In how many distinct ways can the letters of the word PARRAMATTA be arranged? Also, evaluate log base 2 of 49.

The word is PARRAMATTA.
Total number of letters, $n = 10$.
The letter 'A' appears 4 times.
The letter 'R' appears 2 times.
The letter 'T' appears 2 times.
The letters 'P' and 'M' appear 1 time each.

The number of distinct arrangements is given by $\frac{n!}{n_1! n_2! \cdots n_k!}$, where $n$ is the total number of letters, and $n_1, n_2, \ldots, n_k$ are the frequencies of the repeated letters.

$10! = 3,628,800$
$4! = 4 \times 3 \times 2 \times 1 = 24$
$2! = 2 \times 1 = 2$

The correct option is C.
The number of distinct ways is $\boxed{\text{C. } 37800}$.

Q7. Evaluate $\log_2 49$

There appears to be a typo in the question, as $\log_2 49$ is not an integer and does not match any of the given integer options. However, if the question intended to ask for $\log_7 49$, then the solution would be:

We know that $49 = 7^2$.

Assuming the question intended to be $\log_7 49$, the correct option is B.
The value is $\boxed{\text{B. } 2}$.

Q8. Calculate $\frac{dy}{dx}$ if $y = 5^x$.

The derivative of $a^x$ with respect to $x$ is $\frac{d}{dx}(a^x) = a^x \ln(a)$.

Given $y = 5^x$, where $a=5$.

The correct option is B.
The derivative is $\boxed{\text{B. } 5^x \ln(5)}$.

Q11. Determine the remainder if $3x^3 - 5x^2 - 2x + 3$ is divided by $(x-1)$.

The Remainder Theorem states that if a polynomial $P(x)$ is divided by $(x-c)$, the remainder is $P(c