NTA JEE Main 31st August 2021 Shift 2

The number of 4-digit numbers which are neither multiple of 7 nor multiple of 3 is _________.

If $$S = \frac{7}{5} + \frac{9}{5^2} + \frac{13}{5^3} + \frac{19}{5^4} + \ldots$$, then $$160 S$$ is equal to _________.

If the coefficient of $$a^7 b^8$$ in the expansion of $$(a + 2b + 4ab)^{10}$$ is $$K \cdot 2^{16}$$, then $$K$$ is equal to _________.

Let $$B$$ be the centre of the circle $$x^2 + y^2 - 2x + 4y + 1 = 0$$. Let the tangents at two points $$P$$ and $$Q$$ on the circle intersect at the point $$A(3, 1)$$. Then $$8 \cdot \frac{\text{area } \triangle APQ}{\text{area } \triangle BPQ}$$ is equal to _________.

A tangent line $$L$$ is drawn at the point $$(2, -4)$$ on the parabola $$y^2 = 8x$$. If the line $$L$$ is also tangent to the circle $$x^2 + y^2 = a$$, then $$a$$ is equal to _________.

The number of elements in the set $$\{A = \begin{pmatrix} a & b \\ 0 & d \end{pmatrix} : a, b, d \in \{-1, 0, 1\}$$ and $$(I - A)^3 = I - A^3\}$$, where $$I$$ is $$2 \times 2$$ identity matrix, is _________.

Let $$f(x)$$ be a cubic polynomial with $$f(1) = -10$$, $$f(-1) = 6$$, and has a local minima at $$x = 1$$, and $$f'(x)$$ has a local minima at $$x = -1$$. Then $$f(3)$$ is equal to _________.

If $$\int \frac{\sin x}{\sin^3 x + \cos^3 x} dx = \alpha\log_e |1 + \tan x| + \beta\log_e|1 - \tan x + \tan^2 x| + \gamma\tan^{-1}\frac{2\tan x - 1}{\sqrt{3}} + C$$, when $$C$$ is constant of integration, then the value of $$18\alpha + \beta + \gamma^2$$ is _________.

If the line $$y = mx$$ bisects the area enclosed by the lines $$x = 0$$, $$y = 0$$, $$x = \frac{3}{2}$$ and the curve $$y = 1 + 4x - x^2$$, then $$12m$$ is equal to _________.

Suppose the line $$\frac{x-2}{\alpha} = \frac{y-2}{-5} = \frac{z+2}{2}$$ lies on the plane $$x + 3y - 2z + \beta = 0$$. Then $$(\alpha + \beta)$$ is equal to _________.