NTA JEE Mains 5th Apr 2026 Shift 2

Let $$\alpha, \beta$$ be the roots of the equation $$x^2 - x + p = 0$$ and $$\gamma, \delta$$ be the roots of the equation$$x^2 - 4x + q = 0$$, where $$p, q \in \mathbb{Z}$$. If $$\alpha, \beta, \gamma, \delta$$ are in G.P., then $$|p + q|$$  equals :

Let $$z_1, z_2 \in \mathbb{C}$$ be the distinct solutions of the equation $$z^2 + 4z - (1 + 12i) = 0$$. Then $$|z_1|^2 + |z_2|^2$$ is equal to :

Let $$f : \mathbb{N} \to \mathbb{Z}$$ be defined by $$f(n) = \det\begin{bmatrix} n  & -1 & -5\\-2n^2 & 3(2k+1) & 2k+1 \\ -3n^3 & 3k(2k+1) & 3k(k+2)+1 \end{bmatrix}$$, $$k \in \mathbb{N}$$ and  $$\displaystyle\sum_{n=1}^{k} f(n) = 98$$, then $$k$$ is equal to :

Let $$M$$ be a $$3 \times 3$$ matrix such that $$M\begin{bmatrix}1\\0\\0\end{bmatrix} = \begin{bmatrix}1\\2\\3\end{bmatrix}$$, $$M\begin{bmatrix}0\\1\\0\end{bmatrix} = \begin{bmatrix}0\\1\\0\end{bmatrix}$$, $$M\begin{bmatrix}0\\0\\1\end{bmatrix} = \begin{bmatrix}-1\\1\\1\end{bmatrix}$$. If $$M\begin{bmatrix}x\\y\\z\end{bmatrix} = \begin{bmatrix}1\\7\\11\end{bmatrix}$$, then $$x + y + z$$ is equal to :

The sum of the first 10 terms of the series $$\frac{1}{1 + 1^4 \cdot 4} + \frac{2}{1 + 2^4 \cdot 4} + \frac{3}{1 + 3^4 \cdot 4} + \cdots$$ is $$\frac{m}{n}$$, where $$\gcd(m, n) = 1$$. Then $$m + n$$ is equal to :

Let $$A_1, A_2, \ldots, A_{39}$$ be 39 arithmetic means between the numbers 59 and 159. The mean of $$A_{25}, A_{28}, A_{31} and  A_{36}$$ is equal to :

The coefficient of $$x^2$$ in the expansion of $$\left(2x^2 + \frac{1}{x}\right)^{10}$$, $$x \neq 0$$, is :

The probabilities that players A and B of a team are selected for the captaincy for a tournament are 0.6 and 0.4, respectively. If A is selected the captain, the probability that the team wins the tournament is 0.8 and if B is selected the captain, the probability that the team wins the tournament is 0.7. Then the probability, that the team wins the tournament, is :

A box contains 5 blue, 6 yellow, and 4 red balls. The number of ways, of drawing 8 balls containing atleast two balls of each colour, is :

A  variable $$X$$ takes values $$0, 0, 2, 6, 12, 20, \ldots, n(n-1)$$ with frequencies $${^{n} C_{0}}, {^{n} C_{1}}, {^{n} C_{2}}, \ldots, {^{n} C_{n}}$$ respectively. If the mean of the data is 60, then the median is :