NTA JEE Mains 29th Jan 2024 Shift 2 - Mathematics

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

Let $$x = \frac{m}{n}$$ ($$m, n$$ are co-prime natural numbers) be a solution of the equation $$\cos\left(2\sin^{-1}x\right) = \frac{1}{9}$$ and let $$\alpha, \beta (\alpha > \beta)$$ be the roots of the equation $$mx^2 - nx - m + n = 0$$. Then the point $$(\alpha, \beta)$$ lies on the line

Let $$y = \log_e\left(\frac{1 - x^2}{1 + x^2}\right), -1 < x < 1$$. Then at $$x = \frac{1}{2}$$, the value of $$225(y' - y'')$$ is equal to

The function $$f(x) = 2x + 3x^{\frac{2}{3}}, x \in R$$, has

The function $$f(x) = \frac{x}{x^2 - 6x - 16}, x \in \mathbb{R} - \{-2, 8\}$$

If $$\int \frac{\sin^{\frac{3}{2}}x + \cos^{\frac{3}{2}}x}{\sqrt{\sin^3 x \cos^3 x \sin(x - \theta)}} dx = A\sqrt{\cos\theta\tan x - \sin\theta} + B\sqrt{\cos\theta - \sin\theta\cot x} + C$$, where $$C$$ is the integration constant, then $$AB$$ is equal to

If $$\sin\left(\frac{y}{x}\right) = \log_e|x| + \frac{\alpha}{2}$$ is the solution of the differential equation $$x\cos\left(\frac{y}{x}\right)\frac{dy}{dx} = y\cos\left(\frac{y}{x}\right) + x$$ and $$y(1) = \frac{\pi}{3}$$, then $$\alpha^2$$ is equal to

Let $$\vec{OA} = \vec{a}, \vec{OB} = 12\vec{a} + 4\vec{b}$$ and $$\vec{OC} = \vec{b}$$, where $$O$$ is the origin. If $$S$$ is the parallelogram with adjacent sides $$OA$$ and $$OC$$, then $$\frac{\text{area of the quadrilateral } OABC}{\text{area of } S}$$ is equal to _____

Let a unit vector $$\hat{u} = x\hat{i} + y\hat{j} + z\hat{k}$$ make angles $$\frac{\pi}{2}, \frac{\pi}{3}$$ and $$\frac{2\pi}{3}$$ with the vectors $$\frac{1}{\sqrt{2}}\hat{i} + \frac{1}{\sqrt{2}}\hat{k}, \frac{1}{\sqrt{2}}\hat{j} + \frac{1}{\sqrt{2}}\hat{k}$$ and $$\frac{1}{\sqrt{2}}\hat{i} + \frac{1}{\sqrt{2}}\hat{j}$$ respectively. If $$\vec{v} = \frac{1}{\sqrt{2}}\hat{i} + \frac{1}{\sqrt{2}}\hat{j} + \frac{1}{\sqrt{2}}\hat{k}$$, then $$|\hat{u} - \vec{v}|^2$$ is equal to

Let $$P(3, 2, 3), Q(4, 6, 2)$$ and $$R(7, 3, 2)$$ be the vertices of $$\triangle PQR$$. Then, the angle $$\angle QPR$$ is

An integer is chosen at random from the integers $$1, 2, 3, \ldots, 50$$. The probability that the chosen integer is a multiple of at least one of $$4, 6$$ and $$7$$ is

Let the set $$C = \{(x, y) \mid x^2 - 2^y = 2023, x, y \in \mathbb{N}\}$$. Then $$\sum_{(x,y) \in C}(x + y)$$ is equal to ______.

Let $$\alpha, \beta$$ be the roots of the equation $$x^2 - \sqrt{6}x + 3 = 0$$ such that $$\text{Im}(\alpha) > \text{Im}(\beta)$$. Let $$a, b$$ be integers not divisible by $$3$$ and $$n$$ be a natural number such that $$\frac{\alpha^{99}}{\beta} + \alpha^{98} = 3^n(a + ib), i = \sqrt{-1}$$. Then $$n + a + b$$ is equal to ______.

Remainder when $$64^{32^{32}}$$ is divided by $$9$$ is equal to ______.

Let $$P(\alpha, \beta)$$ be a point on the parabola $$y^2 = 4x$$. If $$P$$ also lies on the chord of the parabola $$x^2 = 8y$$ whose mid point is $$\left(1, \frac{5}{4}\right)$$, then $$(\alpha - 28)(\beta - 8)$$ is equal to ______.

Let the slope of the line $$45x + 5y + 3 = 0$$ be $$27r_1 + \frac{9r_2}{2}$$ for some $$r_1, r_2 \in R$$. Then $$\lim_{x \to 3}\left(\int_3^x \frac{8t^2}{\frac{3r_2 x}{2} - r_2 x^2 - r_1 x^3 - 3x} dt\right)$$ is equal to ______.

Let for any three distinct consecutive terms $$a, b, c$$ of an A.P, the lines $$ax + by + c = 0$$ be concurrent at the point $$P$$ and $$Q(\alpha, \beta)$$ be a point such that the system of equations $$x + y + z = 6, 2x + 5y + \alpha z = \beta$$ and $$x + 2y + 3z = 4$$, has infinitely many solutions. Then $$(PQ)^2$$ is equal to ______.

Let $$f(x) = \sqrt{\lim_{r \to x}\left\{\frac{2r^2[(f(r))^2 - f(x)f(r)]}{r^2 - x^2} - r^3 e^{\frac{f(r)}{r}}\right\}}$$ be differentiable in $$(-\infty, 0) \cup (0, \infty)$$ and $$f(1) = 1$$. Then the value of $$ae$$, such that $$f(a) = 0$$, is equal to ______.

If $$\int_{\frac{\pi}{6}}^{\frac{\pi}{3}}\sqrt{1 - \sin 2x} \, dx = \alpha + \beta\sqrt{2} + \gamma\sqrt{3}$$, where $$\alpha, \beta$$ and $$\gamma$$ are rational numbers, then $$3\alpha + 4\beta - \gamma$$ is equal to ______.

Let the area of the region $$\{(x, y) : 0 \leq x \leq 3, 0 \leq y \leq \min\{x^2 + 2, 2x + 2\}\}$$ be $$A$$. Then $$12A$$ is equal to ______.

Let $$O$$ be the origin, and $$M$$ and $$N$$ be the points on the lines $$\frac{x-5}{4} = \frac{y-4}{1} = \frac{z-5}{3}$$ and $$\frac{x+8}{12} = \frac{y+2}{5} = \frac{z+11}{9}$$ respectively such that $$MN$$ is the shortest distance between the given lines. Then $$\vec{OM} \cdot \vec{ON}$$ is equal to ______.