JEE Main 2024 (Online) 30th January Morning Shift

Let \((\alpha, \beta, \gamma)\) be the foot of perpendicular from the point \((1,2,3)\) on the line \(\frac{x+3}{5}=\frac{y-1}{2}=\frac{z+4}{3}\). Then \(19(\alpha+\beta+\gamma)\) is equal to :

Let \(A(2,3,5)\) and \(C(-3,4,-2)\) be opposite vertices of a parallelogram \(A B C D\). If the diagonal \(\overrightarrow{\mathrm{BD}}=\hat{i}+2 \hat{j}+3 \hat{k}\), then the area of the parallelogram is equal to :

If \(\mathrm{d}_1\) is the shortest distance between the lines \(x+1=2 y=-12 z, x=y+2=6 z-6\) and \(\mathrm{d}_2\) is the shortest distance between the lines $$\frac{x-1}{2}=\frac{y+8}{-7}=\frac{z-4}{5}, \frac{x-1}{2}=\frac{y-2}{1}=\frac{z-...

The maximum area of a triangle whose one vertex is at \((0,0)\) and the other two vertices lie on the curve \(y=-2 x^2+54\) at points \((x, y)\) and \((-x, y)\), where \(y>0\), is :

The area (in square units) of the region bounded by the parabola \(y^2=4(x-2)\) and the line \(y=2 x-8\), is :

\(\text { Number of integral terms in the expansion of }\left\{7^{\left(\frac{1}{2}\right)}+11^{\left(\frac{1}{6}\right)}\right\}^{824} \text { is equal to _________. }\)

If the circles \((x+1)^2+(y+2)^2=r^2\) and \(x^2+y^2-4 x-4 y+4=0\) intersect at exactly two distinct points, then

If \(z=x+i y, x y \neq 0\), satisfies the equation \(z^2+i \bar{z}=0\), then \(\left|z^2\right|\) is equal to :

The value of \(\lim _\limits{n \rightarrow \infty} \sum_\limits{k=1}^n \frac{n^3}{\left(n^2+k^2\right)\left(n^2+3 k^2\right)}\) is :

Let \(f:\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \rightarrow \mathbf{R}\) be a differentiable function such that \(f(0)=\frac{1}{2}\). If the $$\lim _\limits{x \rightarrow 0} \frac{x \int_0^x f(\mathrm{t}) \mathrm{dt}}{\mathrm{e}^{x^2}-1}...

The value of \(9 \int_\limits0^9\left[\sqrt{\frac{10 x}{x+1}}\right] \mathrm{d} x\), where \([t]\) denotes the greatest integer less than or equal to \(t\), is

Let \(y=y(x)\) be the solution of the differential equation \(\sec x \mathrm{~d} y+\{2(1-x) \tan x+x(2-x)\} \mathrm{d} x=0\) such that \(y(0)=2\). Then \(y(2)\) is equal to:

Let \(y=y(x)\) be the solution of the differential equation \(\left(1-x^2\right) \mathrm{d} y=\left[x y+\left(x^3+2\right) \sqrt{3\left(1-x^2\right)}\right] \mathrm{d} x, -1< x<1, y(0)=0\). If $$y\left(\frac{1}{2}\right)=\frac{\mathrm{m}}{\...

Let \(g: \mathbf{R} \rightarrow \mathbf{R}\) be a non constant twice differentiable function such that \(\mathrm{g}^{\prime}\left(\frac{1}{2}\right)=\mathrm{g}^{\prime}\left(\frac{3}{2}\right)\). If a real valued function \(f\) is defined a...

