UPSC CSE Prelims CSAT (Paper II) previous year questions on Geometry and Diagrams (Quantitative Ability), asked from 2011–2026, with answers. There are 48 questions in this post. Attempt each question, then click Show Answer to check.
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1] The top of a table is rectangular and its dimensions are 6′ × 10′. Two rectangular portions of the table top are painted in blue colour; both these portions have dimensions 2.5′ × 8′ and each of them has exactly two sides common with two edges of the table top. If the table is fixed to the ground and the remaining portion of the table top is painted in white, how many different patterns are possible when observed from above? [2026/15]
(a) 2
(b) 4
(c) 6
(d) 8
Show Answer
Ans: (b) 4
2] A cut on a solid object divides the object into two parts where the new surfaces thus produced are plane. On the other hand, one single cut can be used to cut more than one object at a time. In an experiment, the total number of pieces produced by applying n cuts is denoted by xₙ. The experiment is performed on a solid cube where pieces remain unmoved after each cut. In this experiment, if after the third cut, the pieces are identical, then which of the following is not a possible value for x₄? [2026/24]
(a) 16
(b) 12
(c) 8
(d) 5
Show Answer
Ans: (a) 16
3] A solid cube is painted yellow on all its faces. The cube is then cut into 60 smaller but equal pieces by making the minimum number of cuts. Which of the following statements is/are correct?
I. The minimum number of cuts is 9.
II. The number of smaller pieces which are not painted on any face is 6.
Select the correct answer using the code given below: [2025/15]
(a) I only
(b) II only
(c) Both I and II
(d) Neither I nor II
Show Answer
Ans: (c) Both I and II
4] What is the least possible number of cuts required to cut a cube into 64 identical pieces? [2024/5]
(a) 8
(b) 9
(c) 12
(d) 16
Show Answer
Ans: (b) 9
5] 125 identical cubes are arranged in the form of a cubical block. How many cubes are surrounded by other cubes form each side? [2023/18]
(a) 27
(b) 25
(c) 21
(d) 18
Show Answer
Ans: (a) 27
6] 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 he drawn using any three points as vertices out of these six points? [2023/30]
(a) 16
(b) 18
(c) 20
(d) 24
Show Answer
Ans: (c) 20
7] A rectangular 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? [2023/67]
(a) 6
(b) 7
(c) 8
(d) 9
Show Answer
Ans: (d) 9
8] A cuboid of dimensions 7 cm X 5 cm X 3 cm is painted red, green and blue colour on each pair of opposite faces of dimensions 7 cm X 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? [2023/79]
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
Show Answer
Ans: (a) 1 only
9] There are eight equidistant points on a circle. How many right-angled triangles can be drawn using these points as vertices and taking the diameter as one side of the triangle? [2022/56]
(a) 24
(b) 16
(c) 12
(d) 8
Show Answer
Ans: (a) 24
10] 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? [2022/59]
(a) 1 only
(b) 2 only
(c) Both 1 and 2
(d) Neither 1 nor 2
Show Answer
Ans: (c) Both 1 and 2
11] Images of consonants of the English alphabet (Capitals) are observed in a mirror. What is the number of images of these which do not look like their original shapes? [2021/17]
(a) 13
(b) 14
(c) 15
(d) 16
Show Answer
Ans: (b) 14
12] A cubical vessel of side 1 m is filled completely with water. How many millilitres of water is contained in it (neglect thickness of the vessel)? [2021/68]
(a) 1000
(b) 10000
(c) 100000
(d) 1000000
Show Answer
Ans: (d) 1000000
13] 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 = n/y. which one of the following is correct? [2020/38]
(a) u = 4v
(b) u = 4v
(c) v = u
(d) v = 4u
Show Answer
Ans: (a) u = 4v
14] Consider the following statements:
