NTA JEE Main 26th June 2022 Shift 2

If $$A = \sum_{n=1}^{\infty} \frac{1}{(3+(- 1)^n)^n}$$ and $$B = \sum_{n=1}^{\infty} \frac{(-1)^n}{(3+(-1)^n)^n}$$, then $$\frac{A}{B}$$ is equal to

$$16\sin(20°)\sin(40°)\sin(80°)$$ is equal to

If $$m$$ is the slope of a common tangent to the curves $$\frac{x^2}{16} + \frac{y^2}{9} = 1$$ and $$x^2 + y^2 = 12$$, then $$12m^2$$ is equal to

The locus of the mid-point of the line segment joining the point $$(4, 3)$$ and the points on the ellipse $$x^2 + 2y^2 = 4$$ is an ellipse with eccentricity

The normal to the hyperbola $$\frac{x^2}{a^2} - \frac{y^2}{9} = 1$$ at the point $$(8, 3\sqrt{3})$$ on it passes through the point

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

Let $$r \in (P, q, \sim p, \sim q)$$ be such that the logical statement $$r \vee (\sim p) \Rightarrow (p \wedge q) \vee r$$ is a tautology. Then $$r$$ is equal to

Let the mean of 50 observations is 15 and the standard deviation is 2. However, one observation was wrongly recorded. The sum of the correct and incorrect observations is 70. If the mean of the correct set of observations is 16, then the variance of the correct set is equal to

If the system of equations $$\alpha x + y + z = 5, x + 2y + 3z = 4, x + 3y + 5z = \beta$$. Has infinitely many solutions, then the ordered pair $$(\alpha, \beta)$$ is equal to

If the inverse trigonometric functions take principal values, then
$$\cos^{-1}\left(\frac{3}{10}\cos\left(\tan^{-1}\left(\frac{4}{3}\right)\right) + \frac{2}{5}\sin\left(\tan^{-1}\left(\frac{4}{3}\right)\right)\right)$$ is equal to