\begin{aligned} & \hat{a} \cdot \hat{b}=\frac{1}{2} \\ & \text { Now }(\lambda \hat{a}+2 \hat{b}) \cdot(3 \hat{a}-\lambda \hat{b})=0 \\ & 3 \lambda \hat{a} \cdot \hat{a}-\lambda^2 \hat{a} \cdot \hat{b}+6 \hat{a} \cdot \hat{b}-2 \lambda \hat{b} \cdot \hat{b}=0 \\ & 3 \lambda-\frac{\lambda^2}{2}+3-2 \lambda=0 \\ & \lambda^2-2 \lambda-6=0 \\ & \lambda=1 \pm \sqrt{7} \\ & \Rightarrow \text { number of values }=0 \end{aligned}