A vector a has components $2 p$ and 1 with respect to a two dimensional rectangular cartesian system. This system is rotated through a certain angle about the origin in the counter-clockwise direction. If a has components $p+\mathrm{l}$ and $\mathrm{l}$ with respect to the new system, then

• Correct answer
  A

  $p=1$ or $p=\frac{-1}{3}$
• B

  $p=-1$ or $p=\frac{1}{3}$
• C

  $p=1$ or $p=-1$
• D

  $p=0$ or $p=\frac{1}{2}$

Solution

Let point $A(2 p, 1)$ A is rotated about origin in the anti-clock wise direction then new coordinate is $(p+1,1)$
Squaring and adding Eqs. (i) and (ii), we get $\begin{gathered} 4 p^2+1=(p+1)^2+1 \Rightarrow 4 p^2=p^2+2 p+1 \\ \Rightarrow 3 p^2-2 p-1=0 \Rightarrow(p-1)(3 p+1)=0 \Rightarrow p=1, \frac{-1}{3} \end{gathered}$

Hence, the correct answer is option: 'A'.