JEE Main 2023 (Online) 30th January Morning Shift

The line \(l_1\) passes through the point (2, 6, 2) and is perpendicular to the plane \(2x+y-2z=10\). Then the shortest distance between the line \(l_1\) and the line \(\frac{x+1}{2}=\frac{y+4}{-3}=\frac{z}{2}\) is :

If the equation of the plane passing through the point \((1,1,2)\) and perpendicular to the line \(x-3 y+ 2 z-1=0=4 x-y+z\) is \(\mathrm{A} x+\mathrm{B} y+\mathrm{C} z=1\), then \(140(\mathrm{C}-\mathrm{B}+\mathrm{A})\) is equal to ________...

If \(\lambda_{1} < \lambda_{2}\) are two values of \(\lambda\) such that the angle between the planes \(P_{1}: \vec{r}(3 \hat{i}-5 \hat{j}+\hat{k})=7\) and
\(P_{2}: \vec{r} \cdot(\lambda \hat{i}+\hat{j}-3 \hat{k})=9\) is $$\sin ^{-1}\left(...

The number of points on the curve \(y=54 x^{5}-135 x^{4}-70 x^{3}+180 x^{2}+210 x\) at which the normal lines are parallel to \(x+90 y+2=0\) is :

Let \(\alpha\) be the area of the larger region bounded by the curve \(y^{2}=8 x\) and the lines \(y=x\) and \(x=2\), which lies in the first quadrant. Then the value of \(3 \alpha\) is equal to ___________.

If the coefficient of \(x^{15}\) in the expansion of \(\left(\mathrm{a} x^{3}+\frac{1}{\mathrm{~b} x^{1 / 3}}\right)^{15}\) is equal to the coefficient of \(x^{-15}\) in the expansion of \(\left(a x^{1 / 3}-\frac{1}{b x^{3}}\right)^{15}\), ...

The coefficient of \({x^{301}}\) in \({(1 + x)^{500}} + x{(1 + x)^{499}} + {x^2}{(1 + x)^{498}}\, + \,...\, + \,{x^{500}}\) is :

Let \(y=x+2,4y=3x+6\) and \(3y=4x+1\) be three tangent lines to the circle \((x-h)^2+(y-k)^2=r^2\). Then \(h+k\) is equal to :

Let \(z=1+i\) and \(z_{1}=\frac{1+i \bar{z}}{\bar{z}(1-z)+\frac{1}{z}}\). Then \(\frac{12}{\pi} \arg \left(z_{1}\right)\) is equal to __________.

If [t] denotes the greatest integer \(\le \mathrm{t}\), then the value of \({{3(e - 1)} \over e}\int\limits_1^2 {{x^2}{e^{[x] + [{x^3}]}}dx}\) is :

\(\lim_\limits{x \rightarrow 0} \frac{48}{x^{4}} \int_\limits{0}^{x} \frac{t^{3}}{t^{6}+1} \mathrm{~d} t\) is equal to ___________.

Let the solution curve \(y=y(x)\) of the differential equation $$
\frac{\mathrm{d} y}{\mathrm{~d} x}-\frac{3 x^{5} \tan ^{-1}\left(x^{3}\right)}{\left(1+x^{6}\right)^{3 / 2}} y=2 x \exp \left\{\frac{x^{3}-\tan ^{-1} x^{3}}{\sqrt{\left(1+x^{...

Let \(f^{1}(x)=\frac{3 x+2}{2 x+3}, x \in \mathbf{R}-\left\{\frac{-3}{2}\right\}\) For \(\mathrm{n} \geq 2\), define \(f^{\mathrm{n}}(x)=f^{1} \mathrm{o} f^{\mathrm{n}-1}(x)\). If $$f^{5}(x)=\frac{\mathrm{a} x+\mathrm{b}}{\mathrm{b} x+\math...

Let \(S=\{1,2,3,4,5,6\}\). Then the number of one-one functions \(f: \mathrm{S} \rightarrow \mathrm{P}(\mathrm{S})\), where \(\mathrm{P}(\mathrm{S})\) denote the power set of \(\mathrm{S}\), such that \(f(n) \subset f(\mathrm{~m})\) where $...

