NTA JEE Main 10th April 2023 Shift 2

Let $$A = \{2, 3, 4\}$$ and $$B = \{8, 9, 12\}$$. Then the number of elements in the relation $$R = \{(a_1, b_1, a_2, b_2) \in A \times B, A \times B: a_1 \text{ divides } b_2 \text{ and } a_2 \text{ divides } b_1\}$$ is

If $$A = \frac{1}{5!6!7!} \begin{vmatrix} 5! & 6! & 7! \\ 6! & 7! & 8! \\ 7! & 8! & 9! \end{vmatrix}$$, then adj $$2A$$ is equal to

Let $$g(x) = f(x) + f(1-x)$$ and $$f''(x) > 0$$, $$x \in (0, 1)$$. If $$g$$ is decreasing in the interval $$(0, \alpha)$$ and increasing in the interval $$(\alpha, 1)$$, then $$\tan^{-1}(2\alpha) + \tan^{-1}\left(\frac{1}{\alpha}\right) + \tan^{-1}\left(\frac{\alpha+1}{\alpha}\right)$$ is equal to

For $$\alpha, \beta, \gamma, \delta \in \mathbb{N}$$, if $$\int \frac{x^{2x}}{e} + \frac{e^{2x}}{x} \log_e x \, dx = \frac{1}{\alpha e} x^{\beta x} - \frac{1}{\gamma x} e^{\delta x} + C$$, where $$e = \sum_{n=0}^\infty \frac{1}{n!}$$ and C is constant of integration, then $$\alpha + 2\beta + 3\gamma - 4\delta$$ is equal to

Let f be a continuous function satisfying $$\int_0^{t^2} f(x) + x^2 dx = \frac{4}{3}t^3$$, $$\forall t > 0$$. Then $$f\left(\frac{\pi^{2}}{4}\right)$$ is equal to

Let $$\vec{a} = 2\hat{i} + 7\hat{j} - \hat{k}$$, $$\vec{b} = 3\hat{i} + 5\hat{k}$$ and $$\vec{c} = \hat{i} - \hat{j} + 2\hat{k}$$. Let $$\vec{d}$$ be a vector which is perpendicular to both $$\vec{a}$$ and $$\vec{b}$$, and $$\vec{c} \cdot \vec{d} = 12$$. Then $$(-\hat{i} + \hat{j} - \hat{k}) \cdot (\vec{c} \times \vec{d})$$ is equal to

If the points $$P$$ and $$Q$$ are respectively the circumcenter and the orthocentre of a $$\triangle ABC$$, then $$\overrightarrow{PA} + \overrightarrow{PB} + \overrightarrow{PC}$$ is equal to

Let the image of the point P(1, 2, 6) in the plane passing through the points A(1, 2, 0) and B(1, 4, 1) C(0, 5, 1) be $$Q(\alpha, \beta, \gamma)$$. Then $$\alpha^2 + \beta^2 + \gamma^2$$ equal to

Let the line $$\frac{x}{1} = \frac{6-y}{2} = \frac{z+8}{5}$$ intersect the lines $$\frac{x-5}{4} = \frac{y-7}{3} = \frac{z+2}{1}$$ and $$\frac{x+3}{6} = \frac{3-y}{3} = \frac{z-6}{1}$$ at the points A and B respectively. Then the distance of the mid-point of the line segment AB from the plane $$2x - 2y + z = 14$$ is

Let a die be rolled n times. Let the probability of getting odd numbers seven times be equal to the probability of getting odd numbers nine times. If the probability of getting even numbers twice is $$\frac{k}{2^{15}}$$, then k is equal to