First slide

Theory of expressions

Question

The number of real solution of $2^{\sin^{2} x} + 5 (2^{\cos^{2} x}) = 7$ is

Moderate

A

zero
B

1
C

finitely many
D

infinitely many

Solution

Put $2^{\sin^{2} x} = t$ so that equation becomes
$t + \frac{5 \times 2}{t} = 7 \Rightarrow t^{2} - 7 t + 10 = 0 \Rightarrow t = 2, 5$
But $1 \leq t \leq 2$ Thus $\sin^{2} x = 1$

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