JEE Main 2023 (Online) 29th January Evening Shift

The plane \(2x-y+z=4\) intersects the line segment joining the points A (\(a,-2,4)\) and B (\(2,b,-3)\) at the point C in the ratio 2 : 1 and the distance of the point C from the origin is \(\sqrt5\). If \(ab < 0\) and P is the point $$(a-b...

If the lines \({{x - 1} \over 1} = {{y - 2} \over 2} = {{z + 3} \over 1}\) and \({{x - a} \over 2} = {{y + 2} \over 3} = {{z - 3} \over 1}\) intersect at the point P, then the distance of the point P from the plane \(z = a\) is :

The shortest distance between the lines \({{x - 1} \over 2} = {{y + 8} \over -7} = {{z - 4} \over 5}\) and \({{x - 1} \over 2} = {{y - 2} \over 1} = {{z - 6} \over { - 3}}\) is :

If the equation of the normal to the curve \(y = {{x - a} \over {(x + b)(x - 2)}}\) at the point (1, \(-\)3) is \(x - 4y = 13\), then the value of \(a + b\) is equal to ___________.

The area of the region \(A = \left\{ {(x,y):\left| {\cos x - \sin x} \right| \le y \le \sin x,0 \le x \le {\pi \over 2}} \right\}\) is

Let K be the sum of the coefficients of the odd powers of \(x\) in the expansion of \((1+x)^{99}\). Let \(a\) be the middle term in the expansion of \({\left( {2 + {1 \over {\sqrt 2 }}} \right)^{200}}\). If $${{{}^{200}{C_{99}}K} \over a} =...

A circle with centre (2, 3) and radius 4 intersects the line \(x+y=3\) at the points P and Q. If the tangents at P and Q intersect at the point \(S(\alpha,\beta)\), then \(4\alpha-7\beta\) is equal to ___________.

Let \(\alpha = 8 - 14i,A = \left\{ {z \in c:{{\alpha z - \overline \alpha \overline z } \over {{z^2} - {{\left( {\overline z } \right)}^2} - 112i}}=1} \right\}\) and \(B = \left\{ {z \in c:\left| {z + 3i} \right| = 4} \right\}\). Then $$\...

The value of the integral \(\int_1^2 {\left( {{{{t^4} + 1} \over {{t^6} + 1}}} \right)dt}\) is

The value of the integral \(\int\limits_{1/2}^2 {{{{{\tan }^{ - 1}}x} \over x}dx}\) is equal to :

Let \(y=y(x)\) be the solution of the differential equation \(x{\log _e}x{{dy} \over {dx}} + y = {x^2}{\log _e}x,(x > 1)\). If \(y(2) = 2\), then \(y(e)\) is equal to

Let \(f\) and \(g\) be the twice differentiable functions on \(\mathbb{R}\) such that
\(f''(x)=g''(x)+6x\)
\(f'(1)=4g'(1)-3=9\)
\(f(2)=3g(2)=12\).
Then which of the following is NOT true?

Consider a function \(f:\mathbb{N}\to\mathbb{R}\), satisfying \(f(1)+2f(2)+3f(3)+....+xf(x)=x(x+1)f(x);x\ge2\) with \(f(1)=1\). Then \(\frac{1}{f(2022)}+\frac{1}{f(2028)}\) is equal to

The statement \(B \Rightarrow \left( {\left( { \sim A} \right) \vee B} \right)\) is equivalent to :

The set of all values of \(\mathrm{t\in \mathbb{R}}\), for which the matrix $$\left[ {\matrix{
{{e^t}} & {{e^{ - t}}(\sin t - 2\cos t)} & {{e^{ - t}}( - 2\sin t - \cos t)} \cr
{{e^t}} & {{e^{ - t}}(2\sin t + \cos t)} & {{e^{ - t}}(\...

