Let $O$ be the vertex of the parabola $x^2 = 4y$ and $Q$ be any point on it. Let the locus of the point $P$, which divides the segment $OQ$ internally in the ratio $2 : 3$, be the conic $C$. Then equation of the chord of $C$, which is bisected at the point $(1, 2)$, is:

1

The sum of all the roots of the equation $(x - 1)^2 - 5|x - 1| + 6 = 0$, is:

Topic: JEE Main 2026 (21 January Morning Shift)

2

The value of $\csc 10^\circ - \sqrt{3} \sec 10^\circ$ is equal to:

Topic: JEE Main 2026 (21 January Morning Shift)

3

If $x^2 + x + 1 = 0$, then the value of $\left(x + \frac{1}{x}\right)^4 + \left(x^2 + \frac{1}{x^2}\right)^4 + \left(x^…

Topic: JEE Main 2026 (21 January Morning Shift)

4

If the coefficient of $x$ in the expansion of $(ax^2 + bx + c)(1 - 2x)^{26}$ is $-56$ and the coefficients of $x^2$ and…

Topic: JEE Main 2026 (21 January Morning Shift)

5

Let PQ and MN be two straight lines touching the circle $x^2 + y^2 - 4x - 6y - 3 = 0$ at the points $A$ and $B$ respect…

Topic: JEE Main 2026 (21 January Morning Shift)

6

Let $(\alpha, \beta, \gamma)$ be the co-ordinates of the foot of the perpendicular drawn from the point $(5,4,2)$ on th…

Topic: JEE Main 2026 (21 January Morning Shift)

7

Let the mean and variance of $7$ observations $2, 4, 10, x, 12, 14, y$, $x > y$, be $8$ and $16$ respectively. Two numb…

Topic: JEE Main 2026 (21 January Morning Shift)

8

The value of $\int_{-\pi/6}^{\pi/6} \left( \frac{\pi + 4x}{1 - \sin(x + \pi/6)} \right) dx$ is equal to:

Topic: JEE Main 2026 (21 January Morning Shift)

9

Let the foci of hyperbola coincide with the foci of the ellipse $\frac{x^2}{36} + \frac{y^2}{16} = 1$. If the eccentric…

Topic: JEE Main 2026 (21 January Morning Shift)

10

Let $f : R \to (0, \infty)$ be a twice differentiable function such that $f(3) = 18$, $f'(3) = 0$ and $f''(3) = 4$. The…

Topic: JEE Main 2026 (21 January Morning Shift)

1

Let $f : R \to R$ be a twice differentiable function such that the quadratic equation $f(x)m^2 - 2f(x)m + f'(x) = 0$ in…

Topic: JEE Main 2026 (21 January Morning Shift)

2

Let $a_1 = 1$ and for $n \ge 1$, $a_{n+1} = \frac{1}{2}a_n + \frac{n^2 - 2n - 1}{n^2(n+1)^2}$. Then $\sum_{n=1}^{\infty…

Topic: JEE Main 2026 (21 January Morning Shift)

3

Let $S = {(m,n) : m,n \in {1,2,3,\ldots,50}}$. If the number of elements $(m,n)$ in $S$ such that $6^m + 9^n$ is a mult…

Topic: JEE Main 2026 (21 January Morning Shift)

4

For some $\alpha, \beta \in R$, let $A = \begin{bmatrix} \alpha & 2 \ 1 & 2 \end{bmatrix}$ and $B = \begin{bmatrix} 1 &…

Topic: JEE Main 2026 (21 January Morning Shift)

5

$6\int_{0}^{\pi/2} (\sin 3x + \sin 2x + \sin x) dx$ is equal to ______

Topic: JEE Main 2026 (21 January Morning Shift)