Mathematics

The area of the rectangle is 125 cm × 64 cm = 8000 cm². The area of the square is five times that, so 5 × 8000 = 40000 cm². The side length of the square is √40000 = 200 cm. The perimeter is 4 × 200 = 800 cm.

Let the total number of students be N. The total mean score is 52, so total sum = 52N. The brightest 20% (0.2N students) have a mean score of 80, so their total sum = 0.2N × 80 = 16N. The dullest 25% (0.25N students) have a mean score of 31, so their total sum = 0.25N × 31 = 7.75N. The remaining 55% (0.55N students) have a total sum = 52N - 16N - 7.75N = 28.25N. Therefore, the mean score of the remaining 55% = 28.25N / 0.55N = 51.36 approximately, which corresponds to option C (51.4 approx.). The initial confusion arose from assuming N=100 but miscalculating sums. The algebraic approach confirms option C is correct.

Solving the first quadratic equation a² - 9a + 20 = 0 gives roots a = 4 or 5. Solving the second quadratic 2b² - 5b - 12 = 0 gives roots b = 3 or -2.5. The possible values of a are greater than or equal to the possible values of b, so the correct relationship is a ≥ b.

The area of a parallelogram is calculated by multiplying the base by the height. Given the base is 24 cm and the height is 16 cm, the area is 24 cm × 16 cm = 384 cm². Therefore, the correct answer is option B (384 cm²). The original question had a mismatch between the correct answer and the explanation; the explanation was correct but the answer key was incorrect.

The garden has a diameter of 4 meters, so its radius is 2 meters. The path is 25 cm (0.25 meters) wide, making the outer radius 2 + 0.25 = 2.25 meters. The area of the path is the difference between the areas of the larger and smaller circles: π(2.25² - 2²) = π(5.0625 - 4) = π(1.0625) ≈ 3.34 square meters.

If the roots of a quadratic equation are 20 and -7, the equation can be formed using the factorized form (x - 20)(x + 7) = 0. Expanding this gives x² + 7x - 20x - 140 = 0, which simplifies to x² - 13x - 140 = 0. Therefore, the correct quadratic equation is x² - 13x - 140 = 0, which corresponds to option C.

The volume of the large cube is 1 m³ (since side = 1 m). Each small cube has a side of 10 cm = 0.1 m, so its volume is (0.1 m)³ = 0.001 m³. The number of small cubes is the volume ratio: 1 m³ / 0.001 m³ = 1000.

The average speed for a round trip when speeds differ is given by the formula: 2 * speed1 * speed2 / (speed1 + speed2). Here, speed1 = 40 km/h and speed2 = 40 + 50% of 40 = 60 km/h. So, average speed = 2 * 40 * 60 / (40 + 60) = 4800 / 100 = 48 km/h.

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.

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.

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.

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).

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 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.

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.

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.

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 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.

The ratio of volumes of two cubes is 729 : 1331, which corresponds to the cubes of their side lengths. Taking cube roots, the ratio of their sides is 9 : 11. Since surface area scales with the square of the side length, the ratio of their total surface areas is (9^2) : (11^2) = 81 : 121.

Let the number of papers be n. The total marks initially are 63n. After adding 20 marks to the Geography paper and 2 marks to the History paper, the total marks become 63n + 22. The new average is 65, so (63n + 22)/n = 65. Solving this gives 63n + 22 = 65n, or 22 = 2n, so n = 11. Therefore, the number of papers in the examination is 11, which corresponds to option D.

The total marks for each batch are calculated by multiplying the number of students by their average marks: (55 × 50) = 2750, (60 × 55) = 3300, and (45 × 60) = 2700. Adding these gives the combined total marks: 2750 + 3300 + 2700 = 8750. The total number of students is 55 + 60 + 45 = 160. The average marks of all students combined is the total marks divided by the total number of students: 8750 ÷ 160 = 54.6875, which rounds to 54.68. Therefore, option B (54.68) is the correct answer, not option A (53.33).

Given the quadratic equation x² + 8x + 4 = 0, the sum of roots a + b = -8 and the product ab = 4. The expression a/b + b/a can be rewritten as (a² + b²)/ab. Using the identity a² + b² = (a + b)² - 2ab, we get ( (-8)² - 2*4 ) / 4 = (64 - 8) / 4 = 56 / 4 = 14.

The discriminant of the quadratic equation 3x² – 12x + 10 = 0 is Δ = (-12)² - 4*3*10 = 144 - 120 = 24, which is positive but not a perfect square. This means the roots are real, unequal, and irrational.

Solving the quadratic equation a² + 11a + 30 = 0 by factoring gives (a + 5)(a + 6) = 0, so the roots are a = -5 and a = -6. Solving the quadratic equation b² + 6b + 5 = 0 by factoring gives (b + 1)(b + 5) = 0, so the roots are b = -1 and b = -5. Comparing the roots, the largest root of a is -5 and the largest root of b is -1, so a < b. Therefore, the correct relationship is a < b, which corresponds to option A.

The quadratic equation x² + 2x - 15 = 0 can be factored as (x + 5)(x - 3) = 0. Setting each factor equal to zero gives the roots x = -5 and x = 3. Therefore, the correct roots are -5 and 3, which corresponds to option A.