NTA JEE Main 8th January 2020 Shift 1

For which of the following ordered pairs $$(\mu, \delta)$$, the system of linear equations
$$x + 2y + 3z = 1$$
$$3x + 4y + 5z = \mu$$
$$4x + 4y + 4z = \delta$$
is inconsistent?

The inverse function of $$f(x) = \frac{8^{2x} - 8^{-2x}}{8^{2x} + 8^{-2x}}$$, $$x \in (-1, 1)$$, is

Let $$f(x) = (\sin(\tan^{-1} x) + \sin(\cot^{-1} x))^2 - 1$$, $$|x| \gt 1$$. If $$\frac{dy}{dx} = \frac{1}{2}\frac{d}{dx}(\sin^{-1}(f(x)))$$ and $$y(\sqrt{3}) = \frac{\pi}{6}$$, then $$y(-\sqrt{3})$$ is equal to:

If $$c$$ is a point at which Rolle's theorem holds for the function, $$f(x) = \log_e\left(\frac{x^2 + \alpha}{7x}\right)$$ in the interval [3, 4], where $$\alpha \in R$$, then $$f''(c)$$ is equal to

Let $$f(x) = x\cos^{-1}(-\sin|x|)$$, $$x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$$, then which of the following is true?

If $$\int \frac{\cos x \, dx}{\sin^3 x (1+\sin^6 x)^{2/3}} = f(x)(1 + \sin^6 x)^{1/\lambda} + c$$, where c is a constant of integration, then $$\lambda f\left(\frac{\pi}{3}\right)$$ is equal to

Let $$y = y(x)$$ be a solution of the differential equation, $$\sqrt{1 - x^2}\frac{dy}{dx} + \sqrt{1 - y^2} = 0$$, $$|x| < 1$$. If $$y\left(\frac{1}{2}\right) = \frac{\sqrt{3}}{2}$$, then $$y\left(\frac{-1}{\sqrt{2}}\right)$$ is equal to

Let the volume of a parallelepiped whose coterminous edges are given by $$\vec{u} = \hat{i} + \hat{j} + \lambda\hat{k}$$, $$\vec{v} = \hat{i} + \hat{j} + 3\hat{k}$$ and $$\vec{w} = 2\hat{i} + \hat{j} + \hat{k}$$ be 1 cu. unit. If $$\theta$$ be the angle between the edges $$\vec{u}$$ and $$\vec{w}$$, then the value of $$\cos\theta$$ can be

The shortest distance between the lines $$\frac{x-3}{3} = \frac{y-8}{-1} = \frac{z-3}{1}$$ and $$\frac{x+3}{-3} = \frac{y+7}{2} = \frac{z-6}{4}$$ is

Let A and B be two independent events such that $$P(A) = \frac{1}{3}$$ and $$P(B) = \frac{1}{6}$$. Then, which of the following is true?