NTA JEE Main 5th September 2020 Shift 2

If the mean and the standard deviation of the data $$3, 5, 7, a, b$$ are $$5$$ and $$2$$ respectively, then $$a$$ and $$b$$ are the roots of the equation:

If the system of linear equations
$$x + y + 3z = 0$$
$$x + 3y + k^2z = 0$$
$$3x + y + 3z = 0$$
has a non-zero solution $$(x, y, z)$$ for some $$k \in \mathbb{R}$$, then $$x + \left(\frac{y}{z}\right)$$ is equal to:

If $$a + x = b + y = c + z + 1$$, where $$a, b, c, x, y, z$$ are non-zero distinct real numbers, then $$\begin{vmatrix} x & a+y & x+a \\ y & b+y & y+b \\ z & c+y & z+c \end{vmatrix}$$ is equal to:

The derivative of $$\tan^{-1}\left(\frac{\sqrt{1+x^2}-1}{x}\right)$$ with respect to $$\tan^{-1}\left(\frac{2x\sqrt{1-x^2}}{1-2x^2}\right)$$ at $$x = \frac{1}{2}$$ is:

If $$x = 1$$ is a critical point of the function $$f(x) = (3x^2 + ax - 2 - a)e^x$$, then:

Which of the following points lies on the tangent to the curve $$x^4 e^y + 2\sqrt{y+1} = 3$$ at the point $$(1, 0)$$?

If $$\int \frac{\cos\theta}{5 + 7\sin\theta - 2\cos^2\theta}\,d\theta = A\log_e|B(\theta)| + C$$, where $$C$$ is a constant of integration, then $$\frac{B(\theta)}{A}$$ can be:

The area (in sq. units) of the region $$A = \{(x,y) : (x-1)[x] \leq y \leq 2\sqrt{x},\; 0 \leq x \leq 2\}$$, where $$[t]$$ denotes the greatest integer function, is:

Let $$y = y(x)$$ be the solution of the differential equation $$\cos x\frac{dy}{dx} + 2y\sin x = \sin 2x$$, $$x \in \left(0, \frac{\pi}{2}\right)$$. If $$y(\pi/3) = 0$$, then $$y(\pi/4)$$ is equal to:

If for some $$\alpha \in \mathbb{R}$$, the lines $$L_1 : \frac{x+1}{2} = \frac{y-2}{-1} = \frac{z-1}{1}$$ and $$L_2 : \frac{x+2}{\alpha} = \frac{y+1}{5-\alpha} = \frac{z+1}{1}$$ are coplanar, then the line $$L_2$$ passes through the point: