JEE Main 2023 (Online) 13th April Morning Shift

Let the equation of plane passing through the line of intersection of the planes \(x+2 y+a z=2\) and \(x-y+z=3\) be \(5 x-11 y+b z=6 a-1\). For \(c \in \mathbb{Z}\), if the distance of this plane from the point \((a,-c, c)\) is $$\frac{2}{\...

The distance of the point \((-1,2,3)\) from the plane \(\vec{r} \cdot(\hat{i}-2 \hat{j}+3 \hat{k})=10\) parallel to the line of the shortest distance between the lines \(\vec{r}=(\hat{i}-\hat{j})+\lambda(2 \hat{i}+\hat{k})\) and $$\vec{r}=(...

Let the image of the point \(\left(\frac{5}{3}, \frac{5}{3}, \frac{8}{3}\right)\) in the plane \(x-2 y+z-2=0\) be P. If the distance of the point \(Q(6,-2, \alpha), \alpha > 0\), from \(\mathrm{P}\) is 13 , then \(\alpha\) is equal to _____...

\(\max _\limits{0 \leq x \leq \pi}\left\{x-2 \sin x \cos x+\frac{1}{3} \sin 3 x\right\}=\)

The area of the region enclosed by the curve
\(f(x)=\max \{\sin x, \cos x\},-\pi \leq x \leq \pi\) and the \(x\)-axis is

Fractional part of the number \(\frac{4^{2022}}{15}\) is equal to

Let \(\alpha\) be the constant term in the binomial expansion of \(\left(\sqrt{x}-\frac{6}{x^{\frac{3}{2}}}\right)^{n}, n \leq 15\). If the sum of the coefficients of the remaining terms in the expansion is 649 and the coefficient of $$x^{-...

Let \(w=z \bar{z}+k_{1} z+k_{2} i z+\lambda(1+i), k_{1}, k_{2} \in \mathbb{R}\). Let \(\operatorname{Re}(w)=0\) be the circle \(\mathrm{C}\) of radius 1 in the first quadrant touching the line \(y=1\) and the \(y\)-axis. If the curve $$\ope...

Among
(S1): \(\lim_\limits{n \rightarrow \infty} \frac{1}{n^{2}}(2+4+6+\ldots \ldots+2 n)=1\)
(S2) : \(\lim_\limits{n \rightarrow \infty} \frac{1}{n^{16}}\left(1^{15}+2^{15}+3^{15}+\ldots \ldots+n^{15}\right)=\frac{1}{16}\)

\(\int_\limits{0}^{\infty} \frac{6}{e^{3 x}+6 e^{2 x}+11 e^{x}+6} d x=\)

Let for \(x \in \mathbb{R}, S_{0}(x)=x, S_{k}(x)=C_{k} x+k \int_{0}^{x} S_{k-1}(t) d t\), where
\(C_{0}=1, C_{k}=1-\int_{0}^{1} S_{k-1}(x) d x, k=1,2,3, \ldots\) Then \(S_{2}(3)+6 C_{3}\) is equal to ____________.

Let \(y=y_{1}(x)\) and \(y=y_{2}(x)\) be the solution curves of the differential equation \(\frac{d y}{d x}=y+7\) with initial conditions \(y_{1}(0)=0\) and \(y_{2}(0)=1\) respectively. Then the curves \(y=y_{1}(x)\) and \(y=y_{2}(x)\) inte...

For the differentiable function \(f: \mathbb{R}-\{0\} \rightarrow \mathbb{R}\), let \(3 f(x)+2 f\left(\frac{1}{x}\right)=\frac{1}{x}-10\), then \(\left|f(3)+f^{\prime}\left(\frac{1}{4}\right)\right|\) is equal to

Let the tangent and normal at the point \((3 \sqrt{3}, 1)\) on the ellipse \(\frac{x^{2}}{36}+\frac{y^{2}}{4}=1\) meet the \(y\)-axis at the points \(A\) and \(B\) respectively. Let the circle \(C\) be drawn taking \(A B\) as a diameter and...

