Let $I(x) = \int \dfrac{3dx}{(4x+6)\sqrt{4x^2 + 8x + 3}}$ and $I(0) = \dfrac{\sqrt{3}}{4} + 20$. If $I\left(\dfrac{1}{2}\right) = \dfrac{a\sqrt{2}}{b} + c$, where $a,b,c \in \mathbb{N}$, $\gcd(a,b)=1$, then $a + b + c$ is equal to:

1

The least value of$\cos^2\theta - 6\sin\theta\cos\theta + 3\sin^2\theta + 2$is:

Topic: JEE Main 2026 (23 January Evening Shift)

2

The area of the region enclosed between the circles$x^2 + y^2 = 4$ and $x^2 + (y-2)^2 = 4$ is:

Topic: JEE Main 2026 (23 January Evening Shift)

3

Bag $A$ contains $9$ white and $8$ black balls, while bag $B$ contains $6$ white and $4$ black balls. One ball is rando…

Topic: JEE Main 2026 (23 January Evening Shift)

4

$\vec{a}, \vec{b}, \vec{c}$ be three vectors such that$\vec{a} \times \vec{b} = 2(\vec{a} \times \vec{c})$.If $|\vec{a}…

Topic: JEE Main 2026 (23 January Evening Shift)

5

Let $PQ$ be a chord of the hyperbola$\frac{x^2}{4} - \frac{y^2}{b^2} = 1$,perpendicular to the x-axis such that triangl…

Topic: JEE Main 2026 (23 January Evening Shift)

6

If the points of intersection of the ellipses $x^2 + 2y^2 - 6x - 12y + 23 = 0$ and $4x^2 + 2y^2 - 20x - 12y + 35 = 0$ l…

Topic: JEE Main 2026 (23 January Evening Shift)

7

The sum of all the real solutions of the equation $\log_{x+3}(6x^2 + 28x + 30) - 5 = 2\log_{6x+10}(x^2 + 6x + 9)$ is eq…

Topic: JEE Main 2026 (23 January Evening Shift)

8

The system of linear equations $x + y + z = 6$ $2x + 5y + az = 36$ $x + 2y + 3z = b$ has

Topic: JEE Main 2026 (23 January Evening Shift)

9

Consider two sets $A = {x \in \mathbb{Z} : |x - 3| - 3 \le 1}$ and $B = {x \in \mathbb{R} - {1,2} : \dfrac{(x-2)(x-4…

Topic: JEE Main 2026 (23 January Evening Shift)

10

Let $A(1, 2)$ and $C(-3, -6)$ be two diagonally opposite vertices of a rhombus, whose sides $AD$ and $BC$ are parallel …

Topic: JEE Main 2026 (23 January Evening Shift)

1

The number of ways, in which $16$ oranges can be distributed to four children such that each child gets at least one or…

Topic: JEE Main 2026 (23 January Evening Shift)

2

Let $A = {0,1,2,\ldots,9}$. Let $R$ be a relation on $A$ defined by $(x,y) \in R$ iff $|x-y|$ is a multiple of $3$. Gi…

Topic: JEE Main 2026 (23 January Evening Shift)

3

If $z = \dfrac{\sqrt{3}}{2} + \dfrac{i}{2},; i = \sqrt{-1}$, then $(z^{201} - i)^5$ is equal to:

Topic: JEE Main 2026 (23 January Evening Shift)

4

Question

Topic: JEE Main 2026 (23 January Evening Shift)

5

An equilateral triangle $OAB$ is inscribed in the parabola $y^2 = 4x$ with vertex at origin. Then the minimum distance …

Topic: JEE Main 2026 (23 January Evening Shift)

6

Let $\sum a_k = \alpha n^2 + \beta n$. If $a_{10} = 59$ and $a_6 = 7a_4$, then $\alpha + \beta$ is equal to:

Topic: JEE Main 2026 (23 January Evening Shift)

7

If the solution curve $y = f(x)$ of the differential equation $(x^2 - 4)y' - 2xy + 2x(4 - x)^2 = 0,; x > 2$, passes t…

Topic: JEE Main 2026 (23 January Evening Shift)

8

If the image of the point $P(a, 2, a)$ in the line $\dfrac{x}{2} = \dfrac{y + a}{1} = \dfrac{z}{1}$ is $Q$ and the im…

Topic: JEE Main 2026 (23 January Evening Shift)

9

The number of elements in the set\( S = \{ x : x \in [0,100], \int_{0}^{x} t^2 \sin(x-t)\,dt = x^2 \} \)is ______.

Topic: JEE Main 2026 (23 January Evening Shift)

10

Let \( A = \begin{bmatrix} 0 & 2 & -3 \\ -2 & 0 & 1 \\ 3 & -1 & 0 \end{bmatrix} \) and \( B \) be a matrix such that…

Topic: JEE Main 2026 (23 January Evening Shift)