JEE Main 2024 Feb 1 Shift 2 — Mathematics

If the domain of the function $f(x)=\frac{\sqrt{x^2-25}}{(4-x^2)}+\log_{10}(x^2+2x-15)$ is $(-\infty, \alpha) \cup [\beta, \infty)$, then $\alpha^2+\beta^3$ is equal to :

Consider the relations $R_1$ and $R_2$ defined as $aR_1b \Leftrightarrow a^2+b^2=1$ for all $a, b \in \mathbf{R}$ and $(a, b)R_2(c, d) \Leftrightarrow a+d=b+c$ for all $(a, b), (c, d) \in \mathbf{N} \times \mathbf{N}$. Then

If $z$ is a complex number such that $|z| \geq 1$, then the minimum value of $\left|z+\frac{1}{2}(3+4i)\right|$ is :

Let $\alpha$ and $\beta$ be the roots of the equation $px^2+qx-r=0$, where $p \neq 0$. If $p$, $q$ and $r$ be the consecutive terms of a non constant G.P. and $\frac{1}{\alpha}+\frac{1}{\beta}=\frac{3}{4}$, then the value of $(\alpha-\beta)^2$ is :

Let the system of equations $x+2y+3z=5$, $2x+3y+z=9$, $4x+3y+\lambda z=\mu$ have infinite number of solutions. Then $\lambda+2\mu$ is equal to :

Let $m$ and $n$ be the coefficients of seventh and thirteenth terms respectively in the expansion of $\left(\frac{1}{3}x^{\frac{1}{3}}+\frac{1}{2x^{\frac{2}{3}}}\right)^{18}$. Then $\left(\frac{n}{m}\right)^{\frac{1}{3}}$ is :

Let $S_n$ denote the sum of the first $n$ terms of an arithmetic progression. If $S_{10}=390$ and the ratio of the tenth and the fifth terms is $15:7$, then $S_{15}-S_5$ is equal to :

Let $f(x)=|2x^2+5|x|-3|$, $x \in \mathbf{R}$. If $m$ and $n$ denote the number of points where $f$ is not continuous and not differentiable respectively, then $m+n$ is equal to :

Let $f(x)=\begin{cases} x-1, & x \text{ is even,} \\ 2x, & x \text{ is odd,} \end{cases} \; x \in \mathbf{N}$. If for some $a \in \mathbf{N}$, $f(f(f(a)))=21$, then $\lim_{x \to a^-}\left\{\frac{|x|^3}{a}-\left[\frac{x}{a}\right]\right\}$, where $[t]$ denotes the greatest integer less than or equal to $t$, is equal to :

If $\int_0^{\frac{\pi}{3}} \cos^4 x \, \mathrm{d}x = a\pi + b\sqrt{3}$, where $a$ and $b$ are rational numbers, then $9a+8b$ is equal to :

The value of $\int_0^1 (2x^3-3x^2-x+1)^{\frac{1}{3}} \, \mathrm{d}x$ is equal to :

Let $\alpha$ be a non-zero real number. Suppose $f: \mathbf{R} \to \mathbf{R}$ is a differentiable function such that $f(0)=2$ and $\lim_{x \to -\infty} f(x)=1$. If $f'(x)=\alpha f(x)+3$, for all $x \in \mathbf{R}$, then $f(-\log_e 2)$ is equal to ________.

Let P be a point on the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$. Let the line passing through P and parallel to $y$-axis meet the circle $x^2+y^2=9$ at point Q such that P and Q are on the same side of the $x$-axis. Then, the eccentricity of the locus of the point R on PQ such that $PR:RQ=4:3$ as P moves on the ellipse, is :

Let the locus of the midpoints of the chords of the circle $x^2+(y-1)^2=1$ drawn from the origin intersect the line $x+y=1$ at P and Q. Then, the length of PQ is :

If the mirror image of the point $P(3, 4, 9)$ in the line $\frac{x-1}{3}=\frac{y+1}{2}=\frac{z-2}{1}$ is $(\alpha, \beta, \gamma)$, then $14(\alpha+\beta+\gamma)$ is :

Let P and Q be the points on the line $\frac{x+3}{8}=\frac{y-4}{2}=\frac{z+1}{2}$ which are at a distance of 6 units from the point $R(1, 2, 3)$. If the centroid of the triangle PQR is $(\alpha, \beta, \gamma)$, then $\alpha^2+\beta^2+\gamma^2$ is :

