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?

1. 40 cm

2. 30 cm

3. 45 cm

4. 60 cm

5. 20 cm

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