$\frac{\Delta \mathrm{T}}{\mathrm{T}}=\frac{1}{2} \frac{\Delta \ell}{\ell}$

When clock gain $12\, sec$

$\frac{12}{\mathrm{T}}=\frac{1}{2} \alpha(40-\theta)$             $...(1)$

When clock lose $4\, sec.$

$\frac{4}{\mathrm{T}}=\frac{1}{2} \alpha(\theta-20)$             $...(2)$

From equation $( 1)\&(2)$

$3=\frac{40-\theta}{\theta-20}$

$3 \theta-60=40-\theta$

$4 \theta=100$

$\theta=25^{\circ} \mathrm{C}$

from equation $( 1 )$

$\frac{12}{\mathrm{T}}=\frac{1}{2} \alpha(40-25)$

$\frac{12}{24 \times 3600}=\frac{1}{2} \alpha \times 15$

$\alpha=\frac{24}{24 \times 3600 \times 15}$

$\alpha=1.85 \times 10^{-15} /^{\circ} \mathrm{C}$