UPSC NDA I 2015 Mathematics Paper

Question Numbers: 70-71

For the next two (2) items that follow:

The integral

∫\frac{dx}{acosx+bsinx}

is of the form

\frac{1}{r}ln[tan(\frac{x+a}{2})]

Question : 71 of 120

Marks: +1, -0

What is α equal to?

Solution:

Concept:

Given

a = r \sin \alpha

and

b = r \cos \alpha

, the ratio

\frac{a}{b}

expresses

\tan \alpha

Explanation:

Start with the given equations:

a = r \sin \alpha

and

b = r \cos \alpha

Divide the first equation by the second:

\frac{a}{b} = \frac{r \sin \alpha}{r \cos \alpha}

Cancel

r

(assuming

r \neq 0

\frac{a}{b} = \tan \alpha

Take the inverse tangent on both sides:

\alpha = \tan^{-1}\left(\frac{a}{b}\right)

Thus,

\alpha

is expressed solely in terms of

a

and

b

Answer:

\alpha = \tan^{-1}\left(\frac{a}{b}\right)

, which matches option A.

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