Explanation

(a)
We have
$1 = a(1 - 2x)(1 - 3x) + b(1 - x)(1 - 3x) + c(1 - x)(1 - 2x)$
On putting $x = \frac{1}{2}$, we get
$1 = 0 + b\left( 1 - \frac{1}{2} \right)\left( 1 - \frac{3}{2} \right) + 0$
$\Rightarrow \ \ 1 = b\left( \frac{1}{2} \right)\left( - \frac{1}{2} \right)$
$\Rightarrow \ \ \ b = - 4$
On putting $x = 1$, we get
$1 = a(1 - 2)(1 - 3) + 0 + 0$
$\Rightarrow \ \ 1 = a( - 1)( - 2)\ \ \Rightarrow \ \ a = \frac{1}{2}$
On putting $x = \frac{1}{3}$, we get
$1 = 0 + 0 + c\left( 1 - \frac{1}{3} \right)\left( 1 - \frac{2}{3} \right)$
$\Rightarrow \ \ 1 = c\left( \frac{2}{3} \right)\left( \frac{1}{3} \right)\ \ \Rightarrow \ \ c = \frac{9}{2}$
Now, $\frac{a}{1} + \frac{b}{3} + \frac{c}{5} = \frac{1}{2} + \frac{( - 4)}{3} + \frac{9}{5.2} = \frac{15 - 40 + 27}{30} = \frac{1}{15}$