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RELATIONS AND FUNCTIONS — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSRELATIONS AND FUNCTIONS1 Mark
MCQ
If $\text{f(x)}=\frac{2^{\text{x}}+2^{-\text{x}}}{2},$ then f(x + y)f(x - y) is equal to:
  • ✓
    $\frac{1}{2}\big[\text{f(2}\text{x})+\text{f}(2\text{y})\big]$
  • B
    $\frac{1}{2}\big[\text{f(2}\text{x})-\text{f}(2\text{y})\big]$
  • C
    $\frac{1}{4}\big[\text{f(2}\text{x})+\text{f}(2\text{y})\big]$
  • D
    $\frac{1}{4}\big[\text{f(2}\text{x})-\text{f}(2\text{y})\big]$
✓

Answer

Correct option: A.
$\frac{1}{2}\big[\text{f(2}\text{x})+\text{f}(2\text{y})\big]$
Given,
$\text{f(x)}=\frac{2^{\text{x}}+2^{-\text{x}}}{2}$
Now,
$\text{f}(\text{x}+\text{y})\text{f}(\text{x}-\text{y})=\Big(\frac{2^{\text{x}+\text{y}}+2^{-\text{x}-\text{y}}}{2}\Big)\Big(\frac{2^{\text{x}-\text{y}}+2^{-\text{x}+\text{y}}}{2}\Big)$
$\Rightarrow\ \text{f}(\text{x}+\text{y})\text{f}(\text{x}-\text{y})=\frac{1}{4}\big(2^{2\text{x}}+2^{-2\text{y}}+2^{2\text{y}}+2^{-2\text{x}}\big)$
$\Rightarrow\ \text{f}(\text{x}+\text{y})\text{f}(\text{x}-\text{y})=\frac{1}{2}\Big(\frac{2^{2\text{x}}+2^{-2\text{x}}}{2}+\frac{2^{2\text{y}}+2^{-2\text{y}}}{2}\Big)$
$\Rightarrow\text{f}(\text{x}+\text{y})\text{f}(\text{x}-\text{y})=\frac{1}{2}\big[\text{f}(2\text{x})+\text{f}(2\text{y})\big]$

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