Daily Question 112

Daily Question 112

$\textbf{Q112.}$ If $x^{15}-x^{13}+x^{11}-x^9+x^7-x^5+x^3-x=7$, where $x>0$, then

(A) $x^{16}$ is equal to 15

(B) $x^{16}$ is less than 15

(C) $x^{16}$ is greater than 15

(D) none of these

$\textbf{Ans.} (C)$

$\textbf{Sol.}$ Given $x^{15}-x^{13}+x^{11}-x^9+x^7-x^5+x^3-x=7$

$\implies(x^2-1)(x^4+1)(x^8+1)x=7\qquad\ldots(1)$

Now, $x^{16}-1=(x^{2}-1)(x^{2}+1)(x^{4}+1)(x^8+1)$

Using Eq. $(1)$

$\implies x^{16}-1=\dfrac{7}{x}(x^2+1)$

$\implies x^{16}-1=7\left[\left(\sqrt{x}-\dfrac{1}{\sqrt{x}}\right)^2+2\right]> 14$

$\therefore x^{16}>15$

Daily Question 111

Compiled Daily Questions 1-100