Ajaysaini231/JEE-Main-2025-Math huggingface dataset:
Question: Let \( y = y(x) \) be the solution curve of the differential equation \( x(x^2 + e^x)dy + (e^x(x - 2)y - x^3)dx = 0,\ x > 0 \), passing through the point \((1, 0)\). Then \( y(2) \) is equal to:
Options:
\( \frac{4}{4 - e^2} \)
\( \frac{2}{2 + e^2} \)
\( \frac{2}{2 - e^2} \)
\( \frac{4}{4 + e^2} \)
Answer: \( \frac{4}{4 + e^2} \)
Question: From a group of 7 batsmen and 6 bowlers, 10 players are to be chosen for a team, which should include at least 4 batsmen and at least 4 bowlers. One batsman and one bowler who are captain and vice-captain respectively of the team should be included. Then the total number of ways such a selection can be made, is:
Options:
165
155
145
135
Answer: 155
Question: If for \( \theta \in \left[ -\frac{\pi}{3}, 0 \right] \), the points \( (x, y) = \left( 3 \tan\left(\theta + \frac{\pi}{3}\right), 2 \tan\left(\theta + \frac{\pi}{6}\right) \right) \) lie on \( xy + \alpha x + \beta y + \gamma = 0 \), then \( \alpha^2 + \beta^2 + \gamma^2 \) is equal to:
Options:
80
72
96
75
Answer: 75
Question: Let \( C_1 \) be the circle in the third quadrant of radius 3, that touches both coordinate axes. Let \( C_2 \) be the circle with centre \((1, 3)\) that touches \( C_1 \) externally at the point \((\alpha, \beta)\). If \( (\beta - \alpha)^2 = \frac{m}{n} \), \( \gcd(m, n) = 1 \), then \( m + n \) is equal to:
Options:
9
13
22
31
Answer: 22
Question: The integral \( \int_{0}^{\frac{\pi}{2}} \frac{(x + 3) \sin x}{1 + 3 \cos^2 x} \, dx \) is equal to:
Options:
\( \frac{\pi}{\sqrt{3}} (\pi + 1) \)
\( \frac{\pi}{\sqrt{3}} (\pi + 2) \)
\( \frac{\pi}{3\sqrt{3}} (\pi + 6) \)
\( \frac{\pi}{2\sqrt{3}} (\pi + 4) \)
Answer: \( \frac{\pi}{3\sqrt{3}} (\pi + 6) \)
Question: Among the statements:
(S1): The set \( \{ z \in \mathbb{C} \setminus \{-i\} : |z| = 1 \text{ and } \frac{z - i}{z + i} \text{ is purely real} \} \) contains exactly two elements, and
(S2): The set \( \{ z \in \mathbb{C} \setminus \{-1\} : |z| = 1 \text{ and } \frac{z - 1}{z + 1} \text{ is purely imaginary} \} \) contains infinitely many elements.
Which of the following is correct?
Options:
both are incorrect
only (S1) is correct
only (S2) is correct
both are correct
Answer: only (S2) is correct
Question: The mean and standard deviation of 100 observations are 40 and 5.1, respectively. By mistake, one observation is taken as 50 instead of 40. If the correct mean and the correct standard deviation are \( \mu \) and \( \sigma \) respectively, then \( 10(\mu + \sigma) \) is equal to:
Options:
445
451
447
449
Answer: 449
Question: Let \( x_1, x_2, x_3, x_4 \) be in a geometric progression. If 2, 7, 9, 5 are subtracted respectively from \( x_1, x_2, x_3, x_4 \), then the resulting numbers are in an arithmetic progression. Then the value of \( \frac{1}{24} x_1 x_2 x_3 x_4 \) is:
Options:
72
18
36
216
Answer: 216
Question: Let the set of all values of \( p \in \mathbb{R} \), for which both the roots of the equation \( x^2 - (p + 2)x + (2p + 9) = 0 \) are negative real numbers, be the interval \( (\alpha, \beta] \). Then \( \beta - 2\alpha \) is equal to:
Options:
0
9
5
20
Answer: 5
Question: Let \( A \) be a \( 3 \times 3 \) matrix such that \( |\operatorname{adj} (\operatorname{adj}(\operatorname{adj} A))| = 81 \). If \[ S = \left\{ n \in \mathbb{Z} : |\operatorname{adj}(\operatorname{adj} A)|^{\frac{(n-1)^2}{2}} = |A|^{3n^2 - 5n - 4} \right\}, \] then \( \sum_{n \in S} \left| A^{n^2 + n} \right| \) is equal to:
Options:
866
750
820
732
Answer: 732
Question: If the area of the region bounded by the curves \( y = 4 - \frac{x^2}{4} \) and \( y = \frac{x - 4}{2} \) is equal to \( \alpha \), then \( 6\alpha \) equals:
Options:
250
210
240
220
Answer: 250
Question: Let the system of equations:
\[\begin{aligned}
2x + 3y + 5z &= 9, \\
7x + 3y - 2z &= 8, \\
12x + 3y - (4 + \lambda)z &= 16 - \mu
\end{aligned}\]
have infinitely many solutions. Then the radius of the circle centred at \((\lambda, \mu)\) and touching the line \(4x = 3y\) is:
Options:
\( \frac{17}{5} \)
\( \frac{7}{5} \)
7
\( \frac{21}{5} \)
Answer: \( \frac{7}{5} \)
Question: Let the line \( L \) pass through \((1, 1, 1)\) and intersect the lines \[ \frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z - 1}{4} \quad \text{and} \quad \frac{x - 3}{1} = \frac{y - 4}{2} = \frac{z}{1}. \] Then, which of the following points lies on the line \( L \)?
