NTA JEE Main 8th April2023 Shift 1

Let the number of elements in sets $$A$$ and $$B$$ be five and two respectively. Then the number of subsets of $$A \times B$$ each having at least 3 and at most 6 elements is

Let $$\begin{bmatrix} 2 & 1 & 0 \\ 1 & 2 & -1 \\ 0 & -1 & 2 \end{bmatrix}$$. If $$|adj(adj(adj(2A)))| = (16)^n$$, then $$n$$ is equal to

Let $$P = \begin{bmatrix} \dfrac{\sqrt{3}}{2} & \dfrac{1}{2} \\ -\dfrac{1}{2} & \dfrac{\sqrt{3}}{2} \end{bmatrix}$$, $$A = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}$$ and $$Q = PAP^T$$. If $$P^TQ^{2007}P = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$$ then $$2a + b - 3c - 4d$$ is equal to

Let $$f(x) = \dfrac{\sin x + \cos x - \sqrt{2}}{\sin x - \cos x}$$, $$x \in [0, \pi] - \{\dfrac{\pi}{4}\}$$, then $$f\left(\dfrac{7\pi}{12}\right) f''\left(\dfrac{7\pi}{12}\right)$$ is equal to

Let $$I(x) = \int \dfrac{x+1}{x(1+xe^x)^2} dx$$, $$x > 0$$. If $$\lim_{x \to \infty} I(x) = 0$$ then $$I(1)$$ is equal to

The area of the region $$\{(x,y): x^2 \le y \le 8-x^2, y \le 7\}$$ is

If the points with position vectors $$\alpha\hat{i} + 10\hat{j} + 13\hat{k}$$, $$6\hat{i} + 11\hat{j} + 11\hat{k}$$, $$\dfrac{9}{2}\hat{i} + \beta\hat{j} - 8\hat{k}$$ are collinear, then $$(19\alpha - 6\beta)^2$$ is equal to

The shortest distance between the lines $$\dfrac{x-4}{4} = \dfrac{y+2}{5} = \dfrac{z+3}{3}$$ and $$\dfrac{x-1}{3} = \dfrac{y-3}{4} = \dfrac{z-4}{2}$$ is

If the equation of the plane containing the line $$x + 2y + 3z - 4 = 0 = 2x + y - z + 5$$ and perpendicular to the plane $$\vec{r} = (\hat{i} - \hat{j}) + \lambda(\hat{i} + \hat{j} + \hat{k}) + \mu(\hat{i} - 2\hat{j} + 3\hat{k})$$ is $$ax + by + cz = 4$$ then $$(a - b + c)$$ is equal to

In a bolt factory, machines A, B and C manufacture respectively 20%, 30% and 50% of the total bolts. Of their output 3, 4 and 2 percent are respectively defective bolts. A bolt is drawn at random from the product. If the bolt drawn is found defective then the probability that it is manufactured by the machine C is