JEE Main 2026 Question Paper (28-Jan-2026) (Shift 2)

Question 1

Let $P_{1} : y = 4x^{2}$ and $P_{2} : y = x^{2} + 27$ be two parabolas. If the area of the bounded region enclosed between $P_{1}$ and $P_{2}$ is six times the area of the bounded region enclosed between the line $y = \alpha x, \alpha > 0$ and $P_{1}$, then $\alpha$ is equal to :

Accepted Answer

$6$

Answer

$8$

Answer

$15$

Answer

$12$

Question 2

Let $f(x) = \int \frac{dx}{x^{\left(\frac{2}{3}\right)} + 2x^{\left(\frac{1}{2}\right)}}$ be such that $f(0) = -26 + 24\log_{e}(2)$. If $f(1) = a + b\log_{e}(3)$, where $a, b \in \mathbf{Z}$, then $a + b$ is equal to :

Accepted Answer

$-26$

Answer

$-11$

Answer

$-5$

Answer

$-18$

Question 3

Given below are two statements :Statement I : $25^{13} + 20^{13} + 8^{13} + 3^{13}$ is divisible by $7$.Statement II : The integral part of $(7 + 4\sqrt{3})^{25}$ is an odd number.In the light of the above statements, choose the correct answer from the options given below :

Accepted Answer

Statement I is false but Statement II is true

Answer

Statement I is true but Statement II is false

Answer

Both Statement I and Statement II are false

Answer

Both Statement I and Statement II are true

Question 4

Let the ellipse $	ext{E} : \frac{x^{2}}{144} + \frac{y^{2}}{169} = 1$ and the hyperbola $	ext{H} : \frac{x^{2}}{16} - \frac{y^{2}}{\lambda^{2}} = -1$ have the same foci. If $e$ and $L$ respectively denote the eccentricity and the length of the latus rectum of $	ext{H}$, then the value of $24(e + L)$ is :

Accepted Answer

$126$

Answer

$67$

Answer

$296$

Answer

$148$

Question 5

Let the arithmetic mean of $\frac{1}{a}$ and $\frac{1}{b}$ be $\frac{5}{16}, a > 2$. If $\alpha$ is such that $a, 4, \alpha, b$ are in A.P., then the equation $\alpha x^{2} - ax + 2(\alpha - 2b) = 0$ has :

Accepted Answer

both roots in the interval $(-2, 0)$

Answer

one root in $(1, 4)$ and another in $(-2, 0)$

Answer

complex roots of magnitude less than $2$

Answer

one root in $(0, 2)$ and another in $(-4, -2)$