Question

Let f(x) = cos(p)x where p =[a] (where [.] is the G.I.F).If the period of f(x) is 2, then a can lie in
(4,51
[4.5)
(4.5)
(14.15)​

Answer 1

Period of a function $f(x)$ is the least possible positive real number $T$ such that $f(T+x)=f(x).$

The period of the function $f(x)=\cos(mx)$ is,

$\longrightarrow T=\dfrac{2\pi}{|m|}$

According to the question,

$\longrightarrow f(x)=\cos((p)x)$

where $p=[a],$ i.e., greatest integer function of $a.$

$\longrightarrow f(x)=\cos([a]x)$

Here, $m=[a].$

Given that the period of $f(x)$ is $\dfrac{\pi}{2}.$.

$\longrightarrow T=\dfrac{\pi}{2}$

$\longrightarrow \dfrac{2\pi}{|m|}=\dfrac{\pi}{2}$

$\longrightarrow \dfrac{2}{|\,[a]\,|}=\dfrac{1}{2}$

$\longrightarrow |\,[a]\,|=4$

$\longrightarrow[a]=\pm4$

But $[a]=\alpha\ \iff\ a\in[\alpha,\ \alpha+1)$

• If $[a]=-4,\ \ a\in[-4,\ -3).$

• If $[a]=4,\ \ a\in[4,\ 5).$

Therefore,

$\Longrightarrow\underline{\underline{a\in[-4,\ -3)\cup[4,\ 5)}}$