Sets, Relations & Functions
Signum Function — Range and Sum
nta_pyq_2026_jan
Grade 11
Question:
The sum of all the elements in the range of $f(x)=\text{Sgn}(\sin x)+\text{Sgn}(\cos x)+\text{Sgn}(\tan x)+\text{Sgn}(\cot x)$, $x\neq\dfrac{n\pi}{2}$, $n\in\mathbb{Z}$, where $\text{Sgn}(t)=\begin{cases}1,&t>0\\-1,&t<0\end{cases}$, is:
Step-by-Step Solution
Key Concept: Evaluate $f$ in each quadrant: $x\in(0,\pi/2)$: all four positive, $f=4$. $x\in(\pi/2,\pi)$: $\sin>0$, $\cos<0$, $\tan<0$, $\cot<0$, $f=-2$. $x\in(\pi,3\pi/2)$: $\sin<0$, $\cos<0$, $\tan>0$, $\cot>0$, $f=0$. $x\in(3\pi/2,2\pi)$: $\sin<0$, $\cos>0$, $\tan<0$, $\cot<0$, $f=-2$.
Range $=\{-2,0,4\}$. Sum $=2$.
Correct Answer: 1