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Functions — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSFunctions1 Mark
MCQ
If $\text{f(x)}=\frac{\sin^{4}\text{x}+\cos^2\text{x}}{\sin^2\text{x}+\cos^4\text{x}}$ for $\text{x}\in\text{R},$ then f(2002) =
  • ✓
    1
  • B
    2
  • C
    3
  • D
    4
✓

Answer

Correct option: A.
1
Given,
$\text{f(x)}=\frac{\sin^{4}\text{x}+\cos^2\text{x}}{\sin^2\text{x}+\cos^4\text{x}}$
On dividing the numerator and denominator by $\cos^4\text{x},$ we get
$\text{f(x)}=\frac{\tan^4\text{x}+\sec^2\text{x}}{1+\tan^2\text{x}\sec^2\text{x}}$
$=\frac{1+\tan^4\text{x}+\tan^2\text{x}}{1+\tan^2\text{x}(1+\tan^2\text{x})}$
$=\frac{1+\tan^{4}\text{x}+\tan^{2}\text{x}}{1+\tan^{4}\text{x}+\tan^{2}\text{x}}=1$ $(\text{For every x}\in\text{R})$
$\text{For x}=2002,$
We have,
$\text{f}(2002)=1$

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