JEE Mains · Maths · STD 11 - 4.2 Quadratic equations and inequations
Let \(\alpha\) and \(\beta\) be two real numbers such that \(\alpha+\beta=1\) and \(\alpha \beta=-1 .\) Let \(p _{ n }=(\alpha)^{ n }+(\beta)^{ n },p _{ n -1}=11\) and \(p _{ n +1}=29\) for some integer \(n \geq 1 .\) Then, the value of \(p _{ n }^{2}\) is .... .
- A \(162\)
- B \(324\)
- C \(648\)
- D \(424\)
Answer & Solution
Correct Answer
(B) \(324\)
Step-by-step Solution
Detailed explanation
\(x ^{2}- x -1=0 \quad\) roots \(=\alpha, \beta\) \(\alpha^{2}-\alpha-1=0 \Rightarrow \alpha^{ n +1}=\alpha^{ n }+\alpha^{ n -1}\) \(\beta^{2}-\beta-1=0 \Rightarrow \beta^{ n +1}=\beta^{ n }+\beta^{ n -1}\) \(\quad\quad\quad\quad\quad\quad\quad\quad+\)_______________________…
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