The roots α and β of a quadratic equation, satisfy the relations α+β=α^2+β^2 and α β=α^2 β^2 . What is the number of such quadratic equations?

Correct answer: B. 2 aligned & α β=α^2 β^2 α β(1-α β)=0 & α β=0 or 1 aligned When α β=1 then α=(1)/(β) aligned & Again from α+β=α^2+β^2 & (1)/(β)+β=(1)/(β^2)+β^2 β^2-β=(1)/(β)-(1)/(β^2) & β(β-1)=((β-1))/(β^2) (β-1)(β-(1)/(β^2))=0 aligned aligned & (β-1)(β^3-1)=0 (β-1)^2(β^2+β+1)=0 & β=1 and β= -1 ± 3 i 2 aligned Again, when β=1 , then α=(1)/(β)=1 , roots (α, β)=(1,1) When β= -1+ 3 i 2 , then α= -1- 3 i 2 Roots (α, β)= ( -1 ± 3 i 2 , -1 3 i 2 ) Thus, number of different quadratic equations=2 .

NDA 2020

Exam: NDA
Year: 2020
Subject: Mathematics
Chapter: Quadratic Equation

NDA 2020 · Mathematics · Quadratic Equation

The roots α and β of a quadratic equation, satisfy the relations...

The roots

\alpha

and

\beta

of a quadratic equation, satisfy the relations

\alpha+\beta=\alpha^2+\beta^2

and

\alpha \beta=\alpha^2 \beta^2

. What is the number of such quadratic equations?

Options

Correct answer

B. 2

Step-by-step solution

\begin{aligned} & \alpha \beta=\alpha^2 \beta^2 \Rightarrow \alpha \beta(1-\alpha \beta)=0 \\ & \therefore \alpha \beta=0 \text{\text{ or } } 1 \end{aligned}

When

\alpha \beta=1

then

\alpha=\frac{1}{\beta}

\begin{aligned} & \text{ Again\text{ from } } \alpha+\beta=\alpha^2+\beta^2 \\ & \Rightarrow \frac{1}{\beta}+\beta=\frac{1}{\beta^2}+\beta^2 \Rightarrow \beta^2-\beta=\frac{1}{\beta}-\frac{1}{\beta^2} \\ & \Rightarrow \beta(\beta-1)=\frac{(\beta-1)}{\beta^2} \Rightarrow(\beta-1)\left(\beta-\frac{1}{\beta^2}\right)=0\end{aligned}

$\begin{aligned} & \Rightarrow(\beta-1)\left(\beta^3-1\right)=0 \Rightarrow(\beta-1)^2\left(\beta^2+\beta+1\right)=0 \\ & \therefore \beta=1 \text{ and } \beta=\frac{-1 \pm \sqrt{3} i}{2}\end{aligned}

Again, when

\beta=1

, then

\alpha=\frac{1}{\beta}=1

, roots

(\alpha, \beta)=(1,1)

When

\beta=\frac{-1+\sqrt{3} i}{2}

, then

\alpha=\frac{-1-\sqrt{3} i}{2}

Roots

(\alpha, \beta)= \left(\frac{-1 \pm \sqrt{3} i}{2}, \frac{-1 \mp \sqrt{3} i}{2}\right) Thus, number

of different quadratic equations=2

.

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