JEE Main 2023 (Online) 6th April Morning Shift

If the equation of the plane passing through the line of intersection of the planes \(2 x-y+z=3,4 x-3 y+5 z+9=0\) and parallel to the line \(\frac{x+1}{-2}=\frac{y+3}{4}=\frac{z-2}{5}\) is \(a x+b y+c z+6=0\), then \(a+b+c\) is equal to :

One vertex of a rectangular parallelopiped is at the origin \(\mathrm{O}\) and the lengths of its edges along \(x, y\) and \(z\) axes are \(3,4\) and \(5\) units respectively. Let \(\mathrm{P}\) be the vertex \((3,4,5)\). Then the shortest ...

Let the image of the point \(\mathrm{P}(1,2,3)\) in the plane \(2 x-y+z=9\) be \(\mathrm{Q}\). If the coordinates of the point \(\mathrm{R}\) are \((6,10,7)\), then the square of the area of the triangle \(\mathrm{PQR}\) is _____________.

If the area of the region \(S=\left\{(x, y): 2 y-y^{2} \leq x^{2} \leq 2 y, x \geq y\right\}\) is equal to \(\frac{n+2}{n+1}-\frac{\pi}{n-1}\), then the natural number \(n\) is equal to ___________.

If \({ }^{2 n} C_{3}:{ }^{n} C_{3}=10: 1\), then the ratio \(\left(n^{2}+3 n\right):\left(n^{2}-3 n+4\right)\) is :

If the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of \(\left(\sqrt[4]{2}+\frac{1}{\sqrt[4]{3}}\right)^{\mathrm{n}}\) is \(\sqrt{6}: 1\), then the third term from the beginning is :

The coefficient of \(x^{18}\) in the expansion of \(\left(x^{4}-\frac{1}{x^{3}}\right)^{15}\) is __________.

Let the point \((p, p+1)\) lie inside the region \(E=\left\{(x, y): 3-x \leq y \leq \sqrt{9-x^{2}}, 0 \leq x \leq 3\right\}\). If the set of all values of \(\mathrm{p}\) is the interval \((a, b)\), then \(b^{2}+b-a^{2}\) is equal to _______...

A circle passing through the point \(P(\alpha, \beta)\) in the first quadrant touches the two coordinate axes at the points \(A\) and \(B\). The point \(P\) is above the line \(A B\). The point \(Q\) on the line segment \(A B\) is the foot...

Let \(5 f(x)+4 f\left(\frac{1}{x}\right)=\frac{1}{x}+3, x > 0\). Then \(18 \int_\limits{1}^{2} f(x) d x\) is equal to :

Let \(y=y(x)\) be a solution of the differential equation \((x \cos x) d y+(x y \sin x+y \cos x-1) d x=0,0 < x < \frac{\pi}{2}\). If \(\frac{\pi}{3} y\left(\frac{\pi}{3}\right)=\sqrt{3}\), then $$\left|\frac{\pi}{6} y^{\prime \prime}\left(\...

If \(2 x^{y}+3 y^{x}=20\), then \(\frac{d y}{d x}\) at \((2,2)\) is equal to :

From the top \(\mathrm{A}\) of a vertical wall \(\mathrm{AB}\) of height \(30 \mathrm{~m}\), the angles of depression of the top \(\mathrm{P}\) and bottom \(\mathrm{Q}\) of a vertical tower \(\mathrm{PQ}\) are \(15^{\circ}\) and $$60^{\circ...

Let \(I(x)=\int \frac{x^{2}\left(x \sec ^{2} x+\tan x\right)}{(x \tan x+1)^{2}} d x\). If \(I(0)=0\), then \(I\left(\frac{\pi}{4}\right)\) is equal to :

Let \(a_{1}, a_{2}, a_{3}, \ldots, a_{\mathrm{n}}\) be \(\mathrm{n}\) positive consecutive terms of an arithmetic progression. If \(\mathrm{d} > 0\) is its common difference, then
$$\lim_\limits{n \rightarrow \infty} \sqrt{\frac{d}{n}}\left...

