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UPSC NDA 2 2016 Mathematics Solved Paper

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Consider the following for the next three (03) items that follow:

Let

f(x)=[x]

, where [.] is the greatest integer function and

g(x)=\sin x

be two real valued functions over R

[NDA II 2016]

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Question : 83 of 120

Marks: +1, -0

Which of the following statements are correct?

(f \circ f)(x) = f(x)

(g \circ g)(x) = g(x)

x=0

(g \circ (f \circ g))(x)

can take only three values.

Select the correct answer using the code given below:

1 and 2 only
2 and 3 only
1 and 3 only
1,2 and 3

Solution:

(f \circ f)(x) = \lfloor \lfloor x \rfloor \rfloor

and

f(x) = \lfloor x \rfloor

Suppose

x=0.2

\Rightarrow (f \circ f)(x) = \lfloor \lfloor 0.2 \rfloor \rfloor = \lfloor 0 \rfloor = 0

f(x) = \lfloor 0.2 \rfloor = 0

\Rightarrow\; \text{fof}\; (x) = f(x)

Now

(g \circ g)(x) = \sin\; \sin\; x

and

g(x) = \sin\; x

At

x=0

(g \circ g)(x) = \sin \sin 0 = \sin 0 = 0

g(x) = \sin 0 = 0

\Rightarrow (g \circ g)(x) = g(x)

at

x=0

\because (f \circ g)(x) = \lfloor \sin x \rfloor

(g \circ (f \circ g))(x) = \sin \lfloor \sin x \rfloor

\because \sin\; x

has value from

-1

to 1

If

-1 \leq \sin x < 0

(g \circ (f \circ g))(x) = \sin(-1) - \sin(-1)

If

0 \leq \sin x < 1

\therefore (g \circ (f \circ g))(x) = \sin(0) = \sin 0

and

If

\sin x = 1

\therefore (g \circ (f \circ g))(x) = \sin(1) = \sin 1

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