JEE Main 2023 (Online) 13th April Evening Shift

The line, that is coplanar to the line \(\frac{x+3}{-3}=\frac{y-1}{1}=\frac{z-5}{5}\), is :

The plane, passing through the points \((0,-1,2)\) and \((-1,2,1)\) and parallel to the line passing through \((5,1,-7)\) and \((1,-1,-1)\), also passes through the point :

Let \(\mathrm{N}\) be the foot of perpendicular from the point \(\mathrm{P}(1,-2,3)\) on the line passing through the points \((4,5,8)\) and \((1,-7,5)\). Then the distance of \(N\) from the plane \(2 x-2 y+z+5=0\) is :

The area of the region \(\left\{(x, y): x^{2} \leq y \leq\left|x^{2}-4\right|, y \geq 1\right\}\) is

The coefficient of \(x^{5}\) in the expansion of \(\left(2 x^{3}-\frac{1}{3 x^{2}}\right)^{5}\) is :

The remainder, when \(7^{103}\) is divided by 17, is __________

Let the centre of a circle C be \((\alpha, \beta)\) and its radius \(r < 8\). Let \(3 x+4 y=24\) and \(3 x-4 y=32\) be two tangents and \(4 x+3 y=1\) be a normal to C. Then \((\alpha-\beta+r)\) is equal to :

Let \(S=\left\{z \in \mathbb{C}: \bar{z}=i\left(z^{2}+\operatorname{Re}(\bar{z})\right)\right\}\). Then \(\sum_\limits{z \in \mathrm{S}}|z|^{2}\) is equal to :

The value of \({{{e^{ - {\pi \over 4}}} + \int\limits_0^{{\pi \over 4}} {{e^{ - x}}{{\tan }^{50}}xdx} } \over {\int\limits_0^{{\pi \over 4}} {{e^{ - x}}({{\tan }^{49}}x + {{\tan }^{51}}x)dx} }}\) is

Let \(f_{n}=\int_\limits{0}^{\frac{\pi}{2}}\left(\sum_\limits{k=1}^{n} \sin ^{k-1} x\right)\left(\sum_\limits{k=1}^{n}(2 k-1) \sin ^{k-1} x\right) \cos x d x, n \in \mathbb{N}\). Then \(f_{21}-f_{20}\) is equal to _________

If \(y=y(x)\) is the solution of the differential equation \(\frac{d y}{d x}+\frac{4 x}{\left(x^{2}-1\right)} y=\frac{x+2}{\left(x^{2}-1\right)^{\frac{5}{2}}}, x > 1\) such that $$y(2)=\frac{2}{9} \log _{e}(2+\sqrt{3}) \text { and } y(\sqrt...

Let \(f(x)=\sum_\limits{k=1}^{10} k x^{k}, x \in \mathbb{R}\). If \(2 f(2)+f^{\prime}(2)=119(2)^{\mathrm{n}}+1\) then \(\mathrm{n}\) is equal to ___________

The range of \(f(x)=4 \sin ^{-1}\left(\frac{x^{2}}{x^{2}+1}\right)\) is

The foci of a hyperbola are \(( \pm 2,0)\) and its eccentricity is \(\frac{3}{2}\). A tangent, perpendicular to the line \(2 x+3 y=6\), is drawn at a point in the first quadrant on the hyperbola. If the intercepts made by the tangent on the...

For \(x \in(-1,1]\), the number of solutions of the equation \(\sin ^{-1} x=2 \tan ^{-1} x\) is equal to __________.

If \(\lim_\limits{x \rightarrow 0} \frac{e^{a x}-\cos (b x)-\frac{cx e^{-c x}}{2}}{1-\cos (2 x)}=17\), then \(5 a^{2}+b^{2}\) is equal to

The statement \((p \wedge(\sim q)) \vee((\sim p) \wedge q) \vee((\sim p) \wedge(\sim q))\) is equivalent to _________.

Let for \(A = \left[ {\matrix{ 1 & 2 & 3 \cr \alpha & 3 & 1 \cr 1 & 1 & 2 \cr } } \right],|A| = 2\). If \(\mathrm{|2\,adj\,(2\,adj\,(2A))| = {32^n}}\), then \(3n + \alpha\) is equal to

If the system of equations
\(2 x+y-z=5\)
\(2 x-5 y+\lambda z=\mu\)
\(x+2 y-5 z=7\)
has infinitely many solutions, then \((\lambda+\mu)^{2}+(\lambda-\mu)^{2}\) is equal to

All words, with or without meaning, are made using all the letters of the word MONDAY. These words are written as in a dictionary with serial numbers. The serial number of the word MONDAY is :

Total numbers of 3-digit numbers that are divisible by 6 and can be formed by using the digits \(1,2,3,4,5\) with repetition, is _________.

The random variable \(\mathrm{X}\) follows binomial distribution \(\mathrm{B}(\mathrm{n}, \mathrm{p})\), for which the difference of the mean and the variance is 1 . If \(2 \mathrm{P}(\mathrm{X}=2)=3 \mathrm{P}(\mathrm{X}=1)\), then $$n^{2}...

Let \(\alpha, \beta\) be the roots of the equation \(x^{2}-\sqrt{2} x+2=0\). Then \(\alpha^{14}+\beta^{14}\) is equal to

Let \([\alpha]\) denote the greatest integer \(\leq \alpha\). Then \([\sqrt{1}]+[\sqrt{2}]+[\sqrt{3}]+\ldots+[\sqrt{120}]\) is equal to __________

Let a\(_1\), a\(_2\), a\(_3\), .... be a G.P. of increasing positive numbers. Let the sum of its 6th and 8th terms be 2 and the product of its 3rd and 5th terms be \(\frac{1}{9}\). Then \(6(a_2+a_4)(a_4+a_6)\) is equal to

Let \(\mathrm{A}=\{-4,-3,-2,0,1,3,4\}\) and \(\mathrm{R}=\left\{(a, b) \in \mathrm{A} \times \mathrm{A}: b=|a|\right.\) or \(\left.b^{2}=a+1\right\}\) be a relation on \(\mathrm{A}\). Then the minimum number of elements, that must be added ...

The mean and standard deviation of the marks of 10 students were found to be 50 and 12 respectively. Later, it was observed that two marks 20 and 25 were wrongly read as 45 and 50 respectively. Then the correct variance is _________

Let \((\alpha, \beta)\) be the centroid of the triangle formed by the lines \(15 x-y=82,6 x-5 y=-4\) and \(9 x+4 y=17\). Then \(\alpha+2 \beta\) and \(2 \alpha-\beta\) are the roots of the equation :

Let \(|\vec{a}|=2,|\vec{b}|=3\) and the angle between the vectors \(\vec{a}\) and \(\vec{b}\) be \(\frac{\pi}{4}\). Then \(|(\vec{a}+2 \vec{b}) \times(2 \vec{a}-3 \vec{b})|^{2}\) is equal to :

Let for a triangle \(\mathrm{ABC}\),
\(\overrightarrow{\mathrm{AB}}=-2 \hat{i}+\hat{j}+3 \hat{k}\)
\(\overrightarrow{\mathrm{CB}}=\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k}\)
$$\overrightarrow{\mathrm{CA}}=4 \hat{i}+3 \hat{j}+\delta \hat{k...