If the area of the region $\left\{(x, y): 1-2 x \leqslant y \leqslant 4-x^{2}, x \geqslant 0, y \geqslant 0\right\}$ is $\frac{\alpha}{\beta}, \alpha, \beta \in \mathbf{N}, \operatorname{gcd}(\alpha, \beta)=1$, then the value of $(\alpha+\beta)$ is :

• A

  $67$
• B

  $91$
• Correct answer
  C

  73
• D

  85

Solution

Required area $= \dfrac{2}{3} \times 8 - \dfrac{1}{2} \times \dfrac{1}{2} \times 1$
$= \dfrac{16}{3} - \dfrac{1}{4} = \dfrac{61}{12} = \dfrac{\alpha}{\beta}$
$\Rightarrow \alpha + \beta = 73$

Hence, the correct answer is option: 'C'.