p2 - (√5)p + 1 = 0 হয়, তবে p3 + (1/p3) এর মান কত?
সমাধান
সঠিক উত্তর: Option D
দেওয়া আছে,
p2 - (√5)p + 1 = 0
বা, p2 + 1 = (√5)p
বা, (p2 + 1)/p = √5
বা, (p2/p) + (1/p) = √5
∴ p + (1/p) = √5
এখন,
p3 + (1/p3)
= {p+ (1/p)}3 - [3 ⋅ p ⋅ (1/p) ⋅ {p + (1/p)}]
= (√5)3 - (3 ⋅ 1 ⋅ √5)
= 5√5 - 3√5
= 2√5
p2 - (√5)p + 1 = 0
বা, p2 + 1 = (√5)p
বা, (p2 + 1)/p = √5
বা, (p2/p) + (1/p) = √5
∴ p + (1/p) = √5
এখন,
p3 + (1/p3)
= {p+ (1/p)}3 - [3 ⋅ p ⋅ (1/p) ⋅ {p + (1/p)}]
= (√5)3 - (3 ⋅ 1 ⋅ √5)
= 5√5 - 3√5
= 2√5
