NIMCET 2024 Paper

Section: Mathematics

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Question : 37 of 120

Marks: +1, -0

If one AM (Arithmetic mean)

a

and two GM's (Geometric means)

p

and

q

be inserted between any two positive numbers, the value of

p^3+q^3

is:

$2 a p q$
$2 p q \/ a$
$p q \/ a$
$p+q+a$

Solution:

Let the two numbers are

x, y

, then

x, a, y

are in AP and

x, p, q, y

are in GP.
Hence

2 a=x+y

and

\;{p}/{x}=\;{q}/{p}=\;{y}/{q}

⇒ x=\;{p^2}/{q}, y=\;{q^2}/{p}

Putting these values, we get

2 a=\;{p^2}/{q}+\;{q^2}/{p}

Or

p^3+q^3=2 a p q

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