JEE Main 2024 (Online) 29th January Evening Shift

Let \(\mathrm{P}(3,2,3), \mathrm{Q}(4,6,2)\) and \(\mathrm{R}(7,3,2)\) be the vertices of \(\triangle \mathrm{PQR}\). Then, the angle \(\angle \mathrm{QPR}\) is

Let O be the origin, and M and \(\mathrm{N}\) be the points on the lines \(\frac{x-5}{4}=\frac{y-4}{1}=\frac{z-5}{3}\) and \(\frac{x+8}{12}=\frac{y+2}{5}=\frac{z+11}{9}\) respectively such that \(\mathrm{MN}\) is the shortest distance betwe...

The function \(f(x)=\frac{x}{x^2-6 x-16}, x \in \mathbb{R}-\{-2,8\}\)

The function \(f(x)=2 x+3(x)^{\frac{2}{3}}, x \in \mathbb{R}\), has

Let the area of the region \(\left\{(x, y): 0 \leq x \leq 3,0 \leq y \leq \min \left\{x^2+2,2 x+2\right\}\right\}\) be A. Then \(12 \mathrm{~A}\) is equal to __________.

Remainder when \(64^{32^{32}}\) is divided by 9 is equal to ________.

Let \(\mathrm{r}\) and \(\theta\) respectively be the modulus and amplitude of the complex number \(z=2-i\left(2 \tan \frac{5 \pi}{8}\right)\), then \((\mathrm{r}, \theta)\) is equal to

Let \(\alpha, \beta\) be the roots of the equation \(x^2-\sqrt{6} x+3=0\) such that \(\operatorname{Im}(\alpha)>\operatorname{Im}(\beta)\). Let \(a, b\) be integers not divisible by 3 and \(n\) be a natural number such that $$\frac{\alpha^{...

Let the slope of the line \(45 x+5 y+3=0\) be \(27 r_1+\frac{9 r_2}{2}\) for some \(r_1, r_2 \in \mathbb{R}\). Then \(\lim _\limits{x \rightarrow 3}\left(\int_3^x \frac{8 t^2}{\frac{3 r_2 x}{2}-r_2 x^2-r_1 x^3-3 x} d t\right)\) is equal to ...

If \(\int_\limits{\frac{\pi}{6}}^{\frac{\pi}{3}} \sqrt{1-\sin 2 x} d x=\alpha+\beta \sqrt{2}+\gamma \sqrt{3}\), where \(\alpha, \beta\) and \(\gamma\) are rational numbers, then
\(3 \alpha+4 \beta-\gamma\) is equal to _________.

If \(\sin \left(\frac{y}{x}\right)=\log _e|x|+\frac{\alpha}{2}\) is the solution of the differential equation \(x \cos \left(\frac{y}{x}\right) \frac{d y}{d x}=y \cos \left(\frac{y}{x}\right)+x\) and \(y(1)=\frac{\pi}{3}\), then $$\alpha^2$...

\(\text { Let } y=\log _e\left(\frac{1-x^2}{1+x^2}\right),-1 < x<1 \text {. Then at } x=\frac{1}{2} \text {, the value of } 225\left(y^{\prime}-y^{\prime \prime}\right) \text { is equal to }\)

If \(\int \frac{\sin ^{\frac{3}{2}} x+\cos ^{\frac{3}{2}} x}{\sqrt{\sin ^3 x \cos ^3 x \sin (x-\theta)}} d x=A \sqrt{\cos \theta \tan x-\sin \theta}+B \sqrt{\cos \theta-\sin \theta \cot x}+C\), where \(C\) is the integration constant, then ...

Let \(x=\frac{m}{n}\) (\(m, n\) are co-prime natural numbers) be a solution of the equation \(\cos \left(2 \sin ^{-1} x\right)=\frac{1}{9}\) and let \(\alpha, \beta(\alpha >\beta)\) be the roots of the equation \(m x^2-n x-m+ n=0\). Then th...

Let \(f(x)=\sqrt{\lim _\limits{r \rightarrow x}\left\{\frac{2 r^2\left[(f(r))^2-f(x) f(r)\right]}{r^2-x^2}-r^3 e^{\frac{f(r)}{r}}\right\}}\) be differentiable in \((-\infty, 0) \cup(0, \infty)\) and \(f(1)=1\). Then the value of ea, such th...

Let $$A=\left[\begin{array}{ccc}2 & 1 & 2 \\ 6 & 2 & 11 \\ 3 & 3 & 2\end{array}\right]$$ and $$P=\left[\begin{array}{lll}1 & 2 & 0 \\ 5 & 0 & 2 \\ 7 & 1 & 5\end{array}\right]$$. The sum of the prime factors of $$\left|P^{-1} A P-2 I\right|$...

Let for any three distinct consecutive terms \(a, b, c\) of an A.P, the lines \(a x+b y+c=0\) be concurrent at the point \(P\) and \(Q(\alpha, \beta)\) be a point such that the system of equations
$$\begin{aligned}
& x+y+z=6, \\
& 2 x+5 y+\...

Let \(P(\alpha, \beta)\) be a point on the parabola \(y^2=4 x\). If \(P\) also lies on the chord of the parabola \(x^2=8 y\) whose mid point is \(\left(1, \frac{5}{4}\right)\), then \((\alpha-28)(\beta-8)\) is equal to _________.

Number of ways of arranging 8 identical books into 4 identical shelves where any number of shelves may remain empty is equal to

An integer is chosen at random from the integers \(1,2,3, \ldots, 50\). The probability that the chosen integer is a multiple of atleast one of 4, 6 and 7 is

Let the set \(C=\left\{(x, y) \mid x^2-2^y=2023, x, y \in \mathbb{N}\right\}\). Then \(\sum_\limits{(x, y) \in C}(x+y)\) is equal to _________.

If \(\log _e \mathrm{a}, \log _e \mathrm{~b}, \log _e \mathrm{c}\) are in an A.P. and \(\log _e \mathrm{a}-\log _e 2 \mathrm{~b}, \log _e 2 \mathrm{~b}-\log _e 3 \mathrm{c}, \log _e 3 \mathrm{c} -\log _e\) a are also in an A.P, then $$a: b:...

If each term of a geometric progression \(a_1, a_2, a_3, \ldots\) with \(a_1=\frac{1}{8}\) and \(a_2 \neq a_1\), is the arithmetic mean of the next two terms and \(S_n=a_1+a_2+\ldots . .+a_n\), then \(S_{20}-S_{18}\) is equal to

If R is the smallest equivalence relation on the set \(\{1,2,3,4\}\) such that \(\{(1,2),(1,3)\} \subset \mathrm{R}\), then the number of elements in \(\mathrm{R}\) is __________.

If the mean and variance of five observations are \(\frac{24}{5}\) and \(\frac{194}{25}\) respectively and the mean of the first four observations is \(\frac{7}{2}\), then the variance of the first four observations in equal to

Let \(\mathrm{A}\) be the point of intersection of the lines \(3 x+2 y=14,5 x-y=6\) and \(\mathrm{B}\) be the point of intersection of the lines \(4 x+3 y=8,6 x+y=5\). The distance of the point \(P(5,-2)\) from the line \(\mathrm{AB}\) is

The distance of the point \((2,3)\) from the line \(2 x-3 y+28=0\), measured parallel to the line \(\sqrt{3} x-y+1=0\), is equal to

The sum of the solutions \(x \in \mathbb{R}\) of the equation \(\frac{3 \cos 2 x+\cos ^3 2 x}{\cos ^6 x-\sin ^6 x}=x^3-x^2+6\) is

Let a unit vector \(\hat{u}=x \hat{i}+y \hat{j}+z \hat{k}\) make angles \(\frac{\pi}{2}, \frac{\pi}{3}\) and \(\frac{2 \pi}{3}\) with the vectors $$\frac{1}{\sqrt{2}} \hat{i}+\frac{1}{\sqrt{2}} \hat{k}, \frac{1}{\sqrt{2}} \hat{j}+\frac{1}{\...

Let \(\overrightarrow{O A}=\vec{a}, \overrightarrow{O B}=12 \vec{a}+4 \vec{b} \text { and } \overrightarrow{O C}=\vec{b}\), where O is the origin. If S is the parallelogram with adjacent sides OA and OC, then $$\mathrm{{{area\,of\,the\,quad...