The number of integral values of $\lambda$ for which the 

equation $x^2+y^2-2\lambda x+2\lambda y+14=0$ represents a 

circle whose radius cannot exceed $6$, is

We have,

Radius $\le 6$

$\begin{array}{l}\Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}(\text{ Radius }{)}^2\le 36\\ \Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{\lambda }^2+{\lambda }^2-14\le 36\\ \Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}{\lambda }^2-25\le 0\\ \Rightarrow \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}-5\le \lambda \le 5\Rightarrow \lambda =0,\pm 1,\pm 2,\pm 3,\pm 4,\pm 5\end{array}$

Hence, there are $11$ integral values of $\lambda$.