JEE Main 2025 Jan 23 Shift 1 — Mathematics

Let $f(x) = \log_e x$ and $g(x) = \frac{x^4 - 2x^3 + 3x^2 - 2x + 2}{2x^2 - 2x + 1}$. Then the domain of $fog$ is

Let $R = \{(1, 2), (2, 3), (3, 3)\}$ be a relation defined on the set $\{1, 2, 3, 4\}$. Then the minimum number of elements, needed to be added in $R$ so that $R$ becomes an equivalence relation, is:

Let $\left|\frac{\bar{z} - i}{2\bar{z} + i}\right| = \frac{1}{3}$, $z \in \mathbb{C}$, be the equation of a circle with center at $C$. If the area of the triangle, whose vertices are at the points $(0, 0)$, $C$ and $(\alpha, 0)$ is 11 square units, then $\alpha^2$ equals:

If the system of equations

$(\lambda - 1)x + (\lambda - 4)y + \lambda z = 5$

$\lambda x + (\lambda - 1)y + (\lambda - 4)z = 7$

$(\lambda + 1)x + (\lambda + 2)y - (\lambda + 2)z = 9$

has infinitely many solutions, then $\lambda^2 + \lambda$ is equal to

If $A$, $B$, and $(\mathrm{adj}(A^{-1}) + \mathrm{adj}(B^{-1}))$ are non-singular matrices of same order, then the inverse of $A(\mathrm{adj}(A^{-1}) + \mathrm{adj}(B^{-1}))^{-1}B$, is equal to

If the first term of an A.P. is 3 and the sum of its first four terms is equal to one-fifth of the sum of the next four terms, then the sum of the first 20 terms is equal to

The number of words, which can be formed using all the letters of the word “DAUGHTER”, so that all the vowels never come together, is

One die has two faces marked 1, two faces marked 2, one face marked 3 and one face marked 4. Another die has one face marked 1, two faces marked 2, two faces marked 3 and one face marked 4. The probability of getting the sum of numbers to be 4 or 5, when both the dice are thrown together, is

Let the area of a $\triangle PQR$ with vertices $P(5, 4)$, $Q(-2, 4)$ and $R(a, b)$ be 35 square units. If its orthocenter and centroid are $O\left(2, \frac{14}{5}\right)$ and $C(c, d)$ respectively, then $c + 2d$ is equal to

If $\frac{\pi}{2} \le x \le \frac{3\pi}{4}$, then $\cos^{-1}\left(\frac{12}{13}\cos x + \frac{5}{13}\sin x\right)$ is equal to

The value of $(\sin 70°)(\cot 10° \cot 70° - 1)$ is

If the line $3x - 2y + 12 = 0$ intersects the parabola $4y = 3x^2$ at the points $A$ and $B$, then at the vertex of the parabola, the line segment $AB$ subtends an angle equal to

Let the arc $AC$ of a circle subtend a right angle at the centre $O$. If the point $B$ on the arc $AC$, divides the arc $AC$ such that $\frac{\text{length of arc } AB}{\text{length of arc } BC} = \frac{1}{5}$, and $\vec{OC} = \alpha \vec{OA} + \beta \vec{OB}$, then $\alpha + \sqrt{2}(\sqrt{3} - 1)\beta$ is equal to

Let the position vectors of the vertices $A$, $B$ and $C$ of a tetrahedron $ABCD$ be $\hat{i} + 2\hat{j} + \hat{k}$, $\hat{i} + 3\hat{j} - 2\hat{k}$ and $2\hat{i} + \hat{j} - \hat{k}$ respectively. The altitude from the vertex $D$ to the opposite face $ABC$ meets the median line segment through $A$ of the triangle $ABC$ at the point $E$. If the length of $AD$ is $\frac{\sqrt{110}}{3}$ and the volume of the tetrahedron is $\frac{\sqrt{805}}{6\sqrt{2}}$, then the position vector of $E$ is [Note: stem recovered from a low-resolution embedded thumbnail; the English and Hindi stem images were destroyed by PDF corruption — numeric values should be verified against another copy of this paper]

Let $P$ be the foot of the perpendicular from the point $Q(10, -3, -1)$ on the line $\frac{x-3}{7} = \frac{y-2}{-1} = \frac{z+1}{-2}$. Then the area of the right angled triangle $PQR$, where $R$ is the point $(3, -2, 1)$, is

If the function

$$f(x) = \begin{cases} \frac{2}{x}\{\sin(k_1 + 1)x + \sin(k_2 - 1)x\}, & x < 0 \\ 4, & x = 0 \\ \frac{2}{x}\log_e\left(\frac{2 + k_1 x}{2 + k_2 x}\right), & x > 0 \end{cases}$$

is continuous at $x = 0$, then $k_1^2 + k_2^2$ is equal to

The marks obtained by all the students of class 12 are presented in a frequency distribution with classes of equal width. Let the median class interval be 12-18 with median class frequency 12, and the median of these grouped data be 14. If the number of students whose marks are less than 12 is 18, then the total number of students is

Let $I(x) = \int \frac{dx}{(x-11)^{\frac{11}{13}}(x+15)^{\frac{15}{13}}}$. If $I(37) - I(24) = \frac{1}{4}\left(\frac{1}{b^{\frac{1}{13}}} - \frac{1}{c^{\frac{1}{13}}}\right)$, $b, c \in \mathbb{N}$, then $3(b + c)$ is equal to

The value of

$$\int_{e^2}^{e^4} \frac{1}{x}\left(\frac{e^{((\log_e x)^2 + 1)^{-1}}}{e^{((\log_e x)^2 + 1)^{-1}} + e^{((6 - \log_e x)^2 + 1)^{-1}}}\right) dx$$ is

Let a curve $y = f(x)$ pass through the points $(0, 5)$ and $(\log_e 2, k)$. If the curve satisfies the differential equation $2(3 + y)e^{2x}dx - (7 + e^{2x})dy = 0$, then $k$ is equal to

If the equation $a(b - c)x^2 + b(c - a)x + c(a - b) = 0$ has equal roots, where $a + c = 15$ and $b = \frac{36}{5}$, then $a^2 + c^2$ is equal to____

If the area of the larger region bounded between the curves $x^2 + y^2 = 25$ and $y = |x - 1|$ is $\frac{1}{4}(b\pi + c)$, $b, c \in \mathbb{N}$, then $b + c$ is equal to ________.

The sum of all rational terms in the expansion of $(1 + 2^{\frac{1}{3}} + 3^{\frac{1}{2}})^6$ is equal to ______