Consider a $\Delta ABC$ where $A(1, 3, 2)$, $B(-2, 8, 0)$ and $C(3, 6, 7)$. If the angle bisector of $\angle BAC$ meets the line BC at D, then the length of the projection of the vector $\vec{AD}$ on the vector $\vec{AC}$ is :

Let Ajay will not appear in JEE exam with probability $p=\frac{2}{7}$, while both Ajay and Vijay will appear in the exam with probability $q=\frac{1}{5}$. Then the probability, that Ajay will appear in the exam and Vijay will not appear is :

Consider 10 observations $x_1, x_2, \ldots, x_{10}$ such that $\sum_{i=1}^{10}(x_i-\alpha)=2$ and $\sum_{i=1}^{10}(x_i-\beta)^2=40$, where $\alpha, \beta$ are positive integers. Let the mean and the variance of the observations be $\frac{6}{5}$ and $\frac{84}{25}$ respectively. Then $\frac{\beta}{\alpha}$ is equal to :

The number of solutions of the equation $4\sin^2 x - 4\cos^3 x + 9 - 4\cos x = 0$; $x \in [-2\pi, 2\pi]$ is :

Let $A = I_2 - 2MM^T$, where $M$ is a real matrix of order $2 \times 1$ such that the relation $M^T M = I_1$ holds. If $\lambda$ is a real number such that the relation $AX = \lambda X$ holds for some non-zero real matrix $X$ of order $2 \times 1$, then the sum of squares of all possible values of $\lambda$ is equal to ________.

The lines $L_1, L_2, \ldots, L_{20}$ are distinct. For $n=1, 2, 3, \ldots, 10$ all the lines $L_{2n-1}$ are parallel to each other and all the lines $L_{2n}$ pass through a given point P. The maximum number of points of intersection of pairs of lines from the set $\{L_1, L_2, \ldots, L_{20}\}$ is equal to ________.

If three successive terms of a G.P. with common ratio $r$ ($r>1$) are the lengths of the sides of a triangle and $[r]$ denotes the greatest integer less than or equal to $r$, then $3[r]+[-r]$ is equal to ________.

Let $f:(0, \infty) \to \mathbf{R}$ and $F(x)=\int_0^x t f(t) \, \mathrm{d}t$. If $F(x^2)=x^4+x^5$, then $\sum_{r=1}^{12} f(r^2)$ is equal to ________.

If $y=\frac{(\sqrt{x}+1)(x^2-\sqrt{x})}{x\sqrt{x}+x+\sqrt{x}}+\frac{1}{15}(3\cos^2 x - 5)\cos^3 x$, then $96\, y'\!\left(\frac{\pi}{6}\right)$ is equal to ________.

The sum of squares of all possible values of $k$, for which area of the region bounded by the parabolas $2y^2=kx$ and $ky^2=2(y-x)$ is maximum, is equal to ________.

If $\frac{\mathrm{d}x}{\mathrm{d}y}=\frac{1+x-y^2}{y}$, $x(1)=1$, then $5x(2)$ is equal to __________.

Three points $O(0, 0)$, $P(a, a^2)$, $Q(-b, b^2)$, $a>0$, $b>0$, are on the parabola $y=x^2$. Let $S_1$ be the area of the region bounded by the line PQ and the parabola, and $S_2$ be the area of the triangle OPQ. If the minimum value of $\frac{S_1}{S_2}$ is $\frac{m}{n}$, $\gcd(m, n)=1$, then $m+n$ is equal to ________.

Let ABC be an isosceles triangle in which A is at $(-1, 0)$, $\angle A = \frac{2\pi}{3}$, $AB=AC$ and B is on the positve $x$-axis. If $BC=4\sqrt{3}$ and the line BC intersects the line $y=x+3$ at $(\alpha, \beta)$, then $\frac{\beta^4}{\alpha^2}$ is ________.

Let $\vec{a}=\hat{i}+\hat{j}+\hat{k}$, $\vec{b}=-\hat{i}-8\hat{j}+2\hat{k}$ and $\vec{c}=4\hat{i}+c_2\hat{j}+c_3\hat{k}$ be three vectors such that $\vec{b} \times \vec{a} = \vec{c} \times \vec{a}$. If the angle between the vector $\vec{c}$ and the vector $3\hat{i}+4\hat{j}+\hat{k}$ is $\theta$, then the greatest integer less than or equal to $\tan^2\theta$ is ________.