Mathematics

The perimeter of a sector of a circle is calculated as the sum of the lengths of the two radii and the arc length. The arc length is given by (central angle/360) × 2π × radius. For a radius of 21 cm and a central angle of 135°, the arc length = (135/360) × 2π × 21 = 49.5 cm (using π ≈ 22/7). Adding the two radii (2 × 21 = 42 cm) gives the perimeter = 42 + 49.5 = 91.5 cm. This matches option A, so the correct answer is A.

The area of an equilateral triangle is given by the formula (√3 / 4) × side². For side 16 cm, area = (√3 / 4) × 16² = (√3 / 4) × 256 = 64√3 cm². However, option B is 128√3 cm², which is double the correct area. The correct area matches option D, 64√3 cm², so the original correct_answer 'D' is correct. The explanation was missing and has been added. The question text had a typo 'are' instead of 'area', which has been corrected.

The total amount after the new member joins is 7 × 20 = Rs. 140. The new member contributes Rs. 50, so the total amount before joining was 140 - 50 = Rs. 90. Since there were 6 members before, the average before joining is 90 ÷ 6 = Rs. 15.

Let the average age of the whole team be x years. The total age of the team is 11x. The captain is 26 years old, and the wicket keeper is 29 years old (3 years older). Excluding these two, the sum of the remaining 9 players' ages is 11x - 26 - 29 = 11x - 55. The average age of these 9 players is (11x - 55)/9, which is given as one year less than the whole team's average, so (11x - 55)/9 = x - 1. Solving this equation gives x = 24 years.

The total area of the four walls is 2 × height × (length + breadth) = 2 × 12 × (25 + 15) = 960 sq ft. The area of the door is 6 × 3 = 18 sq ft, and the total area of the three windows is 3 × (4 × 3) = 36 sq ft. The net area to be whitewashed is 960 - (18 + 36) = 906 sq ft. At Rs. 5 per sq ft, the cost is 906 × 5 = Rs. 4530. However, the correct calculation shows the cost as Rs. 4530, which matches option D, so the original correct answer is correct. Rechecking the calculation: 2 × 12 × (25 + 15) = 2 × 12 × 40 = 960 sq ft; subtract door and windows area (18 + 36 = 54), net area = 906 sq ft; cost = 906 × 5 = Rs. 4530. Therefore, option D is correct.

The distance covered by the wheel is the circumference multiplied by the number of revolutions. Circumference = 2 × π × radius = 2 × 3.14 × 22.4 cm = 140.4 cm = 1.404 m. Distance = 1.404 m × 500 = 702 m approximately. However, the closest option to this calculation is 704 m (option B). On rechecking, the radius is 22.4 cm, so circumference = 2 × π × 22.4 = 140.5 cm or 1.405 m. Distance = 1.405 × 500 = 702.5 m, which matches option B (704 m) closely. Therefore, the original correct answer B is correct. The initial calculation error was due to rounding. The correct answer is B.

The original total marks for 30 students is 30 × 58.5 = 1755. The incorrect mark recorded was 57 instead of 75, so the total should be corrected by adding (75 - 57) = 18. The corrected total is 1755 + 18 = 1773. The corrected mean is 1773 ÷ 30 = 59.1.

The run rate is calculated by dividing the total runs scored by the number of overs faced. In the first 10 overs, the run rate was 3.2, so runs scored = 3.2 × 10 = 32 runs. The target is 282 runs, so remaining runs needed = 282 - 32 = 250 runs. The remaining overs are 40. Therefore, the required run rate for the remaining overs = 250 ÷ 40 = 6.25 runs per over. This corresponds to option A.

The total sum of the four integers is 69 × 4 = 276. Given the highest integer is 93 and the least is 39, their sum is 132. The remaining two integers sum to 276 - 132 = 144, and their difference is 28. Let the two integers be x and y, with x > y. Then x + y = 144 and x - y = 28. Adding these gives 2x = 172, so x = 86. However, 86 is not among the options except B, but option B is 86. Wait, the options are A:58, B:86, C:49, D:None of these. So the higher integer is 86, which is option B. The original correct_answer is B, which is correct. The explanation was missing, so it is added now. Difficulty is medium due to algebraic reasoning required. Tags added accordingly.

When each number is increased by 5, the mean also increases by 5. Since the original mean is 35, the new mean will be 35 + 5 = 40.