Question

Let $S$ denotes the set of all real values of $x$ such that $(x^{2016}+1)(1+x^2+x^4+\cdots +x^{2014})=2016x^{2015}$. Then

• A. Number of elements in $S$ is $2$
• B. Number of elements in $S$ is $1$
• C. Point $(x,2),x\in S$ lies inside the parabola $Y=X^2-2X+2,X\in R$
• D. Image of the point $(x,2)$ about the line $y=x$ lies on the line $x+y=4$

Answer 1

Answer: (B), (C)

The correct options are
B Number of elements in $S$ is $1$
C Point $(x,2),x\in S$ lies inside the parabola $Y=X^2-2X+2,X\in R$

$(x^{2016}+1)(1+x^2+x^4+\cdots +x^{2014})=2016x^{2015}\Rightarrow (x+\frac{1}{x^{2015}})(1+x^2+x^4+\cdots +x^{2014})=2016\Rightarrow x+\frac{1}{x^{2015}}+x^3+\frac{1}{x^{2013}}+\cdots +x^{2015}+\frac{1}{x}=2016\Rightarrow (x+\frac{1}{x})+(x^3+\frac{1}{x^3})+\cdots +(x^{2015}+\frac{1}{x^{2015}})=2016$

When $x>0,x+\frac{1}{x}\ge 2$
When $x<0,x+\frac{1}{x}\le -2$
So, L.H.S. $=$ R.H.S. only if $x=1$

Putting the point $(1,2)$ in the parabola,
$2>1-2+2$
So, the point $(1,2)$ lies inside the parabola.

Image of the point $(1,2)$ about the line $y=x$ is $(2,1)$ which does not lie on the line $x+y=4$