Functions PYQs - Last 10 Years

Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be the function defined by
$$ f(x)= \begin{cases}x^2-4 x-5 & \text { if } x \geq 1, \\ 2 x & \text { if } x<1 .\end{cases} $$
Which one of the following statements is TRUE?

Let $f: Q \rightarrow Q$ be a function such that $f(x+y)=f(x)+f(y)$ for all $x, y \in \mathbf{Q}$, and $f(1)=$ 10. Which one of the following statements is Correct?

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ be two functions such that
$$ g \circ f(x)= \begin{cases}x^2, & \text { if } x \geq 0 \\ e^x-1, & \text { if } x<0\end{cases} $$
Then which of the followi...

Let $[x]$ denote the greatest integer less than or equal to $x$. The number of positive integer solutions of the equation

\(\left[\frac{x}{19}\right]=\left[\frac{x}{20}\right]\)
are:

For what value of $t$ does $f(x)=x^3+t x^2+(2-t) x-3$ have only real roots?