First slide

Binomial theorem for positive integral Index

Question

Let $S (θ)$ denote the sum of coefficients in the expansion of ${(\sqrt{2} - x \sin θ + x^{2} \cos θ)}^{2 n} .$ Maximum value of $S (θ)$ is

Moderate

A

$4^{n}$
B

$8^{n}$
C

$2^{n}$
D

1

Solution

$S (θ) = (\sqrt{2} - \sin θ + \cos θ)^{2 n}$
Maximum possible value of
$\sqrt{2} - \sin θ + \cos θ = \sqrt{2} - \sqrt{2} (\cos \frac{π}{4} \sin θ - \sin \frac{π}{4} \cos θ)$
$\begin{array}{l} = \sqrt{2} - \sqrt{2} \sin (θ - π / 4) \\ is \sqrt{2} + \sqrt{2} = 2 \sqrt{2} \end{array}$
$∴$ Maximum possible value of $S (θ) is {(2^{3 / 2})}^{2 n} = 8^{n}$

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