Mathematics

The sequence given is 9, 16, x, 36. These numbers correspond to perfect squares of consecutive integers: 9 = 3², 16 = 4², x = 5², and 36 = 6². Therefore, the missing number x is 25 (5²). Since 25 corresponds to option D, the correct answer is D.

The sequence consists mostly of perfect squares: 1 (1²), 4 (2²), 9 (3²), 16 (4²), and 36 (6²). The number 26 is not a perfect square, making it the odd one out.

The volume of the pool when half filled is 20 x 20 x 3 = 1200 cubic meters. Since 1 cubic meter of water weighs 1000 kg, the weight of the water is 1200 x 1000 = 1,200,000 kg, which is not listed among the options. Therefore, the correct answer is 'None of these'.

The question asks for the distance traveled when going around a circular path 12 times. The circumference of a circle is π times the diameter, so circumference = π × 400 m ≈ 1256.64 m. Traveling 12 times around means distance = 12 × 1256.64 m ≈ 15079.68 m or about 15.08 km. None of the options A (12 km), B (13 km), or C (14 km) are close to this value, so the correct answer is D (None of these).

First, calculate 12 divided by 8, which equals 1.5. Then add 2.5 to get 4.0. Therefore, the correct answer is 4, which corresponds to option C.

The problem involves finding the straight-line distance (aerial length) from a point on the ground 15 meters away from the base of a 20-meter tall building to the top of the building. This forms a right triangle where the building height is one leg (20 m), the horizontal distance from the building is the other leg (15 m), and the aerial length is the hypotenuse. Using the Pythagorean theorem: hypotenuse = √(20² + 15²) = √(400 + 225) = √625 = 25 meters. Therefore, the correct answer is 25 m, which corresponds to option B.

A binary series consists of only two distinct values alternating or repeating in some pattern. Option A (10, 20, 30) has three distinct values, so it is not a binary series. Options B and C contain only two distinct values each, fitting the definition of a binary series. Option D ('None of these') is incorrect because option A is indeed not a binary series.

Given the system of equations: x + y = 99 and x - y = 9. Adding these two equations results in 2x = 108, so x = 54. Substituting x = 54 into the first equation gives 54 + y = 99, so y = 45. Therefore, the solution is x = 54 and y = 45. However, none of the options list x = 54 and y = 45 in that order. Option A lists (45, 54), which reverses the values. Hence, the correct answer is D (None of these).

The average is calculated by summing the terms and dividing by the number of terms: (4a + (-4a) + 8a + (-8a)) / 4 = (0) / 4 = 0.

The average is calculated by summing the fractions and dividing by the number of terms: (1/2 + 1/4 + 1/6 + 1/8) ÷ 4 = (0.5 + 0.25 + 0.1667 + 0.125) ÷ 4 = 1.0417 ÷ 4 = 0.2604. However, this is approximately 0.260, which matches option C. Rechecking the sum: 0.5 + 0.25 + 0.1667 + 0.125 = 1.0417, average = 1.0417/4 = 0.2604. So option C is correct. The initial confusion was due to misreading the options. Therefore, the correct answer is C.