Explanation

(b) Let $T_{1}$ and $t_{2}$ be the time periods of the pendulums with lengths 1.0 m and 1.21 m respectively
$\frac{T_{2}}{T_{1}} = \sqrt{\frac{l_{2}}{l_{1}}} = \sqrt{\frac{1.21}{1}} = 1.1$ …(i)
Let $v_{1}$ and $v_{2}$ be the vibrations made by two pendulum to swing together
$\therefore\ \ \ v_{1}T_{1} = v_{2}T_{2}$ …(ii)
For the two pendulums to swing together, required condition is
$v_{1} - v_{2} = 1$
or $v_{1} = v_{2} + 1$
$\therefore\ \ \ \left( v_{2} + 1 \right)T_{1} = v_{2}T_{2}$
or $\left( v_{2} + 1 \right)v_{2} = T_{2}/T_{1} = 1.1$
or $1 + \frac{1}{v_{2}} = 1.1$
or $\frac{1}{v_{2}} = 1.1 - 0.1 = \frac{1}{10}$
or $v_{2} = 10$