Mathematics

The power set of a set with n elements contains 2^n elements. Since the set {1, 2, 3} has 3 elements, its power set has 2^3 = 8 elements.

The differences between the numbers are increasing by 2 each time: 5-2=3, 12-5=7, 23-12=11, 38-23=15, 57-38=19. The next difference should be 23 (19+4), so the next number is 57+23=80. Since 80 is not an option, the closest correct option is 'None of these'. However, since 'None of these' is option D, and the original options do not include 80, the correct answer is D.

If 10 men take 12 days, the total work is 10 × 12 = 120 man-days. With 12 men, the time taken is 120 ÷ 12 = 10 days. However, the options given are 12, 10, 9, and 8 days. The correct calculation shows 10 days, which corresponds to option B, not C. Rechecking: 10 men × 12 days = 120 man-days; 120 man-days ÷ 12 men = 10 days. So correct answer is B, not C. Therefore, correct_answer should be B.

Five years ago, the total age of A, B, C, and D was 4 × 45 = 180 years. Their current total age is 180 + (4 × 5) = 200 years. The average age of all five now is 49, so total age of all five is 5 × 49 = 245 years. Therefore, E's age is 245 - 200 = 45 years. However, this contradicts the options, so let's re-check: The question states 'Five years ago, average age was 45', so total age five years ago = 4 × 45 = 180. Current total age of four = 180 + (4 × 5) = 200. Total age of five now = 5 × 49 = 245. So, E's age now = 245 - 200 = 45 years. The correct answer is 45 years, which corresponds to option C, not D. Therefore, correct answer is C.

Let Arif's share be x. Hafiz's share is 4x, and Arif's share is one-third of Haider's, so Haider's share is 3x. Total shares: x + 4x + 3x = 8x = 2560, so x = 2560/8 = 320. Therefore, Arif's share is Rs. 320.

The total age of A, B, C, D, and E is 40 × 5 = 200 years. The total age of A and B is 35 × 2 = 70 years, and the total age of C and D is 42 × 2 = 84 years. Therefore, the age of E = 200 - (70 + 84) = 46 years. However, since 46 is option B, the correct answer is B.

A and B together complete the work in 15 days, so their combined rate is 1/15 work per day. B alone completes the work in 20 days, so B's rate is 1/20 work per day. Therefore, A's rate is (1/15 - 1/20) = (4/60 - 3/60) = 1/60 work per day, meaning A alone can complete the work in 60 days. However, since 60 is option A, the correct answer should be A, not D. Rechecking the calculation: A's rate = 1/15 - 1/20 = (4 - 3)/60 = 1/60, so A alone takes 60 days. So the correct answer is A.

Let the original length be L and the width be W = 2L. The original area is A = L × 2L = 2L². After the increase, the new length is 1.3L and the new width is 1.2 × 2L = 2.4L. The new area is 1.3L × 2.4L = 3.12L². The percentage change in area is ((3.12L² - 2L²) / 2L²) × 100% = (1.12 / 2) × 100% = 56%. Therefore, the area increases by 56%. This matches option A, confirming the original answer is correct.

Twelve men take 6 hours, so total work = 12 × 6 = 72 man-hours. After 1 hour, 12 men complete 12 man-hours, leaving 60 man-hours. Now, 20 men work to finish 60 man-hours, so time = 60 ÷ 20 = 3 hours. However, the question asks how many more hours would 20 men take after the first hour, so the answer is 3 hours (option A). The original options have 3 as A, so correct answer is A, not C.

Let the four consecutive odd numbers be n, n+2, n+4, and n+6. Their sum is 4n + 12. Dividing by 10 gives (4n + 12)/10. For this to be a perfect square, (4n + 12) must be divisible by 10 and the quotient must be a perfect square. Rewrite the sum as (n-3)+(n-1)+(n+1)+(n+3) = 4n, where n is the average of the four numbers. Since the numbers are two-digit odd numbers, n ranges between 14 and 96. Dividing the sum by 10 gives 4n/10 = 2n/5. For this to be an integer perfect square, 2n must be divisible by 5, so n must be a multiple of 5/2, meaning n must be a multiple of 5 and even. Let n = 10m. Then the quotient is 4n/10 = 4(10m)/10 = 4m. For 4m to be a perfect square, m must be a perfect square. The possible values of m within the range give n = 40 or 90, corresponding to sums 160 and 360, which divided by 10 give 16 and 36, both perfect squares. The four numbers with average 40 are 37, 39, 41, 43. Among the options, 41 is included here. Testing the options: - For n=21: sum=21+23+25+27=96, 96/10=9.6 (not perfect square) - For n=25: sum=25+27+29+31=112, 112/10=11.2 (not perfect square) - For n=41: sum=41+43+45+47=176, 176/10=17.6 (not perfect square) - For n=67: sum=67+69+71+73=280, 280/10=28 (not perfect square) However, the average n=40 (numbers 37,39,41,43) gives sum=160, 160/10=16 (perfect square). Since 41 is one of these numbers, option C (41) is correct. Therefore, the correct answer is C (41).