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UPSC NDA I 2015 Mathematics Paper

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Question Numbers: 87-90

Direction: Consider the integral:

I_{m}=∫_{0}^{\pi }\frac{sin2mx}{sinx}dx,

where m is a positive integer.

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Question : 90 of 120

Marks: +1, -0

Consider the following:

I_{m}-I_{m-1}

I_{2m}> I_{m}

Which of the above is/are correct?

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

Solution:

Concept:

The value of the integral

I_m

is zero for all integer

m

, making

I_m - I_{m-1} = 0

always true, but

I_{2m} > I_m

false because both are zero.

Explanation:

From the given definition, it is known that

I_m = 0

for every integer

m

.

Therefore,

I_m - I_{m-1} = 0 - 0 = 0

.

So statement 1 is correct.

Similarly,

I_{2m} = 0

and

I_m = 0

, so

I_{2m} = I_m

, not greater.

Hence statement 2 is incorrect.

Answer:

Option A: 1 only.

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