JEE Main 2023 (Online) 11th April Morning Shift

Let \((\alpha, \beta, \gamma)\) be the image of the point \(\mathrm{P}(2,3,5)\) in the plane \(2 x+y-3 z=6\). Then \(\alpha+\beta+\gamma\) is equal to :

If equation of the plane that contains the point \((-2,3,5)\) and is perpendicular to each of the planes \(2 x+4 y+5 z=8\) and \(3 x-2 y+3 z=5\) is \(\alpha x+\beta y+\gamma z+97=0\) then \(\alpha+\beta+\gamma=\)

Let a line \(l\) pass through the origin and be perpendicular to the lines
\(l_{1}: \vec{r}=(\hat{\imath}-11 \hat{\jmath}-7 \hat{k})+\lambda(\hat{i}+2 \hat{\jmath}+3 \hat{k}), \lambda \in \mathbb{R}\) and
$$l_{2}: \vec{r}=(-\hat{\imath}+\ha...

Let \(f:[2,4] \rightarrow \mathbb{R}\) be a differentiable function such that \(\left(x \log _{e} x\right) f^{\prime}(x)+\left(\log _{e} x\right) f(x)+f(x) \geq 1, x \in[2,4]\) with \(f(2)=\frac{1}{2}\) and \(f(4)=\frac{1}{4}\).
Consider th...

Area of the region \(\left\{(x, y): x^{2}+(y-2)^{2} \leq 4, x^{2} \geq 2 y\right\}\) is

The mean of the coefficients of \(x, x^{2}, \ldots, x^{7}\) in the binomial expansion of \((2+x)^{9}\) is ___________.

The number of integral terms in the expansion of \(\left(3^{\frac{1}{2}}+5^{\frac{1}{4}}\right)^{680}\) is equal to ___________.

Let \(w_{1}\) be the point obtained by the rotation of \(z_{1}=5+4 i\) about the origin through a right angle in the anticlockwise direction, and \(w_{2}\) be the point obtained by the rotation of \(z_{2}=3+5 i\) about the origin through a ...

The value of the integral \(\int_\limits{-\log _{e} 2}^{\log _{e} 2} e^{x}\left(\log _{e}\left(e^{x}+\sqrt{1+e^{2 x}}\right)\right) d x\) is equal to :

For \(m, n > 0\), let \(\alpha(m, n)=\int_\limits{0}^{2} t^{m}(1+3 t)^{n} d t\). If \(11 \alpha(10,6)+18 \alpha(11,5)=p(14)^{6}\), then \(p\) is equal to ___________.

Let \(y=y(x)\) be a solution curve of the differential equation.
\(\left(1-x^{2} y^{2}\right) d x=y d x+x d y\).
If the line \(x=1\) intersects the curve \(y=y(x)\) at \(y=2\) and the line \(x=2\) intersects the curve \(y=y(x)\) at $$y=\alp...

Consider ellipses \(\mathrm{E}_{k}: k x^{2}+k^{2} y^{2}=1, k=1,2, \ldots, 20\). Let \(\mathrm{C}_{k}\) be the circle which touches the four chords joining the end points (one on minor axis and another on major axis) of the ellipse $$\mathrm...

Let R be a rectangle given by the lines \(x=0, x=2, y=0\) and \(y=5\). Let A\((\alpha,0)\) and B\((0,\beta),\alpha\in[0,2]\) and \(\beta\in[0,5]\), be such that the line segment AB divides the area of the rectangle R in the ratio 4 : 1. The...

Let \(\mathrm{H}_{\mathrm{n}}: \frac{x^{2}}{1+n}-\frac{y^{2}}{3+n}=1, n \in N\). Let \(\mathrm{k}\) be the smallest even value of \(\mathrm{n}\) such that the eccentricity of \(\mathrm{H}_{\mathrm{k}}\) is a rational number. If \(l\) is the...

