NTA JEE Main 29th June 2022 Shift 2

Let $$\alpha$$ be a root of the equation $$1 + x^2 + x^4 = 0$$. Then the value of $$\alpha^{1011} + \alpha^{2022} - \alpha^{3033}$$ is equal to:

Let $$(z)$$ represent the principal argument of the complex number $$z$$. The, $$|z| = 3$$ and $$\arg(z-1) - \arg(z+1) = \frac{\pi}{4}$$ intersect:

The sum of the infinite series $$1 + \frac{5}{6} + \frac{12}{6^2} + \frac{22}{6^3} + \frac{35}{6^4} + \frac{51}{6^5} + \frac{70}{6^6} + \ldots$$ is equal to:

Let $$n \geq 5$$ be an integer. If $$9^n - 8n - 1 = 64\alpha$$ and $$6^n - 5n - 1 = 25\beta$$, then $$\alpha - \beta$$ is equal to:

The distance of the origin from the centroid of the triangle whose two sides have the equations $$x - 2y + 1 = 0$$ and $$2x - y - 1 = 0$$ and whose orthocenter is $$\left(\frac{7}{3}, \frac{7}{3}\right)$$ is:

Let a triangle $$ABC$$ be inscribed in the circle $$x^2 - \sqrt{2}(x+y) + y^2 = 0$$ such that $$\angle BAC = \frac{\pi}{2}$$. If the length of side $$AB$$ is $$\sqrt{2}$$, then the area of the $$\triangle ABC$$ is equal to:

Let $$P : y^2 = 4ax$$, $$a > 0$$ be a parabola with focus $$S$$. Let the tangents to the parabola $$P$$ make an angle of $$\frac{\pi}{4}$$ with the line $$y = 3x + 5$$ touch the parabola $$P$$ at $$A$$ and $$B$$. Then the value of $$a$$ for which $$A$$, $$B$$ and $$S$$ are collinear is:

The value of $$\lim_{x \to 1} \frac{(x^2-1)\sin^2(\pi x)}{x^4-2x^3+2x-1}$$ is equal to:

Negation of the Boolean statement $$(p \vee q) \Rightarrow ((\sim r) \vee p)$$ is equivalent to:

The number of values of $$a \in N$$ such that the variance of 3, 7, 12, $$a$$, 43 $$- a$$ is a natural number is: