NTA JEE Main 6th September 2020 Shift 2

For a suitably chosen real constant $$a$$, let a function, $$f : \mathbb{R} - \{-a\} \to \mathbb{R}$$ be defined by $$f(x) = \frac{a-x}{a+x}$$. Further suppose that for any real number $$x \neq -a$$, and $$f(x) \neq -a$$, $$(f \circ f)(x) = x$$. Then $$f\left(-\frac{1}{2}\right)$$ is equal to:

Let $$f : \mathbb{R} \to \mathbb{R}$$ be a function defined by $$f(x) = \max\{x, x^2\}$$. Let $$S$$ denote the set of all points in $$\mathbb{R}$$, where $$f$$ is not differentiable. Then:

The set of all real values of $$\lambda$$ for which the function $$f(x) = (1 - \cos^2 x) \cdot (\lambda + \sin x)$$, $$x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$$, has exactly one maxima and exactly one minima, is:

For all twice differentiable functions $$f : \mathbb{R} \to \mathbb{R}$$, with $$f(0) = f(1) = f'(0) = 0$$,

If the tangent to the curve $$y = f(x) = x\log_e x$$, $$(x > 0)$$ at a point $$(c, f(c))$$ is parallel to the line-segment joining the points $$(1, 0)$$ and $$(e, e)$$, then $$c$$ is equal to:

The integral $$\int_1^2 e^x \cdot x^x(2 + \log_e x)\,dx$$ equals:

The area (in sq. units) of the region enclosed by the curves $$y = x^2 - 1$$ and $$y = 1 - x^2$$ is equal to:

If $$y = \left(\frac{2}{\pi}x - 1\right) \operatorname{cosec} x$$ is the solution of the differential equation, $$\frac{dy}{dx} + p(x)y = -\frac{2}{\pi} \operatorname{cosec} x$$, $$0 < x < \frac{\pi}{2}$$, then the function $$p(x)$$ is equal to:

A plane P meets the coordinate axes at A, B and C respectively. The centroid of $$\triangle ABC$$ is given to be $$(1, 1, 2)$$. Then the equation of the line through this centroid and perpendicular to the plane P is:

The probabilities of three events A, B and C are given $$P(A) = 0.6$$, $$P(B) = 0.4$$ and $$P(C) = 0.5$$. If $$P(A \cup B) = 0.8$$, $$P(A \cap C) = 0.3$$, $$P(A \cap B \cap C) = 0.2$$, $$P(B \cap C) = \beta$$ and $$P(A \cup B \cup C) = \alpha$$, where $$0.85 \leq \alpha \leq 0.95$$, then $$\beta$$ lies in the interval: