CK0607/2025-Jee-Mains-Question huggingface dataset:
JEE Main 2025 (22 Jan Shift 2) — Mathematics, Question 20

Let \( E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, a > b \) and \( H: \frac{x^2}{A^2} - \frac{y^2}{B^2} = 1 \). Let the distance between the foci of \( E \) and the foci of \( H \) be \( 2\sqrt{3} \). If \( a - A = 2 \), and the ratio of the eccentricities of \( E \) and \( H \) is \( \frac{1}{3} \), then the sum of the lengths of their latus rectums is equal to:

(1) 10
(2) 9
(3) 8
(4) 7

Correct option: 3
JEE Main 2025 (22 Jan Shift 2) — Mathematics, Question 21

If \( \sum_{r=1}^{30} \frac{r^3 (\cos \alpha)^2}{30C_r} = \alpha \times 2^{29} \), then \( \alpha \) is equal to _______.

Correct option: 465
JEE Main 2025 (22 Jan Shift 2) — Mathematics, Question 22

Let \( A = \{1, 2, 3\} \). The number of relations on \( A \), containing (1, 2) and (2, 3), which are reflexive and transitive but not symmetric, is _______.

Correct option: 3
JEE Main 2025 (22 Jan Shift 2) — Mathematics, Question 23

Let \( A(6, 8), B(10 \cos \alpha, -10 \sin \alpha), C(-10 \sin \alpha, 10 \cos \alpha) \), be the vertices of a triangle. If \( L(a, 9) \) and \( G(h, k) \) be its orthocenter and centroid respectively, then \( 5a - 3h + 6k + 100 \sin 2\alpha \) is equal to _______.

Correct option: 145
JEE Main 2025 (22 Jan Shift 2) — Mathematics, Question 24

Let \( y = f(x) \) be the solution of the differential equation \( \frac{dy}{dx} + \frac{xy}{x^2 - 1} = \frac{x^2 + 4x}{\sqrt{1-x^2}}, -1 < x < 1 \) such that \( f(0) = 0 \). If \( \int_{-1/2}^{1/2} f(x)dx = 2\pi - \alpha \) then \( \alpha^2 \) is equal to _______.

Correct option: 27
JEE Main 2025 (22 Jan Shift 2) — Mathematics, Question 25

Let the distance between two parallel lines be 5 units and a point \( P \) lie between the lines at a unit distance from one of them. An equilateral triangle \( PQR \) is formed such that \( Q \) lies on one of the parallel lines, while \( R \) lies on the other. Then \( (QR)^2 \) is equal to _______.

Correct option: 28
JEE Main 2025 (23 Jan Shift 1) — Mathematics, Question 1

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

- (1) $-1080$
- (2) $-1020$
- (3) $-1200$
- (4) $-120$

Correct option: 1
JEE Main 2025 (23 Jan Shift 1) — Mathematics, Question 2

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

- (1) $\frac{3}{16}$
- (2) $\frac{1}{4}$
- (3) $\frac{3}{8}$
- (4) $\frac{5}{8}$

Correct option: 2
JEE Main 2025 (23 Jan Shift 1) — Mathematics, Question 3

Let the position vectors of the vertices $A, B$ and $C$ of a tetrahedron $ABCD$ be $\mathbf{i} + 2\mathbf{j} + \mathbf{k}, \mathbf{i} + 3\mathbf{j} = 2\hat{k}$ and $2\mathbf{i} + \mathbf{j} - \mathbf{k}$ 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 $\frac{\sqrt{11}}{3}$ and the volume of the tetrahedron is $\frac{\sqrt{805}}{6}$, then the position vector of $E$ is

- (1) $\frac{1}{3}(7\mathbf{i} + 4\mathbf{j} + 3\mathbf{k})$
- (2) $\frac{1}{3}(i + 4\mathbf{j} + 7\mathbf{k})$
- (3) $\frac{1}{3}(12\mathbf{i} + 12\mathbf{j} + \mathbf{k})$
- (4) $\frac{1}{3}(7\mathbf{i} + 12\mathbf{j} + \mathbf{k})$

