JEE Main 2024 (Online) 9th April Evening Shift

Consider the line \(\mathrm{L}\) passing through the points \((1,2,3)\) and \((2,3,5)\). The distance of the point \(\left(\frac{11}{3}, \frac{11}{3}, \frac{19}{3}\right)\) from the line \(\mathrm{L}\) along the line $$\frac{3 x-11}{2}=\fra...

The square of the distance of the image of the point \((6,1,5)\) in the line \(\frac{x-1}{3}=\frac{y}{2}=\frac{z-2}{4}\), from the origin is __________.

Let the set of all values of \(p\), for which \(f(x)=\left(p^2-6 p+8\right)\left(\sin ^2 2 x-\cos ^2 2 x\right)+2(2-p) x+7\) does not have any critical point, be the interval \((a, b)\). Then \(16 a b\) is equal to _________.

The area (in square units) of the region enclosed by the ellipse \(x^2+3 y^2=18\) in the first quadrant below the line \(y=x\) is

The sum of the coefficient of \(x^{2 / 3}\) and \(x^{-2 / 5}\) in the binomial expansion of \(\left(x^{2 / 3}+\frac{1}{2} x^{-2 / 5}\right)^9\) is

Let \(z\) be a complex number such that the real part of \(\frac{z-2 i}{z+2 i}\) is zero. Then, the maximum value of \(|z-(6+8 i)|\) is equal to

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

The integral \(\int_\limits{1 / 4}^{3 / 4} \cos \left(2 \cot ^{-1} \sqrt{\frac{1-x}{1+x}}\right) d x\) is equal to

\(\lim _\limits{x \rightarrow \frac{\pi}{2}}\left(\frac{\int_{x^3}^{(\pi / 2)^3}\left(\sin \left(2 t^{1 / 3}\right)+\cos \left(t^{1 / 3}\right)\right) d t}{\left(x-\frac{\pi}{2}\right)^2}\right)\) is equal to

Let \(\int_\limits0^x \sqrt{1-\left(y^{\prime}(t)\right)^2} d t=\int_0^x y(t) d t, 0 \leq x \leq 3, y \geq 0, y(0)=0\). Then at \(x=2, y^{\prime \prime}+y+1\) is equal to

For a differentiable function \(f: \mathbb{R} \rightarrow \mathbb{R}\), suppose \(f^{\prime}(x)=3 f(x)+\alpha\), where \(\alpha \in \mathbb{R}, f(0)=1\) and \(\lim _\limits{x \rightarrow-\infty} f(x)=7\). Then \(9 f\left(-\log _e 3\right)\)...

If \(\log _e y=3 \sin ^{-1} x\), then \((1-x^2) y^{\prime \prime}-x y^{\prime}\) at \(x=\frac{1}{2}\) is equal to

Let the range of the function \(f(x)=\frac{1}{2+\sin 3 x+\cos 3 x}, x \in \mathbb{R}\) be \([a, b]\). If \(\alpha\) and \(\beta\) ar respectively the A.M. and the G.M. of \(a\) and \(b\), then \(\frac{\alpha}{\beta}\) is equal to

Let \(A=\{(x, y): 2 x+3 y=23, x, y \in \mathbb{N}\}\) and \(B=\{x:(x, y) \in A\}\). Then the number of one-one functions from \(A\) to \(B\) is equal to _________.

Let the foci of a hyperbola \(H\) coincide with the foci of the ellipse \(E: \frac{(x-1)^2}{100}+\frac{(y-1)^2}{75}=1\) and the eccentricity of the hyperbola \(H\) be the reciprocal of the eccentricity of the ellipse \(E\). If the length of...

Let the inverse trigonometric functions take principal values. The number of real solutions of the equation \(2 \sin ^{-1} x+3 \cos ^{-1} x=\frac{2 \pi}{5}\), is __________.

