NTA JEE Main 4th September 2020 Shift 2

If the system of equations
$$x + y + z = 2$$
$$2x + 4y - z = 6$$
$$3x + 2y + \lambda z = \mu$$
has infinitely many solutions, then:

Suppose the vectors $$x_1, x_2$$ and $$x_3$$ are the solutions of the system of linear equations, $$Ax = b$$ when the vector $$b$$ on the right side is equal to $$b_1, b_2$$ and $$b_3$$ respectively. If $$x_1 = \begin{bmatrix}1\\1\\1\end{bmatrix}$$, $$x_2 = \begin{bmatrix}0\\2\\1\end{bmatrix}$$, $$x_3 = \begin{bmatrix}0\\0\\1\end{bmatrix}$$; $$b_1 = \begin{bmatrix}1\\0\\0\end{bmatrix}$$, $$b_2 = \begin{bmatrix}0\\2\\0\end{bmatrix}$$, $$b_3 = \begin{bmatrix}0\\0\\2\end{bmatrix}$$, then the determinant of $$A$$ is equal to

The minimum value of $$2^{\sin x} + 2^{\cos x}$$ is:

The function $$f(x) = \begin{cases} \frac{\pi}{4} + \tan^{-1}x, & |x| \leq 1 \\ \frac{1}{2}(|x| - 1), & |x| > 1 \end{cases}$$ is:

Let $$f : (0, \infty) \to (0, \infty)$$ be a differentiable function such that $$f(1) = e$$ and $$\lim_{t \to x}\frac{t^2f^2(x) - x^2f^2(t)}{t - x} = 0$$. If $$f(x) = 1$$, then $$x$$ is equal to:

The area (in sq. units) of the largest rectangle $$ABCD$$ whose vertices $$A$$ and $$B$$ lie on the $$x$$-axis and vertices $$C$$ and $$D$$ lie on the parabola, $$y = x^2 - 1$$ below the $$x$$-axis, is:

The integral $$\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \tan^3 x \cdot \sin^2 3x(2\sec^2 x \cdot \sin^2 3x + 3\tan x \cdot \sin 6x)dx$$ is equal to:

The solution of the differential equation $$\frac{dy}{dx} - \frac{y+3x}{\log_e(y+3x)} + 3 = 0$$ is (where C is a constant of integration)

The distance of the point $$(1, -2, 3)$$ from the plane $$x - y + z = 5$$ measured parallel to the line $$\frac{x}{2} = \frac{y}{3} = \frac{z}{-6}$$ is:

In a game two players A and B take turns in throwing a pair of fair dice starting with player A and total of scores on the two dice, in each throw is noted. A wins the game if he throws a total of 6 before B throws a total of 7 and B wins the game if he throws a total of 7 before A throws a total of six. The game stops as soon as either of the players wins. The probability of A winning the game is: