NTA JEE Main 27th June 2022 Shift 1

The number of ways, 16 identical cubes, of which 11 are blue and rest are red, can be placed in a row so that between any two red cubes there should be at least 2 blue cubes, is ______

If the sum of the first ten terms of the series $$\frac{1}{5} + \frac{2}{65} + \frac{3}{325} + \frac{4}{1025} + \frac{5}{2501} + \ldots$$ is $$\frac{m}{n}$$, where $$m$$ and $$n$$ are co-prime numbers, then $$m + n$$ is equal to ______

If the coefficient of $$x^{10}$$ in the binomial expansion of $$\left(\frac{\sqrt{x}}{5^{1/4}} + \frac{\sqrt{5}}{x^{1/3}}\right)^{60}$$ is $$5^k l$$, where $$l, k \in N$$ and $$l$$ is coprime to $$5$$, then $$k$$ is equal to ______

A rectangle $$R$$ with end points of the one of its sides as $$(1, 2)$$ and $$(3, 6)$$ is inscribed in a circle. If the equation of a diameter of the circle is $$2x - y + 4 = 0$$, then the area of $$R$$ is ______

A circle of radius $$2$$ unit passes through the vertex and the focus of the parabola $$y^2 = 2x$$ and touches the parabola $$y = \left(x - \frac{1}{4}\right)^2 + \alpha$$, where $$\alpha > 0$$. Then $$(4\alpha - 8)^2$$ is equal to ______

The positive value of the determinant of the matrix $$A$$, whose $$Adj(Adj(A)) = \begin{bmatrix} 14 & 28 & -14 \\ -14 & 14 & 28 \\ 28 & -14 & 14 \end{bmatrix}$$, is ______

Let $$f : R \to R$$ be a function defined $$f(x) = \frac{2e^{2x}}{e^{2x}+e}$$. Then $$f\left(\frac{1}{100}\right) + f\left(\frac{2}{100}\right) + f\left(\frac{3}{100}\right) + \ldots + f\left(\frac{99}{100}\right)$$ is equal to ______

If the sum of all the roots of the equation $$e^{2x} - 11e^x - 45e^{-x} + \frac{81}{2} = 0$$ is $$\log_e P$$, then $$P$$ is equal to ______

Let $$A_1 = \{(x,y) : |x| \leq y^2, |x| + 2y \leq 8\}$$ and $$A_2 = \{(x,y) : |x| + |y| \leq k\}$$. If $$27$$ (Area $$A_1$$) $$= 5$$ (Area $$A_2$$), then $$k$$ is equal to ______

Let the mirror image of the point $$(a, b, c)$$ with respect to the plane $$3x - 4y + 12z + 19 = 0$$ be $$(a - 6, \beta, \gamma)$$. If $$a + b + c = 5$$, then $$7\beta - 9\gamma$$ is equal to ______