MCQ
If $x + y = 3 - cos4\theta$ and $x - y = 4 \,sin2\theta$ then
- A$x^4 + y^4 = 9$
- B$\sqrt x \, + \,\sqrt y \, = \,16\,$
- C$x^3 + y^3 = 2(x^2 + y^2)$
- ✓$\sqrt x \, + \,\sqrt y \, = \,2$
✓
Answer
Correct option: D.
$\sqrt x \, + \,\sqrt y \, = \,2$
d
On adding and subtracting
On adding and subtracting
$x = \frac{{3 - \cos 4\theta \, + 4\sin 2\theta }}{2}\,$; $y = \frac{{3 - \cos 4\theta \, - 4\sin 2\theta }}{2}\,$
$x =\frac{{4(1 + \sin 2\theta )\, - \,(1 + \cos 4\theta )}}{2}\,$ ; $y =\frac{{4(1 - \sin 2\theta )\, - \,(1 + \cos 4\theta )}}{2}\,$
$x = 2 (1 + sin2\theta ) - cos^22\theta$ ; $y = 2 (1 - sin2\theta ) - cos^22\theta$
$x = 1 + 2\, sin2\theta + sin^22\theta$ ;$ y = 1 - 2 sin2\theta + sin^22\theta$
$x = (1 + sin2\theta )^2$ ;$y = (1 - sin2\theta )^2$ $\Rightarrow \sqrt x \, + \,\sqrt y \, = \,2$
Alternate : Or put $\theta$ = $\frac{\pi }{4}\,$ and verify
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