$$ \begin{aligned} & \mid a+b)+2(a \times b) \mid=2, \theta e(0, \pi) \\ & ((a+b)+2(a \times b))+(a+b)+2(a \times b))=4 \\ & |a+b|+|a \times b|+0=4 \end{aligned} $$ Let the angle be $\theta$ between a and 6 $$ \begin{aligned} & 2+2000 \theta+4 \sin ^2 \theta=4 \\ & 2+2000 \theta-400 a^2 \theta=0 \end{aligned} $$ Let cose = t then $$ \begin{aligned} & 2 t^2-t-1=0 \\ & 2 t^2-2 t+t-1=0 \\ & 2 t(t-1)+(t-1)=0 \\ & (2 t+1)(t-1)=0 \\ & t=-\frac{1}{2} \quad \text { or } \quad t=1 \end{aligned} $$ $\cos \theta=-\frac{1}{2}$ [not possiblic as $\theta$ e ( 0 , xil $$ \theta=\frac{2 \pi}{3} $$ Now, Si $2|\sin |=2 \sin \left(\frac{2 \pi}{3}\right)$ $$ \begin{aligned} & \text { 1-6 } \sqrt{1+1-2 \cos \left(\frac{2 \pi}{3}\right)} \\ & =\sqrt{2-2 \cdot\left(-\frac{1}{2}\right)} \\ & =\sqrt{5} \end{aligned} $$ Si is correct. Se projection of a on $(\vec{a}+\vec{b})$. $$ \begin{aligned} & \frac{a(a+b)}{1+b}=\frac{1+\cos \left(\frac{2 \pi}{3}\right)}{\sqrt{2+2 \cos \frac{2 \pi}{3}}} \\ & =\frac{1-\frac{1}{2}}{\sqrt{1}} \\ & =\frac{1}{2} \end{aligned} $$ C Opton is true.