Geometry and Mensuration

Total Questions: 54

41. Let x, y be the volumes; m, n be the masses of two metallic cubes P and Q respectively. Each side of Q is two times that of P and mass of Q is two times that of P. Let u = m / x and v nly. Which one of the following is correct? [2020-II]

(a) u = 4v
(b) u = 2v
(c) v = u
(d) v = 4u

Correct Answer: (a) u = 4v

Solution:Let side of P = K then side of Q = 2K
Volume of P = X = (K)²

Volume of Q = Y = (2K)³ = 8K³

Mass of P = m and Mass of Q = n

From question, n = 2m

Now, u = m/x = m/(K²)

and v = n/y = (2m)/(8K³) = 1/4 × (m/(K²)) = 1/4 (u) ∴ u = 4v

42. If 1 litre of water weighs 1 kg, then how many cubic millimetres of water will weigh 0.1 gm? [2020-II]

(a) 1
(b) 10
(c) 100
(d) 1000

Correct Answer: (c) 100

Solution:1 Kg = 1000gm = 1 litre

∴ 0.1 gm= 1 × 10⁻⁴ litre=1 × 10⁻¹ cubic meter

= 1 × 10⁻¹ * 1000 cubic millimeter

= 100 cubic millimeter.

43. A vessel full of water weighs 40 kg. If it is one-third filled, its weight becomes 20 kg. What is the weight of the empty vessel? [2020-II]

(a) 10 kg
(b) 15 kg
(c) 20 kg
(d) 25 kg

Correct Answer: (a) 10 kg

Solution:Let vessel weight is B Kg and volume of vessel is V. Full vessel of water weighs = 40 Kg.

1/3 filled vessel of water weighs = 20 Kg.
ATQ,

B + V =40 ...(i)

B + V/3 = 20 ...(ii)
From equation (i) and (ii), we get

(V - V/3) =40-2 ⇒ 2V/3 = 20 ⇒ V=30

and B = 40 - V = 40 - 30 = 10Kg

44. A cubical vesel of side 1 m is filled completely with water. How many millilitres of water is contained in it (neglect thickness of the vessel)? [2020-II]

(a) 1000
(b) 10000
(c) 100000
(d) 1000000

Correct Answer: (d) 1000000

Solution:1m³ = 1000 liter

Volume of cube = 1 ×1 ×1 = 1m³ = 1000 liter

= 1000000 milli liter

45. 125 identical cubes are arranged in the form of a cubical block. How many cubes are surrounded by other cubes from each side? [2023-II]

(a) 27
(b) 25
(c) 21
(d) 18

Correct Answer: (a) 27

Solution:Number of Cubes that touches all

Cubes = (n-2)³ = (5-2)³ = 27.

46. ABCD is a square. One point on each of AB and CD; and two distinct points on each of BC and DA are chosen. How many distinct triangles can be drawn using any three points as vertices out of these six points? [2023-II]

(a) 16
(b) 18
(c) 20
(d) 24

Correct Answer: (c) 20

Solution:

Number of points on different sides = 6
∴ Number of triangles = ⁶C₃ = 20

47. A flag has to be designed with 4 horizontal stripes using some or all of the colours red, green and yellow. What is the number of different ways in which this can be done so that no two adjacent stripes have the same colour? [2023-II]

(a) 12
(b) 18
(c) 24
(d) 36

Correct Answer: (c) 24

Solution:1st strip ← Red or Green or Yellow = 3 way
2nd strip ← Any two colours, other than 1st strip colour = 2way
3rd strip ← Any two colours, other than 2nd strip = 2way
4th strip ← Any two colours, other than 3rd strip = 2way
∴ Total number of ways = 3 × 2 × 2 × 2 = 24

48. Arectangular floor measures 4 m in length and 2.2 m in breadth. Tiles of size 140 cm by 60 cm have to be laid such that the tiles do not overlap. A tile can be placed in any orientation so long as its edges are parallel to the edges of the floor. What is the maximum number of tiles that can be accommodated on the floor? [2023-II]

(a) 6
(b) 7
(c) 8
(d) 9

Correct Answer: (d) 9

Solution:

49. A cuboid of dimensions 7 cm × 5cm x 3 cm is painted red, green and blue colour on each pair of opposite faces of dimensions 7 cm x 5 cm, 5 cm x 3 cm, 7 cm x 3 cm respectively. Then the cuboid is cut and separated into various cubes each of side length 1 cm. Which of the following statements is/are correct? [2023-II]

1. There are exactly 15 small cubes with no paint on any face.

2. There are exactly 6 small cubes with exactly two faces, one painted with blue and the other with green.

Select the correct answer using the code given, below:

(a) 1 only
(b) 2 only
(c) Both 1 and 2
(d) Neither 1 nor 2

Correct Answer: (a) 1 only

Solution:1. Number of small cubes with no paint on any face
= (7 - 2) × (5 - 2) = 5 × 3 = 15
2. Number of cubes with exactly two faces, one painted with blue and other with green.
= 1 + 1 + 1 + 1 = 4
Hence, only statement 1 is correct.

50. P's salary is 20% lower than Q's salary which is 20% lower than R's salary. By how much percent is R's salary more than P's salary? [2024-II]

(a) 48.75%
(b) 56.25%
(c) 60.50%
(d) 62.25%

Correct Answer: (b) 56.25%

Solution:P's salary = 80, Q's salary = 100.
R's salary = 100 × 100/80 = 125

Required = (125 - 80)/80 × 100 = 56.25%

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