NTA JEE Main 5th September 2020 Shift 1

A survey shows that 73% of the persons working in an office like coffee, whereas 65% like tea. If $$x$$ denotes the percentage of them, who like both coffee and tea, then $$x$$ cannot be:

If the minimum and the maximum values of the function $$f : \left[\frac{\pi}{4}, \frac{\pi}{2}\right] \to R$$, defined by$$f(\theta) = \begin{vmatrix} -\sin^2\theta & -1 - \sin^2\theta & 1 \\ -\cos^2\theta & -1 - \cos^2\theta & 1 \\ 12 & 10 & -2 \end{vmatrix}$$ are $$m$$ and $$M$$ respectively, then the ordered pair $$(m, M)$$ is equal to:

Let $$\lambda \in \mathbb{R}$$. The system of linear equations
$$2x_1 - 4x_2 + \lambda x_3 = 1$$
$$x_{1} - 6x_{2} + x_{3} = 2$$
$$\lambda x_1 - 10x_2 + 4x_3 = 3$$
is inconsistent for:

If $$S$$ is the sum of the first 10 terms of the series, $$\tan^{-1}\left(\frac{1}{3}\right) + \tan^{-1}\left(\frac{1}{7}\right) + \tan^{-1}\left(\frac{1}{13}\right) + \tan^{-1}\left(\frac{1}{21}\right) + \ldots$$, then $$\tan(S)$$ is equal to:

If the function $$f(x) = \begin{cases} k_1(x - \pi)^2 - 1, & x \leq \pi \\ k_2 \cos x, & x > \pi \end{cases}$$ is twice differentiable, then the ordered pair $$(k_1, k_2)$$ is equal to:

If $$\int (e^{2x} + 2e^x - e^{-x} - 1)e^{(e^x + e^{-x})}\,dx = g(x)e^{(e^x + e^{-x})} + c$$, where $$c$$ is a constant of integration, then $$g(0)$$ is:

The value of $$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \frac{1}{1 + e^{\sin x}}\,dx$$ is:

If $$y = y(x)$$ is the solution of the differential equation $$\frac{5 + e^x}{2 + y} \cdot \frac{dy}{dx} + e^x = 0$$ satisfying $$y(0) = 1$$, then value of $$y(\log_e 13)$$ is:

If the volume of a parallelepiped, whose coterminous edges are given by the vectors $$\vec{a} = \hat{i} + \hat{j} + n\hat{k}$$, $$\vec{b} = 2\hat{i} + 4\hat{j} - n\hat{k}$$ and $$\vec{c} = \hat{i} + n\hat{j} + 3\hat{k}$$ $$(n \geq 0)$$ is 158 cubic units, then:

If $$(a, b, c)$$ is the image of the point $$(1, 2, -3)$$ in the line, $$\frac{x+1}{2} = \frac{y-3}{-2} = \frac{z}{-1}$$, then $$a + b + c$$ is equal to: