JEE (Advanced) 2020 Paper-2

Let the functions $$f:(-1, 1) \rightarrow R$$ and $$g:(-1, 1) \rightarrow (-1, 1)$$ be defined by
$$f(x) = \mid 2x - 1 \mid + \mid 2x + 1 \mid$$ and $$g(x) = x - [x]$$,
where $$[x]$$ denotes the greatest integer less than or equal to 𝑥. Let $$f \circ g :(−1, 1) \rightarrow R$$ be the composite function defined by $$(f \circ g)(x) = f(g(x))$$. Suppose c is the number of points in the interval (-1, 1) at which $$f \circ g$$ is NOT continuous, and suppose d is the number of points in the interval(-1, 1) at which $$f \circ g$$ is NOT differenciable. Then the value of c + d is ____________

The value of the limit
$$\lim_{x \rightarrow {\frac{\pi}{2}}}\frac{4\sqrt{2}(\sin 3x + \sin x)}{\left(2 \sin 2x \sin \frac{3x}{2} + \cos \frac{5x}{2} \right) - \left(\sqrt{2} + \sqrt{2} \cos 2x + \cos \frac{3x}{2}\right)}$$
is_________.

Let 𝑏 be a nonzero real number. Suppose $$f:R \rightarrow R$$ is a differentiable function such that $$f(0) = 1$$. If the derivative $$f′$$ of $$f$$ satisfies the equation
$$f′(x) = \frac{f(x)}{b^2 + x^2}$$
for all $$x \in R$$, then which of the following statements is/are TRUE?

Let 𝑎 and 𝑏 be positive real numbers such that 𝑎>1 and 𝑏<𝑎. Let 𝑃 be a point in the first quadrant that lies on the hyperbola $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$. Suppose the tangent to the hyperbola at 𝑃 passes through the point (1,0), and suppose the normal to the hyperbola at 𝑃 cuts off equal intercepts on the coordinate axes. Let $$\triangle$$ denote the area of the triangle formed by the tangent at 𝑃, the normal at 𝑃 and the 𝑥-axis. If 𝑒 denotes the eccentricity of the hyperbola, then which of the following statements is/are TRUE?

Let $$f:R \rightarrow R$$ and $$g:R \rightarrow R$$ be functions satisfying
$$f(x + y)=f(x)+f(y)+f(x)f(y)$$ and $$f(x) = xg(x)$$
for all $$𝑥, 𝑦 \in R$$. If $$\lim_{x \rightarrow 0} g(x) = 1$$, then which of the following statements is/are TRUE?

Let $$\alpha, \beta, \gamma, \delta$$ be real numbers such that $$\alpha^2 + \beta^2 + \gamma^2 \neq 0$$ and $$\alpha + \beta = 1$$ Suppose the point (3,2,−1) is the mirror image of the point (1,0,−1) with respect to the plane $$\alpha x + \beta y + \gamma z = \delta$$. Then which of the following statements is/are TRUE?

Let 𝑎 and 𝑏 be positive real numbers. Suppose $$\overrightarrow{PQ} = a\hat{i} + b\hat{j}$$ and $$\overrightarrow{PS} = a\hat{i} - b\hat{j} $$ are adjacent sides of a parallelogram 𝑃𝑄𝑅𝑆. Let $$\overrightarrow{u}$$ and \overrightarrow{v} be the projection vectors of $$\overrightarrow{u} = \hat{i} + \hat{j}$$ along $$\overrightarrow{PQ}$$ and $$\overrightarrow{PS}$$, respectively. If $$\mid \overrightarrow{u}\mid + \mid \overrightarrow{v} \mid = \mid \overrightarrow{w} \mid$$ and if the area of the parallelogram 𝑃𝑄𝑅𝑆 is 8, then which of the following statements is/are TRUE?

For nonnegative integers s and r, let
$$\left(\begin{array}{c}s\\ r\end{array}\right) = \begin{cases}\frac{s!}{r!(s - r)!} & if r \leq s\\0 & if r > s\end{cases}$$
For positive integers 𝑚 and 𝑛, let
$$g(m, n) = \sum_{p=0}^{m+n}\frac{f(m, n, p)}{\left(\begin{array}{c}n+p\\ p\end{array}\right)}$$
where for any nonnegative integer 𝑝,
$$f(m, n, p) = \sum_{i=0}^{p}\left(\begin{array}{c}m\\ i\end{array}\right)\left(\begin{array}{c}n+i\\ p\end{array}\right)\left(\begin{array}{c}p+n\\ p - i\end{array}\right)$$
Then which of the following statements is/are TRUE?

An engineer is required to visit a factory for exactly four days during the first 15 days of every month and it is mandatory that no two visits take place on consecutive days. Then the number of all possible ways in which such visits to the factory can be made by the engineer during 1-15 June 2021 is _____

In a hotel, four rooms are available. Six persons are to be accommodated in these four rooms in such a way that each of these rooms contains at least one person and at most two persons. Then the number of all possible ways in which this can be done is _____