JEE Mains · Maths · STD 11 - 4.2 Quadratic equations and inequations
Let \(\alpha\) and \(\beta\) be the roots of the equation \(\mathrm{x}^{2}-\mathrm{x}-1=0 .\) If \(\mathrm{p}_{\mathrm{k}}=(\alpha)^{\mathrm{k}}+(\beta)^{\mathrm{k}}, \mathrm{k} \geq 1,\) then which one of the following statements is not true?
- A \(\left(p_{1}+p_{2}+p_{3}+p_{4}+p_{5}\right)=26\)
- B \(\mathrm{p}_{5}=11\)
- C \(\mathrm{p}_{3}=\mathrm{p}_{5}-\mathrm{p}_{4}\)
- D \(\mathrm{p}_{5}=\mathrm{p}_{2} \cdot \mathrm{p}_{3}\)
Answer & Solution
Correct Answer
(D) \(\mathrm{p}_{5}=\mathrm{p}_{2} \cdot \mathrm{p}_{3}\)
Step-by-step Solution
Detailed explanation
\(\alpha+\beta=1, \alpha \beta=-1\) \(\mathrm{P}_{\mathrm{k}}=\alpha^{\mathrm{k}}+\beta^{\mathrm{k}}\) \(\alpha^{2}-\alpha-1=0\) \(\Rightarrow \alpha^{\mathrm{k}}-\alpha^{\mathrm{k}-1}-\alpha^{\mathrm{k}-2}=0\) and…
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