Paragraph: Let α and β be the roots of the equation x^2-(1-2 a^2) x+(1-2 a^2)=0 Question: Under what condition is (1)/(α^2)+(1)/(β^2) < 1 ?

Correct answer: A. a^2 < (1)/(2) (1)/(α^2)+(1)/(β^2) < 1 (α^2+β^2)/((α β)^2) < 1 aligned & ((α+β)^2-2 α β)/((α β)^2) < 1 & α+β=(-b)/(a)=(1-2 a^2) & α β=(c)/(a)=(1-2 a^2) & On solving: < 1 aligned aligned & (4 a^4-1)/(4 a^4-4 a^2+1) & 4 a^4-1 < 4 a^4-4 a^2+1 & 4 a^2 < 2 a^2 < (1)/(2) aligned

NDA 2016

Exam: NDA
Year: 2016
Subject: Mathematics
Chapter: Quadratic Equation

NDA 2016 · Mathematics · Quadratic Equation

Paragraph: Let α and β be the roots of the equation x^2-(1-2 a^2)...

Paragraph: Let

\alpha

and

\beta

be the roots of the equation

\mathrm{x}^2-\left(1-2 \mathrm{a}^2\right) \mathrm{x}+\left(1-2 \mathrm{a}^2\right)=0

Question: Under what condition is

\frac{1}{\alpha^2}+\frac{1}{\beta^2} < 1

Options

A $\mathrm{a}^2 < \frac{1}{2}$
B $a^2 > \frac{1}{2}$
C $a^2 > 1$
D $\mathrm{a}^2 \in \left($ \frac{1}{3} $,$ \frac{1}{2}\right) $only$

Correct answer

A. $\mathrm{a}^2 < \frac{1}{2}$

Step-by-step solution

\frac{1}{\alpha^2}+\frac{1}{\beta^2} < 1 \Rightarrow \frac{\alpha^2+\beta^2}{(\alpha \beta)^2} < 1

\begin{aligned} & \Rightarrow \quad \frac{(\alpha+\beta)^2-2 \alpha \beta}{(\alpha \beta)^2} < 1 \\ & \alpha+\beta=\frac{-b}{a}=\left(1-2 \mathrm{a}^2\right) & \alpha \beta=\frac{c}{a}=\left(1-2 \mathrm{a}^2\right) \\ & \text{ On solving: } \quad < 1 \end{aligned}

$\begin{aligned} & \frac{4 a^4-1}{4 a^4-4 a^2+1} \\ & \Rightarrow \quad 4 a^4-1 < 4 a^4-4 a^2+1 \\ & 4 a^2 < 2 \Rightarrow a^2 < \frac{1}{2} \end{aligned}

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