If $$f(x)=\left|\begin{array}{ccc} 2 \cos ^4 x & 2 \sin ^4 x & 3+\sin ^2 2 x \\ 3+2 \cos ^4 x & 2 \sin ^4 x & \sin ^2 2 x \\ 2 \cos ^4 x & 3+2 \sin ^4 x & \sin ^2 2 x \end{array}\right|,$$ then \(\frac{1}{5} f^{\prime}(0)=\) is equal to :

If the length of the minor axis of an ellipse is equal to half of the distance between the foci, then the eccentricity of the ellipse is :

If the domain of the function \(f(x)=\cos ^{-1}\left(\frac{2-|x|}{4}\right)+\left\{\log _e(3-x)\right\}^{-1}\) is \([-\alpha, \beta)-\{\gamma\}\), then \(\alpha+\beta+\gamma\) is equal to :

Let \(\mathrm{A}=\{1,2,3, \ldots, 7\}\) and let \(\mathrm{P}(\mathrm{A})\) denote the power set of \(\mathrm{A}\). If the number of functions \(f: \mathrm{A} \rightarrow \mathrm{P}(\mathrm{A})\) such that $$\mathrm{a} \in f(\mathrm{a}), \fo...

Let the latus rectum of the hyperbola \(\frac{x^2}{9}-\frac{y^2}{b^2}=1\) subtend an angle of \(\frac{\pi}{3}\) at the centre of the hyperbola. If \(\mathrm{b}^2\) is equal to \(\frac{l}{\mathrm{~m}}(1+\sqrt{\mathrm{n}})\), where \(l\) and ...

If the function
$$f(x)= \begin{cases}\frac{1}{|x|}, & |x| \geqslant 2 \\ \mathrm{a} x^2+2 \mathrm{~b}, & |x|<2\end{cases}$$
is differentiable on \(\mathbf{R}\), then \(48(a+b)\) is equal to __________.

Consider the system of linear equations \(x+y+z=4 \mu, x+2 y+2 \lambda z=10 \mu, x+3 y+4 \lambda^2 z=\mu^2+15\) where \(\lambda, \mu \in \mathbf{R}\). Which one of the following statements is NOT correct ?

Two integers \(x\) and \(y\) are chosen with replacement from the set \(\{0,1,2,3, \ldots, 10\}\). Then the probability that \(|x-y|>5\), is :

A group of 40 students appeared in an examination of 3 subjects - Mathematics, Physics and Chemistry. It was found that all students passed in atleast one of the subjects, 20 students passed in Mathematics, 25 students passed in Physics, 16...

Let \(\alpha, \beta \in \mathbf{N}\) be roots of the equation \(x^2-70 x+\lambda=0\), where \(\frac{\lambda}{2}, \frac{\lambda}{3} \notin \mathbf{N}\). If \(\lambda\) assumes the minimum possible value, then $$\frac{(\sqrt{\alpha-1}+\sqrt{\...

Let \(S_n\) denote the sum of first \(n\) terms of an arithmetic progression. If \(S_{20}=790\) and \(S_{10}=145\), then \(\mathrm{S}_{15}-\mathrm{S}_5\) is :

Let \(\alpha=1^2+4^2+8^2+13^2+19^2+26^2+\ldots\) upto 10 terms and \(\beta=\sum_\limits{n=1}^{10} n^4\). If \(4 \alpha-\beta=55 k+40\), then \(\mathrm{k}\) is equal to __________.

Let M denote the median of the following frequency distribution

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A line passing through the point \(\mathrm{A}(9,0)\) makes an angle of \(30^{\circ}\) with the positive direction of \(x\)-axis. If this line is rotated about A through an angle of \(15^{\circ}\) in the clockwise direction, then its equatio...

If \(2 \sin ^3 x+\sin 2 x \cos x+4 \sin x-4=0\) has exactly 3 solutions in the interval \(\left[0, \frac{\mathrm{n} \pi}{2}\right], \mathrm{n} \in \mathrm{N}\), then the roots of the equation \(x^2+\mathrm{n} x+(\mathrm{n}-3)=0\) belong to ...

Let \(\overrightarrow{\mathrm{a}}=\mathrm{a}_1 \hat{i}+\mathrm{a}_2 \hat{j}+\mathrm{a}_3 \hat{k}\) and \(\overrightarrow{\mathrm{b}}=\mathrm{b}_1 \hat{i}+\mathrm{b}_2 \hat{j}+\mathrm{b}_3 \hat{k}\) be two vectors such that $$|\overrightarro...