The rectangular components of a vector lying in xy plane are 1 and $\text{p}+1$ . If co-ordinate system is turned by $30°$ , They are p & 4 respectively the value of ‘p’ is

With respect to old co – ordinate system (xy)

$\overrightarrow{OA}=\overrightarrow{OM}+\overrightarrow{MA}=\hat{i}+\left(\text{P}+1\right)\hat{j}$

w.r.t new co – ordinate system $\left(x^{|},y^{|}\right)$

$\overrightarrow{OA}=\overrightarrow{ON}+\overrightarrow{NA}=\text{P}\hat{i}+4\hat{j}$

through the co – ordinate system is rotated magnitude of $\overrightarrow{OP}$  remains same in both systems

$\sqrt{1+\left(\text{P}+1\right)}=\sqrt{{\text{P}}^{\text{2}}+4^2}$

$1+{\text{P}}^{\text{2}}+1+2\text{P}={\text{P}}^2+16$

$\text{2P}=14$

$\text{P}=7$