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 $\frac{3}{22}, \frac{4}{22}, \frac{5}{22}, \ldots, \frac{11}{22} (9$ values)
When $i=2$
value of t will be $\frac{4}{22}, \frac{5}{22}, \ldots, \frac{11}{22}, \frac{12}{22}$
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 $t$