JEE Main 2023 (Online) 8th April Morning Shift

The shortest distance between the lines \(\frac{x-4}{4}=\frac{y+2}{5}=\frac{z+3}{3}\) and \(\frac{x-1}{3}=\frac{y-3}{4}=\frac{z-4}{2}\) is :

If the equation of the plane containing the line \(x+2 y+3 z-4=0=2 x+y-z+5\) and perpendicular to the plane $\vec{r}=(\hat{i}-\hat{j})+\lambda(\hat{i}+\hat{j}+\hat{k})+\mu(\hat{i}-2 \hat{j}+3 \hat{k})$ is $a x+b y+c z=4$, then \((a-b+c)\) i...

Let \(\lambda_{1}, \lambda_{2}\) be the values of \(\lambda\) for which the points \(\left(\frac{5}{2}, 1, \lambda\right)\) and \((-2,0,1)\) are at equal distance from the plane \(2 x+3 y-6 z+7=0\). If \(\lambda_{1} > \lambda_{2}\), then th...

If \(a_{\alpha}\) is the greatest term in the sequence \(\alpha_{n}=\frac{n^{3}}{n^{4}+147}, n=1,2,3, \ldots\), then \(\alpha\) is equal to _____________.

The area of the region \(\left\{(x, y): x^{2} \leq y \leq 8-x^{2}, y \leq 7\right\}\) is :

If the coefficients of three consecutive terms in the expansion of \((1+x)^{n}\) are in the ratio \(1: 5: 20\), then the coefficient of the fourth term is

Let \([t]\) denote the greatest integer \(\leq t\). If the constant term in the expansion of \(\left(3 x^{2}-\frac{1}{2 x^{5}}\right)^{7}\) is \(\alpha\), then \([\alpha]\) is equal to ___________.

The largest natural number \(n\) such that \(3^{n}\) divides \(66 !\) is ___________.

Consider a circle \(C_{1}: x^{2}+y^{2}-4 x-2 y=\alpha-5\). Let its mirror image in the line \(y=2 x+1\) be another circle \(C_{2}: 5 x^{2}+5 y^{2}-10 f x-10 g y+36=0\). Let \(r\) be the radius of \(C_{2}\). Then \(\alpha+r\) is equal to ___...

If for \(z=\alpha+i \beta,|z+2|=z+4(1+i)\), then \(\alpha+\beta\) and \(\alpha \beta\) are the roots of the equation :

Let \([t]\) denote the greatest integer \(\leq t\). Then \(\frac{2}{\pi} \int_\limits{\pi / 6}^{5 \pi / 6}(8[\operatorname{cosec} x]-5[\cot x]) d x\) is equal to __________.

If the solution curve of the differential equation \(\left(y-2 \log _{e} x\right) d x+\left(x \log _{e} x^{2}\right) d y=0, x > 1\) passes through the points \(\left(e, \frac{4}{3}\right)\) and \(\left(e^{4}, \alpha\right)\), then $$\alpha$...

Let \(f(x)=\frac{\sin x+\cos x-\sqrt{2}}{\sin x-\cos x}, x \in[0, \pi]-\left\{\frac{\pi}{4}\right\}\). Then \(f\left(\frac{7 \pi}{12}\right) f^{\prime \prime}\left(\frac{7 \pi}{12}\right)\) is equal to

Let \(I(x)=\int \frac{(x+1)}{x\left(1+x e^{x}\right)^{2}} d x, x > 0\). If \(\lim_\limits{x \rightarrow \infty} I(x)=0\), then \(I(1)\) is equal to :

\(\lim_\limits{x \rightarrow 0}\left(\left(\frac{\left(1-\cos ^{2}(3 x)\right.}{\cos ^{3}(4 x)}\right)\left(\frac{\sin ^{3}(4 x)}{\left(\log _{e}(2 x+1)\right)^{5}}\right)\right)\) is equal to _____________.

