JEE MAINS 2025 28 Jan

Q1. The number of different 5 digit numbers greater than 50000 that can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, such that the sum of their first and last digits should not be more than 8, is

Q2. Let ABCD be a trapezium whose vertices lie on the parabola $y^2 = 4x$. Let the sides AD and BC of the trapezium be parallel to y-axis. If the diagonal AC is of length $\frac{25}{4}$ and it passes through the point (1, 0), then the area of ABCD is :

Q3. Two number $k_1$ and $k_2$ are randomly chosen from the set of natural numbers. Then, the probability that the value of $i^{k_1} + i^{k_2}$ , $(i = \sqrt{-1})$ is non-zero, equals

Q4. If $f(x) = \frac{2^x}{2^x + \sqrt{2}}$, $x \in R$, then $\sum_{k=1}^{81} f \left( \frac{k}{82} \right)$ is equal to :

Q5. Let $f : R \to R$ be a function defined by $f(x) = (2 + 3a)x^2 + \left( \frac{a+2}{a-1} \right) x + b$, $a \ne 1$. If $f(x + y) = f(x) + f(y) + 1 - \frac{2}{7} xy$, then the value of $\sum_{i=1}^{5} 28 | f(i) |$ is:

Q6. Let A(x, y, z) be a point in xy-plane, which is equidistant from three points (0, 3, 2), (2, 0, 3) and (0, 0, 1).
Let B = (1, 4, –1) and C = (2, 0, –2). Then among the statements
($S_1$) : $\Delta$ABC is an isosceles right angled triangle
and
($S_2$) : the area of $\Delta$ABC is $\frac{9\sqrt{2}}{2}$.

Q7. The relation $R = \{(x, y) : x, y \in Z \text{ and } x + y \text{ is even}\}$ is :

Q8. Let the equation of the circle, which touches x-axis at the point (a, 0), $a > 0$ and cuts off an intercept of length b on y-axis be $x^2 + y^2 - \alpha x + \beta y + \gamma = 0$. If the circle lies below x-axis, then the ordered pair $(2a, b^2)$ is equal to :

Q9. Let $<a_n>$ be a sequence such that $a_0 = 0$, $a_1 = \frac{1}{2}$ and $2a_{n+2} = 5a_{n+1} - 3a_n$, $n = 0, 1, 2, 3, \dots$ Then $\sum_{k=1}^{100} a_k$ is equal to :

Q10. $\cos^{-1} \left( \sin^{-1} \frac{3}{5} + \sin^{-1} \frac{5}{13} + \sin^{-1} \frac{33}{65} \right)$ is equal to :