JEE Main2025Apr 7, Shift 2

JEE Main 2025 — Apr 7, Shift 2

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

Mathematics

Q1If the orthocentre of the triangle formed by the lines $y = x + 1$ , $y = 4 x - 8$ and $y = m x + c$ is at $(3, - 1)$ , then $m - c$ is :
Q2Let $a$ and $b$ be the vectors of the same magnitude such that $\frac{∣ a + b ∣ + ∣ a - b ∣}{∣ a + b ∣ - ∣ a - b ∣} = 2 + 1$ . Then $\frac{∣ a + b ∣ ^{2}}{∣ a ∣ ^{2}}$ is :
Q3Let $A = {(α, β) \in R \times R : ∣ α - 1∣ \leq 4 and ∣ β - 5∣ \leq 6}$ and $B = {(α, β) \in R \times R : 16 (α - 2)^{2} + 9 (β - 6)^{2} \leq 144}$ . Then
Q4If the range of the function $f (x) = \frac{5 - x}{x ^{2} - 3 x + 2}$ , $x \neq = 1, 2$ , is $(- \infty, α] \cup [β, \infty)$ , then $α^{2} + β^{2}$ is equal to :
Q5 A bag contains 19 unbiased coins and one coin with head on both sides. One coin drawn at random is tossed and head turns up. If the probability that the drawn coin was unbiased, is $\frac{m}{n} \dots$
Q6Let a random variable X take values 0, 1, 2, 3 with $P (X = 0) = P (X = 1) = p$ , $P (X = 2) = P (X = 3)$ and $E (X^{2}) = 2 E (X)$ . Then the value of $8 p - 1$ is :
Q7If the area of the region ${(x, y) : 1 + x^{2} \leq y \leq min {x + 7, 11 - 3 x}}$ is A, then 3A is equal to :
Q8Let f : $R \to R$ be a polynomial function of degree four having extreme values at $x = 4$ and $x = 5$ . If $x \to 0 lim \frac{f ( x )}{x ^{2}} = 5$ , then $f (2)$ is equal to :
Q9The number of solutions of the equation $cos 2 θ cos \frac{θ}{2} + cos \frac{5 θ}{2} = 2 cos^{3} \frac{5 θ}{2}$ in $[- \frac{π}{2}, \frac{π}{2}]$ is :
Q10Let $a_{n}$ be the $n^{t h}$ term of an A. P. If $S_{n} = a_{1} + a_{2} + a_{3} + \dots + a_{n} = 700$ , $a_{6} = 7$ and $S_{7} = 7$ , then $a_{n}$ is equal to :
Q11If the locus of $z \in C$ , such that $Re (\frac{z - 1}{2 z + i}) + Re (\frac{z ˉ - 1}{2 z ˉ - i}) = 2$ , is a circle of radius r and center $(a, b)$ then $\frac{15 ab}{r ^{2}}$ is equal to :
Q12 Let the length of a latus rectum of an ellipse $\frac{\dots}{a ^{2}}$
Q13Let $y = y (x)$ be the solution of the differential equation $(x^{2} + 1) y^{'} - 2 x y = (x^{4} + 2 x^{2} + 1) cos x$ , $y (0) = 1$ . Then $\int_{- 3}^{3} y (x) d x$ is :
Q14 If the equation of the line passing through the point $(0, - \frac{1 \dots}{2}$
Q15 Let p be the number of all triangles that can be formed by joining the vertices of a regular polygon P of n sides and q be the number of all quadrilaterals that can be formed by joining the vertices of P. If $p + q = 126$ , then the eccentricity of the ellipse $\dots$
Q16 Consider the lines $L_{1} : x - \dots$
Q17The number of real roots of the equation $x ∣ x - 2∣ + 3∣ x - 3∣ + 1 = 0$ is :
Q18 Let $e_{1}$ and $e_{\dots}$
Q19 Let the system of equations $x + 5 y - z = 1 \dots$
Q20 If the sum of the second, fourth and sixth terms of a G.P. of positive terms is 21 and the sum of its eighth, tenth and twelfth terms is 15309, then the sum of its first nine terms is : …
Q21If the function $f (x) = \frac{tan ( tan x ) - sin ( sin x )}{tan x - sin x}$ is continuous at $x = 0$ , then $f (0)$ is equal to _____.
Q22 If $\int (x ^{2} \dots$
Q23 For $t > - 1$ , let $α_{t \dots}$
Q24 Let the lengths of the transverse and conjugate axes of a hyperbola in standard form be $2 a$ and $2 b$ , respectively, and one focus and the corresponding directrix of this hyperbola be $(- 5, 0)$ and $\dots$
Q25The sum of the series $2 \times 1 \times^{20} C_{4} - 3 \times 2 \times^{20} C_{5} + 4 \times 3 \times^{20} C_{6} - 5 \times 4 \times^{20} C_{7} + \dots + 18 \times 17 \times^{20} C_{20}$ , is equal to _____.

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Chemistry

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