Mathematics

The average of 35 numbers is 25, so their total sum is 35 × 25 = 875. If each number is multiplied by 5, the new sum becomes 875 × 5 = 4375. The new average is then 4375 ÷ 35 = 125.

The average of the first 20 natural numbers is calculated by summing them (1+2+...+20 = 210) and dividing by 20, which equals 10.5.

The first 10 odd numbers are 1, 3, 5, 7, 9, 11, 13, 15, 17, and 19. Their sum is 100. The average is calculated as the sum divided by the number of terms, which is 100 / 10 = 10. Therefore, the correct answer is option B (10).

The average age of the group decreased by 3 years when one person aged 42 was replaced by a new person. Since there are 10 persons, the total decrease in the sum of ages is 3 × 10 = 30 years. The difference between the replaced person's age and the new person's age is therefore 30 years. Hence, the new person's age = 42 - 30 = 12 years. This matches option C.

Let the number of clerks be x. The total average salary is given by (2*600 + x*84)/(2 + x) = 90. Solving this equation gives x = 840 clerks. Therefore, the number of clerks is 840, not officers as the question mistakenly asks.

The average age of 8 men increases by 2 years when two women replace two men aged 20 and 24 years. This means the total age of the group increases by 8 × 2 = 16 years. The combined age of the two men replaced is 20 + 24 = 44 years. Therefore, the combined age of the two women is 44 + 16 = 60 years. The average age of the two women is 60 ÷ 2 = 30 years. Hence, the correct answer is 30 years (Option C).

The total wage paid per day is (20×25) + (15×20) + (5×8) = 500 + 300 + 40 = Rs.840. The total number of workers is 20 + 15 + 5 = 40. Therefore, the average wage per day = 840 ÷ 40 = Rs.21.

The two automobiles are moving toward each other with a combined speed of 60 km/h + 40 km/h = 100 km/h. To cover the 150 km distance between them, the time taken is distance divided by speed, which is 150 km ÷ 100 km/h = 1.5 hours. Therefore, the correct answer is 1.5 hours.

Masood sells the plot to Aftab at a 20% profit, so Aftab's cost price = 8000 + 20% of 8000 = 8000 + 1600 = Rs. 9600. Aftab sells to Zahid at a 20% loss, so Zahid's cost price = 9600 - 20% of 9600 = 9600 - 1920 = Rs. 7680.

Let the total number of students be n. The total marks = 39n. Removing four students with marks 5, 12, 15, and 19 reduces total marks by 51 and students by 4, so new average is (39n - 51)/(n - 4) = 44. Solving gives n = 30.

Let the salaries for January, February, March, April, and May be J, F, M, A, and May respectively. Given the average salary for January to April is Rs. 8000, so (J + F + M + A)/4 = 8000, which implies J + F + M + A = 32000. The average salary for February to May is Rs. 8500, so (F + M + A + May)/4 = 8500, which implies F + M + A + 6500 = 34000, so F + M + A = 27500. Subtracting the two sums, we get J + F + M + A - (F + M + A) = 32000 - 27500, which simplifies to J = 4500. Therefore, the salary for January is Rs. 4500, which corresponds to option C.

Let the four numbers be a, b, c, d. The average of the first three numbers is 16, so a + b + c = 48. The average of the last three numbers is 15, so b + c + d = 45. The sum of the first and last numbers is 13, so a + d = 13. Subtracting the second equation from the first gives a - d = 3. Solving the system of equations a + d = 13 and a - d = 3, we get a = 8 and d = 5. Therefore, the last number is 5, which corresponds to option C.

Let the two consecutive positive integers be n and n+1. The sum of their squares is n^2 + (n+1)^2, and their product is n(n+1). According to the problem, n^2 + (n+1)^2 = n(n+1) + 91. Simplifying, we get 2n^2 + 2n + 1 = n^2 + n + 91, which reduces to n^2 + n - 90 = 0. Solving this quadratic equation gives n = 9 or n = -10 (discard negative). Thus, the integers are 9 and 10. Checking the original condition: 9^2 + 10^2 = 81 + 100 = 181; their product is 90; 181 - 90 = 91, which matches the problem statement. Therefore, the correct answer is A (9, 10).

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.

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.

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 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 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 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.

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 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.

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 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.

Given the sum of five numbers is 655, the average of the first two numbers is 85, so their sum is 2 × 85 = 170. The third number is 125. Therefore, the sum of the first three numbers is 170 + 125 = 295. The sum of the last two numbers is 655 - 295 = 360. The average of the last two numbers is 360 ÷ 2 = 180. Hence, the correct answer is 180, which corresponds to option A.

Let the roots be 2k and 3k since their ratio is 2:3. The sum of roots for the quadratic equation 2x² - 5x + b = 0 is given by -b/a = 5/2 (since sum of roots = - coefficient of x / coefficient of x² = 5/2). So, 2k + 3k = 5k = 5/2, which gives k = 1/2. The product of roots is c/a = b/2. The product of roots also equals (2k)(3k) = 6k² = 6*(1/2)² = 6*(1/4) = 3/2. Equating these, b/2 = 3/2, so b = 3. Therefore, the correct value of b is 3, which corresponds to option A.