NTA JEE Main 24th January 2023 Shift 2

The minimum number of elements that must be added to relation $$R = \{(a,b), (b,c), (b,d)\}$$ on the set $$\{a, b, c, d\}$$, so that it is an equivalence relation is

$$\displaystyle\int_{\frac{3\sqrt{2}}{4}}^{\frac{3\sqrt{3}}{4}} \frac{48}{\sqrt{9-4z^2}} dz$$ is equal to

Let $$f$$ be a differentiable function defined on $$\left[0, \frac{\pi}{2}\right]$$ such that $$f(x) > 0$$ and $$f(x) + \int_0^x f(t)\sqrt{1 - (\log_e(f(t)))^2} dt = e$$ $$\forall x \in \left[0, \frac{\pi}{2}\right]$$, then $$\left\{6\log_e\left(f\left(\frac{\pi}{6}\right)\right)\right\}^2$$ is equal to

If the area of the region bounded by the curves $$y^2 - 2y = -x$$ and $$x + y = 0$$ is $$A$$, then $$8A =$$

Let $$\vec{\alpha} = 4\hat{i} + 3\hat{j} + 5\hat{k}$$ and $$\vec{\beta} = \hat{i} + 2\hat{j} - 4\hat{k}$$. Let $$\vec{\beta_1}$$ be parallel to $$\vec{\alpha}$$ and $$\vec{\beta_2}$$ be perpendicular to $$\vec{\alpha}$$. If $$\vec{\beta} = \vec{\beta_1} + \vec{\beta_2}$$, then the value of $$5\vec{\beta_2} \cdot (\hat{i} + \hat{j} + \hat{k})$$ is

Let $$\vec{a} = \hat{i} + 2\hat{j} + \lambda\hat{k}$$, $$\vec{b} = 3\hat{i} - 5\hat{j} - \lambda\hat{k}$$, $$\vec{a} \cdot \vec{c} = 7$$, $$2(\vec{b} \cdot \vec{c}) + 43 = 0$$, $$\vec{a} \times \vec{c} = \vec{b} \times \vec{c}$$, then $$|\vec{a} \cdot \vec{b}|$$ is equal to

Let the plane containing the line of intersection of the planes $$P_1: x + (\lambda + 4)y + z = 1$$ and $$P_2: 2x + y + z = 2$$ pass through the points $$(0, 1, 0)$$ and $$(1, 0, 1)$$. Then the distance of the point $$(2\lambda, \lambda, -\lambda)$$ from the plane $$P_2$$ is

If the foot of the perpendicular drawn from $$(1, 9, 7)$$ to the line passing through the point $$(3, 2, 1)$$ and parallel to the planes $$x + 2y + z = 0$$ and $$3y - z = 3$$ is $$(\alpha, \beta, \gamma)$$, then $$\alpha + \beta + \gamma$$ is equal to

If the shortest distance between the lines $$\frac{x+\sqrt{6}}{2} = \frac{y-\sqrt{6}}{3} = \frac{z-\sqrt{6}}{4}$$ and $$\frac{x-\lambda}{3} = \frac{y-2\sqrt{6}}{4} = \frac{z+2\sqrt{6}}{5}$$ is 6, then square of sum of all possible values of $$\lambda$$ is

The urns $$A$$, $$B$$ and $$C$$ contains 4 red, 6 black; 5 red, 5 black and $$\lambda$$ red, 4 black balls respectively. One of the urns is selected at random and a ball is drawn. If the ball drawn is red and the probability that it is drawn from urn $$C$$ is 0.4, then the square of length of the side of largest equilateral triangle, inscribed in the parabola $$y^2 = \lambda x$$ with one vertex at vertex of parabola is