JEE Mains · Maths · STD 11 - 4.2 Quadratic equations and inequations
Let \(P_n=\alpha^n+\beta^n, n \in \mathbf{N}\). If \(P_{10}=123, P_9=76\), \(P_8=47\) and \(P_1=1\), then the quadratic equation having roots \(\frac{1}{\alpha}\) and \(\frac{1}{\beta}\) is :
- A \(x^2-x+1=0\)
- B \(x^2+x-1=0\)
- C \(x^2-x-1=0\)
- D \(x^2+x+1=0\)
Answer & Solution
Correct Answer
(B) \(x^2+x-1=0\)
Step-by-step Solution
Detailed explanation
\begin{aligned} & \alpha^{10}+\beta^{10}=123 \\ & \alpha+\beta=1 \\ & \alpha^9+\beta^9=76 \\ & \alpha^8+\beta^8=47 \\ & P_{10}=P_9+P_8 \\ & x^2=x+1 \Rightarrow x^2-x-1=0 \\ & \alpha+\beta=1, \alpha \beta=-1 \\ & \frac{1}{\alpha}+\frac{1}{\beta}=\frac{\alpha+\beta}{\alpha…
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