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The Range of F ( X ) = 1 1 − 2 Cos X is (A) [1/3, 1] (B) [−1, 1/3] (C) (−∞, −1) ∪ [1/3, ∞) (D) [−1/3, 1] - Mathematics

MCQ

The range of  \[f\left( x \right) = \frac{1}{1 - 2\cos x}\] is 

 

Options

  • (a) [1/3, 1]    

  •   (b) [−1, 1/3]    

  •   (c) (−∞, −1) ∪ [1/3, ∞)   

  •    (d) [−1/3, 1]   

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Solution

We know that −1 ≤ cosx ≤ 1 for all x ∈ R.

Now, 

\[- 1 \leq \cos x \leq 1\]
\[ \Rightarrow - 1 \leq - \cos x \leq 1\]
\[ \Rightarrow - 2 \leq - 2\cos x \leq 2\]
\[ \Rightarrow - 1 \leq 1 - 2\cos x \leq 3 \left( \text{ Adding 1 to each term }  \right)\]

But,

\[\cos x \neq \frac{1}{2}\]
\[\Rightarrow 1 - 2\cos x \in \left[ - 1, 3 \right] - \left\{ 0 \right\}\]
\[ \Rightarrow \frac{1}{1 - 2\cos x} \in ( - \infty , - 1] \cup [\frac{1}{3}, \infty )\]
∴ Range of f(x) = (−∞, −1] ∪[ \[\frac{1}{3}\] 
 

Disclaimer: The range of the function does not matches with either of the given options. The range matches with option (c) if it is given as "(−∞, −1] ∪ [1/3, ∞)".

 

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APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 3 Functions
Q 45 | Page 45
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