Answer

Correct Option: (iv) $\{00 + (11)^*0\}^*$

Explanation

Test $w=1101$ against each regex:

(i) $1^*0^*(0+1)$: = $1^*$ then $0^*$ then single $0$ or $1$. $1101 = \underbrace{11}_{1^*} \underbrace{0}_{0^*} \underbrace{1}_{(0+1)}$ → belongs.

(ii) $1(0+1)^*101$: Must start with $1$ and end with $101$. $1101 = 1 \underbrace{1}_{(0+1)^*}101$? Actually $1101$ = $1$ + $\epsilon$ + $101$ with middle $1$? $1(0+1)^*101 = 1\,x\,101$ where $x=0$? For $w=1101$, $x=0$? Wait $1+0+101 =1101$ exactly, so $w = 1\cdot0\cdot101$? No $1\cdot101 =1101$ with $x=\epsilon$? $1(0+1)^* =1$, remaining $101 =101$ → $1101$ → belongs.

(iii) $(1(0)^*(0+1)^*(0+1))^*$: This is essentially $\Sigma^*$ variant (can generate any string) → belongs.

(iv) $\{00 + (11)^*0\}^*$: Generators: $00$ (ends 0) and $(11)^*0$ = even 1's followed by 0, also ends 0. Any concatenation of blocks ending in 0 always ends in 0. $1101$ ends in 1 → does not belong.

Final Answer

$1101
otin {00 + (11)^0}^$ → Option (iv).