Let A be a symmetric matrix such that \(\mathrm{|A|=2}\) and \(\left[ {\matrix{ 2 & 1 \cr 3 & {{3 \over 2}} \cr } } \right]A = \left[ {\matrix{ 1 & 2 \cr \alpha & \beta \cr } } \right]\). If the sum of the diagonal...

If the tangent at a point P on the parabola \(y^2=3x\) is parallel to the line \(x+2y=1\) and the tangents at the points Q and R on the ellipse \(\frac{x^2}{4}+\frac{y^2}{1}=1\) are perpendicular to the line \(x-y=2\), then the area of the ...

A triangle is formed by the tangents at the point (2, 2) on the curves \(y^2=2x\) and \(x^2+y^2=4x\), and the line \(x+y+2=0\). If \(r\) is the radius of its circumcircle, then \(r^2\) is equal to ___________.

The letters of the word OUGHT are written in all possible ways and these words are arranged as in a dictionary, in a series. Then the serial number of the word TOUGH is :

The number of 3 digit numbers, that are divisible by either 3 or 4 but not divisible by 48, is :

The total number of 4-digit numbers whose greatest common divisor with 54 is 2, is __________.

Let \(\mathrm{S} = \{ {w_1},{w_2},......\}\) be the sample space associated to a random experiment. Let \(P({w_n}) = {{P({w_{n - 1}})} \over 2},n \ge 2\). Let \(A = \{ 2k + 3l:k,l \in N\}\) and \(B = \{ {w_n}:n \in A\}\). Then P(B) is eq...

Let \(\alpha_1,\alpha_2,....,\alpha_7\) be the roots of the equation \({x^7} + 3{x^5} - 13{x^3} - 15x = 0\) and \(|{\alpha _1}| \ge |{\alpha _2}| \ge \,...\, \ge \,|{\alpha _7}|\). Then \(\alpha_1\alpha_2-\alpha_3\alpha_4+\alpha_5\alpha_6\)...

Let \(a_1=b_1=1\) and \({a_n} = {a_{n - 1}} + (n - 1),{b_n} = {b_{n - 1}} + {a_{n - 1}},\forall n \ge 2\). If \(S = \sum\limits_{n = 1}^{10} {{{{b_n}} \over {{2^n}}}}\) and \(T = \sum\limits_{n = 1}^8 {{n \over {{2^{n - 1}}}}}\), then $${...

Let \(\{ {a_k}\}\) and \(\{ {b_k}\} ,k \in N\), be two G.P.s with common ratios \({r_1}\) and \({r_2}\) respectively such that \({a_1} = {b_1} = 4\) and \({r_1} < {r_2}\). Let \({c_k} = {a_k} + {b_k},k \in N\). If \({c_2} = 5\) and $${c_3}...

Let R be a relation defined on \(\mathbb{N}\) as \(a\mathrm{R}b\) if \(2a+3b\) is a multiple of \(5,a,b\in \mathbb{N}\). Then R is

Let \(X=\{11,12,13,....,40,41\}\) and \(Y=\{61,62,63,....,90,91\}\) be the two sets of observations. If \(\overline x\) and \(\overline y\) are their respective means and \(\sigma^2\) is the variance of all the observations in $$\mathrm{X...

The set of all values of \(\lambda\) for which the equation \({\cos ^2}2x - 2{\sin ^4}x - 2{\cos ^2}x = \lambda\) has a real solution \(x\), is :

If $$\overrightarrow a = \widehat i + 2\widehat k,\overrightarrow b = \widehat i + \widehat j + \widehat k,\overrightarrow c = 7\widehat i - 3\widehat j + 4\widehat k,\overrightarrow r \times \overrightarrow b + \overrightarrow b \tim...

Let \(\overrightarrow a = 4\widehat i + 3\widehat j\) and \(\overrightarrow b = 3\widehat i - 4\widehat j + 5\widehat k\). If \(\overrightarrow c\) is a vector such that $$\overrightarrow c .\left( {\overrightarrow a \times \overrightar...