The half-life of a radioisotope is \(4\) h. If the initial mass of the isotope was \(200\) g, then the mass remaining after \(24\) h will be:

1. \(1.042\) g
2. \(2.084\) g
3. \(3.125\) g
4. \(4.167\) g

Two boys are standing at the ends \(A\) and \(B\) of a ground, where \(AB = a.\) The boy at \(B\) starts running in a direction perpendicular to \(AB\) with velocity \(v_1.\) The boy \(A\) starts running simultaneously with velocity \(v\) and catches the other boy in a time \(t,\) where \(t\) is:
1. \(\frac{a}{\sqrt{v^2+v_1~^2}}\)
2. \(\sqrt{\frac{a^2}{v^2-v_1~^2}}\)
3. \(\frac{a}{(v-v_1)}\)
4. \(\frac{a}{(v+v_1)}\)

A force \(F\) is given \(F=at+bt^2,\)where \(t\) is time. What are the dimensions of \(a\) and \(b\)?
1. \([MLT^{-1}]\) and \([MLT^{0}]\)
2. \([MLT^{-3}]\) and \([ML^2T^{4}]\)
3. \([MLT^{-4}]\) and \([MLT^{1}]\)
4. \([MLT^{-3}]\) and \([MLT^{-4}]\)