NTA JEE Mains 29th Jan 2024 Shift 2

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