JEE Main2026Jan 23, Shift 1

JEE Main 2026 — Jan 23, Shift 1

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

Mathematics

Q1 Let the domain of the function $f (x) = lo g_{3}$
Q2Let $f (x) = \int \frac{( 2 - x ^{2} ) e ^{x}}{( 1 + x ) ( 1 - x ) ^{3/2}} d x$ . If $f (0) = 0$ , then $f (\frac{1}{2})$ is equal to :
Q3Let $a = - \hat{i} + \hat{j} + 2 \hat{k}$ , $b = \hat{i} - \hat{j} - 3 \hat{k}$ , $c = a \times b$ and $d = c \times a$ . Then $(a - b) \cdot d$ is equal to :
Q4 A rectangle is formed by the lines $x = 0$ , $y = 0$ , $\dots$
Q5 Let the line $y - x = 1$ intersect the ellipse $\dots$
Q6 Let $A = {- 2, - 1, 0, 1, 2, 3 \dots$
Q7Let $y = y (x)$ be the solution of the differential equation $x^{4} d y + (4 x^{3} y + 2 sin x) d x = 0$ , $x > 0$ , $y (\frac{π}{2}) = 0$ . Then $π^{4} y (\frac{π}{3})$ is equal to :
Q8If $α$ and $β$ ( $α < β$ ) are the roots of the equation $(- 2 + 3) (∣ x - 3∣) + (x - 6 x) + (9 - 23) = 0$ , $x \geq 0$ , then $\frac{β}{α} + α β$ is equal to :
Q9Let $f (x) = ⎩ ⎨ ⎧ \frac{a x ^{2} + 2 a x + 3}{4 x ^{2} + 4 x - 3}, x \neq = - \frac{3}{2}, \frac{1}{2} b, x = - \frac{3}{2}, \frac{1}{2}$ be continuous at $x = - \frac{3}{2}$ . If $f \circ f (x) = \frac{7}{5}$ , then $x$ is equal to :
Q10The value of the integral $\int_{π /24}^{5 π /24} \frac{d x}{1 + 3 tan 2 x}$ is :
Q11 Among the statements : I : If $1 cos \dots$
Q12The value of $\frac{^{100} C _{50}}{51} + \frac{^{100} C _{51}}{52} + \dots + \frac{^{100} C _{100}}{101}$ is :
Q13Let the mean and variance of 8 numbers $- 10, - 7, - 1, x, y, 9, 2, 16$ be $\frac{7}{2}$ and $\frac{293}{4}$ respectively. Then the mean of 4 numbers $x, y, x + y + 1, ∣ x - y ∣$ is :
Q14 The sum of all possible values of $n \in N$ , so that the coefficients of $x$ , $x^{\dots}$
Q15 The vertices B and C of a triangle ABC lie on the line $\frac{x}{1} = \dots$
Q16Number of solutions of $3 cos 2 θ + 8 cos θ + 33 = 0$ , $θ \in [- 3 π, 2 π]$ is :
Q17Let $S = {z : 3 \leq ∣2 z - 3 (1 + i) ∣ \leq 7}$ be a set of complex numbers. Then $z \in S min z + \frac{1}{2} (5 + 3 i)$ is equal to :
Q18 Let $α$ and $β$ respectively be the maximum and the minimum values of the function $f (θ) = \dots$
Q19 A building construction work can be completed by two masons A and B together in $22.5$ days. Mason A alone can complete the construction work in 24 days less than mason B alone. Then mason A alone will complete the construction work in : …
Q20 Let the direction cosines of two lines satisfy the equations : $4 ℓ + m - n = 0$ and …
Q21Let $∣ A ∣ = 6$ , where A is a $3 \times 3$ matrix. If $∣ adj (3 adj (A^{2} \cdot adj (2 A))) ∣ = 2^{m} \cdot 3^{n}$ , $m, n \in N$ , then $m + n$ is equal to ______.
Q22Let the area of the region bounded by the curve $y = max {sin x, cos x}$ , lines $x = 0$ , $x = \frac{3 π}{2}$ , and the x-axis be A. Then $A + A^{2}$ is equal to ______.
Q23 Let f be a twice differentiable non-negative function such that $(f (x))^{2} = 25 \dots$
Q24 From the first 100 natural numbers, two numbers first $a$ and then $b$ are selected randomly without replacement. If the probability that $a - b \geq \dots$
Q25The number of 4-letter words, with or without meaning, which can be formed using the letters PQRPQRSTUVP, is ______.

Physics

Chemistry

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