If the range of $\frac{1}{2 - c o s 3 x}$ is [a, b] then which of the following statement(s) is/are false?

9 a^{2} + b^{2} = 2

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9 a^{2} + b^{2} = 1

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27 a^{2} + 4 b^{2} = 7

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54 a^{2} + b^{2} = 7

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Solution

The correct options are
A $9 a^{2} + b^{2} = 2$
C $27 a^{2} + 4 b^{2} = 7$
D $54 a^{2} + b^{2} = 7$
We know that $- 1 \leq - c o s 3 x \leq 1$
$1 \leq 2 - c o s 3 x \leq 3$
$\Rightarrow \frac{1}{3} \leq \frac{1}{2 - c o s x} \leq 1$
$a = \frac{1}{3}, b = 1$ .

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