NTA JEE Main 24th January 2023 Shift 2

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