Correct option: 4
JEE Main 2025 (23 Jan Shift 1) — Mathematics, Question 4

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

- (1) $AB^{-1} + A^{-1}B$
- (2) $\text{adj} (B^{-1}) + \text{adj} (A^{-1})$
- (3) $\frac{AB^{-1}}{|A|} + \frac{BA^{-1}}{|B|}$
- (4) $\frac{1}{|A|}(\text{adj}(B) + \text{adj}(A))$

Correct option: 4
JEE Main 2025 (23 Jan Shift 1) — Mathematics, Question 5

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

- (1) 52
- (2) 48
- (3) 44
- (4) 40

Correct option: 3
JEE Main 2025 (23 Jan Shift 1) — Mathematics, Question 6

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

- (1) 4
- (2) 32
- (3) 8
- (4) 16

Correct option: 3
JEE Main 2025 (23 Jan Shift 1) — Mathematics, Question 7

If the function $f(x) = \begin{cases} \frac{2}{x} \sin (k_1 x + k_2 - 1) x, & x < 0 \\ 4, & x = 0 \\ \frac{2}{x} \log_e (\frac{2 + k_2 x}{2 + k_2 x}), & x > 0 \end{cases}$ is continuous at $x = 0$, then $k_1^2 + k_2^2$ is equal to

- (1) 20
- (2) 5
- (3) 8
- (4) 10

Correct option: 4
JEE Main 2025 (23 Jan Shift 1) — Mathematics, Question 8

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 : (1) \( \tan^{-1} \left(\frac{4}{5}\right) \)  
(2) \( \tan^{-1} \left(\frac{9}{7}\right) \)  
(3) \( \tan^{-1} \left(\frac{11}{9}\right) \)  
(4) \( \frac{\pi}{2} - \tan^{-1} \left(\frac{3}{7}\right) \)  

Correct option: 2
JEE Main 2025 (23 Jan Shift 1) — Mathematics, Question 9

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

\begin{align*}
(1) \ 9\sqrt{15} & & \quad (2) \ \sqrt{30} \\
(3) \ 8\sqrt{15} & & \quad (4) \ 3\sqrt{30}
\end{align*}

Correct option: 4
JEE Main 2025 (23 Jan Shift 1) — Mathematics, Question 10

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 \( \overrightarrow{OC} = \alpha\overrightarrow{OA} + \beta\overrightarrow{OB} \), then \( \alpha + \sqrt{2(\sqrt{3} - 1)}\beta \) is equal to

\begin{align*}
(1) \ 2\sqrt{3} & & \quad (2) \ 2 - \sqrt{3} \\
(3) \ 5\sqrt{3} & & \quad (4) \ 2 + \sqrt{3}
\end{align*}

Correct option: 2
JEE Main 2025 (23 Jan Shift 1) — Mathematics, Question 11

Let \( f(x) = \log_2 x \) and \( g(x) = \frac{x^4 - 2x^3 + 3x^2 - 2x + 2}{2x^2 - 2x + 1} \). Then the domain of \( f \circ g \) is

\begin{align*}
(1) \ [0, \infty) & & \quad (2) \ [1, \infty) \\
(3) \ (0, \infty) & & \quad (4) \ \mathbb{R}
\end{align*}

Correct option: 4
JEE Main 2025 (23 Jan Shift 1) — Mathematics, Question 12

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

If the system of equations \( \lambda x + (\lambda - 1)y + (\lambda - 4)z = 7 \) has infinitely many solutions, then \( \lambda^2 + \lambda \) is equal to

\begin{align*}
(1) \ 6 & & \quad (2) \ 10 \\
(3) \ 20 & & \quad (4) \ 12
\end{align*}

Correct option: 4
JEE Main 2025 (23 Jan Shift 1) — Mathematics, Question 13

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

\begin{align*}
(1) \ 36000 & & \quad (2) \ 37000 \\
(3) \ 34000 & & \quad (4) \ 35000
\end{align*}

Correct option: 1
JEE Main 2025 (23 Jan Shift 1) — Mathematics, Question 14

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

\begin{align*}
(1) \ 10 & & \quad (2) \ 7 \\
(3) \ 8 & & \quad (4) \ 9
\end{align*}

Correct option: 2
JEE Main 2025 (23 Jan Shift 1) — Mathematics, Question 15

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 \( O \left(2, \frac{12}{7}\right) \) and \( C(c, d) \) respectively, then \( c + 2d \) is equal to

\begin{align*}
(1) \ \frac{8}{3} & & \quad (2) \ \frac{7}{3} \\
(3) \ 2 & & \quad (4) \ 3
\end{align*}

