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UPSC NDA I 2016 Mathematics Solved Paper

Question Numbers: 56-57

For the next two (2) items that follow:

Let

\alpha

and

\beta

(

\alpha < \beta

) be the roots of the equation

x^{2} + bx + c = 0

, where

b > 0

and

c < 0

Question : 57 of 120

Marks: +1, -0

Consider the following

1. α + β + αβ > 0

2. α^{2}β + β^{2}α > 0

Which of the above is/are correct?

Solution:

Concept:

Evaluate expressions using sum and product of roots of a quadratic equation.

Explanation:

For

x^{2}+bx+c=0

with

b>0

and

c<0

, let

\alpha

and

\beta

be the roots.

Sum:

\alpha+\beta = -b

Product:

\alpha\beta = c

Statement 1:

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

Since

c<0

and

b>0

c - b < 0

Thus

\alpha+\beta+\alpha\beta < 0

, so statement 1 is incorrect.

Statement 2:

\alpha^{2}\beta + \beta^{2}\alpha = \alpha\beta(\alpha+\beta) = c \cdot (-b) = -bc

Given

c<0

and

b>0

, product

bc<0

, so

-bc > 0

Thus

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

, so statement 2 is correct.

Answer:

Only statement 2 is correct. Option B.

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