Question:
Let $\alpha$ and $\beta$ be the roots of the equation $x^{2}-x-1=0$. If $p_{k}=(\alpha)^{k}+(\beta)^{k}, k \geq 1$, then which one of the following statements is not true ?
Correct Option: , 4
Solution:
$\alpha^{5}=5 \alpha+3$
$\beta^{5}=5 \beta+3$
$p_{5}=5(\alpha+\beta)+6=5(1)+6$
$\left[\because\right.$ from $\left.x^{2}-x-1=0, \alpha+\beta=\frac{-b}{a}=1\right]$
$p_{5}=11$ and $p_{5}=\alpha^{2}+\beta^{2}=\alpha+1+\beta+1$
$p_{2}=3$ and $p_{3}=\alpha^{3}+\beta^{3}=2 \alpha+1+2 \beta+1$
$=2(1)+2=4$
$p_{2} \times p_{3}=12$ and $p_{5}=11 \Rightarrow p_{5} \neq p_{2} \times p_{3}$
Click here to get exam-ready with eSaral
For making your preparation journey smoother of JEE, NEET and Class 8 to 10, grab our app now.