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### Find area of mobility in Argand plane for variable z such that, $3 \le \left| z \right| \le 5$ & $\frac{\pi }{8} \le \arg z \le \frac{\pi }{4}$
The required area = shaded area in the figure = $\frac{1}{2}({5^2} - {3^2})\left( {\frac{\pi }{4} - \frac{\pi }{8}} \right)$
$= \frac{1}{2} \times 16 \times \frac{\pi }{8} = \pi$ square unit.
### $f(x)=x^6+2x^4+x^3+2x+3 $$\mathop {\lim }\limits_{x \to 1} \frac{{{x^n}f(1) - f(x)}}{{x - 1}} = 44$$n=?$
Let $f(x)=x^6+2x^4+x^3+2x+3,x \in R$. Then the natural number n for which $\mathop {\lim }\limits_{x \to 1} \frac{{{x^n}f(1) - f(x)}}{{x - 1}} = 44$ is _ _ _ _ . Solution Since the limit has $\left[ {\frac{0}{0}} \right]$ form, L.H. Rule is applicable. Thus, $\mathop {\lim }\limits_{x \to 1} n{x^{n - 1}}f(1) - f'(x) = 44$ $\therefore nf(1) - f'(1) = 44$ $\therefore n.9 - ({6.1^5} + {8.1^3} + {3.1^2} + 2.1) = 44$ $\Rightarrow 9n - 19 = 44$ $\Rightarrow n=7$