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Study Stuff for MCA Examinations - NIMCET
Study Stuff for MCA Examinations - NIMCET
Previous Year Question (PYQs)
If a, b, c, d are in HP and arithmetic mean of ab, bc, cd is 9 then which of the following number is the value of ad?
Solution
$a, b, c, d$ are in H.P.$\Rightarrow \frac{1}{a}, \frac{1}{b}, \frac{1}{c}, \frac{1}{d}$ are in A.P.
$\Rightarrow \frac{2}{b} = \frac{1}{a} + \frac{1}{c}$ and $\frac{2}{c} = \frac{1}{b} + \frac{1}{d}$
Given A.M. of $ab, bc, cd$ is $9$:
$\frac{ab + bc + cd}{3} = 9$
$\Rightarrow ab + bc + cd = 27$
Now multiply:
$\frac{2}{b} = \frac{1}{a} + \frac{1}{c} $
$\Rightarrow 2ac = b(a + c)$
$\frac{2}{c} = \frac{1}{b} + \frac{1}{d} \Rightarrow 2bd = c(b + d)$
Multiply both:
$4abcd = bc(a + c)(b + d)$
Cancel $bc$:
$4ad = (a + c)(b + d)$
Expand RHS:
$4ad = ab + ad + bc + cd$
$\Rightarrow 3ad = ab + bc + cd$
But $ab + bc + cd = 27$:
$\Rightarrow 3ad = 27$
$\Rightarrow ad = 9$
$\boxed{9}$
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