Let \(a \in \mathbb{Z}\) and \([\mathrm{t}]\) be the greatest integer \(\leq \mathrm{t}\). Then the number of points, where the function \(f(x)=[a+13 \sin x], x \in(0, \pi)\) is not differentiable, is __________.

Statement \(\mathrm{(P \Rightarrow Q) \wedge(R \Rightarrow Q)}\) is logically equivalent to :

If the system of equations
\(x+y+a z=b\)
\(2 x+5 y+2 z=6\)
\(x+2 y+3 z=3\)
has infinitely many solutions, then \(2 a+3 b\) is equal to :

Let \(\mathrm{A}=\left[\mathrm{a}_{\mathrm{ij}}\right]_{2 \times 2}\), where \(\mathrm{a}_{\mathrm{ij}} \neq 0\) for all \(\mathrm{i}, \mathrm{j}\) and \(\mathrm{A}^{2}=\mathrm{I}\). Let a be the sum of all diagonal elements of $$\mathrm{A}...

Let the tangent to the curve \(x^{2}+2 x-4 y+9=0\) at the point \(\mathrm{P}(1,3)\) on it meet the \(y\)-axis at \(\mathrm{A}\). Let the line passing through \(\mathrm{P}\) and parallel to the line \(x-3 y=6\) meet the parabola $$y^{2}=4 x$...

The number of ways of giving 20 distinct oranges to 3 children such that each child gets at least one orange is ___________.

A pair of dice is thrown 5 times. For each throw, a total of 5 is considered a success. If the probability of at least 4 successes is \(\frac{k}{3^{11}}\), then \(k\) is equal to :

Let \(A = \{ x \in R:[x + 3] + [x + 4] \le 3\} ,\)
\(B = \left\{ {x \in R:{3^x}{{\left( {\sum\limits_{r = 1}^\infty {{3 \over {{{10}^r}}}} } \right)}^{x - 3}} < {3^{ - 3x}}} \right\},\) where [t] denotes greatest integer function. Then,

The sum of all the roots of the equation \(\left|x^{2}-8 x+15\right|-2 x+7=0\) is :

The sum of the first \(20\) terms of the series \(5+11+19+29+41+\ldots\) is :

Let \(\mathrm{A}=\{1,2,3,4, \ldots ., 10\}\) and \(\mathrm{B}=\{0,1,2,3,4\}\). The number of elements in the relation \(R=\left\{(a, b) \in A \times A: 2(a-b)^{2}+3(a-b) \in B\right\}\) is ___________.

The mean and variance of a set of 15 numbers are 12 and 14 respectively. The mean and variance of another set of 15 numbers are 14 and \(\sigma^{2}\) respectively. If the variance of all the 30 numbers in the two sets is 13 , then $$\sigma^...

The straight lines \(\mathrm{l_{1}}\) and \(\mathrm{l_{2}}\) pass through the origin and trisect the line segment of the line L : \(9 x+5 y=45\) between the axes. If \(\mathrm{m}_{1}\) and \(\mathrm{m}_{2}\) are the slopes of the lines $$\m...

Let the position vectors of the points A, B, C and D be
\(5 \hat{i}+5 \hat{j}+2 \lambda \hat{k}, \hat{i}+2 \hat{j}+3 \hat{k},-2 \hat{i}+\lambda \hat{j}+4 \hat{k}\) and \(-\hat{i}+5 \hat{j}+6 \hat{k}\). Let the set $$S=\{\lambda \in \mathbb{...

Let \(\vec{a}=2 \hat{i}+3 \hat{j}+4 \hat{k}, \vec{b}=\hat{i}-2 \hat{j}-2 \hat{k}\) and \(\vec{c}=-\hat{i}+4 \hat{j}+3 \hat{k}\).

If \(\vec{d}\) is a vector perpendicular to both \(\vec{b}\) and \(\vec{c}\), and \(\vec{a} \cdot \vec{d}=18\)...