Correct option: 4
JEE Main 2025 (23 Jan Shift 1) — Mathematics, Question 16

The value of \( \int_{\mathbb{R}} \frac{1}{x} \left( e^{(\log_2 x)^2 + 1} - e^{(\log_2 x)^2 - 1} \right) dx \) is

\begin{align*}
(1) \ 2 & & \quad (2) \ \log_2 2 \\
(3) \ 1 & & \quad (4) \ e^2
\end{align*}

Correct option: 3
JEE Main 2025 (23 Jan Shift 1) — Mathematics, Question 17

Let \( \frac{x^2}{16} + \frac{y^2}{25} = 1 \), \( z \in C \), 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:

- (1) 50
- (2) 100
- (3) \( \frac{81}{25} \)
- (4) \( \frac{121}{25} \)

Correct option: 2
JEE Main 2025 (23 Jan Shift 1) — Mathematics, Question 18

The value of \((\sin 70^\circ)(\cot 10^\circ \cot 70^\circ - 1)\) is

- (1) \( 2/3 \)
- (2) 0
- (3) \( 3/2 \)
- (4) 1

Correct option: 2
JEE Main 2025 (23 Jan Shift 1) — Mathematics, Question 19

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

- (1) 22
- (2) 39
- (3) 40
- (4) 26

Correct option: 2
JEE Main 2025 (23 Jan Shift 1) — Mathematics, Question 20

If \( \frac{\pi}{6} \leq x \leq \frac{3\pi}{4} \), then \( \cos^{-1}\left(\frac{12}{13}\cos x + \frac{5}{13}\sin x\right) \) is equal to

- (1) \( x - \tan^{-1}\frac{4}{3} \)
- (2) \( x + \tan^{-1}\frac{4}{5} \)
- (3) \( x - \tan^{-1}\frac{5}{12} \)
- (4) \( x + \tan^{-1}\frac{5}{12} \)

Correct option: 3
JEE Main 2025 (23 Jan Shift 1) — Mathematics, Question 21

Let the circle \( C \) touch the line \( x - y + 1 = 0 \), have the centre on the positive \( x \)-axis, and cut off a chord of length \( \frac{4}{\sqrt{13}} \) along the line \( -3x + 2y = 1 \). Let \( H \) be the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), whose one of the foci is the centre of \( C \) and the length of the transverse axis is the diameter of \( C \). Then \( 2a^2 + 3b^2 \) is equal to

Correct option: 19
JEE Main 2025 (23 Jan Shift 1) — Mathematics, Question 22

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

Correct option: 117
JEE Main 2025 (23 Jan Shift 1) — Mathematics, Question 23

If the set of all values of \( a \), for which the equation \( 5x^3 - 15x - a = 0 \) has three distinct real roots, is the interval \((\alpha, \beta)\), then \( \beta - 2\alpha \) is equal to

Correct option: 30
JEE Main 2025 (23 Jan Shift 1) — Mathematics, Question 24

The sum of all rational terms in the expansion of \( \left(1 + 2^{1/2} + 3^{1/2}\right)^6 \) is equal to

Correct option: 612
JEE Main 2025 (23 Jan Shift 1) — Mathematics, Question 25

If the area of the larger portion bounded between the curves \( x^2 + y^2 = 25 \) and \( y = |x - 1| \) is \( \frac{1}{3}(b\pi + c) \), \( b, c \in \mathbb{N} \), then \( b + c \) is equal to

Correct option: 77
JEE Main 2025 (23 Jan Shift 2) — Mathematics, Question 1

The distance of the line \( \frac{x^2}{2} = \frac{y^6}{3} = \frac{z^3}{4} \) from the point \((1, 4, 0)\) along the line \( \frac{x}{4} = \frac{y^2}{2} = \frac{z^3}{3} \) is:

(1) \( \sqrt{17} \)
(2) \( \sqrt{15} \)
(3) \( \sqrt{14} \)
(4) \( \sqrt{13} \)

