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JEE Main 20252 AprilMorning Shift

Exam: JEE Main
Year: 2025
Date: 2 April
Shift: Morning Shift
Subject: Mathematics
Chapter: Quadratic Equation

JEE Main 2025 · 2 April · Morning Shift · Mathematics · Quadratic Equation

Let P_n=α^n+β^n, n ∈ N . If P_ 10 =123, P_9=76 , P_8=47 and P_1=1 ,...

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 :

Options

A $x^2-x+1=0$
B $x^2+x-1=0$
C $x^2-x-1=0$
D $x^2+x+1=0$

Correct answer

B. $x^2+x-1=0$

Step-by-step solution

\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 \beta}=\frac{1}{-1}=-1, \frac{1}{\alpha \beta}=-1\end{aligned}

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