NTA JEE Main 28th July 2022 Shift 2

Let $$S = \{x \in [-6, 3] - \{-2, 2\} : \frac{|x+3|-1}{|x|-2} \geq 0\}$$ and $$T = \{x \in \mathbb{Z} : x^2 - 7|x| + 9 \leq 0\}$$. Then the number of elements in $$S \cap T$$ is

Let $$\alpha, \beta$$ be the roots of the equation $$x^2 - \sqrt{2}x + \sqrt{6} = 0$$ and $$\frac{1}{\alpha^2+1}, \frac{1}{\beta^2+1}$$ be the roots of the equation $$x^2 + ax + b = 0$$. Then the roots of the equation $$x^2 - (a+b-2)x + (a+b+2) = 0$$ are:

Let the tangents at two points A and B on the circle $$x^2 + y^2 - 4x + 3 = 0$$ meet at origin $$O(0,0)$$. Then the area of the triangle OAB is

Let the hyperbola $$H: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$ pass through the point $$(2\sqrt{2}, -2\sqrt{2})$$. A parabola is drawn whose focus is same as the focus of H with positive abscissa and the directrix of the parabola passes through the other focus of H. If the length of the latus rectum of the parabola is $$e$$ times the length of the latus rectum of H, where $$e$$ is the eccentricity of H, then which of the following points lies on the parabola?

Let
$$p$$: Ramesh listens to music.
$$q$$: Ramesh is out of his village
$$r$$: It is Sunday
$$s$$: It is Saturday
Then the statement "Ramesh listens to music only if he is in his village and it is Sunday or Saturday" can be expressed as

A horizontal park is in the shape of a triangle OAB with $$AB = 16$$. A vertical lamp post OP is erected at the point O such that $$\angle PAO = \angle PBO = 15^\circ$$ and $$\angle PCO = 45^\circ$$, where C is the midpoint of AB. Then $$(OP)^2$$ is equal to

Let A and B be any two $$3 \times 3$$ symmetric and skew symmetric matrices respectively. Then which of the following is NOT true?

Let $$f(x) = ax^2 + bx + c$$ be such that $$f(1) = 3, f(-2) = \lambda$$ and $$f(3) = 4$$. If $$f(0) + f(1) + f(-2) + f(3) = 14$$, then $$\lambda$$ is equal to

The function $$f: \mathbb{R} \to \mathbb{R}$$ defined by $$f(x) = \lim_{n \to \infty} \frac{\cos(2\pi x) - x^{2n}\sin(x-1)}{1 + x^{2n+1} - x^{2n}}$$ is continuous for all $$x$$ in

Let $$x(t) = 2\sqrt{2}\cos t\sqrt{\sin 2t}$$ and $$y(t) = 2\sqrt{2}\sin t\sqrt{\sin 2t}$$, $$t \in (0, \frac{\pi}{2})$$. Then $$\frac{1 + \left(\frac{dy}{dx}\right)^2}{\frac{d^2y}{dx^2}}$$ at $$t = \frac{\pi}{4}$$ is equal to