If one AM 'A' and two GM p and q are inserted between two given numbers, then find the value of (p2/q)+(q2/p)

Question

If one AM 'A' and two GM p and q are inserted between two given numbers, then find the value of (p2/q)+(q2/p) (A) A (B) 2A (C) 3A (D) 4A

Tardigrade Admin · Accepted Answer

Let a and b are two numbers. Then, A=(a+b/2) ldots(i) Also, a, p, q, b are in G P. (p/a)=(q/a)=(b/q) ⇒ (p2/q)=a and (q2/p)=b ∴ (p2/q)+(q2/p)=a+b=2 A [Using eq. (i)]

Q. If one $A M^{'} A^{'}$ and two $GM$ $p$ and $q$ are inserted between two given numbers, then find the value of $\frac{p ^{2}}{q} + \frac{q ^{2}}{p}$

$A$

$2 A$

$3 A$

$4 A$

Let $a$ and $b$ are two numbers.
Then, $A = \frac{a + b}{2} \dots (i)$
Also, $a, p, q, b$ are in $GP$ .
$\frac{p}{a} = \frac{q}{a} = \frac{b}{q}$
$\Rightarrow \frac{p ^{2}}{q} = a$ and $\frac{q ^{2}}{p} = b$
$∴ \frac{p ^{2}}{q} + \frac{q ^{2}}{p} = a + b = 2 A$ [Using eq. (i)]

Q. If one AM′A′ and two GM p and q are inserted between two given numbers, then find the value of qp2​+pq2​

A

2A

3A

4A

Let a and b are two numbers. Then, A=2a+b​…(i) Also, a,p,q,b are in GP. ap​=aq​=qb​ ⇒qp2​=a and pq2​=b ∴qp2​+pq2​=a+b=2A [Using eq. (i)]

Q. If one $A M^{'} A^{'}$ and two $GM$ $p$ and $q$ are inserted between two given numbers, then find the value of $\frac{p ^{2}}{q} + \frac{q ^{2}}{p}$

$A$

$2 A$

$3 A$

$4 A$

Let $a$ and $b$ are two numbers.
Then, $A = \frac{a + b}{2} \dots (i)$
Also, $a, p, q, b$ are in $GP$ .
$\frac{p}{a} = \frac{q}{a} = \frac{b}{q}$
$\Rightarrow \frac{p ^{2}}{q} = a$ and $\frac{q ^{2}}{p} = b$
$∴ \frac{p ^{2}}{q} + \frac{q ^{2}}{p} = a + b = 2 A$ [Using eq. (i)]