JEE Main 2023 (Online) 12th April Morning Shift

Let the lines \(l_{1}: \frac{x+5}{3}=\frac{y+4}{1}=\frac{z-\alpha}{-2}\) and \(l_{2}: 3 x+2 y+z-2=0=x-3 y+2 z-13\) be coplanar. If the point \(\mathrm{P}(a, b, c)\) on \(l_{1}\) is nearest to the point \(\mathrm{Q}(-4,-3,2)\), then $$|a|+|b...

Let the plane P: \(4 x-y+z=10\) be rotated by an angle \(\frac{\pi}{2}\) about its line of intersection with the plane \(x+y-z=4\). If \(\alpha\) is the distance of the point \((2,3,-4)\) from the new position of the plane \(\mathrm{P}\), t...

Let the plane \(x+3 y-2 z+6=0\) meet the co-ordinate axes at the points A, B, C. If the orthocenter of the triangle \(\mathrm{ABC}\) is \(\left(\alpha, \beta, \frac{6}{7}\right)\), then \(98(\alpha+\beta)^{2}\) is equal to ___________.

If the local maximum value of the function \(f(x)=\left(\frac{\sqrt{3 e}}{2 \sin x}\right)^{\sin ^{2} x}, x \in\left(0, \frac{\pi}{2}\right)\) , is \(\frac{k}{e}\), then \(\left(\frac{k}{e}\right)^{8}+\frac{k^{8}}{e^{5}}+k^{8}\) is equal to

The area of the region enclosed by the curve \(y=x^{3}\) and its tangent at the point \((-1,-1)\) is :

If \(\frac{1}{n+1}{ }^{n} \mathrm{C}_{n}+\frac{1}{n}{ }^{n} \mathrm{C}_{n-1}+\ldots+\frac{1}{2}{ }^{n} \mathrm{C}_{1}+{ }^{n} \mathrm{C}_{0}=\frac{1023}{10}\) then \(n\) is equal to :

The sum, of the coefficients of the first 50 terms in the binomial expansion of \((1-x)^{100}\), is equal to

Two circles in the first quadrant of radii \(r_{1}\) and \(r_{2}\) touch the coordinate axes. Each of them cuts off an intercept of 2 units with the line \(x+y=2\). Then \(r_{1}^{2}+r_{2}^{2}-r_{1} r_{2}\) is equal to ___________.

Let \(\mathrm{C}\) be the circle in the complex plane with centre \(\mathrm{z}_{0}=\frac{1}{2}(1+3 i)\) and radius \(r=1\). Let \(\mathrm{z}_{1}=1+\mathrm{i}\) and the complex number \(z_{2}\) be outside the circle \(C\) such that $$\left|z...

If \(\int_\limits{-0.15}^{0.15}\left|100 x^{2}-1\right| d x=\frac{k}{3000}\), then \(k\) is equal to ___________.

Let \(y=y(x), y > 0\), be a solution curve of the differential equation \(\left(1+x^{2}\right) \mathrm{d} y=y(x-y) \mathrm{d} x\). If \(y(0)=1\) and \(y(2 \sqrt{2})=\beta\), then

Let \(\mathrm{P}\left(\frac{2 \sqrt{3}}{\sqrt{7}}, \frac{6}{\sqrt{7}}\right), \mathrm{Q}, \mathrm{R}\) and \(\mathrm{S}\) be four points on the ellipse \(9 x^{2}+4 y^{2}=36\). Let \(\mathrm{PQ}\) and \(\mathrm{RS}\) be mutually perpendicula...

Let \(\mathrm{D}\) be the domain of the function \(f(x)=\sin ^{-1}\left(\log _{3 x}\left(\frac{6+2 \log _{3} x}{-5 x}\right)\right)\). If the range of the function \(\mathrm{g}: \mathrm{D} \rightarrow \mathbb{R}\) defined by $$\mathrm{g}(x)...

Let \(I(x)=\int \sqrt{\frac{x+7}{x}} \mathrm{~d} x\) and \(I(9)=12+7 \log _{e} 7\). If \(I(1)=\alpha+7 \log _{e}(1+2 \sqrt{2})\), then \(\alpha^{4}\) is equal to _________.

Let \([x]\) be the greatest integer \(\leq x\). Then the number of points in the interval \((-2,1)\), where the function \(f(x)=|[x]|+\sqrt{x-[x]}\) is discontinuous, is ___________.

Among the two statements
\((\mathrm{S} 1):(p \Rightarrow q) \wedge(p \wedge(\sim q))\) is a contradiction and
\((\mathrm{S} 2):(p \wedge q) \vee((\sim p) \wedge q) \vee(p \wedge(\sim q)) \vee((\sim p) \wedge(\sim q))\) is a tautology

Let $$A=\left[\begin{array}{cc}1 & \frac{1}{51} \\ 0 & 1\end{array}\right]$$. If $$\mathrm{B}=\left[\begin{array}{cc}1 & 2 \\ -1 & -1\end{array}\right] A\left[\begin{array}{cc}-1 & -2 \\ 1 & 1\end{array}\right]$$, then the sum of all the el...

Let $$\mathrm{D}_{\mathrm{k}}=\left|\begin{array}{ccc}1 & 2 k & 2 k-1 \\ n & n^{2}+n+2 & n^{2} \\ n & n^{2}+n & n^{2}+n+2\end{array}\right|$$. If \(\sum_\limits{k=1}^{n} \mathrm{D}_{\mathrm{k}}=96\), then \(n\) is equal to _____________.

The number of five digit numbers, greater than 40000 and divisible by 5 , which can be formed using the digits \(0,1,3,5,7\) and 9 without repetition, is equal to :

Let the digits a, b, c be in A. P. Nine-digit numbers are to be formed using each of these three digits thrice such that three consecutive digits are in A.P. at least once. How many such numbers can be formed?

Two dice A and B are rolled. Let the numbers obtained on A and B be \(\alpha\) and \(\beta\) respectively. If the variance of \(\alpha-\beta\) is \(\frac{p}{q}\), where \(p\) and \(q\) are co-prime, then the sum of the positive divisors of...

A fair \(n(n > 1)\) faces die is rolled repeatedly until a number less than \(n\) appears. If the mean of the number of tosses required is \(\frac{n}{9}\), then \(n\) is equal to ____________.

In a triangle ABC, if \(\cos \mathrm{A}+2 \cos \mathrm{B}+\cos C=2\) and the lengths of the sides opposite to the angles A and C are 3 and 7 respectively, then \(\mathrm{\cos A-\cos C}\) is equal to

Let \(\alpha, \beta\) be the roots of the quadratic equation \(x^{2}+\sqrt{6} x+3=0\). Then \(\frac{\alpha^{23}+\beta^{23}+\alpha^{14}+\beta^{14}}{\alpha^{15}+\beta^{15}+\alpha^{10}+\beta^{10}}\) is equal to :

Let \(< a_{\mathrm{n}} >\) be a sequence such that \(a_{1}+a_{2}+\ldots+a_{n}=\frac{n^{2}+3 n}{(n+1)(n+2)}\). If \(28 \sum_\limits{k=1}^{10} \frac{1}{a_{k}}=p_{1} p_{2} p_{3} \ldots p_{m}\), where $$\mathrm{p}_{1}, \mathrm{p}_{2}, \ldots ....

The number of relations, on the set \(\{1,2,3\}\) containing \((1,2)\) and \((2,3)\), which are reflexive and transitive but not symmetric, is __________.

Let the positive numbers \(a_{1}, a_{2}, a_{3}, a_{4}\) and \(a_{5}\) be in a G.P. Let their mean and variance be \(\frac{31}{10}\) and \(\frac{m}{n}\) respectively, where \(m\) and \(n\) are co-prime. If the mean of their reciprocals is $$...

If the point \(\left(\alpha, \frac{7 \sqrt{3}}{3}\right)\) lies on the curve traced by the mid-points of the line segments of the lines \(x \cos \theta+y \sin \theta=7, \theta \in\left(0, \frac{\pi}{2}\right)\) between the co-ordinates axes...

Let \(a, b, c\) be three distinct real numbers, none equal to one. If the vectors \(a \hat{i}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \hat{\mathrm{i}}+b \hat{j}+\hat{\mathrm{k}}\) and \(\hat{\mathrm{i}}+\hat{\mathrm{j}}+c \hat{\mathrm{k}}\) are ...

Let \(\lambda \in \mathbb{Z}, \vec{a}=\lambda \hat{i}+\hat{j}-\hat{k}\) and \(\vec{b}=3 \hat{i}-\hat{j}+2 \hat{k}\). Let \(\vec{c}\) be a vector such that $$(\vec{a}+\vec{b}+\vec{c}) \times \vec{c}=\overrightarrow{0}, \vec{a} \cdot \vec{c}=...