JEE Mains 2025

Q1: The value of $\int_{e^2}^{e^4} \frac{1}{x} \left\{ \frac{e^{\left((\log_e x)^2+1\right)^{-1}}}{e^{\left((\log_e x)^2+1\right)^{-1}} + e^{\left((6-\log_e x)^2+1\right)^{-1}}} \right\} dx$ is

Q2: Let $S$ be the set of all $(a, b)$ for which the points of local minimum of $f(x)$ and $g(x)$ are same, where $f(x) = 2x^3 - 3x^2 - 12x + 6$ and $g(x) = 2x^3 - 3ax^2 + b$. Then

Q3: A group of 4 men and 5 women is to be formed from 7 men and 10 women. The number of ways of forming the group is

Q4: If the length of the latus rectum of a hyperbola, whose transverse axis is along the x-axis and the center is at the origin, is 4 and the distance between the foci is $4\sqrt{3}$, then the equation of the hyperbola is

Q5: The value of $\lim_{x \to 0} \frac{e^{3x^2} - \cos(x)}{\sin^2(x)}$ is

Q6: Let $A$ be a $2 \times 2$ matrix such that the sum of its elements is 4. If the matrix $B = A - \alpha I$ is a singular matrix, then the value of $|A - 2B|$ is

Q7: Let the image of the point $(1, 2, 3)$ in the plane $x - 2y + 3z = 10$ be $(\alpha, \beta, \gamma)$. Then the value of $\alpha + \beta + \gamma$ is

Q8: If the sum of the first $n$ terms of an arithmetic progression is $S_n = 3n^2 + 5n$, then the common difference of the A.P. is

Q9: The area of the region bounded by the curves $y = x^2$ and $y = 2x$ is

Q10: The number of real solutions of the equation $x|x| - 5|x| + 6 = 0$ is

Q11: A bag contains 5 red balls and 3 black balls. Two balls are drawn at random. The probability that they are of different colors is

Q12: The solution of the differential equation $(x^2 - y^2)dx + 2xy dy = 0$ is

Q13: The angle between the lines whose direction cosines are proportional to $(2, 3, -6)$ and $(3, -4, 5)$ is

Q14: The value of the integral $\int \frac{dx}{x(x^5 + 1)}$ is

Q15: If $A$ and $B$ are two events such that $P(A) = 0.4$, $P(B) = 0.8$ and $P(B|A) = 0.6$, then $P(A \cup B)$ is

Q16: If $f(x) = \sin^{-1}(\frac{2x}{1+x^2})$, then $f'(2)$ is

Q17: The number of integers greater than 6000 that can be formed using the digits 3, 5, 6, 7 and 8, without repetition, is

Q18: Let $f(x)$ be a function satisfying $f(x+y) = f(x)f(y)$ for all $x, y \in R$. If $f(1) = 2$ and $\sum_{k=1}^n f(k) = 1022$, then the value of $n$ is

Q19: The set of all values of $x$ for which the function $f(x) = \frac{\ln x}{x}$ is increasing is

Q20: The distance between the parallel planes $2x - y + 2z + 3 = 0$ and $4x - 2y + 4z + 5 = 0$ is

Q21: If $y = (\tan^{-1}x)^2$, then $(x^2+1)^2 y_2 + 2x(x^2+1)y_1$ is equal to

Q22: If $A = \{1, 2, 3, 4\}$, then the number of subsets of $A$ containing the element 3 is

Q23: The value of $\cot^{-1}(21) + \cot^{-1}(13) + \cot^{-1}(-8)$ is

Q24: The eccentricity of the ellipse $4x^2 + 9y^2 = 36$ is

Q25: The complex number $z = x + iy$ which satisfies the equation $|\frac{z - 3i}{z + 3i}| = 1$ lies on

Q26: If the system of equations $x + y + z = 1$, $x + 2y + 3z = 2$ and $x + 3y + \lambda z = 3$ has a unique solution, then the value of $\lambda$ can not be

Q27: The variance of the first 10 natural numbers is

Q28: The argument of the complex number $\frac{1 + 2i}{1 - 3i}$ is

Q29: Let $A$ and $B$ be two $3 \times 3$ matrices such that $AB = O$. Then

Q30: The number of ways in which 5 boys and 5 girls can sit on a round table so that no two girls sit together is

Q31: The number of terms in the expansion of $(x+y+z)^{10}$ is

Q32: If the mean of the numbers $27+x, 31+x, 89+x, 107+x, 156+x$ is 82, then the mean of $130+x, 126+x, 68+x, 50+x, 1+x$ is

Q33: The coefficient of $x^n$ in the expansion of $(1+x)(1-x)^n$ is

Q34: If $x$ is real, the maximum value of $3\sin x + 4\cos x$ is

Q35: The negation of the statement 'If it is raining, then I will not come' is

Q36: The sum of the series $1 + \frac{1^2+2^2}{2!} + \frac{1^2+2^2+3^2}{3!} + \dots$ to $n$ terms is

Q37: The domain of the function $f(x) = \sqrt{\log_e(x^2 - 6x + 6)}$ is

Q38: If $\omega$ is a complex cube root of unity, then the value of $(1+\omega - \omega^2)^7$ is

