NTA JEE Main 18th March 2021 Shift 2

Let a complex number be $$w = 1 - \sqrt{3}i$$. Let another complex number $$z$$ be such that $$|zw| = 1$$ and $$\arg(z) - \arg(w) = \frac{\pi}{2}$$. Then the area of the triangle (in sq. units) with vertices origin, $$z$$ and $$w$$ is equal to

Let $$S_1$$ be the sum of first $$2n$$ terms of an arithmetic progression. Let $$S_2$$ be the sum of first $$4n$$ terms of the same arithmetic progression. If $$(S_2 - S_1)$$ is 1000, then the sum of the first $$6n$$ terms of the arithmetic progression is equal to:

If $$15 \sin^4 \alpha + 10 \cos^4 \alpha = 6$$, for some $$\alpha \in R$$, then the value of $$27 \sec^6 \alpha + 8\operatorname{cosec}^6 \alpha$$ is equal to :

Let the centroid of an equilateral triangle $$ABC$$ be at the origin. Let one of the sides of the equilateral triangle be along the straight line $$x + y = 3$$. If $$R$$ and $$r$$ be the radius of circumcircle and incircle respectively of $$\triangle ABC$$, then $$(R + r)$$ is equal to :

Let $$S_1 : x^2 + y^2 = 9$$ and $$S_2 : (x-2)^2 + y^2 = 1$$. Then the locus of center of a variable circle $$S$$ which touches $$S_1$$ internally and $$S_2$$ externally always passes through the points :

Let a tangent be drawn to the ellipse $$\frac{x^2}{27} + y^2 = 1$$ at $$(3\sqrt{3}\cos\theta, \sin\theta)$$ where $$\theta \in \left(0, \frac{\pi}{2}\right)$$. Then the value of $$\theta$$ such that the sum of intercepts on axes made by this tangent is minimum is equal to :

Consider a hyperbola $$H : x^2 - 2y^2 = 4$$. Let the tangent at a point $$P(4, \sqrt{6})$$ meet the x-axis at $$Q$$ and latus rectum at $$R(x_1, y_1)$$, $$x_1 > 0$$. If $$F$$ is a focus of $$H$$ which is nearer to the point $$P$$, then the area of $$\triangle QFR$$ (in sq. units) is equal to

If $$P$$ and $$Q$$ are two statements, then which of the following compound statement is a tautology?

Let in a series of $$2n$$ observations, half of them are equal to $$a$$ and remaining half are equal to $$-a$$. Also by adding a constant $$b$$ in each of these observations, the mean and standard deviation of new set become 5 and 20, respectively. Then the value of $$a^2 + b^2$$ is equal to :

A pole stands vertically inside a triangular park $$ABC$$. Let the angle of elevation of the top of the pole from each corner of the park be $$\frac{\pi}{3}$$. If the radius of the circumcircle of $$\triangle ABC$$ is 2, then the height of the pole is equal to :