JEE Main 2023 (Online) 8th April Evening Shift

For \(\mathrm{a}, \mathrm{b} \in \mathbb{Z}\) and \(|\mathrm{a}-\mathrm{b}| \leq 10\), let the angle between the plane \(\mathrm{P}: \mathrm{ax}+y-\mathrm{z}=\mathrm{b}\) and the line \(l: x-1=\mathrm{a}-y=z+1\) be $$\cos ^{-1}\left(\frac{1...

Let \(\mathrm{P}\) be the plane passing through the line \(\frac{x-1}{1}=\frac{y-2}{-3}=\frac{z+5}{7}\) and the point \((2,4,-3)\). If the image of the point \((-1,3,4)\) in the plane P is \((\alpha, \beta, \gamma)\) then $$\alpha+\beta+\ga...

Let \(\mathrm{P}_{1}\) be the plane \(3 x-y-7 z=11\) and \(\mathrm{P}_{2}\) be the plane passing through the points \((2,-1,0),(2,0,-1)\), and \((5,1,1)\). If the foot of the perpendicular drawn from the point \((7,4,-1)\) on the line of in...

Let the area enclosed by the lines \(x+y=2, \mathrm{y}=0, x=0\) and the curve \(f(x)=\min \left\{x^{2}+\frac{3}{4}, 1+[x]\right\}\) where \([x]\)
denotes the greatest integer \(\leq x\), be \(\mathrm{A}\). Then the value of $$12 \mathrm{~A...

\(25^{190}-19^{190}-8^{190}+2^{190}\) is divisible by :

The absolute difference of the coefficients of \(x^{10}\) and \(x^{7}\) in the expansion of \(\left(2 x^{2}+\frac{1}{2 x}\right)^{11}\) is equal to :

Let O be the origin and OP and OQ be the tangents to the circle \(x^2+y^2-6x+4y+8=0\) at the points P and Q on it. If the circumcircle of the triangle OPQ passes through the point \(\left( {\alpha ,{1 \over 2}} \right)\), then a value of $$...

Let \(A=\left\{\theta \in(0,2 \pi): \frac{1+2 i \sin \theta}{1-i \sin \theta}\right.\) is purely imaginary \(\}\). Then the sum of the elements in \(\mathrm{A}\) is :

Let \([t]\) denote the greatest integer function. If \(\int_\limits{0}^{2.4}\left[x^{2}\right] d x=\alpha+\beta \sqrt{2}+\gamma \sqrt{3}+\delta \sqrt{5}\), then \(\alpha+\beta+\gamma+\delta\) is equal to __________.

Let the solution curve \(x=x(y), 0 < y < \frac{\pi}{2}\), of the differential equation \(\left(\log _{e}(\cos y)\right)^{2} \cos y \mathrm{~d} x-\left(1+3 x \log _{e}(\cos y)\right) \sin \mathrm{y} d y=0\) satisfy $$x\left(\frac{\pi}{3}\rig...

Let \(\mathrm{R}=\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}\}\) and \(\mathrm{S}=\{1,2,3,4\}\). Total number of onto functions \(f: \mathrm{R} \rightarrow \mathrm{S}\)
such that \(f(\mathrm{a}) \neq 1\), is equal to ______...

If domain of the function \(\log _{e}\left(\frac{6 x^{2}+5 x+1}{2 x-1}\right)+\cos ^{-1}\left(\frac{2 x^{2}-3 x+4}{3 x-5}\right)\) is \((\alpha, \beta) \cup(\gamma, \delta]\), then $$18\left(\alpha^{2}+\beta^{2}+\gamma^{2}+\delta^{2}\right)...

The integral \(\int\left[\left(\frac{x}{2}\right)^x+\left(\frac{2}{x}\right)^x\right] \ln \left(\frac{e x}{2}\right) d x\) is equal to :

If \(\alpha > \beta > 0\) are the roots of the equation \(a x^{2}+b x+1=0\), and $$\lim_\limits{x \rightarrow \frac{1}{\alpha}}\left(\frac{1-\cos \left(x^{2}+b x+a\right)}{2(1-\alpha x)^{2}}\right)^{\frac{1}{2}}=\frac{1}{k}\left(\frac{1}{\b...

Let \(\mathrm{k}\) and \(\mathrm{m}\) be positive real numbers such that the function $$f(x)=\left\{\begin{array}{cc}3 x^{2}+k \sqrt{x+1}, & 0 < x < 1 \\ m x^{2}+k^{2}, & x \geq 1\end{array}\right.$$ is differentiable for all \(x > 0\). The...

