JEE Main2026Jan 28, Shift 1

JEE Main 2026 — Jan 28, Shift 1

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

Mathematics

Q1If $g (x) = 3 x^{2} + 2 x - 3$ , $f (0) = - 3$ and $4 g (f (x)) = 3 x^{2} - 32 x + 72$ , then $f (g (2))$ is equal to:
Q2The value of $k = 1 \sum \infty (- 1)^{k + 1} (\frac{k ( k + 1 )}{k !})$ is:
Q3 Let $y = x$ be the equation of a chord of the circle $C_{1\dots}$
Q4If $\frac{tan ( A - B )}{tan A} + \frac{sin ^{2} C}{sin ^{2} A} = 1$ , $A, B, C \in (0, \frac{π}{2})$ , then
Q5Let $z$ be a complex number such that $∣ z - 6∣ = 5$ and $∣ z + 2 - 6 i ∣ = 5$ . Then the value of $z^{3} + 3 z^{2} - 15 z + 141$ is equal to
Q6 Let ABC be an equilateral triangle with orthocenter at the origin and the side BC on the line $x + 2 2 \dots$
Q7 Let $S = {1, 2, 3, 4, 5, 6, \dots$
Q8Let $S = {x^{3} + a x^{2} + b x + c : a, b, c \in N$ and $a, b, c \leq 20}$ be a set of polynomials. Then the number of polynomials in $S$ , which are divisible by $x^{2} + 2$ , is
Q9 A bag contains 10 balls out of which $k$ are red and $(10 - k)$ are black, where $0 \dots$
Q10If $α, β$ , where $α < β$ , are the roots of the equation $λ x^{2} - (λ + 3) x + 3 = 0$ such that $\frac{1}{α} - \frac{1}{β} = \frac{1}{3}$ , then the sum of all possible values of $λ$ is :
Q11If $\int (\frac{1 - 5 cos ^{2} x}{sin ^{5} x cos ^{2} x}) d x = f (x) + C$ where $C$ is the constant of integration, then $f (\frac{π}{6}) - f (\frac{π}{4})$ is equal to
Q12Let $f$ be a polynomial function such that $f (x^{2} + 1) = x^{4} + 5 x^{2} + 2$ , for all $x \in R$ . Then $\int_{0}^{3} f (x) d x$ is equal to
Q13The area of the region $R = {(x, y) : x y \leq 8, 1 \leq y \leq x^{2}, x \geq 0}$ is
Q14The value of $x \to 0 lim \frac{lo g _{e} ( sec ( e x ) \cdot sec ( e ^{2} x ) \dots sec ( e ^{10} x ) )}{e ^{2} - e ^{2 c o s x}}$ is equal to
Q15 The mean and variance of 10 observations are 9 and 34.2, respectively. If 8 of these observations are 2, 3, 5, 10, 11, 13, 15, 21, then the mean deviation about the median of all the 10 observations is …
Q16 Let $A$ , $B$ and $C$ be three $2 \times 2 \dots$
Q17 The common difference of the A.P.: $a_{1}, a_{\dots}$
Q18 If the distances of the point $(1, 2, a)$ from the line $1 \dots$
Q19For three unit vectors $a, b, c$ satisfying $∣ a - b ∣^{2} + ∣ b - c ∣^{2} + ∣ c - a ∣^{2} = 9$ and $∣2 a + k b + k c ∣ = 3$ , the positive value of $k$ is :
Q20Let $y = y (x)$ be the solution of the differential equation $x \frac{d y}{d x} - sin 2 y = x^{3} (2 - x^{3}) cos^{2} y, x \neq = 0$ . If $y (2) = 0$ , then $tan (y (1))$ is equal to
Q21 In a G.P., if the product of the first three terms is 27 and the set of all possible values for the sum of its first three terms is $R - (a, b)$ , then $a^{\dots}$
Q22 For some $θ \in (0, 2 \dots$
Q23If $k = tan (\frac{π}{4} + \frac{1}{2} cos^{- 1} (\frac{2}{3})) + tan (\frac{1}{2} sin^{- 1} (\frac{2}{3}))$ then the number of solutions of the equation $sin^{- 1} (k x - 1) = sin^{- 1} x - cos^{- 1} x$ is ____.
Q24The value of $r = 1 \sum 20 ⌊ π (\int_{0}^{r} x ∣ sin π x ∣ d x) ⌋$ is ____.
Q25 Let PQR be a triangle such that $P Q \dots$

JEE Main 2026 — Jan 28, Shift 1

Mathematics

Physics

Chemistry

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