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NDA Math Quadratic Equations and Inequations

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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]

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Question : 57 of 67

Marks: +1, -0

Consider the following

\alpha + \beta + \alpha\beta > 0

\alpha^2\beta + \beta^2\alpha > 0

Which of the above is/are correct?

1 only
2 only
Both 1 and 2
Neither 1 nor 2

Solution:

Sum of roots

=\alpha+\beta=-b

Multiplication of roots

\; \alpha\beta = c

Hence

\alpha+\beta+\alpha\beta=-b+c

\alpha^{2}\beta+\beta^{2}a=\alpha\beta(\alpha+\beta)

=-bc

\because b>0 \& c<0

\therefore -b+c<0 \& -bc>0

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