Correct option: 3
JEE Main 2025 (23 Jan Shift 2) — Mathematics, Question 2

Let \( A = \{(x, y) \in \mathbb{R} \times \mathbb{R} : |x + y| \geq 3\} \) and \( B = \{(x, y) \in \mathbb{R} \times \mathbb{R} : |x| + |y| \leq 3\} \). If \( C = \{(x, y) \in A \cap B : x = 0 \text{ or } y = 0\} \), then \( \sum_{(x, y) \in C} |x + y| \) is:

(1) 15
(2) 24
(3) 18
(4) 12

Correct option: 4
JEE Main 2025 (23 Jan Shift 2) — Mathematics, Question 3

Let \( X = \mathbb{R} \times \mathbb{R} \). Define a relation \( R \) on \( X \) as: \((a_1, b_1)R(a_2, b_2) \iff b_1 = b_2 \)  Statement I : \( R \) is an equivalence relation. Statement II : For some \((a, b) \in X\), the set \( S = \{(x, y) \in X : (x, y)R(a, b)\} \) represents a line parallel to \( y = x \). In the light of the above statements, choose the correct answer from the options given below:

(1) Both Statement I and Statement II are false
(2) Statement I is true but Statement II is false
(3) Both Statement I and Statement II are true
(4) Statement I is false but Statement II is true

Correct option: 2
JEE Main 2025 (23 Jan Shift 2) — Mathematics, Question 4

Let \( \int x^3 \sin x \, dx = g(x) + C \), where \( C \) is the constant of integration. If \( 8 \left( g\left(\frac{\pi}{2}\right) + g'\left(\frac{\pi}{2}\right)\right) = \alpha \pi^3 + \beta \pi^2 + \gamma, \alpha, \beta, \gamma \in \mathbb{Z} \), then \( \alpha + \beta - \gamma \) equals:

(1) 48
(2) 55
(3) 62
(4) 47

Correct option: 2
JEE Main 2025 (23 Jan Shift 2) — Mathematics, Question 5

A rod of length eight units moves such that its ends \( A \) and \( B \) always lie on the lines \( x - y + 2 = 0 \) and \( y + 2 = 0 \), respectively. If the locus of the point \( P \), that divides the rod \( AB \) internally in the ratio \( 2 : 1 \) is \( 9 \left( x^2 + \alpha y^2 + \beta xy + \gamma x + 28y\right) - 76 = 0 \), then \( \alpha - \beta - \gamma \) is equal to:

(1) 22
(2) 21
(3) 23
(4) 24

Correct option: 3
JEE Main 2025 (23 Jan Shift 2) — Mathematics, Question 6

If the square of the shortest distance between the lines \( \frac{x^2}{m} = \frac{y^6}{n} = \frac{z^3}{3} \) and \( \frac{x^3}{m} = \frac{y^3}{n} = \frac{z^3}{3} \) is \( \frac{m}{n} \), where \( m, n \) are coprime numbers, then \( m + n \) is equal to:

(1) 21
(2) 9
(3) 14
(4) 6

Correct option: 2
JEE Main 2025 (23 Jan Shift 2) — Mathematics, Question 7

\( \lim_{x \to \infty} \frac{(2x^2-3x+5)(3x-1)^{\frac{2}{3}}}{(3x^2+5x+4)\sqrt{(3x+2)^3}} \) is equal to:

(1) \( \frac{2}{3} \sqrt{e} \)
(2) \( \frac{3e}{5} \)
(3) \( \frac{2e}{3} \)
(4) \( \frac{3e}{5} \)

Correct option: 3
JEE Main 2025 (23 Jan Shift 2) — Mathematics, Question 8

Let the point \( A \) divide the line segment joining the points \( P(-1,-1,2) \) and \( Q(5,5,10) \) internally in the ratio \( r : 1(r > 0) \). If \( O \) is the origin and \( \overrightarrow{OQ} \cdot \overrightarrow{OA} = \frac{1}{5} |\overrightarrow{OP} \times \overrightarrow{OA}|^2 \) = 10, then the value of \( r \) is:

(1) \( \sqrt{7} \)
(2) 14
(3) 3
(4) 7

Correct option: 4
JEE Main 2025 (23 Jan Shift 2) — Mathematics, Question 9

The length of the chord of the ellipse \( \frac{x^2}{4} + \frac{y^2}{3} = 1 \), whose mid-point is \((1, \frac{1}{2})\), is:
(1) \( \frac{5}{3} \sqrt{15} \)
(2) \( \frac{1}{3} \sqrt{15} \)
(3) \( \frac{2}{3} \sqrt{15} \)
(4) \( \sqrt{15} \)

