If fx-sin-6sin-2sin x - gx-2sin x - x - Then which among the following is-are correct

F(x)=sin(π6sin(π2sin x)) ∵ 1 ≤sin x ≤ 1 ⇒ π2≤π2sin x ≤π2 ⇒ 1 ≤sin(π2sin x) ≤ 1 ⇒ π6≤π6sin(π6sin x) ≤π6 ⇒ 12≤sin(π6sin x(π2sin x)) ≤12 nbsp;∴ Range is[ 12, ...

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Standard XII

Mathematics

Limit

If fx=sinπ6si...

Question

If $f (x) = sin (\frac{π}{6} sin (\frac{π}{2} sin x)), g (x) = \frac{π}{2} sin x \forall x \in R$ . Then which among the following is/are correct

range of

f (x)

[- \frac{1}{2}, \frac{1}{2}]

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range of

f o g (x)

[- 1, 1]

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Number of integers in the range of

f (x)

3

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Number of integers in the range of

f (g (x))

1

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Solution

The correct option is D Number of integers in the range of $f (g (x))$ is $1$
$f (x) = sin (\frac{π}{6} sin (\frac{π}{2} sin x)) ∵ - 1 \leq sin x \leq 1 \Rightarrow - \frac{π}{2} \leq \frac{π}{2} sin x \leq \frac{π}{2} \Rightarrow - 1 \leq sin (\frac{π}{2} sin x) \leq 1 \Rightarrow - \frac{π}{6} \leq \frac{π}{6} sin (\frac{π}{6} sin x) \leq \frac{π}{6} \Rightarrow - \frac{1}{2} \leq sin (\frac{π}{6} sin x (\frac{π}{2} sin x)) \leq \frac{1}{2}$
$∴ Range is [- \frac{1}{2}, \frac{1}{2}]$
Clearly, only one integer is there in the range of $f (x) .$

Now,
$f o g (x) = f (\frac{π}{2} sin x) = sin (\frac{π}{6} sin (\frac{π}{2} sin (\frac{π}{2} sin x))) ∵ - 1 \leq sin x \leq 1 \Rightarrow \frac{- π}{2} \leq \frac{π}{2} sin x \leq \frac{π}{2} \Rightarrow - 1 \leq sin (\frac{π}{2} sin x) \leq 1 \Rightarrow - \frac{π}{2} \leq \frac{π}{2} (sin (\frac{π}{2} sin x)) \leq \frac{π}{2} \Rightarrow - 1 \leq sin (\frac{π}{2} (sin (\frac{π}{2} sin x))) \leq 1 \Rightarrow - \frac{π}{6} \leq \frac{π}{6} sin (\frac{π}{2} (sin (\frac{π}{2} sin x))) \leq \frac{π}{6} \Rightarrow - \frac{1}{2} \leq sin (\frac{π}{6} sin (\frac{π}{2} (sin (\frac{π}{2} sin x)))) \leq \frac{1}{2}$
$∴$ Range of $f o g (x) = [- \frac{1}{2}, \frac{1}{2}]$
Clearly, only one integer is there in the range of $f (g (x)) .$

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Limit

Standard XII Mathematics

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If f(x)=sin(π6sin(π2sinx)),g(x)=π2sinx ∀x∈R. Then which among the following is/are correct

If $f (x) = sin (\frac{π}{6} sin (\frac{π}{2} sin x)), g (x) = \frac{π}{2} sin x \forall x \in R$ . Then which among the following is/are correct