Explanation

(a)
Given equation of curve is $y = a\sqrt{x} + bx$. This curve passes through (1, 2)
$\therefore 2 = a + b$ …(i)
and area bounded by the curve and line $x = 4$ and $x$-axis is 8 sq unit, then
$\int_{0}^{4}{\left( a\sqrt{x} + bx \right)dx = 8}$
$\Rightarrow \frac{2a}{3}\left\lbrack x^{3/2} \right\rbrack_{0}^{4} + \frac{b}{2}\left\lbrack x^{2} \right\rbrack_{0}^{4} = 8$
$\Rightarrow \frac{2a}{3} \bullet 8 + 8b = 8\ \Rightarrow \ 2a + 3b = 3$ …(ii)
On solving Eqs. (i) and (ii), we get
$a = 3$ and $b = - 1$