Options:
(4, 22, 7)
(5, 4, 3)
(10, -29, -50)
(7, 15, 13)
Answer: (7, 15, 13)
Question: Let the angle \( \theta \), \( 0 < \theta < \frac{\pi}{2} \), be the angle between two unit vectors \( \hat{a} \) and \( \hat{b} \), and \( \theta = \sin^{-1}\left(\frac{\sqrt{65}}{9}\right) \). If the vector \( \vec{c} = 3\hat{a} + 6\hat{b} + 9(\hat{a} \times \hat{b}) \), then the value of \( 9(\vec{c} \cdot \hat{a}) - 3(\vec{c} \cdot \hat{b}) \) is:
Options:
31
27
29
24
Answer: 29
Question: Let ABC be the triangle such that the equations of lines AB and AC be \( 3y - x = 2 \) and \( x + y = 2 \), respectively, and the points B and C lie on the x-axis. If P is the orthocentre of the triangle ABC, then the area of the triangle PBC is equal to:
Options:
4
10
8
6
Answer: 6
Question: The number of points of discontinuity of the function \( f(x) = \left\lfloor \frac{x^2}{2} \right\rfloor - \left\lfloor \sqrt{x} \right\rfloor, x \in [0, 4] \), where \( \lfloor \cdot \rfloor \) denotes the greatest integer function, is _____
Options:
Answer: 8
Question: The number of relations on the set \( A = \{1, 2, 3\} \) containing at most 6 elements including \((1, 2)\), which are reflexive and transitive but not symmetric, is _____
Options:
Answer: 6
Question: Consider the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) having one of its foci at \( P = (-3, 0) \). If the latus rectum through its other focus subtends a right angle at P and \( a^2 b^2 = \alpha \sqrt{2} - \beta \), \( \alpha, \beta \in \mathbb{N} \), then
Options:
Answer: 1944
Question: The number of singular matrices of order 2, whose elements are from the set \( \{2, 3, 6, 9\} \), is _____
Options:
Answer: 36
Question: For \( n \geq 2 \), let \( S_n \) denote the set of all subsets of \( \{1, 2, \ldots, n\} \) with no two consecutive numbers. For example, \( \{1, 3, 5\} \in S_6 \), but \( \{1, 2, 4\} \notin S_6 \). Then \( n(S_5) \) is equal to _____
Options:
Answer: 13
Question: Let the values of \( \lambda \) for which the shortest distance between the lines \( \frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4} \) and \( \frac{x - \lambda}{3} = \frac{y - 4}{4} = \frac{z - 5}{5} \) is \( \frac{1}{\sqrt{6}} \) be \( \lambda_1 \) and \( \lambda_2 \). Then the radius of the circle passing through the points (0, 0), \( (\lambda_1, \lambda_2) \) and \( (\lambda_2, \lambda_1) \) is
Options:
5\sqrt{2}/3
4
\sqrt{2}/3
3
Answer: 5\sqrt{2}/3
Question: Let \( \alpha \) be a solution of \( x^2 + x + 1 = 0 \), and for some a and b in \( \mathbb{R} \), \([4 \; a \; b] \begin{bmatrix}1 & 16 & 13 \\ -1 & -1 & 2 \\ -2 & -14 & -8\end{bmatrix} = [0 \; 0 \; 0]\). If \( \frac{4}{\alpha^4} + \frac{m}{\alpha} + \frac{n}{\alpha^2} = 3 \), then m + n is equal to
Options:
3
11
7
8
Answer: 11
Question: Let the function \( f(x) = \frac{x}{3} + \frac{3}{x} + 3, \; x \ne 0 \) be strictly increasing in \( (-\infty, \alpha_1) \cup (\alpha_2, \infty) \) and strictly decreasing in \( (\alpha_1, \alpha_2) \cup (\alpha_4, \alpha_5) \). Then \( \sum_{i=1}^{5} \alpha_i^2 \) is equal to:
Options:
48
28
40
36
Answer: 36
Question: If \( \frac{1}{1^4} + \frac{1}{2^4} + \frac{1}{3^4} + \ldots = \frac{\pi^4}{90} \),

\( \frac{1}{1^4} + \frac{1}{3^4} + \frac{1}{5^4} + \ldots = \alpha \),

\( \frac{1}{2^4} + \frac{1}{4^4} + \frac{1}{6^4} + \ldots = \beta \),

then \( \frac{\alpha}{\beta} \) is equal to
Options:
23
18
15
14
Answer: 15