Correct option: 3
JEE Main 2025 (23 Jan Shift 2) — Mathematics, Question 10

The system of equations \( x + 2y + 5z = 9 \), has no solution if:
(x) \( x + 5y + \lambda z = \mu \),
(1) \( \lambda = 15, \mu \neq 17 \)
(2) \( \lambda \neq 17, \mu = 18 \)
(3) \( \lambda = 17, \mu \neq 18 \)
(4) \( \lambda = 17, \mu = 18 \)

Correct option: 3
JEE Main 2025 (23 Jan Shift 2) — Mathematics, Question 11

Let the range of the function 
\[ f(x) = 6 + 16 \cos x \cdot \cos \left( \frac{x}{3} - x \right) \cdot \cos \left( \frac{x}{3} + x \right) \cdot \sin 3x \cdot \cos 6x, x \in \mathbb{R} \]
be \([\alpha, \beta] \). Then the distance of the point \((\alpha, \beta)\) from the line \(3x + 4y + 12 = 0\) is:
(1) 11
(2) 8
(3) 10
(4) 9

Correct option: 1
JEE Main 2025 (23 Jan Shift 2) — Mathematics, Question 12

Let \( x = x(y) \) be the solution of the differential equation 
\[ y = \left( x - y \frac{dy}{dx} \right) \sin \left( \frac{x}{y} \right), y > 0 \text{ and } x(1) = \frac{\pi}{2} \]. Then
\[ \cos(x(2)) \] is equal to:
(1) \( 1 - 2(\log_2 2)^2 \)
(2) \( 1 - 2(\log_2 2) \)
(3) \( 2(\log_2 2)^2 - 1 \)
(4) \( 2(\log_2 2)^2 - 1 \)

Correct option: 4
JEE Main 2025 (23 Jan Shift 2) — Mathematics, Question 13

A spherical chocolate ball has a layer of ice-cream of uniform thickness around it. When the thickness of the ice-cream layer is 1 cm, the ice-cream melts at the rate of 81 cm³/min and the thickness of the ice-cream layer decreases at the rate of \( \frac{1}{4} \) cm/min. The surface area (in cm²) of the chocolate ball (without the ice-cream layer) is:
(1) 196π
(2) 256π
(3) 225π
(4) 128π

Correct option: 2
JEE Main 2025 (23 Jan Shift 2) — Mathematics, Question 14

The number of complex numbers \( z \), satisfying \(|z| = 1\) and \( |\frac{z}{2} + \frac{\bar{z}}{2}| = 1\), is:
(1) 4
(2) 8
(3) 10
(4) 6

Correct option: 2
JEE Main 2025 (23 Jan Shift 2) — Mathematics, Question 15

Let \( A = [a_{ij}] \) be a \( 3 \times 3 \) matrix such that
\[ A \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}, A \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \text{ and } A \begin{bmatrix} 2 \\ 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \], then \( a_{23} \) equals:
(1) -1
(2) 2
(3) 1
(4) 0

Correct option: 1
JEE Main 2025 (23 Jan Shift 2) — Mathematics, Question 16

If \( I = \int_{0}^{\pi} \frac{\sin \frac{x}{2}}{\sin \frac{x}{2} \cos \frac{x}{2}} \, dx \), then
\[ I = \int_{0}^{21} \frac{x \sin x \cos x}{\sin^4 x + \cos^4 x} \, dx \] equals:
(1) \( \frac{x^2}{12} \)
(2) \( \frac{x^2}{4} \)
(3) \( \frac{x^2}{16} \)
(4) \( \frac{x^2}{8} \)

Correct option: 3
JEE Main 2025 (23 Jan Shift 2) — Mathematics, Question 17

A board has 16 squares as shown in the figure:

Out of these 16 squares, two squares are chosen at random. The probability that they have no side in common is:

(1) \(7/10\)  
(2) \(4/5\)  
(3) \(23/30\)  
(4) \(3/5\)

