If the sum of first 11 terms of an A.P.,

$a_{1}+a_{2}+a_{3}+\ldots+a_{11}=0$

$\Rightarrow\left(a_{1}+a_{11}\right) \times \frac{11}{2}=0$

$\Rightarrow a_{1}+a_{11}=0$

$\Rightarrow a_{1}+a_{1}+10 d=0$

where $d$ is common difference

$\Rightarrow a_{1}=-5 d$

$a_{1}+a_{3}+a_{5}+\ldots \ldots+a_{23}$

$=\left(a_{1}+a_{23}\right) \times \frac{12}{2}=\left(a_{1}+a_{1}+22 d\right) \times 6$

$=\left(2 \mathrm{a}_{1}+22\left(\frac{-\mathrm{a}_{1}}{5}\right)\right) \times 6$

$=-\frac{72}{5} a_{1} \Rightarrow K=\frac{-72}{5}$