If 35 is removed from the data: 30, 34, 35, 36, 37, 38, 39, 40,

The given data set is

$30,34,35,36,37,38,39,40$

$n=8$ (even)

median $=\frac{\left(\frac{n}{2}\right)^{\text {th }} \operatorname{term}+\left(\frac{n}{2}+1\right)^{\text {th }} \text { term }}{2}$

$=\frac{4^{\text {th }} \text { term }+5^{\text {th }} \text { term }}{2}$

$=\frac{36+37}{2}$

$=\frac{73}{2}$

median $=36.5$

If 35 is removed, then the new data set is 

30, 34, 36, 37, 38, 39, 40

n = 7 (odd)

Therefore,

$\operatorname{median}=\left(\frac{7+1}{2}\right)^{\text {th }}$ term

$=\left(\frac{8}{2}\right)^{\text {th }}$ term

$=4^{\text {th }}$ term

median $=37$

Increase in median

$=37-36.5$

$=0.5$

Hence, the correct option is (d).