JEE (Advanced) 2020 Paper-1

Let $$f:[0, 2]\rightarrow R$$ be the function defined by

$$f(x) = (3- \sin(2\pi x)) \sin \left(\pi x - \frac{\pi}{4}\right) - \sin\left(3\pi x + \frac{\pi}{4}\right)$$

If $$\alpha, \beta \epsilon [0, 2]$$ are such that $$\left\{x \epsilon [0,2] : f(x) \geq 0\right\} = [\alpha, \beta]$$, then the value of $$\beta - \alpha$$ is _________

In a triangle 𝑃𝑄𝑅, let $$\overrightarrow{a} = \overrightarrow{QR}, \overrightarrow{b} = \overrightarrow{RP}$$ and $$\overrightarrow{c} = \overrightarrow{PQ}$$. If

$$|\overrightarrow{a}| = 3, |\overrightarrow{b}| = 4$$ and $$\frac{\overrightarrow{a}\cdot(\overrightarrow{c}-\overrightarrow{b})}{\overrightarrow{c}\cdot(\overrightarrow{a}-\overrightarrow{b})} = \frac{|\overrightarrow{a}|}{|\overrightarrow{a}| + |\overrightarrow{b}|}$$,

then the value of $$\left|\overrightarrow{a} \times \overrightarrow{b}\right|^{2}$$ is ___________

For a polynomial $$g(x)$$ with real coefficients, let mg denote the number of distinct real roots of
$$g(x)$$. Suppose 𝑆 is the set of polynomials with real coefficients defined by

$$S = \left\{(x^{2} − 1)^{2}(a_{0} + a_{1}x + a_{2}x^{2} + a_{3}x^{3}) ∶ a_{0}, a_{1}, a_{2}, a_{3} \epsilon R\right\}$$.

For a polynomial f, let $$f{'}$$ and $$f {'}{'}$$ denote its first and second order derivatives, respectively. Then the minimum possible value of $$(m_{f{'}} + m_{f{'}{'}})$$, where $$f \epsilon S$$, is _____

Let e denote the base of the natural logarithm. The value of the real number a for which the right
hand limit

$$\lim_{x \rightarrow 0^{+}}\frac{(1-x)^{\frac{1}{x}}- e^{-1}}{x^{a}}$$

is equal to a nonzero real number, is _____