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STD 12 - 1. relation and function — Mathematics STD 12 Science — Question

CBSE BoardEnglish MediumSTD 12 ScienceMathematicsSTD 12 - 1. relation and function1 Mark
MCQ
Let $f: R \rightarrow R$ be a function defined by $f(x)=\left(2\left(1-\frac{x^{25}}{2}\right)\left(2+x^{25}\right)\right)^{\frac{1}{50}}$. If the function $g(x)=f(f(f(x)))+f(f(x))$, the the greatest integer less than or equal to $g (1)$ is
  • A
    $3$
  • B
    $7$
  • ✓
    $2$
  • D
    $8$
✓

Answer

Correct option: C.
$2$
c
$f(x)=\left[2\left(1-\frac{x^{25}}{2}\right)\left(2+x^{25}\right)\right]^{\frac{1}{50}}$

$f(x)=\left[\left(2-x^{25}\right)\left(2+x^{25}\right)\right]^{\frac{1}{50}}$

$\quad=\left(4-x^{50}\right)^{1 / 50}$

$\begin{aligned} f(f(x)) =\left(4-\left(\left(4-x^{50}\right)^{1 / 50}\right)^{50}\right)^{1 / 50}=x \\ g(x)=f(f(f(x)))+f(f(x)) \\ \quad=f(x)+x \\ g(1)=f(1)+1=3^{1 / 50}+1 \\g(1)]=\left[3^{1 / 50}+1\right]=2 \end{aligned}$

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