Question

Consider the three statements $\text{p} : \forall \text{n} \in \mathbb{N}, 10\text{n} - 3$ is a prime number, when n is not divisible by 3.
$\text{q} : \frac{2}{\sqrt{3}}, \frac{-2}{\sqrt{3}}, \frac{-1}{\sqrt{3}}$ are the direction cosines of a directed line.
$\text{r} : \sin x$ is an increasing function in the interval $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$.

• A. $(p \land q) \leftrightarrow r$
  (B) $(p \rightarrow q) \rightarrow \sim r$
  (C) $(\sim p \lor q) \land r$
  (D) $(\sim p \land \sim q) \leftrightarrow \sim r$
• B.
• C.
• D.

Hint:

Check truth values first, then evaluate expressions.

Answer 1

The Correct Option is C

Step 1: Evaluate statements.

• $p$: False (counterexample exists)
• $q$: False (sum of squares $\neq 1$)
• $r$: True (increasing in given interval)

Step 2: Check options. (C): \[ (\sim p \lor q) \land r = (T \lor F)\land T = T \]
Step 3: Conclusion. Correct option is (C)