Mathematics

Pipe A fills 1/20 of the cistern per minute, pipe B fills 1/30 per minute, and pipe C empties 1/40 per minute. Net fill rate = 1/20 + 1/30 - 1/40 = (6/120) + (4/120) - (3/120) = 7/120 per minute. Time to fill = 1 / (7/120) = 120/7 = 17 1/7 minutes. However, the calculation shows 17 1/7 minutes, which matches option C, so the original answer is correct. Rechecking the calculation: 1/20=0.05, 1/30=0.0333, 1/40=0.025; net=0.05+0.0333-0.025=0.0583; time=1/0.0583=17.14 minutes, which is 17 1/7 minutes. Therefore, the original correct answer C is correct. The initial confusion was in the explanation. The correct answer is C.

The side of the square is the radius of the circle. Since the area of the square is 1225 sq.units, the side length (radius) is √1225 = 35 units. The length of the rectangle is two-fifths of the radius, so length = (2/5) × 35 = 14 units. Given the breadth is 10 units, the area of the rectangle = length × breadth = 14 × 10 = 140 sq.units. However, the options show 140 as A, but the correct calculation gives 140, so the correct answer is A, not C. Rechecking calculations: length = (2/5)*35=14, area=14*10=140. So the original correct_answer 'A' is correct. Therefore, the correct answer is A.

The side of the square is half the radius of the larger circle, so the side length is √196 = 14 cm. Therefore, the radius of the larger circle is 2 × 14 = 28 cm. The radius of the smaller circle is (3/7) × 28 = 12 cm. The circumference of the smaller circle is 2π × 12 = 24π cm.

The total age of all seven persons is 7 × 28 = 196 years. The total age of the first three persons is 3 × 21 = 63 years, and the total age of the last three persons is 3 × 34 = 102 years. Adding these gives 63 + 102 = 165 years for six persons. Subtracting from the total, the middle person's age is 196 - 165 = 31 years. Therefore, the age of the person sitting in the middle of the row is 31 years, which corresponds to option B.

The total weight of 16 boys is 16 × 50.25 = 804 kg, and the total weight of 8 boys is 8 × 45.15 = 361.2 kg. The combined total weight is 804 + 361.2 = 1165.2 kg. The total number of boys is 16 + 8 = 24. Therefore, the average weight is 1165.2 ÷ 24 = 48.55 kg, which matches option C. However, rechecking the calculation shows 1165.2 ÷ 24 = 48.55, so option C is correct. The initial answer was incorrect; the correct average is 48.55 kg (option C).

Using Heron's formula, the semi-perimeter s = (26 + 24 + 10)/2 = 30 cm. The area = √[s(s-a)(s-b)(s-c)] = √[30(30-26)(30-24)(30-10)] = √[30 × 4 × 6 × 20] = √14400 = 120 cm². However, this calculation shows 120 cm², so let's re-check the arithmetic carefully: 30 - 26 = 4, 30 - 24 = 6, 30 - 10 = 20; product inside the root is 30 × 4 × 6 × 20 = 14400; square root of 14400 is 120. So the area is 120 cm², matching option A. Therefore, the original correct_answer 'A' is correct. The explanation has been added accordingly.

Given the quadratic equation 2x² + 5x + 3 = 0, the sum of roots is -5/2 and product of roots is 3/2. The quadratic with roots as reciprocals has sum = 1/(root1) + 1/(root2) = (root1 + root2)/(root1*root2) = (-5/2)/(3/2) = -5/3 and product = 1/(root1*root2) = 1/(3/2) = 2/3. So the new quadratic is x² - (sum)x + product = x² + (5/3)x + (2/3) = 0. Multiplying through by 3 gives 3x² + 5x + 2 = 0, which corresponds to option A after correcting the sign in the original options (see verification notes).

The total age of all 35 students is 35 × 16 = 560 years. The total age of 21 students is 21 × 14 = 294 years. The total age of the remaining 14 students is 560 - 294 = 266 years. Therefore, the average age of the remaining 14 students is 266 ÷ 14 = 19 years.

Let the total number of workers be x. The total salary is 8000x. The salary of 7 technicians is 7 × 12000 = 84000, and the salary of the rest (x - 7) workers is (x - 7) × 6000. So, 8000x = 84000 + 6000(x - 7). Solving gives x = 21.

The total sum of the five numbers is 5 × 27 = 135. When one number is excluded, the average of the remaining four numbers is 25, so their total sum is 4 × 25 = 100. The excluded number is the difference: 135 - 100 = 35.