Question

Let $P$ and $Q$ are two points in the $x y$ plane on the curve $y=x^{11}-2 x^{7}+7 x^{3}+11 x+6$ such that $\overrightarrow{O P} \cdot \hat{i}=5, \overrightarrow{O Q} \cdot \hat{i}=-5$, then the magnitude of $\overrightarrow{O P}+\overrightarrow{O Q}$ 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 $\left(x_{2} , y_{2}\right)$
$\overset{ \rightarrow }{O P}\cdot \hat{i}=5\Rightarrow x_{1}=5$
$\overset{ \rightarrow }{O Q}\cdot \hat{i}=-5\Rightarrow x_{2}=-5$
Let $y=f\left(x\right)=x^{11}-2x^{7}+7x^{3}+11x+6$
$\therefore y_{1}=f\left(x_{1}\right)=f\left(5\right)$
$y_{2}=f\left(x_{2}\right)=f\left(- 5\right)$
$\overset{ \rightarrow }{O P}+\overset{ \rightarrow }{O Q}=\left(x_{1} + x_{2}\right)\hat{i}+\left(y_{1} + y_{2}\right)\hat{j}$
$=(f(5)+f(-5)) \hat{j}=12 \hat{j}$
$|\overrightarrow{O P}+\overrightarrow{O Q}|=12$