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UPSC NDA 2 2019 Mathematics Solved Paper

Question Numbers: 79-80

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

Let

f(x) = x^{2}

g(x) = \tan x

and

h(x) = \ln x

Question : 79 of 120

Marks: +1, -0

For

x=\frac{\sqrt{\pi }}{2}

, what is the value of [ho(gof)](x)?

Solution:

Concept:

The composite function

h \circ (g \circ f)

means first apply

f

, then

g

, then

h

to the input.

Explanation:

Given

f(x)=x^{2}

g(x)=\tan x

, and

h(x)=\ln x

First, compute

(g\circ f)(x) = g(f(x)) = g(x^{2}) = \tan(x^{2})

Then,

[h\circ (g\circ f)](x) = h(\tan(x^{2})) = \ln(\tan(x^{2}))

Substitute

x = \frac{\sqrt{\pi}}{2}

x^{2} = \left(\frac{\sqrt{\pi}}{2}\right)^{2} = \frac{\pi}{4}

\tan\left(\frac{\pi}{4}\right) = 1

Thus

\ln(1) = 0

Answer:

0

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