Question:
If \(p + q = 20\) and \(p^2 + q^2 = 260\), then find the value of \((p^4 + p^2q^2 + q^4)\).
यदि \(p + q = 20\) और \(p^2 + q^2 = 260\) है, तो \((p^4 + p^2q^2 + q^4)\) का मान ज्ञात कीजिए।
Options:
- 50,000
- 62,700
- 43,600
- 66,800
- 50,000
- 62,700
- 43,600
- 66,800
Correct Answer: b)
Explanation:
\((p + q)^2 = p^2 + q^2 + 2pq \implies 20^2 = 260 + 2pq \implies 400 – 260 = 2pq \implies 2pq = 140 \implies pq = 70\).
Now, \(p^4 + p^2q^2 + q^4 = (p^2 + q^2)^2 – p^2q^2 = 260^2 – 70^2 = 67,600 – 4,900 = 62,700\).
\((p + q)^2 = p^2 + q^2 + 2pq \implies 400 = 260 + 2pq \implies 2pq = 140 \implies pq = 70\)।
अब, \(p^4 + p^2q^2 + q^4 = (p^2 + q^2)^2 – p^2q^2 = 260^2 – 70^2 = 67,600 – 4,900 = 62,700\)।