NTA JEE Main 31st August 2021 Shift 1

The sum of 10 terms of the series $$\frac{3}{1^2 \times 2^2} + \frac{5}{2^2 \times 3^2} + \frac{7}{3^2 \times 4^2} + \ldots$$ is:

Three numbers are in an increasing geometric progression with common ratio $$r$$. If the middle number is doubled, then the new numbers are in an arithmetic progression with common difference $$d$$. If the fourth term of GP is $$3r^2$$, then $$r^2 - d$$ is equal to:

$$\text{cosec } 18°$$ is a root of the equation:

If $$p$$ and $$q$$ are the lengths of the perpendiculars from the origin on the lines, $$x\text{cosec}\alpha - y\sec\alpha = k\cot 2\alpha$$ and $$x\sin\alpha + y\cos\alpha = k\sin 2\alpha$$ respectively, then $$k^2$$ is equal to:

The length of the latus rectum of a parabola, whose vertex and focus are on the positive $$x$$-axis at a distance $$R$$ and $$S (> R)$$ respectively from the origin, is:

The line $$12x\cos\theta + 5y\sin\theta = 60$$ is tangent to which of the following curves?

$$\lim_{x \to 0} \frac{\sin^2(\pi\cos^4 x)}{x^4}$$ is equal to:

Let $$*, \square \in \{\wedge, \vee\}$$ be such that the Boolean expression $$(p * \sim q) \Rightarrow (p \square q)$$ is a tautology. Then:

A vertical pole fixed to the horizontal ground is divided in the ratio 3 : 7 by a mark on it with lower part shorter than the upper part. If the two parts subtend equal angles at a point on the ground 18 m away from the base of the pole, then the height of the pole (in meters) is:

Which of the following is not correct for relation $$R$$ on the set of real numbers?