Geometry & Mensuration MCQs for UPSC Prelims

69 practice questions on Geometry & Mensuration from the Quantitative Aptitude section of the UPSC Prelims syllabus. 69 come with a written explanation and 31 are actual previous year questions. Try the sample set below - the answer stays hidden until you ask for it.

16 Easy 38 Medium 15 Hard 31 from past papers

Sample questions

Q1
Previous year question medium

A village having a population of 4000 requires 150 litres of water per head per day. It has a tank measuring 20 m x 15 m x 6 m. The water of this tank will last for

  1. A 2 days
  2. B 3 days
  3. C 4 days
  4. D 5 days
Show answer and explanation

Correct answer: B - 3 days

Tank volume = 20 x 15 x 6 = 1800 cubic metres. Since 1 cubic metre = 1000 litres, the tank holds 1800 x 1000 = 18,00,000 litres. Daily requirement = 4000 people x 150 litres = 6,00,000 litres per day. Number of days the water lasts = 18,00,000 / 6,00,000 = 3 days.

Q2
Previous year question medium

Consider the following statements in respect of a rectangular sheet of length 20 cm and breadth 8 cm : 1. It is possible to cut the sheet exactly into 4 square sheets. 2. It is possible to cut the sheet into 10 triangular sheets of equal area. Which of the above statements is/are correct?

  1. A 1 only
  2. B 2 only
  3. C Both 1 and 2
  4. D Neither 1 nor 2
Show answer and explanation

Correct answer: C - Both 1 and 2

Statement 1: from the 20 cm x 8 cm sheet cut two 8 cm x 8 cm squares (using 16 cm of the length); the remaining 4 cm x 8 cm strip splits into two 4 cm x 4 cm squares, giving exactly 4 squares. Statement 1 is correct. Statement 2: total area is 160 sq cm, so 10 triangles need 16 sq cm each; cut the sheet into five 4 cm x 8 cm rectangles and halve each along a diagonal to get 10 triangles of equal area. Statement 2 is also correct. Hence option (c); options (a), (b) and (d) each deny a construction that is clearly possible.

Q3
Previous year question easy

AB is a vertical trunk of a huge tree with A being the point where the base of the trunk touches the ground. Due to a cyclone, the trunk has been broken at C which is at a height of 12 meters, broken part is partially attached to the vertical portion of the trunk at C. If the end of the broken part B touches the ground at D which is at a distance of 5 meters from A, then the original height of the trunk is:

  1. A 20 m
  2. B 25 m
  3. C 30 m
  4. D 35 m
Show answer and explanation

Correct answer: B - 25 m

The standing lower part AC is vertical with AC = 12 m. The broken upper part CB swings down so its tip touches the ground at D, with AD = 5 m. Triangle ACD is right-angled at A, so the broken length CD = sqrt(AC^2 + AD^2) = sqrt(12^2 + 5^2) = sqrt(144 + 25) = sqrt(169) = 13 m. The original height of the trunk is the standing part plus the broken part: 12 + 13 = 25 m. Options 20, 30 and 35 do not arise from the 12-5-13 right triangle. Hence 25 m.

Q4
Previous year question medium

A round archery target of diameter 1 m is marked with four scoring regions from the centre outwards as red, blue, yellow and white. The radius of the red band is 0.20 m. The width of all the remaining bands is equal. If archers throw arrows towards the target, what is the probability that the arrows fall in the red region of the archery target?

  1. A 0.40
  2. B 0.20
  3. C 0.16
  4. D 0.04
Show answer and explanation

Correct answer: C - 0.16

The target has diameter 1 m, so its radius is 0.5 m and its area is pi x (0.5)^2. The central red region has radius 0.20 m, so its area is pi x (0.20)^2. The probability of landing in red = area of red / total area = (pi x 0.20^2) / (pi x 0.5^2) = (0.04) / (0.25) = 0.16. Equivalently (0.20/0.50)^2 = (0.4)^2 = 0.16. The value 0.40 is just the radius ratio, and 0.04 is the red area without dividing by total. Hence 0.16.

Q5
Previous year question easy

The outer surface of a 4 cm x 4 cm x 4 cm cube is painted completely in red. It is sliced parallel to the faces to yield sixty four 1 cm x 1 cm x 1 cm small cubes. How many small cubes do not have painted faces? (a) 8 (b) 16 (c) 24 (d) 36

  1. A 8
  2. B 16
  3. C 24
  4. D 36
Show answer and explanation

Correct answer: A - 8

Only the small cubes that lie completely inside, away from every outer face, have no paint. Removing the one outer layer from each of the three dimensions leaves an inner cube of side (4 - 2) = 2 along each edge. The number of unpainted cubes = 2 x 2 x 2 = 8. (The other counts for an n=4 cube are: 3 painted faces = 8 corners, 2 painted faces = 12(n-2) = 24 edges, 1 painted face = 6(n-2)^2 = 24 faces.) Answer: 8.

Q6
Previous year question medium

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?

  1. A 5/7 sq cm
  2. B 7/5 sq cm
  3. C 1 sq cm
  4. D 25/12 sq cm
Show answer and explanation

Correct answer: C - 1 sq cm

Let the side of each small square be s. With 3 rows of 4 squares each, the rectangle measures 4s in length and 3s in width. Its diagonal is sqrt((4s)^2 + (3s)^2) = sqrt(16s^2 + 9s^2) = sqrt(25s^2) = 5s. Given the diagonal is 5 cm, 5s = 5, so s = 1 cm. The area of each square is s^2 = 1 sq cm. Hence (c) is correct. The 3-4-5 right triangle makes the arithmetic clean; the other options do not arise from a consistent 4s-by-3s rectangle with diagonal 5.

Q7
Previous year question medium

A cube has all its faces painted with different colours. It is cut into smaller cubes of equal sizes such that the side of the small cube is one-fourth the big cube. The number of small cubes with only one of the sides painted is:

  1. A 32
  2. B 24
  3. C 16
  4. D 8
Show answer and explanation

Correct answer: B - 24

Cutting each edge into 4 gives 4 x 4 x 4 = 64 small cubes. A small cube with exactly one painted face comes from the interior of a face of the big cube, away from the edges. On each face there are (n-2) x (n-2) such cubes, where n = 4, giving (4-2)^2 = 2 x 2 = 4 single-painted cubes per face. With 6 faces, the total is 4 x 6 = 24. The value 8 is the number of corner cubes (three faces painted), and 32 would be wrong for n = 4. Hence 24.

Q8
Previous year question easy

Location of B is north of A and location of C is east of A. The distances AB and AC are 5 km and 12 km respectively. The shortest distance (in km) between the locations B and C is

  1. A 60
  2. B 13
  3. C 17
  4. D 7
Show answer and explanation

Correct answer: B - 13

North and east directions are perpendicular, so triangle ABC has a right angle at A with legs AB = 5 km and AC = 12 km. The shortest distance BC is the hypotenuse = square root of (5 squared + 12 squared) = square root of (25 + 144) = square root of 169 = 13 km.

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More Quantitative Aptitude topics

Number System 188 Permutation, Combination & Probability 94 Percentage 85 Time, Speed & Distance 66 Algebra 66 Basic Numeracy & Word Problems 66 Ratio & Proportion 60 Clocks & Calendars 56 Sequences & Series 56 Mixtures & Alligation 52 Time & Work 51 Averages 45 Simple & Compound Interest 43 Profit, Loss & Discount 39 Ages 33