Periodic Functions MCQ - Practice Questions & Answers

Solve by difficulty

EASY

The period of $4 x+1-[4 x+1]_{\text {is }}([x]$ is the Greatest Integer Function)

1/4

Not periodic

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The period of the function $f\left ( x \right )=Sin^{4}x+Cos^{4}x\, \, \, is$

$\pi$

$\frac{\pi}{2}$

$2\pi$

None of these

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Concepts Covered - 1

Periodic Functions

A function f(x) is called a periodic function, if there exists a +ve real number T such that f(x+T) = f(x) ∀ x? The domain of f(x).

Here, T is called the period of f(x), where T is the least +ve value.

Graphically: if the graph repeats at a fixed interval, the function is said to be periodic and its period is the width of that interval.

Graph of sin(x) is repeated at an interval of 2π

Some standard results

Functions	Period
sin(x), cos(x), sec(x), cosec(x)	2π
sin²(x), cos²(x)	π
tan(x), cot(x)	π
\|sin x\|, \|cos x\|, \|tan x\|, \|cot x\|, \|cosec x\|, \|sec x\|	π
{x}	1
Algebric function, eg. x², x³+ 6	Not Periodic

Properties of the periodic function

i) if f(x) is periodic with period T, then

cf(x) is periodic with period T
f(x+c) is periodic with period T
f(x) ± c is periodic with period T, where c is any constant.

ii) if $\mathrm{f}(\mathrm{x})$ is periodic with period T , then $\mathrm{kf}(\mathrm{cx}+\mathrm{d})$ has period $\frac{T}{|c|}$ i.e. period is only affected by the coefficient of x ,
iii) if $f_1(x), f_2(x)$ are periodic functions with periods $T_1, T_2$ respectively, then $h(x)=f_1(x)+f_2(x)$ has period
a). LCM of $\left\{T_1, T_2\right\}$, if $\mathrm{h}(\mathrm{x})$ is not an even function.
or
b) 0.5 LCM of $\left\{T_1, T_{2\}}\right.$, if $f_1(x)$ and $f_2(x)$ are complementary pairwise comparable functions.