Q82 / 180 · KCET 2021 · Mathematics

If $a$ and $b$ are fixed non-zero constants, then the derivative of $\frac{a}{x^{4}}-\frac{b}{x^{2}}+\cos x$ is $m a+n b-p$, where

  • A
    $m=4 x^{3}, n=\frac{-2}{x^{3}}$ and $p=\sin x$
  • Correct answer
    B
    $m=\frac{-4}{x^{5}}, n=\frac{2}{x^{3}}$ and $p=\sin x$
  • C
    $m=\frac{-4}{x^{5}}, n=\frac{-2}{x^{3}}$ and $p=\sin x$
  • D
    $m=4 x^{3}, n=\frac{2}{x^{3}}$ and $p=-\sin x$

Solution

Given, function is $\frac{a}{x^{4}}-\frac{b}{x^{2}}+\cos x$. Let $f(x)=\frac{a}{x^{4}}-\frac{b}{x^{2}}+\cos x$ Differentiating w.r.t. $x$, $\begin{gathered} \frac{d}{d x} f(x)=\frac{d}{d x}\left(\frac{a}{x^{4}}-\frac{b}{x^{2}}+\cos x\right) \\ =-\frac{4 a}{x^{5}}+\frac{2 b}{x^{3}}-\sin x \\ =m a+n b-P \\ m=-\frac{4}{x^{5}}, n=\frac{2}{x^{3}} \text { and } p=\sin x \end{gathered}$

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

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