JEE PYQ: Sequence And Series Question 11
Question 11 - 2021 (25 Feb Shift 1)
If $0 < \theta, \phi < \frac{\pi}{2}$, $x = \sum_{n=0}^{\infty} \cos^{2n}\theta$, $y = \sum_{n=0}^{\infty} \sin^{2n}\phi$ and $z = \sum_{n=0}^{\infty} \cos^{2n}\theta \cdot \sin^{2n}\phi$, then
(1) $xyz = 4$
(2) $xy - z = (x + y)z$
(3) $xy + yz + zx = z$
(4) $xy + z = (x + y)z$
Show Answer
Answer: (4)
Solution
$x = \frac{1}{1 - \cos^2\theta} = \frac{1}{\sin^2\theta}$
$y = \frac{1}{1 - \sin^2\phi} = \frac{1}{\cos^2\phi}$
$z = \frac{1}{1 - \cos^2\theta\sin^2\phi} = \frac{xy}{xy - (x-1)(y-1)}$
$xz + yz - z = xy$
$xy + z = (x + y)z$
BETA
