Test Index

UPSC NDA I and II 2020 Math Solved Paper

Show Para Hide Para

Question Numbers: 49-51

Directions: Read the following information and answer three items that follow:

Let

t_{n} = \sin^{n} \theta + \cos^{n} \theta

.

© examsnet.com

Question : 51 of 120

Marks: +1, -0

What is the value of

t_{10}

\theta = 45^{\circ}

?

1
$\frac{1}{4}$
$\frac{1}{16}$
$\frac{1}{32}$

Solution:

Concept:

Use the known values:

\sin 45^\circ = \cos 45^\circ = \frac{1}{\sqrt{2}}

.

Explanation:

We are asked for

t_{10} = \sin^{10}\theta + \cos^{10}\theta

with

\theta = 45^\circ

.

Substitute the values:

t_{10} = \left(\frac{1}{\sqrt{2}}\right)^{10} + \left(\frac{1}{\sqrt{2}}\right)^{10}

.

That equals

2 \times \left(\frac{1}{\sqrt{2}}\right)^{10}

.

Now

\left(\frac{1}{\sqrt{2}}\right)^{10} = \frac{1}{(\sqrt{2})^{10}} = \frac{1}{2^{5}}

because

(\sqrt{2})^{10} = 2^5

.

Thus

t_{10} = 2 \times \frac{1}{2^5} = \frac{2}{32} = \frac{1}{16}

.

Answer:

Option C:

\frac{1}{16}

.

© examsnet.com

Go to Question:

Prev Question

Next Question