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UPSC CDS 2 2019 Math Paper

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Question : 33 of 100

Marks: +1, -0

If

0 < \theta < 90^{\circ}

\sin \theta = \frac{3}{5}

x = \cot \theta

, then what is the value of

1 + 3x + 9x^2 + 27x^3 + 81x^4 + 243x^5

?

941
1000
1220
1365

Solution:

Concept:

Given

\sin\theta = \frac{3}{5}

for an acute angle

\theta

, compute

\cot\theta

and then evaluate the given geometric series.

Explanation:

First, find

\cot\theta

.

Since

\theta

is acute,

\cos\theta

is positive.

\cos\theta = \sqrt{1 - \sin^2\theta} = \sqrt{1 - \frac{9}{25}} = \sqrt{\frac{16}{25}} = \frac{4}{5}

.

Then

\cot\theta = \frac{\cos\theta}{\sin\theta} = \frac{4/5}{3/5} = \frac{4}{3}

.

Thus

x = \frac{4}{3}

.

The expression is

1 + 3x + 9x^2 + 27x^3 + 81x^4 + 243x^5

.

This is a geometric series with first term

1

and common ratio

r = 3x = 3 \times \frac{4}{3} = 4

.

There are 6 terms, so the sum is

S = \frac{r^6 - 1}{r - 1} = \frac{4^6 - 1}{4 - 1} = \frac{4096 - 1}{3} = \frac{4095}{3} = 1365

.

Alternatively, compute directly:

1 + 4 + 16 + 64 + 256 + 1024 = 1365

.

Answer:

Option D,

1365

.

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