Q.1. The area of a rectangle is four times of a square. The length of the rectangle is 80 cm and the breadth of the rectangle is 3 times that of the side of the square. What is the side of the square?

Sol : Option 4
L = 80 cm. B = 3a, where a is the side of the square. ∴ Area of rectangle = LB = 4a^{2} ⇒ 80 x 3a = 4 a^{2} ⇒ a = 60 cm.

Q.2. Two small circular parks of diameters 6 m and 8 m are to be replaced by a bigger circular park. What would be the radius of this new park, in meter, if the new park occupies the same space as the two small parks (in meter)?

1. 5

2. 10

3. 15

4. 20

5. 25

Sol : Option 1
Area of the new circular park = sum of the areas of the 2 smaller parks
⇒ π (6/2)^{2} + π (8/2)^{2} = π(9+ 16) = 25π ⇒ 25 π = π R^{2}. ∴ R^{2} = 25 ⇒ R = 5 m

Q.3. The diameter of a sphere is 10 inches. What are its volume and surface area?

1. 50, 25

2. 10π/3, 15π

3. 500π/3, 100π

4. 75π/2, 35π^{2}/3

Sol : Option 3
The radius is half the diameter or 5 inches.
We use r = 5 in the formulas for volume and surface area.
Volume: V = 4/3π5^{3} = 500π / 3 cubic inches
Surface area: S = 4π5^{2} = 100π square inches.

Q.4. How many coins 3 mm thick and 1.2 cm in diameter should be melted in order to form a right circular cylinder, having base diameter 4 cm and height 27 cm?

1. 850

2. 950

3. 980

4. 1000

5. 900

Sol : Option 4
Let the number of coins be n. We have
n x π x (1.2/2)^{2} x 0.3 = π (4/2)^{2} x 27
⇒ n = 1000

Q.5. The length and breadth of rectangle Park is 56m and 44 m respectively. If a concrete path of width 4 m is made outside and along the length & breadth of rectangle. Find the area of concrete path.

1. 864 m^{2}

2. 432 m^{2}

3. 216 m^{2}

4. 108 m^{2}

5. None of these

Sol : Option 1
Given that ; Length of park= 56m, breadth of park= 44m
Area of park = 56×44=2464
Length of rectangle after including concrete path= 56+8= 64,
Breadth of rectangle after including concrete path= 44+8= 52,
Area of rectangle including concrete path= 64×52=3328
Area of concrete path= 3328-2464=864m^{2}

Q.6. A triangle inscribed in a circle with shorter sides is 3 and 4 units long. If the longer side is the diameter, find the length of the diameter.

1. 5

2. 20

3. 30

4. 40

5. 50

Sol : Option 1
If the longer side is the diameter, then this triangle is a right–angled triangle.
Hence length of diameter = √4^{2} + 3^{2} = √16+9 = √25 = 5units

Q.7. An open rectangular tank is made of concrete, the sides and base being 30 cm thick. Internally the tank is 8 m long, 4 m broad and 3 m high. Find its weight in kg, if concrete weighs 1 kg per 1000 cubic centimeter.

1. 34,548 kg

2. 44,416 kg

3. 39,416 kg

4. 40,000 kg.

5. None of these

Sol : Option 1
The outer dimensions are 8.6 x 4.6 x 3.3 m.
So volume of the block = 8.6 x 4.6 x 3.3 – 8 x 4 x 3 = 130.548 - 96 = 34.548 cu. m = 34548000 cu cm, weight of the block =34548000/1000 = 34548 kg.

Q8. A lawn, 40 m long and 35 m wide, is surrounded by a path 2 m wide. How many cubic metres of gravel are required to cover the path to a depth of 10 cm?

1. 11.8 cu m

2. 31.6 cu m

3. 62 cu m

4. 13.6 cu m

5. None of these

Sol : Option 2
The path is formed by two rectangles, which have dimensions as 40 × 35 and 44 × 39. The required quantity of gravel
= area of the path x depth of gravel = [(44 x 39) – (40 x 35)] x 0.10 = 31.6 cu. m.

Q9. The rainwater from a flat roof 16 m long and 10 m wide is collected in a tank whose internal measurements are 2 m long, 2.4 m wide and 3 m deep. Before it started raining it was half-full and after the rain it was full. Find the height of rainfall.

1. 1.8 cm

2. 0.9 cm

3. 0.09 cm

4. 9 cm

5. 4.5 cm

Sol : Option 5
Let h be the height of the rainfall = h
1.6 × 10 × h = 1/2(2 × 2.4 × 3)
h = .045m = 4.5cm

Q10. A cylinder will exactly fit into a rectangular box whose internal dimensions are: length 14 cm, breadth 4 cm and height 4 cm. What is the volume of the cylinder?

1. 167 cu cm

2. 176 cu cm

3. 200 cu cm

4. 220 cu cm

5. None of these

Sol : Option 2
The diameter of the cylinder = 4 cm and its height is 14 cm.
Volume of the cylinder = π x 2 x 2 x 14 = 176 cc