JEE Main2026Jan 23, Shift 2

JEE Main 2026 — Jan 23, Shift 2

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

Mathematics

Q1If $f (x) = ⎩ ⎨ ⎧ \frac{a ∣ x ∣ + x ^{2} - 2 ( sin ∣ x ∣ ) ( cos ∣ x ∣ )}{x}, x \neq = 0 b, x = 0$ is continuous at $x = 0$ , then $a + b$ is equal to :
Q2Let $\frac{π}{2} < θ < π$ and $cot θ = - \frac{1}{2 2}$ . Then the value of $sin (\frac{15 θ}{2}) (cos 8 θ + sin 8 θ) + cos (\frac{15 θ}{2}) (cos 8 θ - sin 8 θ)$ is equal to :
Q3 Let $a = \dots$
Q4 Let $A (1, 2)$ and $C (- 3, - 6)$ be two diagonally opposite vertices of a rhombus, whose sides $\dots$
Q5Consider two sets $A = {x \in Z : ∣∣ x - 3∣ - 3∣ \leq 1}$ and $B = {x \in R - {1, 2} : \frac{( x - 2 ) ( x - 4 )}{x - 1} lo g_{e} (∣ x - 2∣) = 0}$ . Then the number of onto functions $f : A \to B$ is equal to :
Q6The system of linear equations $x + y + z = 6$ $2 x + 5 y + a z = 36$ $x + 2 y + 3 z = b$ has
Q7The sum of all the real solutions of the equation $lo g_{(x + 3)} (6 x^{2} + 28 x + 30) = 5 - 2 lo g_{(6 x + 10)} (x^{2} + 6 x + 9)$ is equal to :
Q8 If the points of intersection of the ellipses $x^{2} + 2 y^{\dots}$
Q9 Let $P Q$ be a chord of the hyperbola $\frac{x ^{\dots}}{4}$
Q10Let $a, b, c$ be three vectors such that $a \times b = 2 (a \times c)$ . If $∣ a ∣ = 1$ , $∣ b ∣ = 4$ , $∣ c ∣ = 2$ , and the angle between $b$ and $c$ is $6 0^{\circ}$ , then $∣ a \cdot c ∣$ is :
Q11 Bag A contains 9 white and 8 black balls, while bag B contains 6 white and 4 black balls. One ball is randomly picked up from the bag B and mixed up with the balls in the bag A. Then a ball is randomly drawn from the bag A. If the probability, that the ball drawn is white, is $\frac{p}{q}$
Q12The area of the region enclosed between the circles $x^{2} + y^{2} = 4$ and $x^{2} + (y - 2)^{2} = 4$ is :
Q13The least value of $(cos^{2} θ - 6 sin θ cos θ + 3 sin^{2} θ + 2)$ is
Q14Let $I (x) = \int \frac{3 d x}{( 4 x + 6 ) ( 4 x ^{2} + 8 x + 3 )}$ and $I (0) = \frac{3}{4} + 20$ . If $I (\frac{1}{2}) = \frac{a 2}{b} + c$ , where $a, b, c \in N$ , $g cd (a, b) = 1$ , then $a + b + c$ is equal to :
Q15The number of ways, in which 16 oranges can be distributed to four children such that each child gets at least one orange, is
Q16 Let $A = {0, 1, 2, \dots, 9}$ . Let $\dots$
Q17If $z = \frac{3}{2} + \frac{i}{2}$ , $i = - 1$ , then $(z^{201} - i)^{8}$ is equal to
Q18If the mean and the variance of the data are $μ$ and $19$ respectively, then the value of $λ + μ$ is
Q19 An equilateral triangle $O A B$ is inscribed in the parabola $y^{2} = \dots$
Q20Let $k = 1 \sum n a_{k} = α n^{2} + β n$ . If $a_{10} = 59$ and $a_{6} = 7 a_{1}$ , then $α + β$ is equal to
Q21 If the solution curve $y = f (x)$ of the differential equation $(x^{\dots}$
Q22 If the image of the point $P (a, 2, a)$ in the line $2 \dots$
Q23The number of elements in the set $S = {x : x \in [0, 100] and \int_{0}^{x} t^{2} sin (x - t) d t = x^{2}}$ is ________.
Q24Let $A = 02 - 3 - 201 3 - 10$ and $B$ be a matrix such that $B (I - A) = I + A$ . Then the sum of the diagonal elements of $B^{T} B$ is equal to _______.
Q25 Let $S$ denote the set of 4-digit numbers $ab c d$ such that $a > b >\dots$

JEE Main 2026 — Jan 23, Shift 2

Mathematics

Physics

Chemistry

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