Negation of \((p \Rightarrow q) \Rightarrow(q \Rightarrow p)\) is :

Let $$A=\left[\begin{array}{ccc}2 & 1 & 0 \\ 1 & 2 & -1 \\ 0 & -1 & 2\end{array}\right]$$. If \(|\operatorname{adj}(\operatorname{adj}(\operatorname{adj} 2 A))|=(16)^{n}\), then \(n\) is equal to :

Let $$P=\left[\begin{array}{cc}\frac{\sqrt{3}}{2} & \frac{1}{2} \\ -\frac{1}{2} & \frac{\sqrt{3}}{2}\end{array}\right], A=\left[\begin{array}{ll}1 & 1 \\ 0 & 1\end{array}\right]$$ and \(Q=P A P^{T}\). If $$P^{T} Q^{2007} P=\left[\begin{arra...

Let \(R\) be the focus of the parabola \(y^{2}=20 x\) and the line \(y=m x+c\) intersect the parabola at two points \(P\) and \(Q\).
Let the point \(G(10,10)\) be the centroid of the triangle \(P Q R\). If \(c-m=6\), then \((P Q)^{2}\) is ...

The number of arrangements of the letters of the word "INDEPENDENCE" in which all the vowels always occur together is :

The number of ways, in which 5 girls and 7 boys can be seated at a round table so that no two girls sit together, is :

Let the number of elements in sets \(A\) and \(B\) be five and two respectively. Then the number of subsets of \(A \times B\) each having at least 3 and at most 6 elements is :

In a bolt factory, machines \(A, B\) and \(C\) manufacture respectively \(20 \%, 30 \%\) and \(50 \%\) of the total bolts. Of their output 3, 4 and 2 percent are respectively defective bolts. A bolt is drawn at random from the product. If t...

Let \(\alpha, \beta, \gamma\) be the three roots of the equation \(x^{3}+b x+c=0\). If \(\beta \gamma=1=-\alpha\), then \(b^{3}+2 c^{3}-3 \alpha^{3}-6 \beta^{3}-8 \gamma^{3}\) is equal to :

Let \(S_{K}=\frac{1+2+\ldots+K}{K}\) and \(\sum_\limits{j=1}^{n} S_{j}^{2}=\frac{n}{A}\left(B n^{2}+C n+D\right)\), where \(A, B, C, D \in \mathbb{N}\) and \(A\) has least value. Then

Let \(A=\{0,3,4,6,7,8,9,10\}\) and \(R\) be the relation defined on \(A\) such that \(R=\{(x, y) \in A \times A: x-y\) is odd positive integer or \(x-y=2\}\). The minimum number of elements that must be added to the relation \(R\), so that ...

Let the mean and variance of 8 numbers \(x, y, 10,12,6,12,4,8\) be \(9\) and \(9.25\) respectively. If \(x > y\), then \(3 x-2 y\) is equal to _____________.

Let \(C(\alpha, \beta)\) be the circumcenter of the triangle formed by the lines
\(4 x+3 y=69\)
\(4 y-3 x=17\), and
\(x+7 y=61\).
Then \((\alpha-\beta)^{2}+\alpha+\beta\) is equal to :

If the points with position vectors \(\alpha \hat{i}+10 \hat{j}+13 \hat{k}, 6 \hat{i}+11 \hat{j}+11 \hat{k}, \frac{9}{2} \hat{i}+\beta \hat{j}-8 \hat{k}\) are collinear, then \((19 \alpha-6 \beta)^{2}\) is equal to :

Let \(\vec{a}=6 \hat{i}+9 \hat{j}+12 \hat{k}, \vec{b}=\alpha \hat{i}+11 \hat{j}-2 \hat{k}\) and \(\vec{c}\) be vectors such that \(\vec{a} \times \vec{c}=\vec{a} \times \vec{b}\). If $$\vec{a} \cdot \vec{c}=-12, \vec{c} \cdot(\hat{i}-2 \hat...