Question

Let $P$ and $Q$ are two points in the $xy$ plane on the curve $y=x^{11}-2x^7+7x^3+11x+6$ such that $\overrightarrow{OP}\cdot \hat{i}=5,\overrightarrow{OQ}\cdot \hat{i}=-5$, then the magnitude of $\overrightarrow{OP}+\overrightarrow{OQ}$ is

• A. $10$
• B. $12$
• C. $14$
• D. $8$

Answer 1

Answer: (B)

Let $P$ and $Q$ be $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$
$\overrightarrow{\mathrm{OP}}\cdot \hat{i}=5\Rightarrow x_1=5$
$\overrightarrow{\mathrm{OQ}}\cdot \hat{i}=-5\Rightarrow x_2=-5$
Let $y=f\left(x\right)=x^{11}-2x^7+7x^3+11x+6$
$\therefore y_1=f\left(x_1\right)=f\left(5\right)$
$y_2=f\left(x_2\right)=f\left(-5\right)$
$\overrightarrow{\mathrm{OP}}+\overrightarrow{\mathrm{OQ}}=\left(x_1+x_2\right)\hat{i}+\left(y_1+y_2\right)\hat{j}$
$=(f(5)+f(-5))\hat{j}=12\hat{j}$
$|\overrightarrow{OP}+\overrightarrow{OQ}|=12$