First slide

Geometric progression

Question

For $0 < x < π$ the values of $x$ which satisfy the relation $9^{1 + | \cos x | + \cos^{2} x + |\cos^{3} x| + \dots upto \infty} = 3^{4}$ are given by

Moderate

A

$\frac{π}{3}, \frac{2 π}{3}$
B

$\frac{π}{3}, \frac{3 π}{4}$
C

$\frac{π}{4}, \frac{3 π}{4}$
D

$\frac{π}{3}, \frac{π}{4}$

Solution

Since $0 < x < π, - 1 < \cos x < 1 \Rightarrow 0 \leq |\cos x| < 1 .$ We can write the given expression as $9^{1 / (1 - | \cos x |)} = 81 = 9^{2}$
$\Rightarrow \frac{1}{1 - | \cos x |} = 2 \Rightarrow 1 - | \cos x | = \frac{1}{2}$
$\Rightarrow | \cos x | = 1 / 2 \Rightarrow \cos x = \pm 1 / 2$
$\Rightarrow x = π / 3$ or $2 π / 3$

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