Q39: Let $f: R \to R$ be a function defined by $f(x) = \frac{x^2 - 8}{x^2 + 2}$. Then $f$ is

Q40: The foci of the ellipse $\frac{x^2}{16} + \frac{y^2}{b^2} = 1$ and the hyperbola $\frac{x^2}{144} - \frac{y^2}{81} = \frac{1}{25}$ coincide. Then the value of $b^2$ is

Q41: If the vectors $\vec{a} = 2\hat{i} + 3\hat{j} - \hat{k}$, $\vec{b} = \hat{i} + 2\hat{j} - 3\hat{k}$ and $\vec{c} = 3\hat{i} + \lambda\hat{j} + 2\hat{k}$ are coplanar, then the value of $\lambda$ is

Q42: If $A$ is a square matrix of order 3 such that $|A| = 5$, then the value of $|2A'|$ is

Q43: The number of common tangents to the circles $x^2 + y^2 = 4$ and $x^2 + y^2 - 6x - 8y = 24$ is

Q44: The value of $\tan(1^\circ) \tan(2^\circ) \tan(3^\circ) \dots \tan(89^\circ)$ is

Q45: The general solution of the differential equation $y' = \frac{y}{x} + \tan(\frac{y}{x})$ is

Q46: The area bounded by the parabola $y^2 = 4x$ and the line $y = 2x - 4$ is (in sq. units)

Q47: A fair coin is tossed 10 times. The probability of getting exactly 6 heads is

Q48: The system of equations $kx + y + z = 1$, $x + ky + z = 1$ and $x + y + kz = 1$ has no solution if

Q49: The term independent of $x$ in the expansion of $(x^2 - \frac{1}{x^3})^{10}$ is

Q50: Let $g(x) = \int_0^x f(t) dt$, where $f$ is such that $\frac{1}{2} \leq f(t) \leq 1$ for $t \in [0, 1]$ and $0 \leq f(t) \leq \frac{1}{2}$ for $t \in [1, 2]$. Then

Q51: The value of $i^{2n} + i^{2n+1} + i^{2n+2} + i^{2n+3}$, where $n \in Z$, is

Q52: The equation of the plane passing through the point $(1, 1, 1)$ and perpendicular to the planes $2x + y - 2z = 5$ and $3x - 6y - 2z = 7$ is

Q53: The position vectors of the vertices of a triangle are $3\hat{i} + 4\hat{j} + 5\hat{k}$, $\hat{i} + 7\hat{k}$ and $5\hat{i} + 5\hat{j}$. The distance between the orthocenter and circumcenter is

Q54: A line makes angles $\alpha, \beta, \gamma$ with the positive axes. Then $\cos(2\alpha) + \cos(2\beta) + \cos(2\gamma)$ is equal to

Q55: If $y = \log_e(\tan x)$, then $\frac{dy}{dx}$ is

Q56: The value of $\int_0^{\pi/2} \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} dx$ is

Q57: The maximum value of the function $f(x) = x^3 - 6x^2 + 9x + 15$ in the interval $[0, 6]$ is

Q58: The probability that a leap year will have 53 Fridays or 53 Saturdays is

Q59: The value of $\lim_{n \to \infty} \sum_{r=1}^n \frac{1}{n} e^{r/n}$ is

Q60: The projection of the vector $\vec{a} = 2\hat{i} + 3\hat{j} + 2\hat{k}$ on the vector $\vec{b} = \hat{i} + 2\hat{j} + \hat{k}$ is

Q61: The number of values of $k$ for which the system of equations $(k+1)x + 8y = 4k$ and $kx + (k+3)y = 3k-1$ has infinitely many solutions is

Q62: The equation of the tangent to the curve $y=x+\frac{4}{x^2}$ that is parallel to the x-axis is

Q63: If $f(x) = |x-2|$, then $f'(2)$ is

Q64: The value of $\int_0^1 \frac{\tan^{-1}x}{1+x^2} dx$ is

Q65: A man is known to speak the truth 3 out of 4 times. He throws a die and reports that it is a six. The probability that it is actually a six is

Q66: Let $f(x) = \begin{cases} \frac{k \cos x}{\pi - 2x} & \text{if } x \neq \pi/2 \\ 3 & \text{if } x = \pi/2 \end{cases}$. If $f$ is continuous at $x = \pi/2$, then the value of $k$ is

Q67: The area of the triangle formed by the complex numbers $z, iz$ and $z+iz$ in the Argand plane is

Q68: The value of $\lim_{x \to \infty} (\frac{x+6}{x+1})^{x+4}$ is

Q69: The solution of the differential equation $x \frac{dy}{dx} + y = x \log x$ is

Q70: The number of diagonals of a polygon with 20 sides is

Q71: If $\int_0^a \frac{1}{1+4x^2} dx = \frac{\pi}{8}$, then $a$ is

Q72: The contrapositive of the statement 'If you are born in India, then you are a citizen of India' is

Q73: The equation of the normal to the curve $y = \sin x$ at $(0, 0)$ is