The negation of \((p \wedge(\sim q)) \vee(\sim p)\) is equivalent to :

If $$A=\left[\begin{array}{cc}1 & 5 \\ \lambda & 10\end{array}\right], \mathrm{A}^{-1}=\alpha \mathrm{A}+\beta \mathrm{I}$$ and \(\alpha+\beta=-2\), then \(4 \alpha^{2}+\beta^{2}+\lambda^{2}\) is equal to :

Let S be the set of all values of \(\theta \in[-\pi, \pi]\) for which the system of linear equations
\(x+y+\sqrt{3} z=0\)
\(-x+(\tan \theta) y+\sqrt{7} z=0\)
\(x+y+(\tan \theta) z=0\)
has non-trivial solution. Then $$\frac{120}{\pi} \sum_\l...

Let \(\mathrm{A}(0,1), \mathrm{B}(1,1)\) and \(\mathrm{C}(1,0)\) be the mid-points of the sides of a triangle with incentre at the point \(\mathrm{D}\). If the focus of the parabola \(y^{2}=4 \mathrm{ax}\) passing through \(\mathrm{D}\) is ...

The ordinates of the points P and \(\mathrm{Q}\) on the parabola with focus \((3,0)\) and directrix \(x=-3\) are in the ratio \(3: 1\). If \(\mathrm{R}(\alpha, \beta)\) is the point of intersection of the tangents to the parabola at $$\math...

If the number of words, with or without meaning, which can be made using all the letters of the word MATHEMATICS in which \(\mathrm{C}\) and \(\mathrm{S}\) do not come together, is \((6 !) \mathrm{k}\), then \(\mathrm{k}\) is equal to :

If the probability that the random variable \(\mathrm{X}\) takes values \(x\) is given by \(\mathrm{P}(\mathrm{X}=x)=\mathrm{k}(x+1) 3^{-x}, x=0,1,2,3, \ldots\), where \(\mathrm{k}\) is a constant, then \(\mathrm{P}(\mathrm{X} \geq 2)\) is ...

Let m and \(\mathrm{n}\) be the numbers of real roots of the quadratic equations \(x^{2}-12 x+[x]+31=0\) and \(x^{2}-5|x+2|-4=0\) respectively, where \([x]\) denotes the greatest integer \(\leq x\). Then $$\mathrm{m}^{2}+\mathrm{mn}+\mathr...

Let \(\mathrm{a}_{\mathrm{n}}\) be the \(\mathrm{n}^{\text {th }}\) term of the series \(5+8+14+23+35+50+\ldots\) and \(\mathrm{S}_{\mathrm{n}}=\sum_\limits{k=1}^{n} a_{k}\). Then \(\mathrm{S}_{30}-a_{40}\) is equal to :

Let \(0 < z < y < x\) be three real numbers such that \(\frac{1}{x}, \frac{1}{y}, \frac{1}{z}\) are in an arithmetic progression and \(x, \sqrt{2} y, z\) are in a geometric progression. If \(x y+y z+z x=\frac{3}{\sqrt{2}} x y z\) , then $$3...

Let \(\mathrm{A}=\{1,2,3,4,5,6,7\}\). Then the relation \(\mathrm{R}=\{(x, y) \in \mathrm{A} \times \mathrm{A}: x+y=7\}\) is :

Let the mean and variance of 12 observations be \(\frac{9}{2}\) and 4 respectively. Later on, it was observed that two observations were considered as 9 and 10 instead of 7 and 14 respectively. If the correct variance is \(\frac{m}{n}\), wh...

The value of \(36\left(4 \cos ^{2} 9^{\circ}-1\right)\left(4 \cos ^{2} 27^{\circ}-1\right)\left(4 \cos ^{2} 81^{\circ}-1\right)\left(4 \cos ^{2} 243^{\circ}-1\right)\) is :

Let the vectors \(\vec{u}_{1}=\hat{i}+\hat{j}+a \hat{k}, \vec{u}_{2}=\hat{i}+b \hat{j}+\hat{k}\) and \(\vec{u}_{3}=c \hat{i}+\hat{j}+\hat{k}\) be coplanar. If the vectors $$\vec{v}_{1}=(a+b) \hat{i}+c \hat{j}+c \hat{k}, \vec{v}_{2}=a \hat{i...

The area of the quadrilateral \(\mathrm{ABCD}\) with vertices \(\mathrm{A}(2,1,1), \mathrm{B}(1,2,5), \mathrm{C}(-2,-3,5)\) and \(\mathrm{D}(1,-6,-7)\) is equal to :