3 questions from 2026
Consider the function \( f : \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \rightarrow (-\infty, \infty) \) defined by
\( f(x) = (|x| + |x - 1|)\sin x + [ x\sin x ]\), where \( [x\sin x ]\) is the greatest integer less than or equal to \( x\sin x \). Let \( \alpha \) be the total number of points in the interval \( \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \) at which \( f \) is NOT continuous and let \( \beta \) be the total number of points in the interval \( \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \) at which \( f \) is NOT differentiable. Then the value of \( \alpha + \beta \) is __________.
For a real number \( \alpha \), let \( [\alpha] \) denote the greatest integer less than or equal to \( \alpha \). For a finite set \( S \), let \( |S| \) denote the number of elements in the set \( S \).
Consider the functions \( f:(-3,3)\rightarrow(-\infty,\infty) \) and \( g:(-3,3)\rightarrow(-\infty,\infty) \) defined by \( f(x)=[x^{3}] \log_{e}\!\big(1+\sin^{2}(\pi(x-[x]))\big) \) and \( g(x)=x^{3}\sin^{2}\!\big(\pi\log_{e}(1+x-[x])\big) \).
Let \( A=\{x\in(-3,3): f \text{ is discontinuous at } x\} \) and \( B=\{x\in(-3,3): g \text{ is discontinuous at } x\} \). Then the value of \( |A|+2|B|-|A\cap B| \) is ____.
Let \( \mathbb{R} \) denote the set of all real numbers. Let \( f : \mathbb{R} \to \mathbb{R} \) be an arbitrary function and let \( g : \mathbb{R} \to \mathbb{R} \) be the function defined by \( g(x) = x f(x) \), for all \( x \in \mathbb{R} \).
Then which of the following statements is (are) TRUE?