For \(x \in \mathbb{R}\), two real valued functions \(f(x)\) and \(g(x)\) are such that, \(g(x)=\sqrt{x}+1\) and \(f \circ g(x)=x+3-\sqrt{x}\). Then \(f(0)\) is equal to

Let \(m_{1}\) and \(m_{2}\) be the slopes of the tangents drawn from the point \(\mathrm{P}(4,1)\) to the hyperbola \(H: \frac{y^{2}}{25}-\frac{x^{2}}{16}=1\). If \(\mathrm{Q}\) is the point from which the tangents drawn to \(\mathrm{H}\) h...

If \(S=\left\{x \in \mathbb{R}: \sin ^{-1}\left(\frac{x+1}{\sqrt{x^{2}+2 x+2}}\right)-\sin ^{-1}\left(\frac{x}{\sqrt{x^{2}+1}}\right)=\frac{\pi}{4}\right\}\), then $$\sum_\limits{x \in s}\left(\sin \left(\left(x^{2}+x+5\right) \frac{\pi}{2}...

The negation of the statement \(((A \wedge(B \vee C)) \Rightarrow(A \vee B)) \Rightarrow A\) is

For the system of linear equations
\(2 x+4 y+2 a z=b\)
\(x+2 y+3 z=4\)
\(2 x-5 y+2 z=8\)
which of the following is NOT correct?

Let $$B=\left[\begin{array}{lll}1 & 3 & \alpha \\ 1 & 2 & 3 \\ \alpha & \alpha & 4\end{array}\right], \alpha > 2$$ be the adjoint of a matrix \(A\) and \(|A|=2\). Then
$$\left[\begin{array}{ccc}\alpha & -2 \alpha & \alpha\end{array}\right]...

The number of symmetric matrices of order 3, with all the entries from the set \(\{0,1,2,3,4,5,6,7,8,9\}\) is :

Let \(\mathrm{PQ}\) be a focal chord of the parabola \(y^{2}=36 x\) of length 100 , making an acute angle with the positive \(x\)-axis. Let the ordinate of \(\mathrm{P}\) be positive and \(\mathrm{M}\) be the point on the line segment PQ su...

The number of seven digit positive integers formed using the digits \(1,2,3\) and \(4\) only and sum of the digits equal to \(12\) is ___________.

A coin is biased so that the head is 3 times as likely to occur as tail. This coin is tossed until a head or three tails occur. If \(\mathrm{X}\) denotes the number of tosses of the coin, then the mean of \(\mathrm{X}\) is :

The set of all \(a \in \mathbb{R}\) for which the equation \(x|x-1|+|x+2|+a=0\) has exactly one real root, is :

Let \(s_{1}, s_{2}, s_{3}, \ldots, s_{10}\) respectively be the sum to 12 terms of 10 A.P. s whose first terms are \(1,2,3, \ldots .10\) and the common differences are \(1,3,5, \ldots \ldots, 19\) respectively. Then $$\sum_\limits{i=1}^{10}...

The sum to \(20\) terms of the series \(2 \cdot 2^{2}-3^{2}+2 \cdot 4^{2}-5^{2}+2 \cdot 6^{2}-\ldots \ldots\) is equal to __________.

Let the mean of the data

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Let \(\vec{a}=\hat{i}+4 \hat{j}+2 \hat{k}, \vec{b}=3 \hat{i}-2 \hat{j}+7 \hat{k}\) and \(\vec{c}=2 \hat{i}-\hat{j}+4 \hat{k}\). If a vector \(\vec{d}\) satisfies \(\vec{d} \times \vec{b}=\vec{c} \times \vec{b}\) and $$\vec{d} \cdot \vec{a}=...

Let \(\vec{a}=3 \hat{i}+\hat{j}-\hat{k}\) and \(\vec{c}=2 \hat{i}-3 \hat{j}+3 \hat{k}\). If \(\vec{b}\) is a vector such that \(\vec{a}=\vec{b} \times \vec{c}\) and \(|\vec{b}|^{2}=50\), then \(|72-| \vec{b}+\left.\vec{c}\right|^{2} \mid\) ...