Question

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

• A. $x+z=18$
• B. $x+z=24$
• C. maximum value of $\sqrt{(x-3)\sin \alpha -(y-10)\cos \beta +2}$ is 4 where $\alpha ,\beta \in R$.
• D. maximum value of $\sqrt{(x-3)\sin \alpha -(y-10)\sin \alpha }$ is 10 where $\alpha \in R$.

Answer 1

Answer: (B), (C)

$x,12,y$ in H.P.
So, $12=\frac{2xy}{x+y}\Rightarrow 12(x+y)=2xy$.....(1)
$x,12,z,y$ are in increasing A.P.
So, $24=x+z$ (option B)...(2)
$2z=12+y$.....(3)
$12+z=x+y$.....(4)
From (2), (3) and (4) put values in (1)
$12(12+z)=2(24-z)(2z-12)$
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 .