JEE Main 2024 (Online) 6th April Evening Shift

Let \(\mathrm{P}(\alpha, \beta, \gamma)\) be the image of the point \(\mathrm{Q}(3,-3,1)\) in the line \(\frac{x-0}{1}=\frac{y-3}{1}=\frac{z-1}{-1}\) and \(\mathrm{R}\) be the point \((2,5,-1)\). If the area of the triangle \(\mathrm{PQR}\)...

If the shortest distance between the lines \(\frac{x-\lambda}{3}=\frac{y-2}{-1}=\frac{z-1}{1}\) and \(\frac{x+2}{-3}=\frac{y+5}{2}=\frac{z-4}{4}\) is \(\frac{44}{\sqrt{30}}\), then the largest possible value of \(|\lambda|\) is equal to ___...

If the area of the region \(\left\{(x, y): \frac{\mathrm{a}}{x^2} \leq y \leq \frac{1}{x}, 1 \leq x \leq 2,0<\mathrm{a}<1\right\}\) is \(\left(\log _{\mathrm{e}} 2\right)-\frac{1}{7}\) then the value of \(7 \mathrm{a}-3\) is equal to :

If \(\mathrm{P}(6,1)\) be the orthocentre of the triangle whose vertices are \(\mathrm{A}(5,-2), \mathrm{B}(8,3)\) and \(\mathrm{C}(\mathrm{h}, \mathrm{k})\), then the point \(\mathrm{C}\) lies on the circle :

If \(z_1, z_2\) are two distinct complex number such that \(\left|\frac{z_1-2 z_2}{\frac{1}{2}-z_1 \bar{z}_2}\right|=2\), then

Let \([t]\) denote the largest integer less than or equal to \(t\). If \(\int_\limits0^3\left(\left[x^2\right]+\left[\frac{x^2}{2}\right]\right) \mathrm{d} x=\mathrm{a}+\mathrm{b} \sqrt{2}-\sqrt{3}-\sqrt{5}+\mathrm{c} \sqrt{6}-\sqrt{7}\), w...

Suppose the solution of the differential equation \(\frac{d y}{d x}=\frac{(2+\alpha) x-\beta y+2}{\beta x-2 \alpha y-(\beta \gamma-4 \alpha)}\) represents a circle passing through origin. Then the radius of this circle is :

If the solution \(y(x)\) of the given differential equation \(\left(e^y+1\right) \cos x \mathrm{~d} x+\mathrm{e}^y \sin x \mathrm{~d} y=0\) passes through the point \(\left(\frac{\pi}{2}, 0\right)\), then the value of $$e^{y\left(\frac{\pi}...

Suppose for a differentiable function \(h, h(0)=0, h(1)=1\) and \(h^{\prime}(0)=h^{\prime}(1)=2\). If \(g(x)=h\left(\mathrm{e}^x\right) \mathrm{e}^{h(x)}\), then \(g^{\prime}(0)\) is equal to:

If the function \(f(x)=\left(\frac{1}{x}\right)^{2 x} ; x>0\) attains the maximum value at \(x=\frac{1}{\mathrm{e}}\) then :

Let \(f(x)=\frac{1}{7-\sin 5 x}\) be a function defined on \(\mathbf{R}\). Then the range of the function \(f(x)\) is equal to :

The length of the latus rectum and directrices of hyperbola with eccentricity e are 9 and \(x= \pm \frac{4}{\sqrt{3}}\), respectively. Let the line \(y-\sqrt{3} x+\sqrt{3}=0\) touch this hyperbola at \(\left(x_0, y_0\right)\). If $$\mathrm{...

If \(\int \frac{1}{\mathrm{a}^2 \sin ^2 x+\mathrm{b}^2 \cos ^2 x} \mathrm{~d} x=\frac{1}{12} \tan ^{-1}(3 \tan x)+\) constant, then the maximum value of \(\mathrm{a} \sin x+\mathrm{b} \cos x\), is :

\(\lim _\limits{n \rightarrow \infty} \frac{\left(1^2-1\right)(n-1)+\left(2^2-2\right)(n-2)+\cdots+\left((n-1)^2-(n-1)\right) \cdot 1}{\left(1^3+2^3+\cdots \cdots+n^3\right)-\left(1^2+2^2+\cdots \cdots+n^2\right)}\) is equal to :

Let \([t]\) denote the greatest integer less than or equal to \(t\). Let \(f:[0, \infty) \rightarrow \mathbf{R}\) be a function defined by \(f(x)=\left[\frac{x}{2}+3\right]-[\sqrt{x}]\). Let \(\mathrm{S}\) be the set of all points in the in...

If \(A\) is a square matrix of order 3 such that \(\operatorname{det}(A)=3\) and $$\operatorname{det}\left(\operatorname{adj}\left(-4 \operatorname{adj}\left(-3 \operatorname{adj}\left(3 \operatorname{adj}\left((2 \mathrm{~A})^{-1}\right)\r...

If the system of equations
$$\begin{aligned} & 2 x+7 y+\lambda z=3 \\ & 3 x+2 y+5 z=4 \\ & x+\mu y+32 z=-1 \end{aligned}$$
has infinitely many solutions, then \((\lambda-\mu)\) is equal to ______ :

If all the words with or without meaning made using all the letters of the word "NAGPUR" are arranged as in a dictionary, then the word at \(315^{\text {th }}\) position in this arrangement is :

Let \(0 \leq r \leq n\). If \({ }^{n+1} C_{r+1}:{ }^n C_r:{ }^{n-1} C_{r-1}=55: 35: 21\), then \(2 n+5 r\) is equal to :

If three letters can be posted to any one of the 5 different addresses, then the probability that the three letters are posted to exactly two addresses is :

From a lot of 12 items containing 3 defectives, a sample of 5 items is drawn at random. Let the random variable \(X\) denote the number of defective items in the sample. Let items in the sample be drawn one by one without replacement. If va...

In a triangle \(\mathrm{ABC}, \mathrm{BC}=7, \mathrm{AC}=8, \mathrm{AB}=\alpha \in \mathrm{N}\) and \(\cos \mathrm{A}=\frac{2}{3}\). If \(49 \cos (3 \mathrm{C})+42=\frac{\mathrm{m}}{\mathrm{n}}\), where \(\operatorname{gcd}(m, n)=1\), then ...

Let \(\alpha, \beta\) be roots of \(x^2+\sqrt{2} x-8=0\). If \(\mathrm{U}_{\mathrm{n}}=\alpha^{\mathrm{n}}+\beta^{\mathrm{n}}\), then \(\frac{\mathrm{U}_{10}+\sqrt{2} \mathrm{U}_9}{2 \mathrm{U}_8}\) is equal to ________.

Let \(A B C\) be an equilateral triangle. A new triangle is formed by joining the middle points of all sides of the triangle \(A B C\) and the same process is repeated infinitely many times. If \(\mathrm{P}\) is the sum of perimeters and $$...

A software company sets up m number of computer systems to finish an assignment in 17 days. If 4 computer systems crashed on the start of the second day, 4 more computer systems crashed on the start of the third day and so on, then it took ...

If \(\mathrm{S}(x)=(1+x)+2(1+x)^2+3(1+x)^3+\cdots+60(1+x)^{60}, x \neq 0\), and \((60)^2 \mathrm{~S}(60)=\mathrm{a}(\mathrm{b})^{\mathrm{b}}+\mathrm{b}\), where \(a, b \in N\), then \((a+b)\) equal to _________.

Let \(\mathrm{A}=\{1,2,3,4,5\}\). Let \(\mathrm{R}\) be a relation on \(\mathrm{A}\) defined by \(x \mathrm{R} y\) if and only if \(4 x \leq 5 \mathrm{y}\). Let \(\mathrm{m}\) be the number of elements in \(\mathrm{R}\) and \(\mathrm{n}\) b...

If the locus of the point, whose distances from the point \((2,1)\) and \((1,3)\) are in the ratio \(5: 4\), is \(a x^2+b y^2+c x y+d x+e y+170=0\), then the value of \(a^2+2 b+3 c+4 d+e\) is equal to :

Let \(\vec{a}=2 \hat{i}+\hat{j}-\hat{k}, \vec{b}=((\vec{a} \times(\hat{i}+\hat{j})) \times \hat{i}) \times \hat{i}\). Then the square of the projection of \(\vec{a}\) on \(\vec{b}\) is:

Let \(\overrightarrow{\mathrm{a}}=6 \hat{i}+\hat{j}-\hat{k}\) and \(\overrightarrow{\mathrm{b}}=\hat{i}+\hat{j}\). If \(\overrightarrow{\mathrm{c}}\) is a is vector such that $$|\overrightarrow{\mathrm{c}}| \geq 6, \overrightarrow{\mathrm{a...