Question

Two points $P$ and $Q$ are given in rectangular cartesian coordinates on the curves $y=2^{x+2}$, such that $\overline{OP}.\overline{i}=-1$ and $\overline{OQ}.\overline{i}=2$ where $\overline{i}$ is a unit vector along the $x$-axis. The magnitude of the vector $\overline{OQ}-4\overline{OP}$ is

• A. $2$
• B. $5$
• C. $10$
• D. $12$

Answer 1

The correct option is C $10$

$\overline{OP}.i=-1⟹P_x=-1$

$P$ lies on $y=2^{x+2}y=2^{-1+2}y=2$

$(P_x,P_y)=(-1,2)$

$\overline{OQ}.i=2⟹Q_x=2$

$Q$ lies on $y=2^{x+2}y=2^{2+2}y=16$

$(Q_x,Q_y)=(2,16)$

$\overline{OQ}=2i+16j\overline{OP}=-i+2j$

$\overline{OQ}-4\overline{OP}2i+16j+4i-8j6i+8j$

Magnitude is $\sqrt{6^2+8^2}\sqrt{36+64}\sqrt{100}=10$