Consider the following 1. β 2. β Which of the above is/are correct?

Correct Answer is : Both 1 and 2

Correct Answer is : Both 1 and 2

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UPSC NDA I 2016 Mathematics Solved Paper

For the next two (2) items that follow :

Let α and β

(\alpha < \beta)

be the roots of the equation

x^2 +bx+c=0

, where

b>0

and

c<0

[NDA I 2016]

Question : 56 of 120

Marks: +1, -0

Consider the following

\beta < -\alpha

\beta < |\alpha|

Which of the above is/are correct?

Solution:

Given quadratic equation,

x^{2}+bx+c=0

and roots are α and β where

\alpha < \beta

Hence roots of given quadratic equation are

\beta = \frac{-b + \sqrt{b^{2} - 4c}}{2}

\alpha = \frac{-b - \sqrt{b^{2} - 4c}}{2}

(\because \alpha < \beta)

\Rightarrow -\alpha = \frac{b + \sqrt{b^{2} - 4c}}{2}

and

|\alpha| = \frac{b + \sqrt{b^{2} - 4c}}{2}

\therefore \beta < -\alpha

and

\beta < |\alpha|

both are correct.

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