NTA JEE Main 20th July 2021 Shift 2

The number of solutions of the equation $$\log_{(x+1)}(2x^2 + 7x + 5) + \log_{(2x+5)}(x+1)^2 - 4 = 0$$, $$x > 0$$, is ___.

Let $$\{a_n\}_{n=1}^\infty$$ be a sequence such that $$a_1 = 1$$, $$a_2 = 1$$ and $$a_{n+2} = 2a_{n+1} + a_n$$ for all $$n \ge 1$$. Then the value of $$47\sum_{n=1}^\infty \frac{a_n}{2^{3n}}$$ is equal to ___.

For $$k \in N$$, let $$\frac{1}{\alpha(\alpha+1)(\alpha+2)\ldots(\alpha+20)} = \sum_{K=0}^{20} \frac{A_K}{\alpha+k}$$, where $$\alpha > 0$$. Then the value of $$100\left(\frac{A_{14}+A_{15}}{A_{13}}\right)^2$$ is equal to ___.

Consider a triangle having vertices $$A(-2, 3)$$, $$B(1, 9)$$ and $$C(3, 8)$$. If a line $$L$$ passing through the circumcentre of triangle $$ABC$$, bisects line $$BC$$, and intersects y-axis at point $$\left(0, \frac{\alpha}{2}\right)$$, then the value of real number $$\alpha$$ is ___.

If the point on the curve $$y^2 = 6x$$, nearest to the point $$\left(3, \frac{3}{2}\right)$$ is $$(\alpha, \beta)$$, then $$2(\alpha + \beta)$$ is equal to ___.

If $$\lim_{x \to 0} \left[\frac{\alpha x e^x - \beta \log_e(1+x) + \gamma x^2 e^{-x}}{x \sin^2 x}\right] = 10$$, $$\alpha, \beta, \gamma \in R$$, then the value of $$\alpha + \beta + \gamma$$ is ___.

Let $$A = \{a_{ij}\}$$ be a $$3 \times 3$$ matrix, where $$a_{ij} = \begin{cases} (-1)^{j-i} & \text{if } i < j \\ 2 & \text{if } i = j \\ (-1)^{i+j} & \text{if } i > j \end{cases}$$
then det$$(3 \text{Adj}(2A^{-1}))$$ is equal to ___.

Let a function $$g : [0, 4] \to R$$ be defined as
$$g(x) =\begin{cases}\max\limits_{0 \le t \le x} \{ t^3 - 6t^2 + 9t - 3 \}, & 0 \le x \le 3 \\4 - x, & 3 < x \le 4\end{cases}$$
then the number of points in the interval $$(0, 4)$$ where $$g(x)$$ is NOT differentiable, is ___.

Let a curve $$y = y(x)$$ be given by the solution of the differential equation $$\cos\left(\frac{1}{2}\cos^{-1}(e^{-x})\right)dx = \left(\sqrt{e^{2x}-1}\right)dy$$. If it intersects y-axis at $$y = -1$$, and the intersection point of the curve with x-axis is $$(\alpha, 0)$$, then $$e^\alpha$$ is equal to ___.

For $$p > 0$$, a vector $$\vec{v_2} = 2\hat{i} + (p+1)\hat{j}$$ is obtained by rotating the vector $$\vec{v_1} = \sqrt{3}p\hat{i} + \hat{j}$$ by an angle $$\theta$$ about origin in counter clockwise direction. If $$\tan\theta = \frac{(\alpha\sqrt{3}-2)}{(4\sqrt{3}+3)}$$, then the value of $$\alpha$$ is equal to ___.