Geometry and Mensuration

Total Questions: 54

31. The diameters of two circular coins are in the ratio of 1: 3. The smaller coin is, made to roll around the bigger coin till it returns to the position from where the, process of rolling started. How many times the smaller coin rolled around the bigger coin? [2010]

(a) 9
(b) 6
(c) 3
(d) 1.5

Correct Answer: (c) 3

Solution:Perimeter of bigger coin : Perimeter of smaller coin

= 2 π (3) : 2 π (1) = 3 : 1

Hence, smaller coin will roll 3 times around the bigger coin.

32. A village having a population of 4000 requires 150 litres of water per head per day. It has a tank measuring 20 m × 15 m × 6 m. The water of this tank will last for [2011-II]

(a) 2 days
(b) 3 days
(c) 4 days
(d) 5 days

Correct Answer: (b) 3 days

Solution:Volume of tank

= 20 \times 15 \times 6 m³ = 20 \times 15 \times 6 \times 1000 litre

$∴$ The water of the tank will last for

$20 \times 15 \times 6 \times 1000$ days. i.e.

The water of the tank will last for 3 days.

33. Two glasses of equal volume are respectively half and three-fourths filled with milk. They are then filled to the brim by adding water. Their contents are then poured into another vessel. What will be the ratio of milk to water in this vessel? [2011-II]

(a) 1 : 3
(b) 2 : 3
(c) 3 : 2
(d) 5 : 3

Correct Answer: (d) 5 : 3

Solution:Milk in 1st Glass = 1/2 V
Milk in 2nd Glass = 3/4 V
Now water in 1st glass = 1/2 V
Now water in 2nd glass = 1/4 V
When both glasses are mixed then the ratio of milk to water

= 1/2 + 3/4 : 1/2 + 1/4 = (2+3)/4 : (2+1)/4 = 5 : 3

34. A gardener increased the area of his rectangular garden by increasing its length by 40% and decreasing its width by 20%. The area of the new garden [2014 - II]

(a) has increased by 20%.
(b) has increased by 12%.
(c) has increased by 8%.
(d) is exactly the same as the old area.

Correct Answer: (b) has increased by 12%.

Solution:Let initial dimensions be l & b
Final length is 1.4 l
Final breadth is 0.8 b
Final area is = 1.4l × 0.8 b = 1.12 lb
Area is increased by 12%.

35. In a plane, line X is perpendicular to line Y and parallel to line Z; line U is perpendicular to both lines V and W; line X is perpendicular to line V. [2015-II]

Which one of the following statements is correct?

(a) Z, U and W are parallel.
(b) X, V and Y are parallel.
(c) Z, V and U are all perpendicular to W.
(d) Y, V and W are parallel.

Correct Answer: (b) X, V and Y are parallel.

Solution:

From above diagram, it is clear that Y, V and W are parallel.

36. A cylindrical overhead tank of radius 2 m and height 7 m is to be filled from an underground tank of size 5.5 m x 4m x 6m. How much portion of the underground tank is still filled with water after filling the overhead tank completely? [2016-II]

(a) 1/3
(b) 1/2
(c) 1/4
(d) 1/6

Correct Answer: (a) 1/3

Solution:Volume of cylindrical overhead tank 22 = πr²h =22/7 × (2)² × 7 = 88 m³
Volume of underground tank = (5.5 × 4 × 6)m³ = 132m³

Portion of underground tank still filled with water after filling the overhead tank completely.
Volume of underground tank - Volume of overhead tank/Volume of underground tank

= 132 - 88/132 = 44/132 = 1/3

37. A piece of tin is in the form of a rectangle having length 12 cm and width 8 cm. This is used to construct a closed cube. The side of the cube is: [2016-II]

(a) 2 cm
(b) 3 cm
(c) 4 cm
(d) 6 cm

Correct Answer: (c) 4 cm

Solution:Area of rectangle = 12 x 8 cm² = 96 cm²

Rectangle is used to construct a closed cube.

Surface area of cube formed - Area of given rectangle.

6(side)² =96 cm² ⇒ side= √16 ⇒ side= 4cm .

38. Twelve equal squares are placed to fit in a rectangle of diagonal 5 cm. There are three rows containing four squares each. No gaps are left between adjacent squares. What is the area of each square? [2018-II]

(a) 5/7 sq. cm
(b) 7/5 sq. cm
(c) 1 sq. cm
(d) 25/12 sq. cm

Correct Answer: (c) 1 sq. cm

Solution:Diagonal of rectangle = 5 cm; √(l² + b² ) = 5

then value of / and b must be 4 and 3

Area of rectangle = 4 × 3 = 12 cm²

Area of square be a²

12 × a² = 12 ∴ a² = 1sq . cm.

39. Consider the following three-dimensional figure: [2018-II]

How many triangles does the above figure have?

(a) 18
(b) 20
(c) 22
(d) 24

Correct Answer: (b) 20

Solution:In each of the top and bottom part, there are 5 triangles which are based on a pentagon. In middle part, there are 5 quadrilaterals enclosed between two pentagons. Further each quadrilateral is divided into two triangles. Hence in the middle part, there are 10 triangles.

∴ Total number of triangles in the given 3D-figure.

= (No. of triangles in top part) + (No. of triangle in bottom part) + (No. of triangle in middle part) = 5 + 5 + 10 = 20

40. A solid cube is painted yellow, blue and black such that opposite faces are of same colour. The cube is then cut into 36 cubes of two different sizes such that 32 cubes are small and the other four cubes are big. None of the faces of the bigger cubes is painted blue. How many cubes have only one face painted? [2019-II]

(a) 4
(b) 6
(c) 8
(d) 10

Correct Answer: (c) 8

Solution:Number of cubes have only one face painted = 2 × 4 = 8

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