Question: If A and B are two events such that \( P(A) = 0.7 \), \( P(B) = 0.4 \) and \( P(A \cap \overline{B}) = 0.5 \), where \( \overline{B} \) denotes the complement of B, then \( P(B \mid (A \cup \overline{B})) \) is equal to:
Options:
1/4
1/2
1/6
1/3
Answer: 1/4
Question: The sum of the squares of the roots of \( |x + 2|^2 + |x - 2| - 2 = 0 \) and the squares of the roots of \( x^2 - 2|x - 3| - 5 = 0 \), is:
Options:
26
36
30
24
Answer: 36
Question: Let \( a \) be the length of a side of a square OABC with O being the origin. Its side OA makes an acute angle \( \alpha \) with the positive x-axis and the equations of its diagonals are \( (\sqrt{3} + 1)x + (\sqrt{3} - 1)y = 0 \) and \( (\sqrt{3} - 1)x - (\sqrt{3} + 1)y + 8\sqrt{3} = 0 \). Then \( a^2 \) is equal to:
Options:
48
32
16
24
Answer: 48
Question: Let \( f(x) \) be a positive function and \[ I_1 = \int_{-\frac{1}{2}}^{1} 2x f(2x(1 - 2x)) \, dx \quad \text{and} \quad I_2 = \int_{-1}^{\frac{1}{2}} f(x(1 - x)) \, dx. \] Then the value of \( \frac{I_2}{I_1} \) is equal to:
Options:
9
6
12
4
Answer: 4
Question: Let \( \vec{a} = \hat{i} + 2\hat{j} + \hat{k} \) and \( \vec{b} = 2\hat{i} + \hat{j} - \hat{k} \). Let \( \vec{c} \) be a unit vector in the plane of the vectors \( \vec{a} \) and \( \vec{b} \) and be perpendicular to \( \vec{a} \). Then such a vector \( \vec{c} \) is:
Options:
\( \frac{1}{\sqrt{5}} (\hat{j} - 2\hat{k}) \)
\( \frac{1}{\sqrt{3}} (-\hat{i} + \hat{j} - \hat{k}) \)
\( \frac{1}{\sqrt{3}} (\hat{i} - \hat{j} + \hat{k}) \)
\( \frac{1}{\sqrt{2}} (-\hat{i} + \hat{k}) \)
Answer: \( \frac{1}{\sqrt{2}} (-\hat{i} + \hat{k}) \)
Question: Let the ellipse \( 3x^2 + py^2 = 4 \) pass through the centre C of the circle \( x^2 + y^2 - 2x - 4y - 11 = 0 \) of radius \( r \). Let \( f_1, f_2 \) be the focal distances of the point C on the ellipse. Then \( 6f_1f_2 - r \) is equal to:
Options:
74
68
70
78
Answer: 70
Question: The integral \( \int_{-1}^{\frac{3}{2}} \left| \pi^2 x \sin(\pi x) \right| \, dx \) is equal to:
Options:
3 + 2\pi
4 + \pi
1 + 3\pi
2 + 3\pi
Answer: 1 + 3\pi
Question: A line passing through the point \( P(a, \theta) \) makes an acute angle \( \alpha \) with the positive x-axis. Let this line be rotated about the point P through an angle \( \frac{\alpha}{2} \) in the clockwise direction. If in the new position, the slope of the line is \( 2 - \sqrt{3} \) and its distance from the origin is \( \frac{1}{\sqrt{2}} \), then the value of \( 3a^2 \tan^2 \alpha - 2\sqrt{3} \) is:
Options:
4
6
5
8
Answer: 4
Question: There are 12 points in a plane, no three of which are in the same straight line, except 5 points which are collinear. Then the total number of triangles that can be formed with the vertices at any three of these 12 points is:
Options:
230
220
200
210
Answer: 210
Question: Let \( A = \left\{ \theta \in [0, 2\pi] : 1 + 10\,\text{Re} \left( \frac{2\cos\theta + i\sin\theta}{\cos\theta - 3i\sin\theta} \right) = 0 \right\}. \) Then \( \sum_{\theta \in A} \theta^2 \) is equal to:
Options:
\( \frac{21}{4} \pi^2 \)
\( 8\pi^2 \)
\( \frac{27}{4} \pi^2 \)
\( 6\pi^2 \)
Answer: \( \frac{21}{4} \pi^2 \)
Question: The number of integral terms in the expansion of \( \left( \frac{1}{5^2} + \frac{1}{7^8} \right)^{1016} \) is:
Options:
127
130
129
128
Answer: 128
Question: Let \( A = \{0, 1, 2, 3, 4, 5\} \). Let \( R \) be a relation on \( A \) defined by \( (x, y) \in R \) if and only if \( \max\{x, y\} \in \{3, 4\} \). Then among the statements  
(S₁): The number of elements in R is 18, and  
(S₂): The relation R is symmetric but neither reflexive nor transitive  
Which of the following is correct?
