NTA JEE Mains 5th April Shift 1 2026

Let $$a, b \in \mathbb{C}$$. Let $$\alpha, \beta$$ be the roots of $$x^2 + ax + b = 0$$. If $$\beta - \alpha = \sqrt{11}$$ and $$\beta^2 - \alpha^2 = 3i\sqrt{11}$$, then $$(\beta^3 - \alpha^3)^2$$ is equal to :

Let the sum of first $$n$$ terms of an A.P. is $$3n^2 + 5n$$. The sum of squares of the first 10 terms is :

Let $$A$$ is a $$3 \times 3$$ matrix such that $$A^T \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 5 \\ 2 \\ 2 \end{bmatrix}$$, $$A^T \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 3 \\ 1 \\ 1 \end{bmatrix}$$, $$A \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 3 \\ 4 \\ 4 \end{bmatrix}$$, $$A \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 1 \\ 3 \\ 1 \end{bmatrix}$$. If $$\det(A) = 1$$, then $$\det(\text{adj}(A^2 + A))$$ is equal to :

Consider the system of equations in $$x, y, z$$:
$$x + 2y + tz = 0$$,
$$6x + y + 5tz = 0$$,
$$3x + t^2 y + f(t)z = 0$$,
where $$f: \mathbb{R} \to \mathbb{R}$$ is differentiable function. If this system has infinitely many solutions for all $$t \in \mathbb{R}$$, then $$f$$ is :

$$\displaystyle\sum_{n=1}^{10} \left(\frac{528}{n(n+1)(n+2)}\right)$$ is equal to :

Let $$\tan A$$ and $$\tan B$$, where $$A, B \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$$, be the roots of the quadratic equation $$x^2 - 2x - 5 = 0$$. Then $$20\sin^2\left(\frac{A+B}{2}\right)$$ is equal to :

A letter is known to have arrived by post either from KANPUR or from ANANTPUR. On the envelope just two consecutive letters AN are visible. The probability, that the letter came from ANANTPUR, is :

The mean deviation about the mean for the data

is equal to :

Let a focus of the ellipse $$E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ be $$S(4, 0)$$ and its eccentricity be $$\frac{4}{5}$$. If $$P(3, \alpha)$$ lies on  $$E$$ and $$O$$ is the origin, then the area of $$\triangle POS$$ is equal to:

Let $$P$$ moving point on the circle $$x^2 + y^2 - 6x - 8y + 21 = 0$$. Then,the maximum distance of $$P$$ from the vertex of the parabola $$x^2 + 6x + y + 13 = 0$$ is :