1, 1, 2, 6, 24,?, 720

Let the daughter's age be x years.

Then, father's age = (3x) years.

Mother's age = (3x - 9) years; Son's age = (x + 7) years.

So, x + 7 = (3x-9)/2 ⟺ 2x + 14 = 3x - 9 ⟺ x = 23.

Therefore Mother's age = (3X - 9) = (69 - 9) years = 60 years.

Suppose the boy got x sums right and 2x sums wrong.

Then, x + 2x = 48 ⟺ 3x = 48 ⟺ x = 16.

Clearly, we have to first find two numbers whose difference is 2 and of which the smaller one is a perfect square and the bigger one a perfect cube.

Such numbers are 25 and 27.

Thus, Nitin is now 26 years old. Since the next perfect cube after 27 is 64,

so required time period = (64 - 26) years = 38 years.

Let the fixed charge be Rs. x and variable charge be Rs.y per km. Then,

x + 16y = 156 ...(i) and

x + 24y = 204 ...(ii)

Solving (i) and (ii), we get: x = 60, y = 6.

Therefore Cost of travelling 30 km = 60 + 30 y = Rs. (60 + 30 x 6) = Rs. 240.

Let the missing terms of the series be x₁ and x₂.

Thus, the sequence 2, 15, 4, 12, 6, 7, x₁ x₂ is a combination of two series :

I. 2, 4, 6, x₁ and II. 15, 12, 7, x₂I consists of consecutive even numbers.

So, missing term, x₁ = 8.

The pattern in II is - 3, - 5,......So, missing term, x₂ = 7 - 7 = 0.

The pattern is + 1008.

So, missing term - 6848 + 1008 = 7856.

Each term in the series is obtained by subtracting from the preceding term the number

formed by the first three digits of the preceding term.

So, missing term = 5242 - 524 = 4718.

Each term in the series, except the first two terms, is the product of the preceding two terms.

So, missing term = 16 x 128 = 2048.

The pattern is + 1, x 3, + 1, x 3, + 1, x 3, + 1,.....

So, missing term = 67 x 3 = 201.