Suppose \(f: \mathbb{R} \rightarrow(0, \infty)\) be a differentiable function such that \(5 f(x+y)=f(x) \cdot f(y), \forall x, y \in \mathbb{R}\). If \(f(3)=320\), then \(\sum_\limits{n=0}^{5} f(n)\) is equal to :

If the solution of the equation \(\log _{\cos x} \cot x+4 \log _{\sin x} \tan x=1, x \in\left(0, \frac{\pi}{2}\right)\), is \(\sin ^{-1}\left(\frac{\alpha+\sqrt{\beta}}{2}\right)\), where \(\alpha\), \(\beta\) are integers, then $$\alpha+\b...

Among the statements :
\((\mathrm{S} 1)~((\mathrm{p} \vee \mathrm{q}) \Rightarrow \mathrm{r}) \Leftrightarrow(\mathrm{p} \Rightarrow \mathrm{r})\)
$$(\mathrm{S} 2)~((\mathrm{p} \vee \mathrm{q}) \Rightarrow \mathrm{r}) \Leftrightarrow((\math...

Let the system of linear equations
\(x+y+kz=2\)
\(2x+3y-z=1\)
\(3x+4y+2z=k\)
have infinitely many solutions. Then the system
\((k+1)x+(2k-1)y=7\)
\((2k+1)x+(k+5)y=10\)
has :

Let $$A=\left(\begin{array}{cc}\mathrm{m} & \mathrm{n} \\ \mathrm{p} & \mathrm{q}\end{array}\right), \mathrm{d}=|\mathrm{A}| \neq 0$$ and \(\mathrm{|A-d(A d j A)|=0}\). Then

If \(\mathrm{P}(\mathrm{h}, \mathrm{k})\) be a point on the parabola \(x=4 y^{2}\), which is nearest to the point \(\mathrm{Q}(0,33)\), then the distance of \(\mathrm{P}\) from the directrix of the parabola \(\quad y^{2}=4(x+y)\) is equal t...

Number of 4-digit numbers (the repetition of digits is allowed) which are made using the digits 1, 2, 3 and 5, and are divisible by 15, is equal to ___________.

If an unbiased die, marked with \(-2,-1,0,1,2,3\) on its faces, is thrown five times, then the probability that the product of the outcomes is positive, is :

A straight line cuts off the intercepts \(\mathrm{OA}=\mathrm{a}\) and \(\mathrm{OB}=\mathrm{b}\) on the positive directions of \(x\)-axis and \(y\) axis respectively. If the perpendicular from origin \(O\) to this line makes an angle of $$...

If \({a_n} = {{ - 2} \over {4{n^2} - 16n + 15}}\), then \({a_1} + {a_2}\, + \,....\, + \,{a_{25}}\) is equal to :

Let \(\sum_\limits{n=0}^{\infty} \frac{\mathrm{n}^{3}((2 \mathrm{n}) !)+(2 \mathrm{n}-1)(\mathrm{n} !)}{(\mathrm{n} !)((2 \mathrm{n}) !)}=\mathrm{ae}+\frac{\mathrm{b}}{\mathrm{e}}+\mathrm{c}\), where $$\mathrm{a}, \mathrm{b}, \mathrm{c} \in...

The minimum number of elements that must be added to the relation \(\mathrm{R}=\{(\mathrm{a}, \mathrm{b}),(\mathrm{b}, \mathrm{c})\}\) on the set \(\{a, b, c\}\) so that it becomes symmetric and transitive is :

The mean and variance of 7 observations are 8 and 16 respectively. If one observation 14 is omitted and a and b are respectively mean and variance of remaining 6 observation, then \(\mathrm{a+3 b-5}\) is equal to ___________.

If \(\tan 15^\circ + {1 \over {\tan 75^\circ }} + {1 \over {\tan 105^\circ }} + \tan 195^\circ = 2a\), then the value of \(\left( {a + {1 \over a}} \right)\) is :

If \(\overrightarrow a ,\overrightarrow b ,\overrightarrow c\) are three non-zero vectors and \(\widehat n\) is a unit vector perpendicular to \(\overrightarrow c\) such that $$\overrightarrow a = \alpha \overrightarrow b - \widehat n,(...

Let a unit vector \(\widehat{O P}\) make angles \(\alpha, \beta, \gamma\) with the positive directions of the co-ordinate axes \(\mathrm{OX}\), \(\mathrm{OY}, \mathrm{OZ}\) respectively, where \(\beta \in\left(0, \frac{\pi}{2}\right)\). If ...