JEE Main2025Jan 28, Shift 1

JEE Main 2025 — Jan 28, Shift 1

75 questions with step-by-step solutions. All free.

Mathematics

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 $y^{2} = 4 x$ . Let the sides AD and BC of the trapezium be parallel to y-axis. If the diagonal AC is of length $\dots$
Q3 Two numbers $k_{1}$ and $k_{\dots}$
Q4If $f (x) = \frac{2 ^{x}}{2 ^{x} + 2}$ , $x \in R$ , then $k = 1 \sum 81 f (\frac{k}{82})$ is equal to :
Q5Let $f : R \to R$ be a function defined by $f (x) = (2 + 3 a) x^{2} + (\frac{a + 2}{a - 1}) x + b$ , $a \neq = 1$ . If $f (x + y) = f (x) + f (y) + 1 - \frac{2}{7} x y$ , then the value of $28 i = 1 \sum 5 ∣ f (i) ∣$ is:
Q6 Let $A (x, y, z)$ be a point in xy-plane, which is equidistant from three points $(0, 3, 2\dots$
Q7The relation $R = {(x, y) : x, y \in Z$ and $x + y$ 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 $\dots$
Q9Let $⟨ a_{n} ⟩$ be a sequence such that $a_{0} = 0$ , $a_{1} = \frac{1}{2}$ and $2 a_{n + 2} = 5 a_{n + 1} - 3 a_{n}$ , $n = 0, 1, 2, 3, \dots$ Then $k = 1 \sum 100 a_{k}$ is equal to :
Q10 $cos (sin^{- 1} \frac{3}{5} + sin^{- 1} \frac{5}{13} + sin^{- 1} \frac{33}{65})$ is equal to :
Q11Let $T_{r}$ be the $r$ -th term of an A.P. If for some $m$ , $T_{m} = \frac{1}{25}$ , $T_{25} = \frac{1}{20}$ and $20 r = 1 \sum 25 T_{r} = 13$ , then $5 m r = m \sum 2 m T_{r}$ is equal to :
Q12If the image of the point $(4, 4, 3)$ in the line $\frac{x - 1}{2} = \frac{y - 2}{1} = \frac{z - 1}{3}$ is $(α, β, γ)$ , then $α + β + γ$ is equal to
Q13If $\int_{- π /2}^{π /2} \frac{96 x ^{2} cos ^{2} x}{( 1 + e ^{x} )} d x = π (α π^{2} + β)$ , $α, β \in Z$ , then $(α + β)^{2}$ equals :
Q14The sum of all local minimum values of the function $f (x) = ⎩ ⎨ ⎧ 1 - 2 x, x < - 1 \frac{1}{3} (7 + 2∣ x ∣), - 1 \leq x \leq 2 \frac{11}{18} (x - 4) (x - 5), x > 2$ is
Q15The sum of the squares of all the roots of the equation $x^{2} + ∣2 x - 3∣ - 4 = 0$ , is :
Q16Let for some function $y = f (x)$ , $\int_{0}^{x} t f (t) d t = x^{2} f (x)$ , $x > 0$ and $f (2) = 3$ . Then $f (6)$ is equal to :
Q17 Let $^{n} C_{r - 1 \dots}$
Q18Let $O$ be the origin, the point $A$ be $z_{1} = 3 + 22 i$ , the point $B (z_{2})$ be such that $3 ∣ z_{2} ∣ = ∣ z_{1} ∣$ and $ar g (z_{2}) = ar g (z_{1}) + \frac{π}{6}$ . Then
Q19 Three defective oranges are accidently mixed with seven good ones and on looking at them, it is not possible to differentiate between them. Two oranges are drawn at random from the lot. If $x$ denotes the number of defective oranges, then the variance of $x$ is : …
Q20The area (in sq. units) of the region ${(x, y) : 0 \leq y \leq 2∣ x ∣ + 1, 0 \leq y \leq x^{2} + 1, ∣ x ∣ \leq 3}$ is
Q21 Let $M$ denote the set of all real matrices of order $3 \times 3$ and let $S = \dots$
Q22If $α = 1 + r = 1 \sum 6 (- 3)^{r - 1}^{12} C_{2 r - 1}$ , then the distance of the point $(12, 3)$ from the line $α x - 3 y + 1 = 0$ is _______.
Q23 Let $a = \dots$
Q24 Let $f (x) = ⎩ \dots$
Q25 Let $E_{1} : \dots$

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Chemistry

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