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Q.
If $x , y , z$ are three distinct real numbers such that $x , 12$, $y$ are in H.P. and $x , 12, z , y$ are in increasing A.P. then
Sequences and Series
Solution:
$ x , 12, y$ in H.P.
So, $12=\frac{2 x y}{x+y} \Rightarrow 12(x+y)=2 x y$.....(1)
$x , 12, z , y$ are in increasing A.P.
So, $24= x + z$ (option B)...(2)
$2 z =12+ y$.....(3)
$12+z=x+y$.....(4)
From (2), (3) and (4) put values in (1)
$12(12+z)=2(24-z)(2 z-12)$
we get, $z=15$
Put $z=15$ in equation (2) and (3), we get
$x =9, y =18$
So, maximum value of $\sqrt{(x-3) \sin \alpha-(y-10) \cos \beta+2}=\sqrt{6 \sin \alpha-8 \cos \beta+2}$
Maximum when $\sin \alpha=1$ and $\cos \beta=-1$ is 4 .