Which of the following statements has the truth value T ?

A: cube roots of unity are in Geometric progression and their sum is 1

B: $4+7>10$ iff $2+8<10$

C: $\exists x \in \mathbb{N}$ such that $x^2-3 x+2=0$ and $\exists \mathrm{n} \in \mathbb{N}$ such that n is an odd number

D: $3+\mathrm{i}$ is a complex number or $\sqrt{2}+\sqrt{3}=\sqrt{5}$

If $\{(\mathrm{p} \wedge \sim \mathrm{q}) \wedge(\mathrm{p} \wedge \mathrm{r})\} \rightarrow \sim \mathrm{p} \vee \mathrm{q}$ has truth value false then truth values of the statements $p, q, r$ are respectively

The correct simplified circuit diagram for the logical statement $[\{\mathrm{q} \wedge(\sim \mathrm{q} \vee \mathrm{r})\} \wedge\{\sim \mathrm{p} \vee(\mathrm{p} \wedge \sim \mathrm{r})\}] \vee(\mathrm{p} \wedge \mathrm{r})$ Where $p, q, r$ represents switches $s_1, s_2, s_3$ respectively.

The logical statement

$$ [\sim(\sim p \vee q) \vee(p \wedge r) \wedge(\sim q \wedge r)] $$

is equivalent to