NTA JEE Main 25th July 2022 Shift 1

If $$\alpha, \beta, \gamma, \delta$$ are the roots of the equation $$x^4 + x^3 + x^2 + x + 1 = 0$$, then $$\alpha^{2021} + \beta^{2021} + \gamma^{2021} + \delta^{2021}$$ is equal to

For $$n \in \mathbb{N}$$, let $$S_n = \{z \in \mathbb{C} : |z - 3 + 2i| = \dfrac{n}{4}\}$$ and $$T_n = \{z \in \mathbb{C} : |z - 2 + 3i| = \dfrac{1}{n}\}$$. Then the number of elements in the set $$\{n \in \mathbb{N} : S_n \cap T_n = \phi\}$$ is

The number of solutions of $$\cos x = |\sin x|$$, such that $$-4\pi \le x \le 4\pi$$ is

A line, with the slope greater than one, passes through the point $$A(4, 3)$$ and intersects the line $$x - y - 2 = 0$$ at the point $$B$$. If the length of the line segment $$AB$$ is $$\dfrac{\sqrt{29}}{3}$$, then $$B$$ also lies on the line

Let the locus of the centre $$(\alpha, \beta)$$, $$\beta > 0$$, of the circle which touches the circle $$x^2 + (y - 1)^2 = 1$$ externally and also touches the $$x$$-axis be $$L$$. Then the area bounded by $$L$$ and the line $$y = 4$$ is

If $$\lim_{n \to \infty} \left(\sqrt{n^2 - n - 1} + n\alpha + \beta\right) = 0$$ then $$8(\alpha + \beta)$$ is equal to

Which of the following statements is a tautology?

A tower $$PQ$$ stands on a horizontal ground with base $$Q$$ on the ground. The point $$R$$ divides the tower in two parts such that $$QR = 15 \text{ m}$$. If from a point $$A$$ on the ground the angle of elevation of $$R$$ is $$60°$$ and the part $$PR$$ of the tower subtends an angle of $$15°$$ at $$A$$, then the height of the tower is

The number of $$\theta \in [0, 4\pi]$$ for which the system of linear equations
$$3(\sin 3\theta) x - y + z = 2$$
$$3(\cos 2\theta) x + 4y + 3z = 3$$
$$6x + 7y + 7z = 9$$
has no solution is

The total number of functions, $$f: \{1, 2, 3, 4\} \to \{1, 2, 3, 4, 5, 6\}$$ such that $$f(1) + f(2) = f(3)$$, is equal to