JEE Main 2023 (Online) 25th January Evening Shift

The foot of perpendicular of the point (2, 0, 5) on the line \({{x + 1} \over 2} = {{y - 1} \over 5} = {{z + 1} \over { - 1}}\) is (\(\alpha,\beta,\gamma\)). Then, which of the following is NOT correct?

The shortest distance between the lines \(x+1=2y=-12z\) and \(x=y+2=6z-6\) is :

If the shortest distance between the line joining the points (1, 2, 3) and (2, 3, 4), and the line \({{x - 1} \over 2} = {{y + 1} \over { - 1}} = {{z - 2} \over 0}\) is \(\alpha\), then 28\(\alpha^2\) is equal to ____________.

Let the function \(f(x) = 2{x^3} + (2p - 7){x^2} + 3(2p - 9)x - 6\) have a maxima for some value of \(x < 0\) and a minima for some value of \(x > 0\). Then, the set of all values of p is

The remainder when (2023)\(^{2023}\) is divided by 35 is __________.

Points P(\(-\)3, 2), Q(9, 10) and R(\(\alpha,4\)) lie on a circle C and PR as its diameter. The tangents to C at the points Q and R intersect at the point S. If S lies on the line \(2x-ky=1\), then k is equal to ____________.

Let \(z\) be a complex number such that \(\left| {{{z - 2i} \over {z + i}}} \right| = 2,z \ne - i\). Then \(z\) lies on the circle of radius 2 and centre :

The integral \(16\int\limits_1^2 {{{dx} \over {{x^3}{{\left( {{x^2} + 2} \right)}^2}}}}\) is equal to

If \(\int\limits_{{1 \over 3}}^3 {|{{\log }_e}x|dx = {m \over n}{{\log }_e}\left( {{{{n^2}} \over e}} \right)}\), where m and n are coprime natural numbers, then \({m^2} + {n^2} - 5\) is equal to _____________.

Let \(y=y(t)\) be a solution of the differential equation \({{dy} \over {dt}} + \alpha y = \gamma {e^{ - \beta t}}\) where, \(\alpha > 0,\beta > 0\) and \(\gamma > 0\). Then \(\mathop {\lim }\limits_{t \to \infty } y(t)\)

The number of functions
\(f:\{ 1,2,3,4\} \to \{ a \in Z|a| \le 8\}\)
satisfying \(f(n) + {1 \over n}f(n + 1) = 1,\forall n \in \{ 1,2,3\}\) is

Let \(f:\mathbb{R}\to\mathbb{R}\) be a function defined by \(f(x) = {\log _{\sqrt m }}\{ \sqrt 2 (\sin x - \cos x) + m - 2\}\), for some \(m\), such that the range of \(f\) is [0, 2]. Then the value of \(m\) is _________

Let \(f(x) = 2{x^n} + \lambda ,\lambda \in R,n \in N\), and \(f(4) = 133,f(5) = 255\). Then the sum of all the positive integer divisors of \((f(3) - f(2))\) is

Let T and C respectively be the transverse and conjugate axes of the hyperbola \(16{x^2} - {y^2} + 64x + 4y + 44 = 0\). Then the area of the region above the parabola \({x^2} = y + 4\), below the transverse axis T and on the right of the co...

If the function $$f(x) = \left\{ {\matrix{
{(1 + |\cos x|)^{\lambda \over {|\cos x|}}} & , & {0 < x < {\pi \over 2}} \cr
\mu & , & {x = {\pi \over 2}} \cr
e^{{{\cot 6x} \over {{}\cot 4x}}} & , & {{\pi \over 2} < x < \pi } ...

Let \(\Delta ,\nabla \in \{ \wedge , \vee \}\) be such that \(\mathrm{(p \to q)\Delta (p\nabla q)}\) is a tautology. Then

Let A, B, C be 3 \(\times\) 3 matrices such that A is symmetric and B and C are skew-symmetric. Consider the statements
(S1) A\(^{13}\) B\(^{26}\) \(-\) B\(^{26}\) A\(^{13}\) is symmetric
(S2) A\(^{26}\) C\(^{13}\) \(-\) C\(^{13}\) A$$^{26}...

Let \(A = \left[ {\matrix{ {{1 \over {\sqrt {10} }}} & {{3 \over {\sqrt {10} }}} \cr {{{ - 3} \over {\sqrt {10} }}} & {{1 \over {\sqrt {10} }}} \cr } } \right]\) and $$B = \left[ {\matrix{
1 & { - i} \cr
0 & 1 \cr

} ...

The equations of two sides of a variable triangle are \(x=0\) and \(y=3\), and its third side is a tangent to the parabola \(y^2=6x\). The locus of its circumcentre is :

The number of numbers, strictly between 5000 and 10000 can be formed using the digits 1, 3, 5, 7, 9 without repetition, is :

\(\sum\limits_{k = 0}^6 {{}^{51 - k}{C_3}}\) is equal to :

A triangle is formed by X-axis, Y-axis and the line \(3x+4y=60\). Then the number of points P(a, b) which lie strictly inside the triangle, where a is an integer and b is a multiple of a, is ____________.

Suppose Anil's mother wants to give 5 whole fruits to Anil from a basket of 7 red apples, 5 white apples and 8 oranges. If in the selected 5 fruits, at least 2 oranges, at least one red apple and at least one white apple must be given, then...

Let N be the sum of the numbers appeared when two fair dice are rolled and let the probability that \(N-2,\sqrt{3N},N+2\) are in geometric progression be \(\frac{k}{48}\). Then the value of k is :

25% of the population are smokers. A smoker has 27 times more chances to develop lung cancer than a non smoker. A person is diagnosed with lung cancer and the probability that this person is a smoker is \(\frac{k}{10}%\). Then the value of ...

Let \(\alpha \in\mathbb{R}\) and let \(\alpha,\beta\) be the roots of the equation \({x^2} + {60^{{1 \over 4}}}x + a = 0\). If \({\alpha ^4} + {\beta ^4} = - 30\), then the product of all possible values of \(a\) is ____________.

For the two positive numbers \(a,b,\) if \(a,b\) and \(\frac{1}{18}\) are in a geometric progression, while \(\frac{1}{a},10\) and \(\frac{1}{b}\) are in an arithmetic progression, then \(16a+12b\) is equal to _________.

If m and n respectively are the numbers of positive and negative values of \(\theta\) in the interval \([-\pi,\pi]\) that satisfy the equation \(\cos 2\theta \cos {\theta \over 2} = \cos 3\theta \cos {{9\theta } \over 2}\), then mn is equa...

Let \(\overrightarrow a = - \widehat i - \widehat j + \widehat k,\overrightarrow a \,.\,\overrightarrow b = 1\) and \(\overrightarrow a \times \overrightarrow b = \widehat i - \widehat j\). Then $$\overrightarrow a - 6\overrightarrow ...

If the four points, whose position vectors are \(3\widehat i - 4\widehat j + 2\widehat k,\widehat i + 2\widehat j - \widehat k, - 2\widehat i - \widehat j + 3\widehat k\) and \(5\widehat i - 2\alpha \widehat j + 4\widehat k\) are coplanar, ...