Q74 / 180 · COMEDK 2013 · Mathematics

If $x=A \cos 4 t+B \sin 4 t$, then $\frac{d^{2} x}{d t^{2}}$ is equal to

  • Correct answer
    A
    $-16 x$
  • B
    $16 x$
  • C
    $x$
  • D
    $-x$

Solution

We have, $$ \begin{aligned} x &=A \cos 4 t+B \sin 4 t \\ \Rightarrow \frac{d x}{d t} &=A(-\sin 4 t \cdot 4)+B(\cos 4 t \cdot 4) \\ &=-4 A \sin 4 t+4 B \cos 4 t \end{aligned} $$ Again, $$ \begin{aligned} \frac{d^{2} x}{d t^{2}} &=-4 A(\cos 4 t \cdot 4)+4 B(-\sin 4 t \cdot 4) \\ &=-16 A \cos 4 t-16 B \sin 4 t \\ &=-16[A \cos 4 t+B \sin 4 t]=-16 x \end{aligned} $$

Hence, the correct answer is option: 'A'.

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