JEE Main 2023 (Online) 1st February Evening Shift

Let the plane P pass through the intersection of the planes \(2x+3y-z=2\) and \(x+2y+3z=6\), and be perpendicular to the plane \(2x+y-z+1=0\). If d is the distance of P from the point (\(-\)7, 1, 1), then \(\mathrm{d^{2}}\) is equal to :

The point of intersection \(\mathrm{C}\) of the plane \(8 x+y+2 z=0\) and the line joining the points \(\mathrm{A}(-3,-6,1)\) and \(\mathrm{B}(2,4,-3)\) divides the line segment \(\mathrm{AB}\) internally in the ratio \(\mathrm{k}: 1\). If ...

Let \(\alpha x+\beta y+\gamma z=1\) be the equation of a plane passing through the point \((3,-2,5)\) and perpendicular to the line joining the points \((1,2,3)\) and \((-2,3,5)\). Then the value of \(\alpha \beta y\) is equal to __________...

The sum of the absolute maximum and minimum values of the function \(f(x)=\left|x^{2}-5 x+6\right|-3 x+2\) in the interval \([-1,3]\) is equal to :

The area of the region given by \(\{ (x,y):xy \le 8,1 \le y \le {x^2}\}\) is :

Let the sixth term in the binomial expansion of \({\left( {\sqrt {{2^{{{\log }_2}\left( {10 - {3^x}} \right)}}} + \root 5 \of {{2^{(x - 2){{\log }_2}3}}} } \right)^m}\) in the increasing powers of \(2^{(x-2) \log _{2} 3}\), be 21 . If the ...

If the term without \(x\) in the expansion of \(\left(x^{\frac{2}{3}}+\frac{\alpha}{x^{3}}\right)^{22}\) is 7315 , then \(|\alpha|\) is equal to ___________.

Let \(a,b\) be two real numbers such that \(ab < 0\). IF the complex number \(\frac{1+ai}{b+i}\) is of unit modulus and \(a+ib\) lies on the circle \(|z-1|=|2z|\), then a possible value of \(\frac{1+[a]}{4b}\), where \([t]\) is greatest int...

The value of the integral \(\int\limits_{ - {\pi \over 4}}^{{\pi \over 4}} {{{x + {\pi \over 4}} \over {2 - \cos 2x}}dx}\) is :

If \(\int\limits_0^\pi {{{{5^{\cos x}}(1 + \cos x\cos 3x + {{\cos }^2}x + {{\cos }^3}x\cos 3x)dx} \over {1 + {5^{\cos x}}}} = {{k\pi } \over {16}}}\), then k is equal to _____________.

Let \(\alpha x=\exp \left(x^{\beta} y^{\gamma}\right)\) be the solution of the differential equation \(2 x^{2} y \mathrm{~d} y-\left(1-x y^{2}\right) \mathrm{d} x=0, x > 0,y(2)=\sqrt{\log _{e} 2}\). Then \(\alpha+\beta-\gamma\) equals :

If \(y(x)=x^{x},x > 0\), then \(y''(2)-2y'(2)\) is equal to

The line \(x=8\) is the directrix of the ellipse \(\mathrm{E}:\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) with the corresponding focus \((2,0)\). If the tangent to \(\mathrm{E}\) at the point \(\mathrm{P}\) in the first quadrant passes thro...

Let \(f:\mathbb{R}-{0,1}\to \mathbb{R}\) be a function such that \(f(x)+f\left(\frac{1}{1-x}\right)=1+x\). Then \(f(2)\) is equal to

Let \(\mathrm{P}\left(x_{0}, y_{0}\right)\) be the point on the hyperbola \(3 x^{2}-4 y^{2}=36\), which is nearest to the line \(3 x+2 y=1\). Then \(\sqrt{2}\left(y_{0}-x_{0}\right)\) is equal to :

Let \(S = \left\{ {x \in R:0 < x < 1\,\mathrm{and}\,2{{\tan }^{ - 1}}\left( {{{1 - x} \over {1 + x}}} \right) = {{\cos }^{ - 1}}\left( {{{1 - {x^2}} \over {1 + {x^2}}}} \right)} \right\}\).
If \(\mathrm{n(S)}\) denotes the number of element...

Which of the following statements is a tautology?

For the system of linear equations \(\alpha x+y+z=1,x+\alpha y+z=1,x+y+\alpha z=\beta\), which one of the following statements is NOT correct?

If \(A = {1 \over 2}\left[ {\matrix{ 1 & {\sqrt 3 } \cr { - \sqrt 3 } & 1 \cr } } \right]\), then :

If the \(x\)-intercept of a focal chord of the parabola \(y^{2}=8x+4y+4\) is 3, then the length of this chord is equal to ____________.

Number of integral solutions to the equation \(x+y+z=21\), where \(x \ge 1,y\ge3,z\ge4\), is equal to ____________.

The total number of six digit numbers, formed using the digits 4, 5, 9 only and divisible by 6, is ____________.

Two dice are thrown independently. Let \(\mathrm{A}\) be the event that the number appeared on the \(1^{\text {st }}\) die is less than the number appeared on the \(2^{\text {nd }}\) die, \(\mathrm{B}\) be the event that the number appeared...

The number of integral values of k, for which one root of the equation \(2x^2-8x+k=0\) lies in the interval (1, 2) and its other root lies in the interval (2, 3), is :

The sum \(\sum\limits_{n = 1}^\infty {{{2{n^2} + 3n + 4} \over {(2n)!}}}\) is equal to :

The sum of the common terms of the following three arithmetic progressions.
\(3,7,11,15, \ldots ., 399\),
\(2,5,8,11, \ldots ., 359\) and
\(2,7,12,17, \ldots ., 197\),
is equal to _____________.

Let \(P(S)\) denote the power set of \(S=\{1,2,3, \ldots ., 10\}\). Define the relations \(R_{1}\) and \(R_{2}\) on \(P(S)\) as \(\mathrm{AR}_{1} \mathrm{~B}\) if $$\left(\mathrm{A} \cap \mathrm{B}^{\mathrm{c}}\right) \cup\left(\mathrm{B} \...

Let \(9=x_{1} < x_{2} < \ldots < x_{7}\) be in an A.P. with common difference d. If the standard deviation of \(x_{1}, x_{2}..., x_{7}\) is 4 and the mean is \(\bar{x}\), then \(\bar{x}+x_{6}\) is equal to :

Let \(\vec{a}=5 \hat{i}-\hat{j}-3 \hat{k}\) and \(\vec{b}=\hat{i}+3 \hat{j}+5 \hat{k}\) be two vectors. Then which one of the following statements is TRUE ?

Let \(\vec{a}=2 \hat{i}-7 \hat{j}+5 \hat{k}, \vec{b}=\hat{i}+\hat{k}\) and \(\vec{c}=\hat{i}+2 \hat{j}-3 \hat{k}\) be three given vectors. If \(\overrightarrow{\mathrm{r}}\) is a vector such that $$\vec{r} \times \vec{a}=\vec{c} \times \vec...