Question

Ten ants are on the real line. At time $t=0$, the kth ant starts at the point $k^2$ and travelling at uniform speed, reaches th e point $(11-k{)}^2$ at time $t=1$. The number of distinct times at which at least two ants are at the same location is

• A. 45
• B. 11
• C. 17
• D. 9

Answer 1

Answer: (C)

At time $t=0$, kth. ant starts at point $k^2$ and reaches at time $t=0$ at the point $(11-k{)}^2$
$\therefore$ Velocity of kth ant
$=\frac{x_2-x_1}{t_2-t_1}=\frac{{\left(11-k\right)}^2-k^2}{1-0}$
$u=121-12k$
Now, two ants are at the same location
$\therefore x_i=x_j$
$x_i=x+ut$
$k_i^2-22k_{i}t+121t=k_j-22k_{j}t+121t$
$\Rightarrow t=\frac{k_j^2-k_i^2}{22\left(k_j-k_i\right)}$
$=\frac{k_j+k_i}{22}\left[k_i\ne k_j\right]$
Now, for $i=1$,
values of t will be $\frac{3}{22},\frac{4}{22},\frac{5}{22},\dots ,\frac{11}{22}(9$ values)
When $i=2$
value of t will be $\frac{4}{22},\frac{5}{22},\dots ,\frac{11}{22},\frac{12}{22}$
Only one distinct value
Similarly, for $i=3,4,5,6,7,8,9$, we get only $1$ distinct value
So, in all there are $17$ distinct value of $t$