Mathematics

Let the original average be x. After 10 innings, total runs = 10x. After scoring 108 in the 11th inning, the new average is x + 6. So, (10x + 108)/11 = x + 6. Solving gives x = 46, so new average = 52 runs.

Let the weights of A, B, and C be a, b, and c respectively. From the averages: (a + b + c)/3 = 45, so a + b + c = 135. Also, (a + b)/2 = 40, so a + b = 80, and (b + c)/2 = 43, so b + c = 86. Adding a + b = 80 and b + c = 86 gives a + 2b + c = 166. Subtracting a + b + c = 135 from this yields b = 31. However, this contradicts the options. Rechecking calculations: a + b = 80, b + c = 86, a + b + c = 135. Subtracting a + b = 80 from a + b + c = 135 gives c = 55. Subtracting b + c = 86 from a + b + c = 135 gives a = 49. Using a + b = 80, b = 80 - a = 80 - 49 = 31. So b = 31 kg, which matches option D. Therefore, the original correct answer D is correct. The initial confusion was in the explanation. The correct weight of B is 31 kg.

Let the roots of the quadratic equation x^2 - 12x + a = 0 be r and 3r, where one root is thrice the other. According to Vieta's formulas, the sum of the roots is equal to the coefficient of x with opposite sign, so r + 3r = 4r = 12, which gives r = 3. The product of the roots is equal to the constant term a, so r × 3r = 3 × 9 = 27. Therefore, the value of a is 27, which corresponds to option B.

The total salary of 20 employees is 20 × 1500 = Rs. 30,000. When the manager's salary is added, the number of employees becomes 21 and the average salary increases by Rs. 100, making the new average 1600. Therefore, the total salary for 21 employees is 21 × 1600 = Rs. 33,600. The manager's salary is the difference between the new total and the original total, which is 33,600 - 30,000 = Rs. 3,600. Hence, the manager's monthly salary is Rs. 3,600, which corresponds to option C.

The total initial sum of heights is 35 × 180 = 6300 cm. The incorrect height recorded was 166 cm instead of 106 cm, so the total sum should be corrected by subtracting 166 and adding 106, resulting in 6300 - 166 + 106 = 6240 cm. The actual average height is 6240 ÷ 35 = 178.2857, which rounds to 178.29 cm. However, since the calculation shows 178.29 cm, option B matches this value exactly, but the original correct_answer was B. Rechecking the calculation: 6300 - 166 + 106 = 6240, 6240/35 = 178.2857 ≈ 178.29 cm, so the correct answer is B, not D. The initial verification incorrectly marked D. Therefore, the correct answer is B.

Let the number of boys be B and the number of girls be G. The total age of boys is 16.4B and that of girls is 15.4G. The overall average age is 15.8, so the equation is (16.4B + 15.4G) / (B + G) = 15.8. Multiplying both sides by (B + G) gives 16.4B + 15.4G = 15.8B + 15.8G. Rearranging terms: 16.4B - 15.8B = 15.8G - 15.4G, which simplifies to 0.6B = 0.4G. Dividing both sides by G and then by 0.6 gives B/G = 0.4/0.6 = 2/3. Therefore, the ratio of boys to girls is 2:3, which corresponds to option B.

The side of the square is √4096 = 64 cm. The rectangle's length is twice the side, so 128 cm, and its breadth is 64 - 24 = 40 cm. The ratio of breadth to length is 40:128, which simplifies to 5:16. However, the question asks for the ratio of breadth to length, so the ratio is 5:16, which corresponds to option B. Since option B is 7:16, which is incorrect, and option A is 18:5, also incorrect, the correct ratio is 5:16, which is not listed. Therefore, the correct answer should be "None of these" (D).

For the quadratic equation x² + 20x + 3 = 0, the sum of the roots is -b/a = -20/1 = -20, and the product of the roots is c/a = 3/1 = 3. Therefore, the correct answer is sum = -20 and product = 3.

Let the number of women workers be W. The average age of women is 22 years, and the average age of children is 8 years with 16 children. The combined average age is 15 years. Using the weighted average formula: (22W + 8*16) / (W + 16) = 15. Solving for W gives W = 12. Since 10 women are married, the number of unmarried women is 12 - 10 = 2. However, 2 is not an option, so rechecking the calculation: (22W + 128) = 15(W + 16) => 22W + 128 = 15W + 240 => 7W = 112 => W = 16. Number of unmarried women = 16 - 10 = 6, which matches option D. Therefore, the correct answer is D.

Solving the first quadratic equation a² - 7a + 12 = 0 gives roots a = 3 and a = 4. Solving the second quadratic equation b² - 3b + 2 = 0 gives roots b = 1 and b = 2. Since the smallest value of a (3) is greater than the largest value of b (2), the relationship a > b holds.