NTA JEE Mains 9th April 2024 Shift 1

The sum of the square of the modulus of the elements in the set $$\{z = a + ib : a, b \in \mathbb{Z}, z \in \mathbb{C}, |z - 1| \leq 1, |z - 5| \leq |z - 5i|\}$$ is ________

The remainder when $$428^{2024}$$ is divided by 21 is __________

Let the centre of a circle, passing through the points $$(0, 0)$$, $$(1, 0)$$ and touching the circle $$x^2 + y^2 = 9$$, be $$(h, k)$$. Then for all possible values of the coordinates of the centre $$(h, k)$$, $$4(h^2 + k^2)$$ is equal to _________

Let $$\lim_{n \to \infty} \left(\frac{n}{\sqrt{n^4+1}} - \frac{2n}{(n^2+1)\sqrt{n^4+1}} + \frac{n}{\sqrt{n^4+16}} - \frac{8n}{(n^2+4)\sqrt{n^4+16}} + \ldots + \frac{n}{\sqrt{n^4+n^4}} - \frac{2n \cdot n^2}{(n^2+n^2)\sqrt{n^4+n^4}}\right)$$ be $$\frac{\pi}{k}$$, using only the principal values of the inverse trigonometric functions. Then $$k^2$$ is equal to ________

Let $$A = \{2, 3, 6, 7\}$$ and $$B = \{4, 5, 6, 8\}$$. Let $$R$$ be a relation defined on $$A \times B$$ by $$(a_1, b_1) R (a_2, b_2)$$ if and only if $$a_1 + a_2 = b_1 + b_2$$. Then the number of elements in $$R$$ is _________

Let $$A$$ be a non-singular matrix of order 3. If $$\det(3 \text{ adj}(2 \text{ adj}((\det A)A))) = 3^{-13} \cdot 2^{-10}$$ and $$\det(3 \text{ adj}(2A)) = 2^m \cdot 3^n$$, then $$|3m + 2n|$$ is equal to ________

If a function $$f$$ satisfies $$f(m + n) = f(m) + f(n)$$ for all $$m, n \in \mathbb{N}$$ and $$f(1) = 1$$, then the largest natural number $$\lambda$$ such that $$\sum_{k=1}^{2022} f(\lambda + k) \leq (2022)^2$$ is equal to _________

Let $$f : (0, \pi) \rightarrow \mathbb{R}$$ be a function given by
$$f(x) = \begin{cases} \left(\frac{8}{7}\right)^{\frac{\tan 8x}{\tan 7x}}, & 0 < x < \frac{\pi}{2} \\ a - 8, & x = \frac{\pi}{2} \\ (1 + |\cot x|)^{\frac{b}{|\tan x|}}, & \frac{\pi}{2} < x < \pi \end{cases}$$
where $$a, b \in \mathbb{Z}$$. If $$f$$ is continuous at $$x = \frac{\pi}{2}$$, then $$a^2 + b^2$$ is equal to ________

Let the set of all positive values of $$\lambda$$, for which the point of local minimum of the function $$(1 + x(\lambda^2 - x^2))$$ satisfies $$\frac{x^2 + x + 2}{x^2 + 5x + 6} < 0$$, be $$(\alpha, \beta)$$. Then $$\alpha^2 + \beta^2$$ is equal to _________

Let a, b and c denote the outcome of three independent rolls of a fair tetrahedral die, whose four faces are marked $$1, 2, 3, 4$$. If the probability that $$ax^2 + bx + c = 0$$ has all real roots is $$\frac{m}{n}$$, $$\gcd(m, n) = 1$$, then $$m + n$$ is equal to ________