Mathematics

The expression 2^√64 means 2 raised to the power of the square root of 64. Since √64 = 8, the expression becomes 2^8 = 256, which is not among the options. However, the options suggest a misunderstanding. If the question intended 2^(√64) = 2^8 = 256, none of the options match. If the question meant (2^√64) = 2^8 = 256, then 'None of these' would be correct. But if the question is interpreted as 2^(√64) = 2^8 = 256, none of the options A, B, or C are correct. Therefore, the best correction is to clarify the question and options. Alternatively, if the question is 2^(√64) = 2^8 = 256, none of the options match, so 'D' is correct. Since the original correct_answer is B (2), which is incorrect, the correct answer should be D (None of these).

Given the pth term is q and the qth term is p in an arithmetic progression, solving the equations shows the (p+q)th term equals zero.

If 20% of the bill is deducted, 80% remains to be paid. Since Rs. 100 is 80% of the original bill, the original bill = 100 ÷ 0.8 = Rs. 125.

The sum of n terms of an arithmetic progression (A.P.) is given by S_n = n/2 (a + l), where a is the first term and l is the last term. Here, a = 1/n and l = n^2 - n + 1/n. Thus, S_n = n/2 [1/n + (n^2 - n + 1/n)] = n/2 [n^2 - n + 2/n] = n/2 (n^2 - n) + n/2 * 2/n = (n^3 - n^2)/2 + 1 = (n^3 - n^2 + 2)/2. None of the given options exactly match this expression, so the correct choice is 'None of these'.

The phrase 'subtract eight times of the number b from ten times of the number c' means 10c minus 8b, which corresponds to option C. Option B reverses the order, which is incorrect.

The series 1, 5, 9, 13, ... is an arithmetic progression with first term a = 1 and common difference d = 4. The sum of the first n terms of an arithmetic progression is given by the formula S_n = n/2 [2a + (n-1)d]. For n = 40, the sum is S_40 = 40/2 [2*1 + 39*4] = 20 [2 + 156] = 20 * 158 = 3160. Since 3160 is not among the given options, the correct answer is "None of these" (option D).

Solving the system of equations: From x + 2y = 13 and x + y + z = 12 and 2y + z = 11, substituting and solving yields x = 4, y = 5, and z = 3, which matches option A.

Ali's current age is given as (x - 10) years. After 12 years, his age will be (x - 10) + 12 = x + 2 years. Therefore, the correct answer is option A: x + 2.

If A's salary is 50% more than B's, then A = 1.5 × B. To find by what percent B's salary is less than A's, we calculate the percentage difference relative to A: ((A - B) / A) × 100 = ((1.5B - B) / 1.5B) × 100 = (0.5B / 1.5B) × 100 = 33 1/3%. This matches option C, which correctly represents 33 1/3%. Options A and B represent 33.25% and 33.5% respectively, which are incorrect. Therefore, the correct answer is option C.

5% of Rs. 100 is Rs. 5. Then, 5% of Rs. 5 is Rs. 0.25. Therefore, 5% of 5% of Rs. 100 equals Rs. 0.25.