Mathematics

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.

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.

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 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.

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.

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.

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.

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 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.

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.