Step 1: Understanding the Concept:
We must systematically evaluate the individual truth values (True or False) of the three distinct mathematical statements $p$, $q$, and $r$. Once we have their individual boolean values, we substitute them into the logical structures provided in the options to see which evaluates to True.
Step 2: Key Formula or Approach:
- A universal statement ($\forall n$) is proven False by finding just one counterexample.
- Direction cosines $l, m, n$ must mathematically satisfy the identity $l^2 + m^2 + n^2 = 1$.
- A function is increasing on an interval if its first derivative is positive (or non-negative) throughout that interval.
Step 3: Detailed Explanation:
Let's evaluate statement $p$:
It claims $10n-3$ is prime for all $n \in \mathbb{N}$ not divisible by 3. Let's test values.
$n = 1$: $10(1)-3 = 7$ (Prime)
$n = 2$: $10(2)-3 = 17$ (Prime)
$n = 4$: $10(4)-3 = 37$ (Prime)
$n = 5$: $10(5)-3 = 47$ (Prime)
$n = 7$: $10(7)-3 = 67$ (Prime)
$n = 8$: $10(8)-3 = 77$. Note that $77 = 7 \times 11$, which is a composite number. Since $n=8$ is not divisible by 3, this is a valid counterexample.
Thus, statement $p$ is False ($F$).
Let's evaluate statement $q$:
It claims the values are direction cosines. They must satisfy $l^2 + m^2 + n^2 = 1$.
Calculate the sum of squares:
\[ \left(\frac{2}{\sqrt{3}}\right)^2 + \left(\frac{-2}{\sqrt{3}}\right)^2 + \left(\frac{-1}{\sqrt{3}}\right)^2 = \frac{4}{3} + \frac{4}{3} + \frac{1}{3} = \frac{9}{3} = 3 \]
Since $3 \neq 1$, they are not direction cosines.
Thus, statement $q$ is False ($F$).
Let's evaluate statement $r$:
The function is $f(x) = \sin x$. Its derivative is $f'(x) = \cos x$.
In the interval $[-\pi/2, \pi/2]$ (which corresponds to the 4th and 1st quadrants), the value of $\cos x$ is strictly $\ge 0$.
Since the derivative is non-negative everywhere in the interval, the function $\sin x$ is indeed increasing.
Thus, statement $r$ is True ($T$).
Now, substitute $(p=F, q=F, r=T)$ into the options to find which expression evaluates to True:
(A) $(F \land F) \leftrightarrow T \equiv F \leftrightarrow T \equiv \text{False}$
(B) $(F \rightarrow F) \rightarrow \sim T \equiv T \rightarrow F \equiv \text{False}$
(C) $(\sim F \lor F) \land T \equiv (T \lor F) \land T \equiv T \land T \equiv True$
(D) $(\sim F \land \sim F) \leftrightarrow \sim T \equiv (T \land T) \leftrightarrow F \equiv T \leftrightarrow F \equiv \text{False}$
Step 4: Final Answer:
The true pattern is $(\sim p \lor q) \land r$.