$\text{ If the angle between the vectors }\lambda \widehat{i}-3\widehat{j}+5\widehat{k}\text{ and }2\lambda \widehat{i}-\lambda \widehat{j}-\widehat{k}\text{ is an obtuse angle and }\lambda$ $\text{ is an integer, then the number of value(s) of }\lambda \text{ is/are }$

$\begin{array}{l}\text{ Let }\overline{a}=\lambda \widehat{i}-3\widehat{j}+5\widehat{k}\text{ and }\overline{b}=2\lambda \widehat{i}-\lambda \widehat{j}-\widehat{k}\\ \text{ The angle between the vector }\overline{a}\text{ and }\overline{b}\text{ is an obtuse angle then }\overline{a}\cdot \overline{b}<0\\ \Rightarrow (\lambda \widehat{i}-3\widehat{j}+5\widehat{k})\cdot (2\lambda \widehat{i}-\lambda \widehat{j}-\widehat{k})<0\end{array}$

$\begin{array}{l}\Rightarrow 2{\lambda }^2+3\lambda -5<0\\ \Rightarrow (2\lambda +5)(\lambda -1)<0\\ \Rightarrow \lambda \in \left(\frac{-5}{2},1\right)\end{array}$

$\text{ Integer values of }\lambda \text{ are }-2,-1,0$

$\text{ Hence, the number of values of }\lambda \text{ is }3$

Therefore, the correct answer is(1).