Let $\mathbb{R}$ denote the set of all real numbers. For a real number $x$, let [ x ] denote the greatest integer less than or equal to $x$. Let $n$ denote a natural number.
Match each entry in List-I to the correct entry in List-II and choose the correct option.
| List–I | List–II |
|---|---|
| (P) The minimum value of $n$ for which the function $$ f(x)=\left[\frac{10 x^3-45 x^2+60 x+35}{n}\right] $$ is continuous on the interval $[1,2]$, is | (1) 8 |
| (Q) The minimum value of $n$ for which $g(x)=\left(2 n^2-13 n-15\right)\left(x^3+3 x\right)$, $x \in \mathbb{R}$, is an increasing function on $\mathbb{R}$, is | (2) 9 |
| (R) The smallest natural number $n$ which is greater than 5 , such that $x=3$ is a point of local minima of $$ h(x)=\left(x^2-9\right)^n\left(x^2+2 x+3\right) $$ is | (3) 5 |
| (S) Number of $x_0 \in \mathbb{R}$ such that
$$ l(x)=\sum\limits_{k=0}^4\left(\sin |x-k|+\cos \left|x-k+\frac{1}{2}\right|\right) $$ $x \in \mathbb{R}$, is NOT differentiable at $x_0$, is |
(4) 6 |
| (5) 10 |
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function defined by
$$ f(x)=\left\{\begin{array}{cc} x^2 \sin \left(\frac{\pi}{x^2}\right), & \text { if } x \neq 0, \\ 0, & \text { if } x=0 . \end{array}\right. $$
Then which of the following statements is TRUE?
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ be functions defined by
$$ f(x)=\left\{\begin{array}{ll} x|x| \sin \left(\frac{1}{x}\right), & x \neq 0, \\ 0, & x=0, \end{array} \quad \text { and } g(x)= \begin{cases}1-2 x, & 0 \leq x \leq \frac{1}{2}, \\ 0, & \text { otherwise } .\end{cases}\right. $$
Let $a, b, c, d \in \mathbb{R}$. Define the function $h: \mathbb{R} \rightarrow \mathbb{R}$ by
$$ h(x)=a f(x)+b\left(g(x)+g\left(\frac{1}{2}-x\right)\right)+c(x-g(x))+d g(x), x \in \mathbb{R} . $$
Match each entry in List-I to the correct entry in List-II.
| List-I | List-II |
|---|---|
| (P) If $a = 0$, $b = 1$, $c = 0$, and $d = 0$, then | (1) $h$ is one-one. |
| (Q) If $a = 1$, $b = 0$, $c = 0$, and $d = 0$, then | (2) $h$ is onto. |
| (R) If $a = 0$, $b = 0$, $c = 1$, and $d = 0$, then | (3) $h$ is differentiable on $\mathbb{R}$. |
| (S) If $a = 0$, $b = 0$, $c = 0$, and $d = 1$, then | (4) the range of $h$ is $[0, 1]$. |
| (5) the range of $h$ is $\{0, 1\}$. |
The correct option is