NTA JEE Main 25th July 2021 Shift 1

Let $$S_n$$ be the sum of the first $$n$$ terms of an arithmetic progression. If $$S_{3n} = 3S_{2n}$$, then the value of $$\frac{S_{4n}}{S_{2n}}$$ is:

If $$b$$ is very small as compared to the value of $$a$$, so that the cube and other higher powers of $$\frac{b}{a}$$ can be neglected in the identity
$$\frac{1}{a-b} + \frac{1}{a-2b} + \frac{1}{a-3b} + \ldots + \frac{1}{a-nb} = \alpha n + \beta n^2 + \gamma n^3$$
then the value of $$\gamma$$ is:

The sum of all values of $$x$$ in $$[0, 2\pi]$$, for which $$\sin x + \sin 2x + \sin 3x + \sin 4x = 0$$, is equal to:

Let a parabola $$P$$ be such that its vertex and focus lie on the positive $$x$$-axis at a distance 2 and 4 units from the origin, respectively. If tangents are drawn from $$O(0, 0)$$ to the parabola $$P$$ which meet $$P$$ at $$S$$ and $$R$$, then the area (in sq. units) of $$\triangle SOR$$ is equal to:

Let an ellipse $$E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$, $$a^2 > b^2$$, passes through $$\left(\sqrt{\frac{3}{2}}, 1\right)$$ and has eccentricity $$\frac{1}{\sqrt{3}}$$. If a circle, centered at focus $$F(\alpha, 0)$$, $$\alpha > 0$$, of $$E$$ and radius $$\frac{2}{\sqrt{3}}$$, intersects $$E$$ at two points $$P$$ and $$Q$$, then $$PQ^2$$ is equal to:

The locus of the centroid of the triangle formed by any point P on the hyperbola $$16x^2 - 9y^2 + 32x + 36y - 164 = 0$$ and its foci is

The Boolean expression $$(p \Rightarrow q) \wedge (q \Rightarrow \sim p)$$ is equivalent to:

A spherical gas balloon of radius 16 meter subtends an angle 60$$^\circ$$ at the eye of the observer A while the angle of elevation of its center from the eye of A is 75$$^\circ$$. Then the height (in meter) of the top most point of the balloon from the level of the observer's eye is:

The values of $$a$$ and $$b$$, for which the system of equations
$$2x + 3y + 6z = 8$$
$$x + 2y + az = 5$$
$$3x + 5y + 9z = b$$
has no solution, are:

Let $$g : N \to N$$ be defined as
$$g(3n+1) = 3n+2$$
$$g(3n+2) = 3n+3$$
$$g(3n+3) = 3n+1$$, for all $$n \ge 0$$
Then which of the following statements is true?