JEE Main 2024 (Online) 29th January Morning Shift

Let \(O\) be the origin and the position vectors of \(A\) and \(B\) be \(2 \hat{i}+2 \hat{j}+\hat{k}\) and \(2 \hat{i}+4 \hat{j}+4 \hat{k}\) respectively. If the internal bisector of \(\angle \mathrm{AOB}\) meets the line \(\mathrm{AB}\) at...

Let \(P Q R\) be a triangle with \(R(-1,4,2)\). Suppose \(M(2,1,2)\) is the mid point of \(\mathrm{PQ}\). The distance of the centroid of \(\triangle \mathrm{PQR}\) from the point of intersection of the lines $$\frac{x-2}{0}=\frac{y}{2}=\fr...

A line with direction ratios \(2,1,2\) meets the lines \(x=y+2=z\) and \(x+2=2 y=2 z\) respectively at the points \(\mathrm{P}\) and \(\mathrm{Q}\). If the length of the perpendicular from the point \((1,2,12)\) to the line \(\mathrm{PQ}\) ...

Consider the function \(f:\left[\frac{1}{2}, 1\right] \rightarrow \mathbb{R}\) defined by \(f(x)=4 \sqrt{2} x^3-3 \sqrt{2} x-1\). Consider the statements
(I) The curve \(y=f(x)\) intersects the \(x\)-axis exactly at one point.
(II) The curv...

Let \(f(x)=2^x-x^2, x \in \mathbb{R}\). If \(m\) and \(n\) are respectively the number of points at which the curves \(y=f(x)\) and \(y=f^{\prime}(x)\) intersect the \(x\)-axis, then the value of \(\mathrm{m}+\mathrm{n}\) is ___________.

The area (in sq. units) of the part of the circle \(x^2+y^2=169\) which is below the line \(5 x-y=13\) is \(\frac{\pi \alpha}{2 \beta}-\frac{65}{2}+\frac{\alpha}{\beta} \sin ^{-1}\left(\frac{12}{13}\right)\), where \(\alpha, \beta\) are cop...

If the points of intersection of two distinct conics \(x^2+y^2=4 b\) and \(\frac{x^2}{16}+\frac{y^2}{b^2}=1\) lie on the curve \(y^2=3 x^2\), then \(3 \sqrt{3}\) times the area of the rectangle formed by the intersection points is _________...

\(\text { If } \frac{{ }^{11} C_1}{2}+\frac{{ }^{11} C_2}{3}+\ldots+\frac{{ }^{11} C_9}{10}=\frac{n}{m} \text { with } \operatorname{gcd}(n, m)=1 \text {, then } n+m \text { is equal to }\) _______.

Equations of two diameters of a circle are \(2 x-3 y=5\) and \(3 x-4 y=7\). The line joining the points \(\left(-\frac{22}{7},-4\right)\) and \(\left(-\frac{1}{7}, 3\right)\) intersects the circle at only one point \(P(\alpha, \beta)\). The...

If \(z=\frac{1}{2}-2 i\) is such that \(|z+1|=\alpha z+\beta(1+i), i=\sqrt{-1}\) and \(\alpha, \beta \in \mathbb{R}\), then \(\alpha+\beta\) is equal to

Let \(\alpha, \beta\) be the roots of the equation \(x^2-x+2=0\) with \(\operatorname{Im}(\alpha)>\operatorname{Im}(\beta)\). Then \(\alpha^6+\alpha^4+\beta^4-5 \alpha^2\) is equal to ___________.

\(\mathop {\lim }\limits_{x \to {\pi \over 2}} \left( {{1 \over {{{\left( {x - {\pi \over 2}} \right)}^2}}}\int\limits_{{x^3}}^{{{\left( {{\pi \over 2}} \right)}^3}} {\cos \left( {{t^{{1 \over 3}}}} \right)dt} } \right)\) is equal to

If the value of the integral \(\int_\limits{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(\frac{x^2 \cos x}{1+\pi^x}+\frac{1+\sin ^2 x}{1+e^{\sin x^{2123}}}\right) d x=\frac{\pi}{4}(\pi+a)-2\), then the value of \(a\) is

A function \(y=f(x)\) satisfies \(f(x) \sin 2 x+\sin x-\left(1+\cos ^2 x\right) f^{\prime}(x)=0\) with condition \(f(0)=0\). Then, \(f\left(\frac{\pi}{2}\right)\) is equal to

If the solution curve \(y=y(x)\) of the differential equation \(\left(1+y^2\right)\left(1+\log _{\mathrm{e}} x\right) d x+x d y=0, x > 0\) passes through the point \((1,1)\) and $$y(e)=\frac{\alpha-\tan \left(\frac{3}{2}\right)}{\beta+\tan ...

