} \right) \Rightarrow \left( {p \Rightarrow \left( { \sim q} \right)} \right)\) is

Let \(x,y,z > 1\) and \(A = \left[ {\matrix{ 1 & {{{\log }_x}y} & {{{\log }_x}z} \cr {{{\log }_y}x} & 2 & {{{\log }_y}z} \cr {{{\log }_z}x} & {{{\log }_z}y} & 3 \cr } } \right]\). Then \(\mathrm{|adj~(adj~A^2)|}\) is equal ...

Let S\(_1\) and S\(_2\) be respectively the sets of all \(a \in \mathbb{R} - \{ 0\}\) for which the system of linear equations
\(ax + 2ay - 3az = 1\)
\((2a + 1)x + (2a + 3)y + (a + 1)z = 2\)
\((3a + 5)x + (a + 5)y + (a + 2)z = 3\)
has uniq...

Let \(\mathrm{A_1,A_2,A_3}\) be the three A.P. with the same common difference d and having their first terms as \(\mathrm{A,A+1,A+2}\), respectively. Let a, b, c be the \(\mathrm{7^{th},9^{th},17^{th}}\) terms of \(\mathrm{A_1,A_2,A_3}\), ...

The distance of the point \((6,-2\sqrt2)\) from the common tangent \(\mathrm{y=mx+c,m > 0}\), of the curves \(x=2y^2\) and \(x=1+y^2\) is :

Let \(x\) and \(y\) be distinct integers where \(1 \le x \le 25\) and \(1 \le y \le 25\). Then, the number of ways of choosing \(x\) and \(y\), such that \(x+y\) is divisible by 5, is ____________.

Let M be the maximum value of the product of two positive integers when their sum is 66. Let the sample space \(S = \left\{ {x \in \mathbb{Z}:x(66 - x) \ge {5 \over 9}M} \right\}\) and the event $$\mathrm{A = \{ x \in S:x\,is\,a\,multiple\,...

Let S = {1, 2, 3, 5, 7, 10, 11}. The number of non-empty subsets of S that have the sum of all elements a multiple of 3, is _____________.

The mean and variance of the marks obtained by the students in a test are 10 and 4 respectively. Later, the marks of one of the students is increased from 8 to 12. If the new mean of the marks is 10.2, then their new variance is equal to :

The vector \(\overrightarrow a = - \widehat i + 2\widehat j + \widehat k\) is rotated through a right angle, passing through the y-axis in its way and the resulting vector is \(\overrightarrow b\). Then the projection of $$3\overrightarr...

Let \(\overrightarrow a\), \(\overrightarrow b\) and \(\overrightarrow c\) be three non zero vectors such that \(\overrightarrow b\) . \(\overrightarrow c\) = 0 and $$\overrightarrow a \times (\overrightarrow b \times \overrightarrow...