\(\lim _\limits{x \rightarrow 0} \frac{e-(1+2 x)^{\frac{1}{2 x}}}{x}\) is equal to

Consider the matrices : $$A=\left[\begin{array}{cc}2 & -5 \\ 3 & m\end{array}\right], B=\left[\begin{array}{l}20 \\ m\end{array}\right]$$ and $$X=\left[\begin{array}{l}x \\ y\end{array}\right]$$. Let the set of all \(m\), for which the syst...

Let $$B=\left[\begin{array}{ll}1 & 3 \\ 1 & 5\end{array}\right]$$ and \(A\) be a \(2 \times 2\) matrix such that \(A B^{-1}=A^{-1}\). If \(B C B^{-1}=A\) and \(C^4+\alpha C^2+\beta I=O\), then \(2 \beta-\alpha\) is equal to

Let \(A, B\) and \(C\) be three points on the parabola \(y^2=6 x\) and let the line segment \(A B\) meet the line \(L\) through \(C\) parallel to the \(x\)-axis at the point \(D\). Let \(M\) and \(N\) respectively be the feet of the perpend...

Consider the circle \(C: x^2+y^2=4\) and the parabola \(P: y^2=8 x\). If the set of all values of \(\alpha\), for which three chords of the circle \(C\) on three distinct lines passing through the point \((\alpha, 0)\) are bisected by the p...

The number of integers, between 100 and 1000 having the sum of their digits equals to 14 , is __________.

If an unbiased dice is rolled thrice, then the probability of getting a greater number in the \(i^{\text {th }}\) roll than the number obtained in the \((i-1)^{\text {th }}\) roll, \(i=2,3\), is equal to

Two vertices of a triangle \(\mathrm{ABC}\) are \(\mathrm{A}(3,-1)\) and \(\mathrm{B}(-2,3)\), and its orthocentre is \(\mathrm{P}(1,1)\). If the coordinates of the point \(\mathrm{C}\) are \((\alpha, \beta)\) and the centre of the of the c...

Let \(\alpha, \beta ; \alpha>\beta\), be the roots of the equation \(x^2-\sqrt{2} x-\sqrt{3}=0\). Let \(\mathrm{P}_n=\alpha^n-\beta^n, n \in \mathrm{N}\). Then $$(11 \sqrt{3}-10 \sqrt{2}) \mathrm{P}_{10}+(11 \sqrt{2}+10) \mathrm{P}_{11}-11 ...

Let \(a, a r, a r^2\), ............ be an infinite G.P. If \(\sum_\limits{n=0}^{\infty} a r^n=57\) and \(\sum_\limits{n=0}^{\infty} a^3 r^{3 n}=9747\), then \(a+18 r\) is equal to

If \(\left(\frac{1}{\alpha+1}+\frac{1}{\alpha+2}+\ldots . .+\frac{1}{\alpha+1012}\right)-\left(\frac{1}{2 \cdot 1}+\frac{1}{4 \cdot 3}+\frac{1}{6 \cdot 5}+\ldots \ldots+\frac{1}{2024 \cdot 2023}\right)=\frac{1}{2024}\), then \(\alpha\) is e...

If the variance of the frequency distribution

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Between the following two statements:
Statement I : Let \(\vec{a}=\hat{i}+2 \hat{j}-3 \hat{k}\) and \(\vec{b}=2 \hat{i}+\hat{j}-\hat{k}\). Then the vector \(\vec{r}\) satisfying \(\vec{a} \times \vec{r}=\vec{a} \times \vec{b}\) and $$\vec{a...

Let \(\vec{a}=2 \hat{i}+\alpha \hat{j}+\hat{k}, \vec{b}=-\hat{i}+\hat{k}, \vec{c}=\beta \hat{j}-\hat{k}\), where \(\alpha\) and \(\beta\) are integers and \(\alpha \beta=-6\). Let the values of the ordered pair \((\alpha, \beta)\), for whic...