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\mathrm{\text { If } f(x)=\left|\begin{array}{ccc} x^n & \sin x & \cos x \\ n ! & \sin \frac{n \pi}{2} & \frac{\cos n \pi}{2} \\ a & a^2 & a^3 \end{array}\right| \text {, then } \frac{d^n f(x)}{d x^n} \text { at } x=2 \pi \text { is } }

Option: 1

0


Option: 2

1


Option: 3

a^3


Option: 4

a+a^2


Answers (1)

best_answer

\mathrm{\frac{d^n f(x)}{d x^n}=\left|\begin{array}{ccc}\frac{d^n x^n}{d x^n} & \frac{d^n \sin x}{d x^n} & \frac{d^n \cos x}{d x^n} \\ n ! & \sin \frac{n \pi}{2} & \cos \frac{n \pi}{2} \\ a & a^2 & a^3\end{array}\right| }
\mathrm{=\left|\begin{array}{ccc} n ! & \sin \left(x+\frac{n \pi}{2}\right) & \cos \left(x+\frac{n \pi}{2}\right) \\ n ! & \sin \frac{n \pi}{2} & \cos \frac{n \pi}{2} \\ a & a^2 & a^3 \end{array}\right| }
\mathrm{\therefore \quad \frac{d^n f(x)}{d x^n} \, \, at\, \, x=2 \pi } is zero since row 1 and row 2 are same.

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Shailly goel

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