Let \(f(x)=\left[x^{2}-x\right]+|-x+[x]|\), where \(x \in \mathbb{R}\) and \([t]\) denotes the greatest integer less than or equal to \(t\). Then, \(f\) is :

The number of integral solutions \(x\) of \(\log _{\left(x+\frac{7}{2}\right)}\left(\frac{x-7}{2 x-3}\right)^{2} \geq 0\) is :

The number of ordered triplets of the truth values of \(p, q\) and \(r\) such that the truth value of the statement \((p \vee q) \wedge(p \vee r) \Rightarrow(q \vee r)\) is True, is equal to ___________.

Let \(\mathrm{A}\) be a \(2 \times 2\) matrix with real entries such that \(\mathrm{A}'=\alpha \mathrm{A}+\mathrm{I}\), where \(\alpha \in \mathbb{R}-\{-1,1\}\). If \(\operatorname{det}\left(A^{2}-A\right)=4\), then the sum of all possible ...

Let $$A=\left[\begin{array}{lll}0 & 1 & 2 \\ a & 0 & 3 \\ 1 & c & 0\end{array}\right]$$, where \(a, c \in \mathbb{R}\). If \(A^{3}=A\) and the positive value of \(a\) belongs to the interval \((n-1, n]\), where \(n \in \mathbb{N}\), then $$...

The number of triplets \((x, \mathrm{y}, \mathrm{z})\), where \(x, \mathrm{y}, \mathrm{z}\) are distinct non negative integers satisfying \(x+y+z=15\), is :

In an examination, 5 students have been allotted their seats as per their roll numbers. The number of ways, in which none of the students sits on the allotted seat, is _________.

Let \(S=\left\{M=\left[a_{i j}\right], a_{i j} \in\{0,1,2\}, 1 \leq i, j \leq 2\right\}\) be a sample space and \(A=\{M \in S: M\) is invertible \(\}\) be an event. Then \(P(A)\) is equal to :

If \(a\) and \(b\) are the roots of the equation \(x^{2}-7 x-1=0\), then the value of \(\frac{a^{21}+b^{21}+a^{17}+b^{17}}{a^{19}+b^{19}}\) is equal to _____________.

Let \(x_{1}, x_{2}, \ldots, x_{100}\) be in an arithmetic progression, with \(x_{1}=2\) and their mean equal to 200 . If \(y_{i}=i\left(x_{i}-i\right), 1 \leq i \leq 100\), then the mean of \(y_{1}, y_{2}, \ldots, y_{100}\) is :

Let \(S=109+\frac{108}{5}+\frac{107}{5^{2}}+\ldots .+\frac{2}{5^{107}}+\frac{1}{5^{108}}\). Then the value of \(\left(16 S-(25)^{-54}\right)\) is equal to ___________.

An organization awarded 48 medals in event 'A', 25 in event 'B' and 18 in event 'C'. If these medals went to total 60 men and only five men got medals in all the three events, then, how many received medals in exactly two of three events?

Let sets A and B have 5 elements each. Let the mean of the elements in sets A and B be 5 and 8 respectively and the variance of the elements in sets A and B be 12 and 20 respectively. A new set C of 10 elements is formed by subtracting 3 fr...

The number of elements in the set
\(S=\left\{\theta \in[0,2 \pi]: 3 \cos ^{4} \theta-5 \cos ^{2} \theta-2 \sin ^{6} \theta+2=0\right\}\) is :

For any vector \(\vec{a}=a_{1} \hat{i}+a_{2} \hat{j}+a_{3} \hat{k}\), with \(10\left|a_{i}\right|<1, i=1,2,3\), consider the following statements :
(A): $$\max \left\{\left|a_{1}\right|,\left|a_{2}\right|,\left|a_{3}\right|\right\} \leq|\ve...

Let \(\vec{a}\) be a non-zero vector parallel to the line of intersection of the two planes described by \(\hat{i}+\hat{j}, \hat{i}+\hat{k}\) and \(\hat{i}-\hat{j}, \hat{j}-\hat{k}\). If \(\theta\) is the angle between the vector $$\vec{a}$...