Question:
If \(p – q = 6\) and \(pq = 7\), find the value of \(p^3 – q^3 + 6(p + q)^2\).
यदि \(p – q = 6\) और \(pq = 7\) है, तो \(p^3 – q^3 + 6(p + q)^2\) का मान ज्ञात कीजिए।
Options:
- 756
- 726
- 912
- 1024
- 756
- 726
- 912
- 1024
Correct Answer: b)
Explanation:
Using algebraic identities:
\(p^3 – q^3 = (p – q)^3 + 3pq(p – q) = 6^3 + 3(7)(6) = 216 + 126 = 342\).
Also, \((p + q)^2 = (p – q)^2 + 4pq = 6^2 + 4(7) = 36 + 28 = 64\).
Therefore, \(p^3 – q^3 + 6(p + q)^2 = 342 + 6(64) = 342 + 384 = 726\).
सर्वसमिकाओं का प्रयोग:
\(p^3 – q^3 = (p – q)^3 + 3pq(p – q) = 6^3 + 3 \times 7 \times 6 = 216 + 126 = 342\).
\((p + q)^2 = (p – q)^2 + 4pq = 6^2 + 4(7) = 36 + 28 = 64\).
अतः \(p^3 – q^3 + 6(p + q)^2 = 342 + 6(64) = 342 + 384 = 726\)।