NTA JEE Main 24th February 2021 Shift 2

The number of the real roots of the equation $$(x + 1)^2 + |x - 5| = \frac{27}{4}$$ is ______.

Let $$i = \sqrt{-1}$$. If $$\frac{(-1 + i\sqrt{3})^{21}}{(1 - i)^{24}} + \frac{(1 + i\sqrt{3})^{21}}{(1 + i)^{24}} = k$$, and $$n = [|k|]$$ be the greatest integral part of $$|k|$$. Then $$\sum_{j=0}^{n+5} (j + 5)^2 - \sum_{j=0}^{n+5} (j + 5)$$ is equal to ______.

The students $$S_1, S_2, \ldots, S_{10}$$ are to be divided into 3 groups $$A$$, $$B$$ and $$C$$ such that each group has at least one student and the group $$C$$ has at most 3 students. Then the total number of possibilities of forming such groups is ______.

The sum of first four terms of a geometric progression (G.P.) is $$\frac{65}{12}$$ and the sum of their respective reciprocals is $$\frac{65}{18}$$. If the product of first three terms of the G.P. is 1, and the third term is $$\alpha$$, then $$2\alpha$$ is ______.

For integers $$n$$ and $$r$$, let $$\binom{n}{r} = \begin{cases} {}^nC_r, & \text{if } n \geq r \geq 0 \\ 0, & \text{otherwise} \end{cases}$$. The maximum value of $$k$$ for which the sum $$\sum_{i=0}^{k} \binom{10}{i}\binom{15}{k-i} + \sum_{i=0}^{k+1} \binom{12}{i}\binom{13}{k+1-i}$$ is maximum, is equal to ______.

Let a point $$P$$ be such that its distance from the point (5, 0) is thrice the distance of $$P$$ from the point (-5, 0). If the locus of the point $$P$$ is a circle of radius $$r$$, then $$4r^2$$ (in the nearest integer) is equal to ______.

If the variance of 10 natural numbers 1, 1, 1, ..., 1, $$k$$ is less than 10, then the maximum possible value of $$k$$ is ______.

If $$a + \alpha = 1, b + \beta = 2$$ and $$af(x) + \alpha f\left(\frac{1}{x}\right) = bx + \frac{\beta}{x}, x \neq 0$$, then the value of the expression $$\frac{f(x) + f\left(\frac{1}{x}\right)}{x + \frac{1}{x}}$$ is ______.

If the area of the triangle formed by the $$x$$-axis, the normal and the tangent to the circle $$(x - 2)^2 + (y - 3)^2 = 25$$ at the point (5, 7) is $$A$$, then $$24A$$ is equal to ______.

Let $$\lambda$$ be an integer. If the shortest distance between the lines $$x - \lambda = 2y - 1 = -2z$$ and $$x = y + 2\lambda = z - \lambda$$ is $$\frac{\sqrt{7}}{2\sqrt{2}}$$, then the value of $$|\lambda|$$ is ______.