NTA JEE Main 24th January 2023 Shift 2 - Mathematics

The number of real solutions of the equation $$3\left(x^2 + \frac{1}{x^2}\right) - 2\left(x + \frac{1}{x}\right) + 5 = 0$$, is

The value of $$\left(\frac{1 + \sin\frac{2\pi}{9} + i\cos\frac{2\pi}{9}}{1 + \sin\frac{2\pi}{9} - i\cos\frac{2\pi}{9}}\right)^3$$ is

The number of integers, greater than 7000 that can be formed, using the digits 3, 5, 6, 7, 8 without repetition is

If $$\frac{1^3 + 2^3 + 3^3 + \ldots \text{upto n terms}}{1 \cdot 3 + 2 \cdot 5 + 3 \cdot 7 + \ldots \text{upto n terms}} = \frac{9}{5}$$ then the value of $$n$$ is

If $$\binom{30}{1}^2 + 2\binom{30}{2}^2 + 3\binom{30}{3}^2 + \ldots + 30\binom{30}{30}^2 = \frac{\alpha \cdot 60!}{(30!)^2}$$, then $$\alpha$$ is equal to

The locus of the middle points of the chords of the circle $$C_1: (x-4)^2 + (y-5)^2 = 4$$ which subtend an angle $$\theta_i$$ at the centre of the circle $$C_i$$, is a circle of radius $$r_i$$. If $$\theta_1 = \frac{\pi}{3}$$, $$\theta_3 = \frac{2\pi}{3}$$ and $$r_1^2 = r_2^2 + r_3^2$$, then $$\theta_2$$ is equal to

The equations of sides $$AB$$ and $$AC$$ of a triangle $$ABC$$ are $$(\lambda + 1)x + \lambda y = 4$$ and $$\lambda x + (1 - \lambda)y + \lambda = 0$$ respectively. Its vertex $$A$$ is on the $$y$$-axis and its orthocentre is $$(1, 2)$$. The length of the tangent from the point $$C$$ to the part of the parabola $$y^2 = 6x$$ in the first quadrant is

The set of values of $$a$$ for which $$\lim_{x \to a} ([x-5] - [2x+2]) = 0$$, where $$[\zeta]$$ denotes the greatest integer less than or equal to $$\zeta$$ is equal to

Let $$p$$ and $$q$$ be two statements. Then $$\sim(p \wedge (p \to \sim q))$$ is equivalent to

Let the six numbers $$a_1, a_2, \ldots, a_6$$ be in A.P. and $$a_1 + a_3 = 10$$. If the mean of these six numbers is $$\frac{19}{2}$$ and their variance is $$\sigma^2$$, then $$8\sigma^2$$ is equal to

The number of square matrices of order 5 with entries from the set $$\{0, 1\}$$, such that the sum of all the elements in each row is 1 and the sum of all the elements in each column is also 1, is

Let $$A$$ be a $$3 \times 3$$ matrix such that $$|adj(adj(adj \cdot A))| = 12^4$$. Then $$|A^{-1} adj A|$$ is equal to

If the system of equations $$x + 2y + 3z = 3$$, $$4x + 3y - 4z = 4$$ and $$8x + 4y - \lambda z = 9 + \mu$$ has infinitely many solutions, then the ordered pair $$(\lambda, \mu)$$ is equal to

If $$f(x) = \frac{2^{2x}}{2^{2x}+2}$$, $$x \in \mathbb{R}$$, then $$f\left(\frac{1}{2023}\right) + f\left(\frac{2}{2023}\right) + f\left(\frac{3}{2023}\right) + \ldots + f\left(\frac{2022}{2023}\right)$$ is equal to

Let $$f(x)$$ be a function such that $$f(x + y) = f(x) \cdot f(y)$$ for all $$x, y \in \mathbb{N}$$. If $$f(1) = 3$$ and $$\sum_{k=1}^{n} f(k) = 3279$$, then the value of $$n$$ is

If $$f(x) = x^3 - x^2f'(1) + xf''(2) - f'''(3)$$, $$x \in \mathbb{R}$$, then

Let $$y = y(x)$$ be the solution of the differential equation $$(x^2 - 3y^2)dx + 3xy$$ dy = 0, $$y(1) = 1$$. Then $$6y^2(e)$$ is equal to

Let the sum of the coefficient of first three terms in the expansion of $$\left(x - \frac{3}{x^2}\right)^n$$; $$x \neq 0$$, $$n \in \mathbb{N}$$ be 376. Then, the coefficient of $$x^4$$ is equal to:

Let $$S = \{\theta \in [0, 2\pi) : \tan(\pi\cos\theta) + \tan(\pi\sin\theta) = 0\}$$, then $$\sum_{\theta \in S} \sin^2\left(\theta + \frac{\pi}{4}\right)$$ is equal to

The equations of the sides $$AB$$, $$BC$$ and $$CA$$ of a triangle $$ABC$$ are $$2x + y = 0$$, $$x + py = 21a$$ ($$a \neq 0$$) and $$x - y = 3$$ respectively. Let $$P(2, a)$$ be the centroid of the triangle $$ABC$$, then $$(BC)^2$$ is equal to

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