Find area of mobility in Argand plane for variable z such that, <p></p>$3 \le \left| z \right| \le 5$ & <p></p>$\frac{\pi }{8} \le \arg z \le \frac{\pi }{4}$

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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.

If the sum of the coefficients in the expansion of

$(x+y)^n$ is 4096, then the greatest coefficient in the expansion is _ _ _ _ . Solution

$C_0 + C_1 + C_2 + C_3 + ......................... + C_n =4096$

$\therefore 2^n = 4096 =2^{12}$

$\Rightarrow n = 12$ Greatest coefficient =

${}^{12}{C_6} = 924$

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