1. The minimum number of points of intersection of a square and a circle is 2.
2. The maximum number of points of intersection of a square and a circle is 8.
Which of the above statements is/are correct? [2020/73]
(a) 1 only
(b) 2 only
(c) Both 1 and 2
(d) Neither 1 nor 2
Show Answer
Ans: (b) 2 only
15] If 1 litre of water weighs 1 kg, then how many cubic millilitres of water will weigh 0.1 gm? [2020/78]
(a) 1
(b) 10
(c) 100
(d) 1000
Show Answer
Ans: (c) 100
16] 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? [2019/10]
(a) 4
(b) 6
(c) 8
(d) 10
Show Answer
Ans: (c) 8
17] The number of parallelograms that can be formed from a set of four parallel lines intersecting another set of four parallel lines, is [2019/14]
(a) 18
(b) 24
(c) 32
(d) 36
Show Answer
Ans: (d) 36
18] Each face of a cube can be painted in black or white colours. In how many different ways can the cube be painted? [2019/72]
(a) 9
(b) 10
(c) 11
(d) 12
Show Answer
Ans: (b) 10
19] Consider the following three-dimensional figure:
How many triangles does the above figure have? [2018/1]
(a) 18
(b) 20
(c) 22
(d) 24
Show Answer
Ans: (b) 20
Directions for the following 3 (three) items
Rotated positions of a single solid are shown below. The various faces of the solid are marked with different symbols like dots, cross and line.
Answer the three items that follow the given figures
20] What is the symbol on the face opposite to that containing a single dot? [2018/14]
(a) Four dots
(b) Three dots
(c) Two dots
(d) Cross
Show Answer
Ans: (b) Three dots
21] What is the symbol on the face opposite to that containing two dots? [2018/15]
(a) Single dot
(b) Three dots
(c) Four dots
(d) Line
Show Answer
Ans: (c) Four dots
22] What is the symbol on the face opposite to that containing the cross? [2018/16]
(a) Single dot
(b) Two dots
(c) Line
(d) Four dots
Show Answer
Ans: (c) Line
23] A solid cube of 3 cm side, painted on all its faces, is cut up into small cubes of 1 cm side. How many of the small cubes will have exactly two painted faces? [2018/35]
(a) 12
(b) 8
(c) 6
(d) 4
Show Answer
Ans: (a) 12
24] There are 24 equally spaced points lying on the circumference of a circle. What is the maximum number of equilateral triangles that can be drawn by taking sets of three points as the vertices? [2018/58]
(a) 4
(b) 6
(c) 8
(d) 12
Show Answer
Ans: (c) 8
25] 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? [2018/60]
(a) 5/7 sq cm
(b) 7/5 sq cm
(c) 1 sq cm
(d) 25/12 sq cm
Show Answer
Ans: (c) 1 sq cm
26] There are 4 horizontal and 4 vertical lines, parallel and equidistant to one another on a board. What is the maximum number of rectangles and squares that can be formed? [2017/38]
(a) 16
(b) 24
(c) 36
(d) 42
Show Answer
Ans: (c) 36
27] 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? [2017/58]
(a) 8
(b) 16
(c) 24
(d) 36
Show Answer
Ans: (a) 8
28] Two walls and a ceiling of a room meet at right angles at a point P. A fly is in the air 1 m from one wall, 8 m from the other wall and 9 m from the point P. How many meters is the fly from the ceiling? [2017/75]
(a) 4
(b) 6
(c) 12
(d) 15
Show Answer
Ans: (a) 4
Directions for the following 3 (three) items:
consider the given information and answer the three items that follow.
Eight railway stations A, B, C, D, E, F, G and H are connected either by two-way passages or one-way passages. One-way passages are from C to A, E to G, B to F, D to H, G to C, E to C and H to G. Two-way passages are between A and E, G and B, F and D, and E and D.
29] While travelling from C to H, which one of the following stations must be passed through? [2017/76]
(a) G
(b) E
(c) D
(d) F
Show Answer
Ans: (b) E
30] In how many different ways can a train travel from F to A without passing through any station more than once? [2017/77]
(a) 1
(b) 2
(c) 3
(d) 4
Show Answer
Ans: (d) 4
31] If the route between G and C is closed, which one of the following stations need not be passed through while travelling from H to C? [2017/78]
(a) E
(b) D
(c) A
(d) B
Show Answer
Ans: (c) A
32] 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: [2016/39]
(a) 2 cm
(b) 3 cm
(c) 4 cm
(d) 6 cm
Show Answer
Ans: (c) 4 cm
33] 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: [2016/57]
(a) 20 m
(b) 25 m
(c) 30 m
(d) 35 m
Show Answer
Ans: (b) 25 m
34] 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: [2016/59]
(a) 32
(b) 24
(c) 16
(d) 8
Show Answer
Ans: (b) 24
35] A cylindrical overhead tank of radius 2 m and height 7 m is to be filled from an underground tank of size 5.5m x 4m x 6m. How much portion of the underground tank is still filled with water after filling the overhead tank completely? [2016/73]
(a) 1/3
(b) 1/2
(c) 1/4
(d) 1/6
Show Answer
Ans: (a) 1/3
36] Four cardboard pieces of specific shapes are shown in the following figure.