Options:
both are true
both are false
only (S₂) is true
only (S₁) is true
Answer: only (S₂) is true
Question: Let \( f(x) = x - 1 \) and \( g(x) = e^x \) for \( x \in \mathbb{R} \). If \[ \frac{dy}{dx} = \left( e^{-2\sqrt{x}} g(f(f(f(x)))) - \frac{y}{\sqrt{x}} \right), \quad y(0) = 0, \] then \( y(1) \) is:
Options:
\( \frac{1 - e^2}{e^4} \)
\( \frac{2e - 1}{e^3} \)
\( \frac{e - 1}{e^4} \)
\( \frac{1 - e^3}{e^4} \)
Answer: \( \frac{e - 1}{e^4} \)
Question: The value of \[ \cot^{-1} \left( \frac{\sqrt{1 + \tan^2(2)} - 1}{\tan(2)} \right) - \cot^{-1} \left( \frac{\sqrt{1 + \tan^2\left(\frac{1}{2}\right)} + 1}{\tan\left(\frac{1}{2}\right)} \right) \] is equal to:
Options:
\( \pi - \frac{5}{4} \)
\( \pi - \frac{3}{2} \)
\( \pi + \frac{3}{2} \)
\( \pi + \frac{5}{2} \)
Answer: \( \pi - \frac{5}{4} \)
Question: Let \( A = \begin{bmatrix} 2 & 2 + p & 2 + p + q \\ 4 & 6 + 2p & 8 + 3p + 2q \\ 6 & 12 + 3p & 20 + 6p + 3q \end{bmatrix} \). If \( \det(\text{adj}(\text{adj}(3A))) = 2^m \cdot 3^n \), \( m, n \in \mathbb{N} \), then \( m + n \) is equal to:
Options:
22
24
26
20
Answer: 24
Question: Given below are two statements:

**Statement I:** \[ \lim_{x \to 0} \left( \frac{\tan^{-1}x + \log_e \left( \frac{\sqrt{1 + x} - \sqrt{1 - x}}{x} \right) - 2x}{x^5} \right) = \frac{2}{5} \]

**Statement II:** \[ \lim_{x \to 1} \left( \frac{2}{x^{1 - x}} \right) = \frac{1}{e^2} \]

In the light of the above statements, choose the correct answer from the options given below:
Options:
Statement I is false but Statement II is true
Statement I is true but Statement II is false
Both Statement I and Statement II are false
Both Statement I and Statement II are true
Answer: Both Statement I and Statement II are true
Question: Let the area of the bounded region \( \{(x, y) : 0 \leq 9x \leq y^2, \; y \geq 3x - 6 \} \) be \( A \). Then \( 6A \) is equal to _____
Options:
Answer: 15
Question: Let the domain of the function \( f(x) = \cos^{-1} \left( \frac{4x + 5}{3x - 7} \right) \) be \( [\alpha, \beta] \) and the domain of \( g(x) = \log_2(2 - 6\log_7(2x + 5)) \) be \( [\gamma, \delta] \). Then \( |7(\alpha + \beta) + 4(\gamma + \delta)| \) is equal to _____
Options:
Answer: 96
Question: Let the area of the triangle formed by the lines \( x + 2 = y - 1 = z \), \( \frac{x - 3}{5} = \frac{y}{-1} = \frac{z - 1}{1} \), and \( \frac{x}{-3} = \frac{y - 3}{3} = \frac{z - 2}{1} \) be \( A \). Then \( A^2 \) is equal to _____
Options:
Answer: 56
Question: The product of the last two digits of \( (1919)^{1919} \) is _____
Options:
Answer: 63
Question: Let \( r \) be the radius of the circle, which touches the x-axis at point \( (a, 0), \; a < 0 \) and the parabola \( y^2 = 9x \) at the point \( (4, 6) \). Then \( r \) is equal to _____
Options:
Answer: 30