Question

Roots of the equation $2x^2-5x+1=0$ and $x^2+5x+2=0$ are

• A. Reciprocal and of the same sign
• B. Reciprocal and of opposite sign
• C. Equal in magnitude
• D. Imaginary

Answer 1

Answer: (B)

Reciprocal and of opposite sign

Let roots of $2x^2-5x+1=0$ be $\alpha ,\beta$

$\Rightarrow \alpha +\beta =\frac{5}{2};\alpha \beta =\frac{1}{2}$

Let roots of $x^2+5x+2=0$ be $\gamma ,\delta$

$\Rightarrow \gamma +\delta =-5;\gamma \delta =2$

$-\left(\frac{\alpha +\beta }{\alpha \beta }\right)=-\left(\frac{\frac{3}{2}}{\frac{1}{2}}\right)=-5$

$\Rightarrow -\frac{1}{\alpha }-\frac{1}{\beta }=-5$

and $-\frac{1}{\alpha }\times \frac{-1}{\beta }=\frac{1}{\alpha \beta }=\frac{1}{\frac{1}{2}}=2.$

$\therefore$ roots of $x^2+5x+2$ are reciprocals and of opposite sign.