JEE Main2026Jan 28, Shift 2

JEE Main 2026 — Jan 28, Shift 2

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

Mathematics

Q1 Given below two statements : Statement I : $2 5^{13} + 2 0^{\dots}$
Q2The sum of the coefficients of $x^{499}$ and $x^{500}$ in $(1 + x)^{1000} + x (1 + x)^{999} + x^{2} (1 + x)^{998} + \dots + x^{1000}$ is
Q3 Let A be the focus of the parabola $y^{2} = 8 x$ . Let the line $\dots$
Q4The probability distribution of a random variable X is given below : If $E (X) = \frac{263}{15}$ , then $P (X < 20)$ is equal to :
Q5 Let $f (x) = θ \to 0 \dots$
Q6 Considering the principal values of inverse trigonometric functions, the value of the expression $tan (2 sin^{- 1} \dots$
Q7Let the arithmetic mean of $\frac{1}{a}$ and $\frac{1}{b}$ be $\frac{5}{16}, a > 2$ . If $α$ is such that $a, 4, α, b$ are in A.P., then the equation $α x^{2} - a x + 2 (α - 2 b) = 0$ has :
Q8 Given below are two statements : Statement I : The function $f : R \to R$ defined by $f (\dots$
Q9An ellipse has its center at $(1, - 2)$ , one focus at $(3, - 2)$ and one vertex at $(5, - 2)$ . Then the length of its latus rectum is :
Q10 Let the ellipse $E : \frac{x ^{\dots}}{144}$
Q11 Let $P_{1} : y \dots$
Q12 Let the circle $x^{2} + y^{\dots}$
Q13Let $[\cdot]$ denote the greatest integer function. Then $\int_{- π /2}^{π /2} (\frac{12 ( 3 + [ x ])}{3 + [ sin x ] + [ cos x ]}) d x$ is equal to :
Q14Let $y = y (x)$ be the solution of the differential equation $x \frac{d y}{d x} - y = x^{2} cot x, x \in (0, π)$ . If $y (\frac{π}{2}) = \frac{π}{2}$ , then $6 y (\frac{π}{6}) - 8 y (\frac{π}{4})$ is equal to :
Q15 The sum of all the elements in the range of $f (x) = Sgn (sin x) + \dots$
Q16Let $Q (a, b, c)$ be the image of the point $P (3, 2, 1)$ in the line $\frac{x - 1}{1} = \frac{y}{2} = \frac{z - 1}{1}$ . Then the distance of $Q$ from the line $\frac{x - 9}{3} = \frac{y - 9}{2} = \frac{z - 5}{- 2}$ is
Q17 Let $P$ be a point in the plane of the vector $A B \dots$
Q18Let $A = {z \in C : ∣ z - 2∣ \leq 4}$ and $B = {z \in C : ∣ z - 2∣ + ∣ z + 2∣ = 5}$ . Then the max ${∣ z_{1} - z_{2} ∣ : z_{1} \in A and z_{2} \in B}$ is
Q19Let $f (x) = \int \frac{d x}{x ^{(\frac{2}{3})} + 2 x ^{(\frac{1}{2})}}$ be such that $f (0) = - 26 + 24 lo g_{e} (2)$ . If $f (1) = a + b lo g_{e} (3)$ , where $a, b \in Z$ , then $a + b$ is equal to :
Q20 $\frac{6}{3 ^{26}} + \frac{10 \cdot 1}{3 ^{25}} + \frac{10 \cdot 2}{3 ^{24}} + \frac{10 \cdot 2 ^{2}}{3 ^{23}} + \dots + \frac{10 \cdot 2 ^{24}}{3}$ is equal to
Q21If $r = 1 \sum 25 (\frac{r}{r ^{4} + r ^{2} + 1}) = \frac{p}{q}$ , where $p$ and $q$ are positive integers such that $g cd (p, q) = 1$ , then $p + q$ is equal to ______.
Q22 Three persons enter in a lift at the ground floor. The lift will go upto $1 0^{t h}$ floor. The number of ways, in which the three persons can exit the lift at three different floors, if the lift does not stop at first, second and third floors, is equal to ______ . …
Q23 Let $f$ be a differentiable function satisfying $f (x) = 1 - \dots$
Q24 If the distance of the point $P (43, α, β), β < 0$ , from the line $\dots$
Q25Let $A = [3 - 4 1 - 1]$ and $B$ be two matrices such that $A^{100} = 100 B + I$ . Then the sum of all the elements of $B^{100}$ is ______ .

Physics

Chemistry

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