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UPSC NDA 2 2016 Mathematics Solved Paper

Question Numbers: 39-40

Consider the following for the next two (02) items that follow:

2x^{2} + 3x - \alpha = 0

has roots

-2

and

\beta

while the equation

x^{2} - 3mx + 2m^{2} = 0

has both roots positive, where

\alpha > 0

and

\beta > 0

Question : 39 of 120

Marks: +1, -0

What is the value of α?

Solution:

Concept:

Sum and product of roots of a quadratic equation

ax^{2}+bx+c=0

are given by

-\frac{b}{a}

and

\frac{c}{a}

, respectively.

Explanation:

Given quadratic:

2x^{2}+3x-\alpha=0

with roots

-2

and

\beta

Here

a=2

b=3

c=-\alpha

Sum of roots:

(-2)+\beta = -\frac{b}{a} = -\frac{3}{2}

Solve for

\beta

\beta = -\frac{3}{2}+2 = \frac{1}{2}

Product of roots:

(-2)\times\beta = \frac{c}{a} = \frac{-\alpha}{2}

Substitute

\beta = \frac{1}{2}

(-2)\times\frac{1}{2} = -\frac{\alpha}{2}

Simplify left side:

-1 = -\frac{\alpha}{2}

Multiply both sides by

-2

\alpha = 2

Answer:

\alpha = 2

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