NTA JEE Main 24th February 2021 Shift 1

If the least and the largest real values of $$\alpha$$, for which the equation $$z + \alpha|z - 1| + 2i = 0$$ ($$z \in C$$ and $$i = \sqrt{-1}$$) has a solution, are $$p$$ and $$q$$ respectively; then $$4p^2 + q^2$$ is equal to ______.

If one of the diameters of the circle $$x^2 + y^2 - 2x - 6y + 6 = 0$$ is a chord of another circle $$C$$, whose center is at (2, 1), then its radius is ______.

Let $$A = \{n \in N : n \text{ is a 3-digit number}\}$$, $$B = \{9k + 2 : k \in N\}$$ and $$C = \{9k + l : k \in N\}$$ for some $$l$$ ($$0 < l < 9$$). If the sum of all the elements of the set $$A \cap (B \cup C)$$ is $$274 \times 400$$, then $$l$$ is equal to ______

Let $$P = \begin{pmatrix} 3 & -1 & -2 \\ 2 & 0 & \alpha \\ 3 & -5 & 0 \end{pmatrix}$$, where $$\alpha \in R$$. Suppose $$Q = [q_{ij}]$$ is a matrix satisfying $$PQ = kI_3$$ for some non-zero $$k \in R$$. If $$q_{23} = -\frac{k}{8}$$ and $$Q = \frac{k^2}{2}$$, then $$\alpha^2 + k^2$$ is equal to ______.

Let $$M$$ be any $$3 \times 3$$ matrix with entries from the set $$\{0, 1, 2\}$$. The maximum number of such matrices, for which the sum of diagonal elements of $$M^T M$$ is seven, is ______.

$$\lim_{n \to \infty} \tan \sum_{r=1}^{n} \tan^{-1}\frac{1}{1 + r + r^2}$$ is equal to ______.

The minimum value of $$\alpha$$ for which the equation $$\frac{4}{\sin x} + \frac{1}{1 - \sin x} = \alpha$$ has at least one solution in $$\left(0, \frac{\pi}{2}\right)$$ is ______.

If $$\int_{-a}^{a} (|x| + |x - 2|) dx = 22$$, $$a > 2$$ and $$x$$ denotes the greatest integer $$\leq x$$, then $$\int_{a}^{a} (x + |x|) dx$$ is equal to ______

Let three vectors $$\vec{a}$$, $$\vec{b}$$ and $$\vec{c}$$ be such that $$\vec{c}$$ is coplanar with $$\vec{a}$$ and $$\vec{b}$$, $$\vec{a} \cdot \vec{c} = 7$$ and $$\vec{b}$$ is perpendicular to $$\vec{c}$$, where $$\vec{a} = -\hat{i} + \hat{j} + \hat{k}$$ and $$\vec{b} = 2\hat{i} + \hat{k}$$, then the value of $$|2\vec{a} + \vec{b} + \vec{c}|^2$$ is ______

Let $$B_i$$ ($$i = 1, 2, 3$$) be three independent events in a sample space. The probability that only $$B_1$$ occur is $$\alpha$$, only $$B_2$$ occurs is $$\beta$$ and only $$B_3$$ occurs is $$\gamma$$. Let $$p$$ be the probability that none of the events $$B_i$$ occurs and these 4 probabilities satisfy the equations $$(\alpha - 2\beta)p = \alpha\beta$$ and $$(\beta - 3\gamma)p = 2\beta\gamma$$ (All the probabilities are assumed to lie in the interval (0, 1)). Then $$\frac{P(B_1)}{P(B_3)}$$ is equal to ______.