(a) A telephone company charges initially Rs. 0.50 and then Rs. 0.11 for every minute. Write an expression that gives the cost of a call that lasts N minutes. (b) Find the missing number in the series below: (i) 1, 8, 4, 27, 9, ? (ii) 3, 6, 8, 16, 18, ? (iii) 2, 8, 512, ? (iv) 81, 9, 64, 8, ?, 12 (v) 6, 11, 21, 36, 56, ? (c) The perimeter of the rectangle given below is 114 cm. Find the area of the rectangle. (Top side (3x-y) cm, bottom side (2x+y) cm, right side (2x-3) cm.) (d) Ahmad stands at point D, 2 m in front of a spotlight at point A. He is 1.6 m tall and is facing the wall of a building which is 10.5 m away from him. How tall (BC) is his shadow on the wall of the building?

(a) Cost of a call

The cost has a fixed part of Rs. 0.50 and a variable part of Rs. 0.11 per minute.

Cost, C = 0.50 + 0.11N rupees, where N is the number of minutes.

For example, a 10-minute call costs 0.50 + 1.10 = Rs. 1.60.

(b) Number series

(i) The answer is 64. Cubes and squares alternate: 1, 2³ = 8, 2² = 4, 3³ = 27, 3² = 9, and then 4³ = 64.

(ii) The answer is 36. The pattern alternates × 2 and + 2: 3, 6, 8, 16, 18, and then 18 × 2 = 36.

(iii) The answer is 134,217,728. Each term is the cube of the previous one: 2³ = 8, 8³ = 512, and then 512³ = 134,217,728.

(iv) The answer is 144. The series gives pairs of a square and its square root: (81, 9), (64, 8), and then (144, 12).

(v) The answer is 81. The differences are +5, +10, +15, +20, and then +25: 56 + 25 = 81.

(c) Area of the rectangle

Note: the label on the left side of the figure is cut off in the available copy of the paper, but it is not needed, because the opposite sides of a rectangle are equal.

Top and bottom are equal: 3x − y = 2x + y, so x = 2y.

Length = 3x − y = 6y − y = 5y. Width = 2x − 3 = 4y − 3.

Perimeter = 2(length + width) = 114, so length + width = 57.

5y + 4y − 3 = 57, so 9y = 60, and y = 20/3. Therefore, x = 40/3.

Length = 5 × 20/3 = 100/3 ≈ 33.33 cm. Width = 4 × 20/3 − 3 = 71/3 ≈ 23.67 cm.

Area = 100/3 × 71/3 = 7,100/9 ≈ 788.9 cm².

(d) Height of the shadow

Triangles ADE and ABC are similar, since they share the angle at A, and both have a right angle (at D and at B).

AD = 2 m, DE = 1.6 m, and AB = 2 + 10.5 = 12.5 m.

Corresponding sides are proportional: BC ÷ DE = AB ÷ AD.

BC = 1.6 × 12.5 ÷ 2 = 10 m.