First slide

Functions (XII)

Question

If the fundamental period of function $f (x) = \sin x + \cos (\sqrt{4 - a^{2}}) x is 4 π,$ then the value of a is I are

Moderate

A

$\frac{\sqrt{15}}{2}$
B

$- \frac{\sqrt{15}}{2}$
C

$\frac{\sqrt{7}}{2}$
D

$- \frac{\sqrt{7}}{2}$

Solution

$\begin{array}{l} Period of \sin x is 2 π and period of \cos (\sqrt{4 - a^{2}} x) is \frac{2 π}{\sqrt{4 - a^{2}}} . \\ \Rightarrow L C M (2 π, \frac{2 π}{\sqrt{4 - a^{2}}}) = 4 π (given) \\ i . e ., \sqrt{4 - a^{2}} = \frac{p}{2} where p = 1, 3 \\ Hence a^{2} = \frac{15}{4}, \frac{7}{4}, a = \pm \frac{\sqrt{15}}{2}, \pm \frac{\sqrt{7}}{2} \end{array}$

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