NTA JEE Mains 29th Jan 2024 Shift 2

Let $$r$$ and $$\theta$$ respectively be the modulus and amplitude of the complex number $$z = 2 - i\left(2\tan\frac{5\pi}{8}\right)$$, then $$(r, \theta)$$ is equal to

Number of ways of arranging $$8$$ identical books into $$4$$ identical shelves where any number of shelves may remain empty is equal to

If $$\log_e a, \log_e b, \log_e c$$ are in an A.P. and $$\log_e a - \log_e 2b, \log_e 2b - \log_e 3c, \log_e 3c - \log_e a$$ are also in an A.P., then $$a : b : c$$ is equal to

If each term of a geometric progression $$a_1, a_2, a_3, \ldots$$ with $$a_1 = \frac{1}{8}$$ and $$a_2 \neq a_1$$, is the arithmetic mean of the next two terms and $$S_n = a_1 + a_2 + \ldots + a_n$$, then $$S_{20} - S_{18}$$ is equal to

The sum of the solutions $$x \in R$$ of the equation $$\frac{3\cos 2x + \cos^3 2x}{\cos^6 x - \sin^6 x} = x^3 - x^2 + 6$$ is

Let $$A$$ be the point of intersection of the lines $$3x + 2y = 14, 5x - y = 6$$ and $$B$$ be the point of intersection of the lines $$4x + 3y = 8, 6x + y = 5$$. The distance of the point $$P(5, -2)$$ from the line $$AB$$ is

The distance of the point $$(2, 3)$$ from the line $$2x - 3y + 28 = 0$$, measured parallel to the line $$\sqrt{3}x - y + 1 = 0$$, is equal to

If the mean and variance of five observations are $$\frac{24}{5}$$ and $$\frac{194}{25}$$ respectively and the mean of first four observations is $$\frac{7}{2}$$, then the variance of the first four observations is equal to

If $$R$$ is the smallest equivalence relation on the set $$\{1, 2, 3, 4\}$$ such that $$\{(1, 2), (1, 3)\} \subset R$$, then the number of elements in $$R$$ is ______.

Let $$A = \begin{bmatrix} 2 & 1 & 2 \\ 6 & 2 & 11 \\ 3 & 3 & 2 \end{bmatrix}$$ and $$P = \begin{bmatrix} 1 & 2 & 0 \\ 5 & 0 & 2 \\ 7 & 1 & 5 \end{bmatrix}$$. The sum of the prime factors of $$|P^{-1}AP - 2I|$$ is equal to