JEE Main2026Jan 21, Shift 1

JEE Main 2026 — Jan 21, Shift 1

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

Mathematics

Q1If the domain of the function $f (x) = cos^{- 1} (\frac{2 x - 5}{11 - 3 x}) + sin^{- 1} (2 x^{2} - 3 x + 1)$ is the interval $[α, β]$ , then $α + 2 β$ is equal to :
Q2The area of the region, inside the ellipse $x^{2} + 4 y^{2} = 4$ and outside the region bounded by the curves $y = ∣ x ∣ - 1$ and $y = 1 - ∣ x ∣$ , is :
Q3The number of relations, defined on the set ${a, b, c, d}$ , which are both reflexive and symmetric, is equal to:
Q4 Let a point A lie between the parallel lines $L_{1}$ and $L_{\dots}$
Q5Let $a = - \hat{i} + 2 \hat{j} + 2 \hat{k}, b = 8 \hat{i} + 7 \hat{j} - 3 \hat{k}$ and $c$ be a vector such that $a \times c = b$ . If $c \cdot (\hat{i} + \hat{j} + \hat{k}) = 4$ , then $∣ a + c ∣^{2}$ is equal to :
Q6Let $a_{1}, a_{2}, a_{3}, \dots$ be a G.P. of increasing positive terms such that $a_{2} \cdot a_{3} \cdot a_{4} = 64$ and $a_{1} + a_{3} + a_{5} = \frac{813}{7}$ . Then $a_{3} + a_{5} + a_{7}$ is equal to :
Q7 Let $c$ and $\dots$
Q8Let $y = y (x)$ be the solution curve of the differential equation $(1 + x^{2}) d y + (y - tan^{- 1} x) d x = 0$ , $y (0) = 1$ . Then the value of $y (1)$ is :
Q9The number of strictly increasing functions f from the set ${1, 2, 3, 4, 5, 6}$ to the set $(1, 2, 3, \dots, 9)$ such that $f (i) \neq = i$ for $1 \leq i \leq 6$ , is equal to:
Q10 Let $f : R \to (0, \infty)$ be a twice differentiable function such that $\dots$
Q11 Let the foci of hyperbola coincide with the foci of the ellipse $\frac{x ^{\dots}}{36}$
Q12The value of $\int_{- π /6}^{π /6} (\frac{π + 4 x ^{11}}{1 - sin ( ∣ x ∣ + π /6 )}) d x$ is equal to
Q13 Let the mean and variance of 7 observations $2, 4, 10, x, 12, 14, y$ , $\dots$
Q14 Let $(α, β, γ)$ be the co-ordinates of the foot of the perpendicular drawn from the point $(5, 4, \dots$
Q15 Let PQ and MN be two straight lines touching the circle $x^{2} + y^{\dots}$
Q16 If the coefficient of x in the expansion of $(a x^{2} + b x + \dots$
Q17If $x^{2} + x + 1 = 0$ , then the value of $(x + \frac{1}{x})^{4} + (x^{2} + \frac{1}{x ^{2}})^{4} + (x^{3} + \frac{1}{x ^{3}})^{4} + \dots + (x^{25} + \frac{1}{x ^{25}})^{4}$ is :
Q18The value of $csc 10° - 3 sec 10°$ is equal to:
Q19The sum of all the roots of the equation $(x - 1)^{2} - 5∣ x - 1∣ + 6 = 0$ , is:
Q20 Let O be the vertex of the parabola $x^{2} = 4 y$ and Q be any point on it. Let the locus of the point P, which divides the line segment OQ internally in the ratio $\dots$
Q21 Let $f : R \to R$ be a twice differentiable function such that the quadratic equation $f (\dots$
Q22Let $a_{1} = 1$ and for $n \geq 1$ , $a_{n + 1} = \frac{1}{2} a_{n} + \frac{n ^{2} - 2 n - 1}{n ^{2} ( n + 1 ) ^{2}}$ . Then $n = 1 \sum \infty (a_{n} - \frac{2}{n ^{2}})$ is equal to ______.
Q23 Let $S = {(m, n) : m, n \dots$
Q24For some $α, β \in R$ , let $A = [α 2 12]$ and $B = [11 1 β]$ be such that $A^{2} - 4 A + 2 I = B^{2} - 3 B + I = O$ . Then $(det (adj (A^{3} - B^{3})))^{2}$ is equal to ………
Q25 $6 \int_{0}^{π} ∣ sin 3 x + sin 2 x + sin x ∣ d x$ is equal to ……..

JEE Main 2026 — Jan 21, Shift 1

Mathematics

Physics

Chemistry

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