Correct option: 2
JEE Main 2025 (23 Jan Shift 2) — Mathematics, Question 18

Let the shortest distance from \((a, 0), a > 0\) to the parabola \(y^2 = 4x\) be 4. Then the equation of the circle passing through the point \((a, 0)\) and the focus of the parabola, and having its centre on the axis of the parabola is:

(1) \(x^2 + y^2 - 10x + 9 = 0\)  
(2) \(x^2 + y^2 - 6x + 5 = 0\)  
(3) \(x^2 + y^2 - 4x + 3 = 0\)  
(4) \(x^2 + y^2 - 8x + 7 = 0\)

Correct option: 2
JEE Main 2025 (23 Jan Shift 2) — Mathematics, Question 19

If in the expansion of \((1 + x)^p(1 - x)^q\), the coefficients of \(x\) and \(x^2\) are 1 and -2, respectively, then \(p^2 + q^2\) is equal to:

(1) 18  
(2) 13  
(3) 8  
(4) 20

Correct option: 2
JEE Main 2025 (23 Jan Shift 2) — Mathematics, Question 20

If the area of the region \(\{(x, y) : -1 \leq x \leq 1, 0 \leq y \leq a + e^{x+1} - e^{-x}, a > 0\}\) is \(\frac{e^{x+1} e^{x+1}}{e}\), then the value of \(a\) is:

(1) 8  
(2) 7  
(3) 5  
(4) 6

Correct option: 3
JEE Main 2025 (23 Jan Shift 2) — Mathematics, Question 21

The variance of the numbers 8, 21, 34, 47, \ldots, 320 is

Correct option: 3
JEE Main 2025 (23 Jan Shift 2) — Mathematics, Question 22

The roots of the quadratic equation \(3x^2 - px + q = 0\) are 10th and 11th terms of an arithmetic progression with common difference \(\frac{\alpha}{2}\). If the sum of the first 11 terms of this arithmetic progression is 88, then \(q - 2p\) is equal to

Correct option: 474
JEE Main 2025 (23 Jan Shift 2) — Mathematics, Question 23

The number of ways, 5 boys and 4 girls can sit in a row so that either all the boys sit together or no two boys sit together, is

Correct option: 17280
JEE Main 2025 (23 Jan Shift 2) — Mathematics, Question 24

The focus of the parabola \(y^2 = 4x + 16\) is the centre of the circle \(C\) of radius 5. If the values of \(\lambda\), for which \(C\) passes through the point of intersection of the lines \(3x - y = 0\) and \(x + \lambda y = 4\), are \(\lambda_1\) and \(\lambda_2\), then \(12\lambda_1 + 29\lambda_2\) is equal to

Correct option: 15
JEE Main 2025 (23 Jan Shift 2) — Mathematics, Question 25

Let \(\alpha, \beta\) be the roots of the equation \(x^2 - ax - b = 0\) with \(\text{Im}(\alpha) < \text{Im}(\beta)\). Let \(P_n = \alpha^n - \beta^n\). If \(P_3 = -5\sqrt{7}i, P_4 = -3\sqrt{7}i, P_5 = 11\sqrt{7}i\) and \(P_6 = 45\sqrt{7}i\), then \(|\alpha^4 + \beta^4|\) is equal to

Correct option: 31
JEE Main 2025 (24 Jan Shift 1) — Mathematics, Question 1

Let circle $C$ be the image of $x^2 + y^2 - 2x + 4y - 4 = 0$ in the line $2x - 3y + 5 = 0$ and $A$ be the point on $C$ such that $OA$ is parallel to $x$-axis and $A$ lies on the right hand side of the centre $O$ of $C$. If $B(\alpha, \beta)$, with $\beta < 4$, lies on $C$ such that the length of the arc $AB$ is $(1/6)^{th}$ of the perimeter of $C$, then $\beta - \sqrt{3}\alpha$ is equal to

(1) $3 + \sqrt{3}$  
(2) $4$  
(3) $4 - \sqrt{3}$  
(4) $3$

Correct option: 2
JEE Main 2025 (24 Jan Shift 1) — Mathematics, Question 2

Let in a $\triangle ABC$, the length of the side $AC$ be $6$, the vertex $B$ be $(1, 2, 3)$ and the vertices $A, C$ lie on the line $\frac{x-3}{2} = \frac{y-7}{2} = \frac{z-7}{2}$. Then the area (in sq. units) of $\triangle ABC$ is:

