NTA JEE Main 25th January 2023 Shift 1

The constant term in the expansion of $$\left(2x + \frac{1}{x^7} + 3x^2\right)^5$$ is _____.

The vertices of a hyperbola H are $$(\pm 6, 0)$$ and its eccentricity is $$\frac{\sqrt{5}}{2}$$. Let N be the normal to H at a point in the first quadrant and parallel to the line $$\sqrt{2}x + y = 2\sqrt{2}$$. If $$d$$ is the length of the line segment of N between H and the y-axis then $$d^2$$ is equal to _____.

If the sum of all the solutions of $$\tan^{-1}\left(\frac{2x}{1-x^2}\right) + \cot^{-1}\left(\frac{1-x^2}{2x}\right) = \frac{\pi}{3}$$, $$-1 < x < 1$$, $$x \neq 0$$, is $$\alpha - \frac{4}{\sqrt{3}}$$, then $$\alpha$$ is equal to _____.

For some $$a, b, c \in \mathbb{N}$$, let $$f(x) = ax - 3$$ and $$g(x) = x^b + c$$, $$x \in \mathbb{R}$$. If $$(f \circ g)^{-1}(x) = \left(\frac{x-7}{2}\right)^{1/3}$$, then $$(f \circ g)(ac) + (g \circ f)(b)$$ is equal to _____.

Let $$f(x) = \int \frac{2x}{(x^2+1)(x^2+3)} dx$$. If $$f(3) = \frac{1}{2}(\log_e 5 - \log_e 6)$$, then $$f(4)$$ is equal to

If the area enclosed by the parabolas $$P_1: 2y = 5x^2$$ and $$P_2: x^2 - y + 6 = 0$$ is equal to the area enclosed by $$P_1$$ and $$y = \alpha x$$, $$\alpha > 0$$, then $$\alpha^3$$ is equal to _____.

Let $$\vec{a}, \vec{b}$$ and $$\vec{c}$$ be three non zero vectors such that $$\vec{b} \cdot \vec{c} = 0$$ and $$\vec{a} \times (\vec{b} \times \vec{c}) = \frac{\vec{b} - \vec{c}}{2}$$. If $$\vec{d}$$ be a vector such that $$\vec{b} \cdot \vec{d} = \vec{a} \cdot \vec{b}$$, then $$(\vec{a} \times \vec{b}) \cdot (\vec{c} \times \vec{d})$$ is equal to

The vector $$\vec{a} = -\hat{i} + 2\hat{j} + \hat{k}$$ is rotated through a right angle, passing through the y-axis in its way and the resulting vector is $$\vec{b}$$. Then the projection of $$3\vec{a} + \sqrt{2}\vec{b}$$ on $$\vec{c} = 5\hat{i} + 4\hat{j} + 3\hat{k}$$ is

Consider the lines $$L_1$$ and $$L_2$$ given by
$$L_1: \frac{x-1}{2} = \frac{y-3}{1} = \frac{z-2}{2}$$
$$L_2: \frac{x-2}{1} = \frac{y-2}{2} = \frac{z-3}{3}$$
A line $$L_3$$ having direction ratios $$1, -1, -2$$, intersects $$L_1$$ and $$L_2$$ at the points $$P$$ and $$Q$$ respectively. Then the length of line segment $$PQ$$ is

Let the equation of the plane passing through the line $$x - 2y - z - 5 = 0 = x + y + 3z - 5$$ and parallel to the line $$x + y + 2z - 7 = 0 = 2x + 3y + z - 2$$ be $$ax + by + cz = 65$$. Then the distance of the point $$(a, b, c)$$ from the plane $$2x + 2y - z + 16 = 0$$ is _____.