KVPY SX SB Exam 2016 Question Paper

Velocity of any ant U $={\left(11-\mathit{k}\right)}^2-{\mathit{k}}^2$ $=121-22\mathit{k}$
Now at any time distance travelled by any ant will be
$\mathit{S}={\mathit{S}}_0+\mathit{u}\mathit{t}$
Where ${\mathit{S}}_0$ is the initial position
Now two ants will be at same position
If ${\mathit{S}}_{\mathit{i}}={\mathit{S}}_{\mathit{j}}$
${\mathit{k}}_{\mathit{i}}^2-22{\mathit{k}}_{\mathit{i}}\mathit{t}+121\mathit{t}$ $={\mathit{k}}_{\mathit{j}}^2-22{\mathit{k}}_{\mathit{j}}\mathit{t}+121\mathit{t}$
$\mathit{t}=\frac{{\mathit{k}}_{\mathit{j}}^2-{\mathit{k}}_{\mathit{i}}^2}{22\left({\mathit{k}}_{\mathit{j}}-{\mathit{k}}_{\mathit{i}}\right)}$
$\Rightarrow \mathit{t}=\frac{{\mathit{k}}_{\mathit{j}}+{\mathit{k}}_{\mathit{i}}}{22}$
Now for i = 1
Values of t will be $\frac{3}{22},\frac{4}{22},\frac{5}{22},....\frac{.11}{22}$ (9 values)
for i = 2
values of t will be $\frac{4}{22},\frac{5}{22},....,\frac{11}{22},\frac{12}{22}$
We can see there is only 1 distinct value
Similarly of i = 3,4,5,6,7,8,9 we get only 1 distinct value each.
So in all there 17 distinct values of 't'