(1) $17$  
(2) $21$  
(3) $56$  
(4) $42$

Correct option: 2
JEE Main 2025 (24 Jan Shift 1) — Mathematics, Question 3

Let the product of the focal distances of the point $\left(\sqrt{3}, \frac{1}{3}\right)$ on the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, $(a > b)$, be $\frac{7}{4}$. Then the absolute difference of the eccentricities of two such ellipses is

(1) $\frac{1 - \sqrt{3}}{\sqrt{2}}$  
(2) $\frac{3 - 2\sqrt{2}}{2\sqrt{3}}$  
(3) $\frac{3 - 2\sqrt{2}}{3\sqrt{2}}$  
(4) $\frac{1 - 2\sqrt{2}}{\sqrt{3}}$

Correct option: 2
JEE Main 2025 (24 Jan Shift 1) — Mathematics, Question 4

If the system of equations $5x + \lambda y + 3z = 12$ and $100x - 47y + \mu z = 212$ has infinitely many solutions, then $\mu - 2\lambda$ is equal to

(1) $57$  
(2) $59$  
(3) $55$  
(4) $56$

Correct option: 1
JEE Main 2025 (24 Jan Shift 1) — Mathematics, Question 5

For some $n \neq 10$, let the coefficients of the $5$th, $6$th and $7$th terms in the binomial expansion of $(1 + x)^{n+4}$ be in A.P. Then the largest coefficient in the expansion of $(1 + x)^{n+4}$ is:

(1) $20$  
(2) $10$  
(3) $35$  
(4) $70$

Correct option: 3
JEE Main 2025 (24 Jan Shift 1) — Mathematics, Question 6

The product of all the rational roots of the equation $\left(x^2 - 9x + 11\right)^2 - (x - 4)(x - 5) = 3$, is equal to

(1) $14$  
(2) $21$  
(3) $28$  
(4) $7$

Correct option: 1
JEE Main 2025 (24 Jan Shift 1) — Mathematics, Question 7

Let the line passing through the points $(-1, 2, 1)$ and parallel to the line $\frac{x+1}{2} = \frac{y+1}{3} = \frac{z-1}{3}$ intersect the line $\frac{x+2}{3} = \frac{y-3}{2} = \frac{z+4}{1}$ at the point $P$. Then the distance of $P$ from the point $Q(4, -5, 1)$ is

(1) $5$  
(2) $5\sqrt{5}$  
(3) $5\sqrt{6}$  
(4) $10$

Correct option: 2
JEE Main 2025 (24 Jan Shift 1) — Mathematics, Question 8

Let the lines $3x - 4y - \alpha = 0$, $8x - 11y - 33 = 0$, and $2x - 3y + \lambda = 0$ be concurrent. If the image of the point $(1, 2)$ in the line $2x - 3y + \lambda = 0$ is $\left(\frac{57}{13}, \frac{-40}{13}\right)$, then $|\alpha\lambda|$ is equal to

(1) $84$  
(2) $113$  
(3) $91$  
(4) $101$

Correct option: 3
JEE Main 2025 (24 Jan Shift 1) — Mathematics, Question 9

If $\alpha$ and $\beta$ are the roots of the equation $2x^2 - 3x - 2i = 0$, where $i = \sqrt{-1}$, then $16 \cdot \text{Re} \left(\frac{\alpha^{19} + \beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^{5} + \beta^{5}}\right) \cdot \text{Im} \left(\frac{\alpha^{19} + \beta^{19} + \alpha^{11} + \beta^{11}}{\alpha^{5} + \beta^{5}}\right)$ is equal to : (1) 441  
(2) 398  
(3) 312  
(4) 409  

Correct option: 1
JEE Main 2025 (24 Jan Shift 1) — Mathematics, Question 10

For a statistical data \(x_1, x_2, \ldots, x_{10}\) of 10 values, a student obtained the mean as 5.5 and \(\sum_{i=1}^{10} x_i^2 = 371\). He later found that he had noted two values in the data incorrectly as 4 and 5, instead of the correct values 6 and 8, respectively. The variance of the corrected data is

(1) 9
(2) 5
(3) 7
(4) 4

Correct option: 3