NTA JEE Main 7th January 2020 Shift 2

Let $$A = [a_{ij}]$$ and $$B = [b_{ij}]$$ be two $$3 \times 3$$ real matrices such that $$b_{ij} = (3)^{(i+j-2)} a_{ij}$$, where $$i, j = 1, 2, 3$$. If the determinant of B is 81, then determinant of A is

Let $$y = y(x)$$ be a function of $$x$$ satisfying $$y\sqrt{1 - x^2} = k - x\sqrt{1 - y^2}$$ where $$k$$ is a constant and $$y\left(\frac{1}{2}\right) = -\frac{1}{4}$$. Then $$\frac{dy}{dx}$$ at $$x = \frac{1}{2}$$, is equal to

The value of $$c$$, in the Lagrange's mean value theorem for the function $$f(x) = x^3 - 4x^2 + 8x + 11$$, when $$x \in [0, 1]$$, is

Let $$f(x)$$ be a polynomial of degree 5 such that $$x = \pm 1$$ are its critical points. If $$\lim_{x \to 0}\left(2 + \frac{f(x)}{x^3}\right) = 4$$, then which one of the following is not true?

The value of $$\alpha$$ for which $$4\alpha \int_{-1}^{2} e^{-\alpha|x|}dx = 5$$, is

If $$\theta_1$$ and $$\theta_2$$ be respectively the smallest and the largest values of $$\theta$$ in $$(0, 2\pi) - \{\pi\}$$ which satisfy the equation, $$2\cot^2\theta - \frac{5}{\sin\theta} + 4 = 0$$, then $$\int_{\theta_1}^{\theta_2} \cos^2 3\theta \, d\theta$$ is equal to:

The area (in sq. units) of the region $$\{(x, y) \in R^2 | 4x^2 \le y \le 8x + 12\}$$ is

Let $$y = y(x)$$ be the solution curve of the differential equation, $$(y^2 - x)\frac{dy}{dx} = 1$$, satisfying $$y(0) = 1$$. This curve intersects the X-axis at a point whose abscissa is

Let $$\vec{a}, \vec{b}$$ and $$\vec{c}$$ be three unit vectors such that $$\vec{a} + \vec{b} + \vec{c} = 0$$. If $$\lambda = \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}$$ and $$\vec{d} = \vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a}$$, then the order pair, $$\left(\lambda, \vec{d}\right)$$, is equal to

In a workshop, there are five machines and the probability of any one of them to be out of service on a day is $$\frac{1}{4}$$. If the probability that at most two machines will be out of service on the same day is $$\left(\frac{3}{4}\right)^3 k$$, then $$k$$ is equal to