Suppose \(f(x)=\frac{\left(2^x+2^{-x}\right) \tan x \sqrt{\tan ^{-1}\left(x^2-x+1\right)}}{\left(7 x^2+3 x+1\right)^3}\). Then the value of \(f^{\prime}(0)\) is equal to

If $$f(x)=\left\{\begin{array}{cc}2+2 x, & -1 \leq x < 0 \\ 1-\frac{x}{3}, & 0 \leq x \leq 3\end{array} ; g(x)=\left\{\begin{array}{cc}-x, & -3 \leq x \leq 0 \\ x, & 0 < x \leq 1\end{array}\right.\right.$$, then range of \((f o g)(x)\) is

For \(x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\), if \(y(x)=\int \frac{\operatorname{cosec} x+\sin x}{\operatorname{cosec} x \sec x+\tan x \sin ^2 x} d x\), and \(\lim _\limits{x \rightarrow\left(\frac{\pi}{2}\right)^{-}} y(x)=0\) th...

$$\text { Let } A=\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & \alpha & \beta \\ 0 & \beta & \alpha \end{array}\right] \text { and }|2 \mathrm{~A}|^3=2^{21} \text { where } \alpha, \beta \in Z \text {, Then a value of } \alpha \text { is }$$

Let \(\mathrm{A}\) be a square matrix such that \(\mathrm{AA}^{\mathrm{T}}=\mathrm{I}\). Then \(\frac{1}{2} A\left[\left(A+A^T\right)^2+\left(A-A^T\right)^2\right]\) is equal to

All the letters of the word "GTWENTY" are written in all possible ways with or without meaning and these words are written as in a dictionary. The serial number of the word "GTWENTY" is _________.

A fair die is thrown until 2 appears. Then the probability, that 2 appears in even number of throws, is

Let \(\left(5, \frac{a}{4}\right)\) be the circumcenter of a triangle with vertices \(\mathrm{A}(a,-2), \mathrm{B}(a, 6)\) and \(C\left(\frac{a}{4},-2\right)\). Let \(\alpha\) denote the circumradius, \(\beta\) denote the area and $$\gamma$...

If in a G.P. of 64 terms, the sum of all the terms is 7 times the sum of the odd terms of the G.P, then the common ratio of the G.P. is equal to

In an A.P., the sixth term \(a_6=2\). If the product \(a_1 a_4 a_5\) is the greatest, then the common difference of the A.P. is equal to

Let \(R\) be a relation on \(Z \times Z\) defined by \((a, b) R(c, d)\) if and only if \(a d-b c\) is divisible by 5. Then \(R\) is

If the mean and variance of the data \(65,68,58,44,48,45,60, \alpha, \beta, 60\) where \(\alpha> \beta\), are 56 and 66.2 respectively, then \(\alpha^2+\beta^2\) is equal to _________.

In a \(\triangle A B C\), suppose \(y=x\) is the equation of the bisector of the angle \(B\) and the equation of the side \(A C\) is \(2 x-y=2\). If \(2 A B=B C\) and the points \(A\) and \(B\) are respectively \((4,6)\) and $$(\alpha, \bet...

If \(\alpha,-\frac{\pi}{2}<\alpha<\frac{\pi}{2}\) is the solution of \(4 \cos \theta+5 \sin \theta=1\), then the value of \(\tan \alpha\) is

Let \(\vec{a}, \vec{b}\) and \(\vec{c}\) be three non-zero vectors such that \(\vec{b}\) and \(\vec{c}\) are non-collinear. If \(\vec{a}+5 \vec{b}\) is collinear with \(\vec{c}, \vec{b}+6 \vec{c}\) is collinear with \(\vec{a}\) and $$\vec{a...