Geometry and Mensuration

Total Questions: 54

1. In the Cartesian plane four points P, Q, R, S have co-ordinates (1, 1), (4, 2), (4, 4) and (1, 4) respectively. The area of the quadrilateral PQRS is [1995]

(a) 9
(b) 7.5
(c) 4.5
(d) impossible to find unless diagonals are known the lengths of the

Correct Answer: (b) 7.5

Solution:

2. A rectangular sump of dimension 6m × 5m × 4m is to be built by using bricks to make the outer dimension 6.2m × 5.2m × 4.2m. Approximately how many bricks of size 20cm × 10cm × 5cm are required to build the sump for storing water? [1995]

(a) 15408
(b) 3000
(c) 15000
(d) 30000

Correct Answer: (a) 15408

Solution:Let the number of bricks be

n

. Now, Volume of outer dimension of sump - volume of sump = volume needed to be built by the bricks.

(6.2 \times 5.2 \times 4.2 - 6 \times 5 \times 4) = 15.408 m³

. Volume of each brick =

0.2 \times 0.1 \times 0.05

10 \times 1 0⁻⁴ m³ = 1 \times 1 0 ⁻³ m⁻³

$∴ (1 \times 1 0 ⁻³) n = 15.408$

$n = 15.408/1 \times 10 ⁻³ = 15408$

3. The following figure contains three squares with areas of 100, 16 and 49 sq. units respectively laying side by side as shown. By how much should the area of the middle square be reduced in order that the total length PQ of the resulting three squares is 19? [1996]

(a) 12
(b) 4
(c) 2
(d) 3

Correct Answer: (a) 12

Solution:Let the final length of the side of the smaller square be

a

Now, $a + 10 + 7 = 19$ ; $a = 19 - 17 = 2$

$∴$ Area of the smaller square = $(2)² = 4$

$∴$ Decrease in the area of the smaller square $= 16 - 4 = 12$ sq. units.

4. A rectangle has perimeter of 50 metres. If its length is 13 metres more than its breadth, then its area is: [1996]

(a) 124 m²
(b) 144 m²
(c) 114 m²
(d) 104 m²

Correct Answer: (c) 114 m²

Solution:Let the breadth be

b

Then,

length $= b + 13$

Perimeter $= 50 = 2 (l + b)$

$2 (b + 13 + b) = 50$

$b = 6$ m

$l = 6 + 13 = 19$ m

$∴$ Area $=$ length $\times$ breadth $= 19 (6) = 114$ m²

5. Consider the following figures: [1996]

Which one of the following conclusions can be drawn from these figures?

(a) The areas of the three figures are all different
(b) The areas of all the three figures are equal
(c) The perimeters of the three figures are equal
(d) The perimeters of figures I and II are equal

Correct Answer: (b) The areas of all the three figures are equal

Solution:Area of (rectangle)

= 9 \times 4 = 36

Area of (square) $= 6 \times 6 = 36$

Area (triangle) $=1/2 \times 9 \times 8 = 36$

$∴$ Area of all the three figures are equal.

6. In the following figure [1997]

$P$ is 300 km eastward of $O$ and $Q$ is 400 km north of $O$ . $R$ is exactly in the middle of $Q$ and $P$ . The distance between $Q$ and $R$ is:

(a) 250 km
(b) 300 km
(c) 350 km
(d) 250√2 km

Correct Answer: (a) 250 km

Solution:

7. A rectangular plot of lawn shown in the figure has dimensions 'x' and 'y' and is surrounded by a gravel pathway of width 2 m. What is the total area of the Gravel Pathway? [1997]

Correct Answer: (d)

Solution:Length of the lawn including Gravel pathway

= x + 2 + 2 = x + 4

Breadth of the lawn including Gravel pathway $= y + 2 + 2 = y + 4$

$∴$ Area of the lawn including of Gravel pathway $= (x + 4) (y + 4)$

$∴$ Area of Gravel pathway $=$ Area of the lawn including Gravel pathway $-$ Area of the lawn.

$= (x + 4) (y + 4) - x y = 4 x + 4 y + 16$

8. Consider the figure given below: [1995]

PQRS is a square of side 1 unit and Q, S are the centres of the two circles. The area of the shaded portion is

Correct Answer: (d)

Solution:

9. If the numbers representing volume and surface area of a cube are equal, then the length of the edge of the cube in terms of the unit of measurement will be [1997]

(a) 3
(b) 4
(c) 5
(d) 6

Correct Answer: (d) 6

Solution:Let the length of the edge be

a

Now, a³ = 6a², a = 6

10. The length, breadth and height of a room are l, b and h respectively. The perimeter of the ceiling expressed as a percentage of the total area of the four walls, is [1997]

(a) 100 h %
(b) 100⁄h %
(c) h %
(d) h/100 %

Correct Answer: (b) 100⁄h %

Solution:Total area of four walls

= 2 h (l + b)

Perimeter of the ceiling $= 2 l + 2 b = 2 (l + b)$

Perimeter of the ceiling expressed in percentage of the total area of the four walls $= 2 ( l + b )/2 h ( l + b ) \times 100% = 100/h %$

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Geometry and Mensuration

1. In the Cartesian plane four points P, Q, R, S have co-ordinates (1, 1), (4, 2), (4, 4) and (1, 4) respectively. The area of the quadrilateral PQRS is [1995]

2. A rectangular sump of dimension 6m × 5m × 4m is to be built by using bricks to make the outer dimension 6.2m × 5.2m × 4.2m. Approximately how many bricks of size 20cm × 10cm × 5cm are required to build the sump for storing water? [1995]

3. The following figure contains three squares with areas of 100, 16 and 49 sq. units respectively laying side by side as shown. By how much should the area of the middle square be reduced in order that the total length PQ of the resulting three squares is 19? [1996]

4. A rectangle has perimeter of 50 metres. If its length is 13 metres more than its breadth, then its area is: [1996]

5. Consider the following figures: [1996]

6. In the following figure [1997]

7. A rectangular plot of lawn shown in the figure has dimensions 'x' and 'y' and is surrounded by a gravel pathway of width 2 m. What is the total area of the Gravel Pathway? [1997]

8. Consider the figure given below: [1995]

9. If the numbers representing volume and surface area of a cube are equal, then the length of the edge of the cube in terms of the unit of measurement will be [1997]

10. The length, breadth and height of a room are l, b and h respectively. The perimeter of the ceiling expressed as a percentage of the total area of the four walls, is [1997]

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