JEE Main 2024 (Online) 27th January Evening Shift

Let the image of the point \((1,0,7)\) in the line \(\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}\) be the point \((\alpha, \beta, \gamma)\). Then which one of the following points lies on the line passing through \((\alpha, \beta, \gamma)\) and...

The lines \(\frac{x-2}{2}=\frac{y}{-2}=\frac{z-7}{16}\) and \(\frac{x+3}{4}=\frac{y+2}{3}=\frac{z+2}{1}\) intersect at the point \(P\). If the distance of \(\mathrm{P}\) from the line \(\frac{x+1}{2}=\frac{y-1}{3}=\frac{z-1}{1}\) is \(l\), ...

Let \(g(x)=3 f\left(\frac{x}{3}\right)+f(3-x)\) and \(f^{\prime \prime}(x)>0\) for all \(x \in(0,3)\). If \(g\) is decreasing in \((0, \alpha)\) and increasing in \((\alpha, 3)\), then \(8 \alpha\) is :

If the area of the region \(\left\{(x, y): 0 \leq y \leq \min \left\{2 x, 6 x-x^2\right\}\right\}\) is \(\mathrm{A}\), then \(12 \mathrm{~A}\) is equal to ________.

The coefficient of \(x^{2012}\) in the expansion of \((1-x)^{2008}\left(1+x+x^2\right)^{2007}\) is equal to _________.

Consider a circle \((x-\alpha)^2+(y-\beta)^2=50\), where \(\alpha, \beta>0\). If the circle touches the line \(y+x=0\) at the point \(P\), whose distance from the origin is \(4 \sqrt{2}\), then \((\alpha+\beta)^2\) is equal to __________.

Let the complex numbers \(\alpha\) and \(\frac{1}{\bar{\alpha}}\) lie on the circles \(\left|z-z_0\right|^2=4\) and \(\left|z-z_0\right|^2=16\) respectively, where \(z_0=1+i\). Then, the value of \(100|\alpha|^2\) is __________.

For \(0 < \mathrm{a} < 1\), the value of the integral \(\int_\limits0^\pi \frac{\mathrm{d} x}{1-2 \mathrm{a} \cos x+\mathrm{a}^2}\) is :

Let \(f(x)=\int_\limits0^x g(t) \log _{\mathrm{e}}\left(\frac{1-\mathrm{t}}{1+\mathrm{t}}\right) \mathrm{dt}\), where \(g\) is a continuous odd function.
If $$\int_{-\pi / 2}^{\pi / 2}\left(f(x)+\frac{x^2 \cos x}{1+\mathrm{e}^x}\right) \mat...

If \(y=y(x)\) is the solution curve of the differential equation \(\left(x^2-4\right) \mathrm{d} y-\left(y^2-3 y\right) \mathrm{d} x=0, x>2, y(4)=\frac{3}{2}\) and the slope of the curve is never zero, then the value of \(y(10)\) equals :

If the solution curve, of the differential equation \(\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{x+y-2}{x-y}\) passing through the point \((2,1)\) is $$\tan ^{-1}\left(\frac{y-1}{x-1}\right)-\frac{1}{\beta} \log _{\mathrm{e}}\left(\alpha+\lef...

Let \(f: \mathbf{R}-\left\{\frac{-1}{2}\right\} \rightarrow \mathbf{R}\) and \(g: \mathbf{R}-\left\{\frac{-5}{2}\right\} \rightarrow \mathbf{R}\) be defined as \(f(x)=\frac{2 x+3}{2 x+1}\) and \(g(x)=\frac{|x|+1}{2 x+5}\). Then, the domain ...

Let \(e_1\) be the eccentricity of the hyperbola \(\frac{x^2}{16}-\frac{y^2}{9}=1\) and \(e_2\) be the eccentricity of the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1, \mathrm{a} > \mathrm{b}\), which passes through the foci of the hyperbol...

\(\text { The integral } \int \frac{\left(x^8-x^2\right) \mathrm{d} x}{\left(x^{12}+3 x^6+1\right) \tan ^{-1}\left(x^3+\frac{1}{x^3}\right)} \text { is equal to : }\)

Considering only the principal values of inverse trigonometric functions, the number of positive real values of \(x\) satisfying \(\tan ^{-1}(x)+\tan ^{-1}(2 x)=\frac{\pi}{4}\) is :

Consider the function \(f:(0,2) \rightarrow \mathbf{R}\) defined by \(f(x)=\frac{x}{2}+\frac{2}{x}\) and the function \(g(x)\) defined by
$$g(x)=\left\{\begin{array}{ll}
\min \lfloor f(t)\}, & 0<\mathrm{t} \leq x \text { and } 0 < x \leq 1 ...

\(\text { If } \lim _\limits{x \rightarrow 0} \frac{3+\alpha \sin x+\beta \cos x+\log _e(1-x)}{3 \tan ^2 x}=\frac{1}{3} \text {, then } 2 \alpha-\beta \text { is equal to : }\)

The values of \(\alpha\), for which $$\left|\begin{array}{ccc}1 & \frac{3}{2} & \alpha+\frac{3}{2} \\ 1 & \frac{1}{3} & \alpha+\frac{1}{3} \\ 2 \alpha+3 & 3 \alpha+1 & 0\end{array}\right|=0$$, lie in the interval

Let \(A\) be a \(2 \times 2\) real matrix and \(I\) be the identity matrix of order 2. If the roots of the equation \(|\mathrm{A}-x \mathrm{I}|=0\) be \(-1\) and 3, then the sum of the diagonal elements of the matrix \(\mathrm{A}^2\) is

Let \(\alpha=\frac{(4 !) !}{(4 !)^{3 !}}\) and \(\beta=\frac{(5 !) !}{(5 !)^{4 !}}\). Then :

An urn contains 6 white and 9 black balls. Two successive draws of 4 balls are made without replacement. The probability, that the first draw gives all white balls and the second draw gives all black balls, is :

If \(\alpha, \beta\) are the roots of the equation, \(x^2-x-1=0\) and \(S_n=2023 \alpha^n+2024 \beta^n\), then :

\(\text { The } 20^{\text {th }} \text { term from the end of the progression } 20,19 \frac{1}{4}, 18 \frac{1}{2}, 17 \frac{3}{4}, \ldots,-129 \frac{1}{4} \text { is : }\)

Let \(A\) and \(B\) be two finite sets with \(m\) and \(n\) elements respectively. The total number of subsets of the set \(A\) is 56 more than the total number of subsets of \(B\). Then the distance of the point \(P(m, n)\) from the point ...

The mean and standard deviation of 15 observations were found to be 12 and 3 respectively. On rechecking it was found that an observation was read as 10 in place of 12 . If \(\mu\) and \(\sigma^2\) denote the mean and variance of the correc...

Let \(\mathrm{R}\) be the interior region between the lines \(3 x-y+1=0\) and \(x+2 y-5=0\) containing the origin. The set of all values of \(a\), for which the points \(\left(a^2, a+1\right)\) lie in \(R\), is :

If the sum of squares of all real values of \(\alpha\), for which the lines \(2 x-y+3=0,6 x+3 y+1=0\) and \(\alpha x+2 y-2=0\) do not form a triangle is \(p\), then the greatest integer less than or equal to \(p\) is _________.

If \(2 \tan ^2 \theta-5 \sec \theta=1\) has exactly 7 solutions in the interval \(\left[0, \frac{n \pi}{2}\right]\), for the least value of \(n \in \mathbf{N}\), then \(\sum_\limits{k=1}^n \frac{k}{2^k}\) is equal to:

Let the position vectors of the vertices \(\mathrm{A}, \mathrm{B}\) and \(\mathrm{C}\) of a triangle be \(2 \hat{i}+2 \hat{j}+\hat{k}, \hat{i}+2 \hat{j}+2 \hat{k}\) and \(2 \hat{i}+\hat{j}+2 \hat{k}\) respectively. Let \(l_1, l_2\) and $$l_...

The position vectors of the vertices \(\mathrm{A}, \mathrm{B}\) and \(\mathrm{C}\) of a triangle are \(2 \hat{i}-3 \hat{j}+3 \hat{k}, 2 \hat{i}+2 \hat{j}+3 \hat{k}\) and \(-\hat{i}+\hat{j}+3 \hat{k}\) respectively. Let \(l\) denotes the len...