What is f(π /4) equal to ?

Correct Answer is : 12{1}{{2}}21

Correct Answer is : 12{1}{{2}}21

Test Index

UPSC NDA I 2023 Mathematics Solved Paper

Question Numbers: 79-80

Consider the following for the next two (02) items that follow :

Let

f(x) = \sin(\pi^{2})x + \cos(-\pi^{2})x

where

[.]

is a greatest ‘integer function

Question : 80 of 120

Marks: +1, -0

What is

f(\frac{\pi }{4})

equal to ?

Solution:

Concept:

The greatest integer function

[x]

gives the largest integer less than or equal to

x

Explanation:

Given

f(x) = \sin\left([\pi^2] x\right) + \cos\left([-\pi^2] x\right)

Since

\pi^2 \approx 9.869

, we have

[\pi^2] = 9

and

[-\pi^2] = -10

Thus

f(x) = \sin(9x) + \cos(-10x) = \sin(9x) + \cos(10x)

Now evaluate at

x = \frac{\pi}{4}

f\left(\frac{\pi}{4}\right) = \sin\left(9 \cdot \frac{\pi}{4}\right) + \cos\left(10 \cdot \frac{\pi}{4}\right)

= \sin\left(\frac{9\pi}{4}\right) + \cos\left(\frac{10\pi}{4}\right)

= \sin\left(\frac{9\pi}{4}\right) + \cos\left(\frac{5\pi}{2}\right)

Now

\sin\left(\frac{9\pi}{4}\right) = \sin\left(2\pi + \frac{\pi}{4}\right) = \sin\frac{\pi}{4} = \frac{1}{\sqrt{2}}

and

\cos\left(\frac{5\pi}{2}\right) = \cos\left(2\pi + \frac{\pi}{2}\right) = \cos\frac{\pi}{2} = 0

Therefore

f\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} + 0 = \frac{1}{\sqrt{2}}

Answer:

\frac{1}{\sqrt{2}}

(Option D).

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