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UPSC NDA I 2016 Mathematics Solved Paper

Question Numbers: 38-39

Direction: For the next two (2) items that follow:

Given that

\tan \alpha

and

\tan \beta

are the roots of the equation

x^{2} + bx + c = 0

with

b \neq 0

Question : 39 of 120

Marks: +1, -0

What is sin (α + β) sec α sec β equal to?

Solution:

Concept:

The expression simplifies to the sum of tangents of two angles when the angles are the roots of a given quadratic equation.

Explanation:

Expand

\sin(\alpha+\beta)

\sin\alpha\cos\beta + \cos\alpha\sin\beta

Then the expression becomes

\frac{\sin\alpha\cos\beta + \cos\alpha\sin\beta}{\cos\alpha\cos\beta}

Separate into two fractions:

\frac{\sin\alpha\cos\beta}{\cos\alpha\cos\beta} + \frac{\cos\alpha\sin\beta}{\cos\alpha\cos\beta}

Simplify each term:

\frac{\sin\alpha}{\cos\alpha} + \frac{\sin\beta}{\cos\beta} = \tan\alpha + \tan\beta

Now, given that

\tan\alpha

and

\tan\beta

are the roots of

x^2 + bx + c = 0

For a quadratic

ax^2 + bx + c = 0

, sum of roots =

-\frac{b}{a}

. Here

a=1

, so sum

\tan\alpha + \tan\beta = -b

Thus

\sin(\alpha+\beta)\sec\alpha\sec\beta = \tan\alpha + \tan\beta = -b

Answer:

\boxed{-b}

(Option B)

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