NTA JEE Main 27th July 2021 Shift 1

If $$\log_3 2, \log_3(2^x - 5), \log_3\left(2^x - \frac{7}{2}\right)$$ are in an arithmetic progression, then the value of $$x$$ is equal to _________.

For real numbers $$\alpha$$ and $$\beta$$, consider the following system of linear equations: $$x + y - z = 2$$, $$x + 2y + \alpha z = 1$$ and $$2x - y + z = \beta$$. If the system has infinite solutions, then $$\alpha + \beta$$ is equal to _________.

Let $$f(x) = \begin{vmatrix} \sin^2 x & -2 + \cos^2 x & \cos 2x \\ 2 + \sin^2 x & \cos^2 x & \cos 2x \\ \sin^2 x & \cos^2 x & 1 + \cos 2x \end{vmatrix}$$, $$x \in [0, \pi]$$. Then the maximum value of $$f(x)$$ is equal to _________.

Let the domain of the function $$f(x) = \log_4(\log_5(\log_3(18x - x^2 - 77)))$$ be $$(a, b)$$. Then the value of the integral $$\int_a^b \frac{\sin^3 x}{\sin^3 x + \sin^3(a + b - x)}$$ is equal to _________.

Let $$S = \{1, 2, 3, 4, 5, 6, 7\}$$. Then the number of possible functions $$f : S \rightarrow S$$ such that $$f(m \cdot n) = f(m) \cdot f(n)$$ for every $$m, n \in S$$ and $$m \cdot n \in S$$, is equal to _________.

Let $$f : [0, 3] \rightarrow R$$ be defined by $$f(x) = \min\{x - [x], 1 + [x] - x\}$$ where $$[x]$$ is the greatest integer less than or equal to $$x$$. Let $$P$$ denote the set containing all $$x \in [0, 3]$$ where $$f$$ is discontinuous, and $$Q$$ denote the set containing all $$x \in (0, 3)$$ where $$f$$ is not differentiable. Then the sum of number of elements in $$P$$ and $$Q$$ is equal to _________.

Let $$F : [3, 5] \rightarrow R$$ be a twice differentiable function on $$(3, 5)$$ such that $$F(x) = e^{-x} \int_3^x (3t^2 + 2t + 4F'(t)) \, dt$$. If $$F'(4) = \frac{\alpha e^\beta - 224}{(e^\beta - 4)^2}$$, then $$\alpha + \beta$$ is equal to _________.

If $$y = y(x)$$, $$y \in \left[0, \frac{\pi}{2}\right)$$ is the solution of the differential equation $$\sec y \frac{dy}{dx} - \sin(x + y) - \sin(x - y) = 0$$, with $$y(0) = 0$$, then $$5y'\left(\frac{\pi}{2}\right)$$ is equal to _________.

Let $$\vec{a} = \hat{i} + \hat{j} + \hat{k}$$, $$\vec{b}$$ and $$\vec{c} = \hat{j} - \hat{k}$$ be three vectors such that $$\vec{a} \times \vec{b} = \vec{c}$$ and $$\vec{a} \cdot \vec{b} = 1$$. If the length of projection vector of the vector $$\vec{b}$$ on the vector $$\vec{a} \times \vec{c}$$ is $$l$$, then the value of $$3l^2$$ is equal to _________.

Let a plane $$P$$ pass through the point $$(3, 7, -7)$$ and contain the line, $$\frac{x - 2}{-3} = \frac{y - 3}{2} = \frac{z + 2}{1}$$. If distance of the plane $$P$$ from the origin is $$d$$, then $$d^2$$ is equal to _________.