Which one the following given can be formed by joining these pieces together? [2015/14]
Show Answer
Ans: (d)
37] Each of the six different faces of a cube has been coated with a different colour i.e., V, l, B, G, Y and O. Following information is given;
1. Colours Y, O and B are on Adjacent faces.
2. Colours I, G and Y are on adjacent faces.
3. Colours B, G and Y are on adjacent faces.
4. Colours O, V and B are on adjacent faces.
Which is the colour of the face opposite to the face coloured with O? [2015/54]
(a) B
(b) V
(c) G
(d) I
Show Answer
Ans: (c) G
38] 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.
Which one of the following statements is correct? [2015/76]
(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.
Show Answer
Ans: (d) Y, V and W are parallel.
39] With reference to the figure given below, the number of different routes from S to T without retracing from U and/or V, is [2014/44]
(a) 3
(b) 6
(c) 9
(d) 18
Show Answer
Ans: (d) 18
40] A cube has six numbers marked 1, 2, 3, 4, 5 and 6 on its faces. Three views of the cube are shown below
What possible numbers can exist on the two faces marked A and B, respectively on the cube? [2013/51]
(a) 2 and 3
(b) 6 and 1
(c) 1 and 4
(d) 3 and 1
Show Answer
Ans: (a) 2 and 3
41] Six squares are coloured, front and back, red(R), blue(B), yellow (Y), green (G), white (W), orange (O) and are hinged together as shown in the figure given below. If they were folded to form a cube what would be the face opposite to white face? [2012/47]
(a) R
(b) G
(c) B
(d) O
Show Answer
Ans: (c) B
42] Three views of a cube following a particular motion are given below: [2012/49]
What is letter opposite to A?
(a) H
(b) P
(c) B
(d) M
Show Answer
Ans: (a) H
43] Study the following figure:
A person goes from A to B always moving to the right or downwards along the lines. How many different routes can he adopt? [2011/25]
Select the correct answer from the codes given below:
(a) 4
(b) 5
(c) 6
(d) 7
Show Answer
Ans: (c) 6
44] Consider the following figure and answer the item that follow:
What is total number of triangles in the above grid? [2011/26]
(a) 27
(b) 26
(c) 23
(d) 22
Show Answer
Ans: (c) 23
45] Consider the figure given below and answer the items that follows:
In the figure shown above, OP 1 and OP 2 are two plane mirrors kept perpendicular to each other. S is the direction of a beam of light falling on the mirror OP 1. The direction of the reflected beam of light from the mirror OP 2 will be [2011/36]
(a) Perpendicular to the direction S.
(b) At 45° to the direction S.
(c) Opposite and parallel to the direction S.
(d) At 60° to the direction S.
Show Answer
Ans: (c) Opposite and parallel to the direction S.
46] Consider the following figure and answer the item that follows:
What is the minimum number of different colours required to paint the figure given above such that no two adjacent regions have the same colour? [2011/37]
(a) 3
(b) 4
(c) 5
(d) 6
Show Answer
Ans: (a) 3
47] Consider the following figure and answer the items that follows:
A square is divided into four rectangles as shown above. The lengths of the sides of rectangles are natural numbers. The areas of two rectangles are indicated in the figure. What is the length of each side of the square? [2011/38]
(a) 10
(b) 11
(c) 15
(d) Cannot be determined as the given data are Insufficient
Show Answer
Ans: (b) 11
48] 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 [2011/43]
(a) 2 days
(b) 3 days
(c) 4 days
(d) 5 days
Show Answer
Ans: (b) 3 days
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