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

Here are the solutions to the questions from the image: Q6. In how many distinct ways can the letters of the word PARRAMATTA be arranged? Step 1: Count the total number of letters and the frequency of each repeated letter. 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. Step 2: Use the formula for permutations with repetitions. The number of distinct arrangements is given by (n!)/(n_1! n_2! ·s n_k!), where n is the total number of letters, and n_1, n_2, , n_k are the frequencies of the repeated letters. Number of arrangements = (10!)/(4! · 2! · 2!) Step 3: Calculate the factorials and simplify. 10! = 3,628,800 4! = 4 × 3 × 2 × 1 = 24 2! = 2 × 1 = 2 Number of arrangements = (3,628,800)/(24 · 2 · 2) = (3,628,800)/(96) = 37,800 The correct option is C. The number of distinct ways is C. 37800. Q7. Evaluate _2 49 There appears to be a typo in the question, as _2 49 is not an integer and does not match any of the given integer options. However, if the question intended to ask for _7 49, then the solution would be: Step 1: Rewrite the argument as a power of the base. We know that 49 = 7^2. Step 2: Apply the logarithm property _b b^x = x. _7 49 = _7 7^2 = 2 Assuming the question intended to be _7 49, the correct option is B. The value is B. 2. Q8. Calculate (dy)/(dx) if y = 5^x. Step 1: Recall the differentiation rule for exponential functions. The derivative of a^x with respect to x is (d)/(dx)(a^x) = a^x (a). Step 2: Apply the rule to the given function. Given y = 5^x, where a=5. (dy)/(dx) = 5^x (5) The correct option is B. The derivative is B. 5^x (5). Q11. Determine the remainder if 3x^3 - 5x^2 - 2x + 3 is divided by (x-1). Step 1: Use the Remainder Theorem. The Remainder Theorem states that if a polynomial P(x) is divided by (x-c), the remainder is $P(c