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Determinants — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSDeterminants1 Mark
Question
Let $f(x)=\left|\begin{array}{cc}x^2 & \sin x \\ p & -1\end{array}\right|,$ where $p$ is a constant. The value of $p$ for which $f^{\prime}(0)=1$ is
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Answer

Given, $f(x)=\left|\begin{array}{cc}x^2 & \sin x \\ p & -1\end{array}\right|=-x^2-p \sin x$
$\Rightarrow f^{\prime}(x)=-2 x-p \cos x$
$\text { We have, } f^{\prime}(0)=1 $
$\Rightarrow-2(0)-p \cos (0)=1 $
$\Rightarrow-p=1 $
$\Rightarrow p=-1$

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