CHAPTER

3 SPATIAL APTITUDE
Trmsformations ofShapes 3. [MCQJ [GATE-CE-2023:1M)
Consider a circle with its entre at the origin (O), as
1. [MCQJ [GATE-CS-2023:1M] shown. Two operations are allowed on the circle.
Looking at the surfoce of a smooth 3-dimensional
Operation I: Scale independently along the x and y
object from the outside, which one of
followino

options is TRUE? axes.

the
be concave Operation 2: Rotation in any direction about
(a) The surface of the object must
everywhere. origin.
can be achieved
object must be con vex Which figure among the options
(b) The surface of the operations on the
everywhere. through a combination of these two
nmay be concave in some given circle?
(e) The surface of the object
other places.
places and convex in cdges, but no corners.
have
(d) The object can
(GATE-ME-2023: 2M)
2. [MCQ] square
pyramid (shown below). with a
An opaque in the path of a
isosceles faces, is suspended
hase and on
is cast
beam of light, such that its shadow of the
parallel perpendicular to the direction (b)
oriented reoriented in any
a screen pyramid can be
The
light beam.
the ight
beam. Under these
within and S
direction
the shadows P, Q. R,
one of
condi tions, which
is NOT possible?
Opaque
Pyt (d)
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A block with a
[GATE-ME-2022: IM) (a)The sequence of operations () and (2)
ae
trapezoidal cross-section is placed cquivalent
over a block with
rectangular cross section as shown (b The sequence of
operations (1) and (3) are
above.
equivalent
Which one of the following is the correct drawing of (c) The sequence
the view of the 3D object as viewed in the of operations (2) and (3) are
direction cquivalent
indicated by an arrow in the below figure?
(d) The sequence of operations (1). (2) and (3) are
cquivalent
6 [MCQ) [GATE-CE-2022:1MI
For the picture shown below, which one of the
following is the correct picture representing reflection
with respect to the miror shown as the doted line?

(a)

(c)

5. [MCQJ [GATE-EC-2022:2MI
Consider the following square with
btficky
(a)
the four corners and the center
marked as P. 0. R. S and T
respectively.
Let X, Yand Z represent the following operations:
X: rotation of the square by 180 degree with respect to
the S-Q axis. (b)
Y: rotation of the square by 180 degree with respect tc
the P-R axis.
with
Z: rotation of the square by 90 degree clockwise
respect lo the axis perpendicular, going into the screen
and passing through the point T. (c)
Consider the following three distinct sequences of
order).
operation (which are applied in the left to right
(1) XYZ2

(2) XY (dy
3) ZZL is correct as per
Which one of the following statements
information provided above?
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7. (MCQI [GATE-CS-2021: 1M) 9. [MCQJ (GATE-CE-2021: IM]

A transparent square shet shown below is folded
along the dotted line. The folded sheet will look
like

PHYLAXIS

X
The mirror image of the below ext about the X-axis

(b)

(a)
(c)
dHArviS
10. [MCQ)
Which one of the (GATE-CS-2017:1M)
options best describes the
transformation of the
2-dimensional figure P to Q, and
then to R, as shown?

N P
Operation 1 Operation
8. [MCO1
(GATE-CE-2021:
The mimOr image of the above IMJ
text about the X-axis (a)
Operation 1:A clockwise rotation by 90°
about
Y an axis
perpendicular to the plane of the figure
Operation 2: A
(b) Operation I: A reflection along a horizontal line.
about an axis counterclockwise
perpendicular
to
rotation by 90
the
figure plane of the
TRIANGLE Operation 2: A reflection along a
(c) Operation 1: A clockwise rotationhorizontal
by 90°
line.
to the plane of the about
an axis
perpendicular
Operation 2: A figure
(d) Operation 1: A
reflection along a vertical line.
(b) 180° about an axis
the figure
counterclockwise rotation by
perpendicular to the plane
(c)
of
Operation 2: A reflection along a vertical
LK0VIari
(d)
line.

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Classification and Grouping 13. [MCQ) [GATE-CE-2022:2M]

1L [MCQ (GATE-ME-2022: 1M Consider a cube made by folding a single sheet of
paper of appropriate shape. The interior foces of the
cube are all blank. However, the exterior faces that are
not visible in the above view may not be blank.
Which one of the following represents a possible
unfolding of the cube?
Which one of the groups given below can be
assembled to get the shape that is shown above using
each piece only once without overlapping with each
other? (rotation and translation operations may be
used).

(a (h

(a)
(d

Patlerns in 2) & 3D
12. [MCQI (GATE-CE-2023: IM]
A line of symmetry is defined as a line that divides a (b)
figure into two parts in a way such that each part is a
mirror image of the other part about that line.
The figure below consists of 20-unit squares arranged
as shown. In addition to the given black squares, upto
5more may be coloured black. Which one among the
following options depicts the minimum number of
boxes that must be coloured black to achie ve two lines
of symmetry? (The figure is representative)
b

8 h (d)

(a) d (b) cd,i

(c) c, i (d) c, d, i, f. g
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14. [MCQN (GATE-CE-2021: 2M)
Consider two rectangular sheets, Shecet M and Sheet
N of dimensions 6 cm x 4 cm each.
Folding operation 1: The sheet is folded into half by
joining the short edges of the current shape. (c)
Folding operation 2: The sheet is folded into half by
joining the long edges of the current shape.
Folding operation I is carried out on Sheet M three
times.
Folding operation 2 is carried out on Sheet N three
times (d)
The ratio of perimeters of the final folded shape of
Sheet N to the final folded shape of Sheet M is
16. [MCQ) (GATE-EC-2021:1M]
(a) 13:7 (b) 3:2
(c) 7:5 (d 5:13 The least number of squares that must be added so that
the line P-Q becomes the line of symmetry is
15. [MCQ) IGATE-ME-2021: 2M1

NPpffice
A jigsaw puzzle has 2 pieces. One of the pieces is
shown above. Which one of the given options for the
(a) 4 (b) 3
missing piece when assembled will fom a rectangle?
The piece can be moved, rotated or flipped to (c) 6 (d) 7
ssemble with the above piece.
17. [MCQI [GATE-EE-2021: 1M]

(a)
K
A transparent square sheet shown above is folded
along the dotted line. The folded sheet will look like

(a) (b

(c) (d)
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18 (MCQJ (GATE-CS-2021: IM) 20. [MCQ] [GATE-CS-20021: 2M]

A polygon is convex if, for every pair of points, P and A jigsaw puzzle has 2 pieces. One of the pieces is
Q belon ging to the polygon, the line segment PQ lies sho wn below. Which one of the given options for tho
completely inside or on the polygon. missing picce when assembled will form a rectangle?
Which one of the following is NOT a convex The picce can be moved, rotated or flipped to
polygon? assemble with the below piece.

(a) (b)

(d)

19. [MCQJ [GATE-CS-2021: IM)

Acircular sheet of paper is folded along the lines in
the directions shown. The paper, after being punched
in the final folded sate as shown and unfolded in the
reverse order of folding, will look like

(a) 0

(b (d: (0OD)
DOO
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ANSWER KEY

1 (C) 2 3. (a) 4. (a)

5. (b) 6. 7. (b) 8 (b
9. (b) 10 11. (b, c) 12. (b)
13. (MTA) 14. (a) 15. 16. (c)

17. (c) 18. (a) 19, (a) 20. (a)

SOLUTIONS

1. (c)
If the object has edges, then it surely shall have corners: Scan for Video solution
option D wrong
Just secing the surface of SMOOTH 3D object from
outside, we cannot say it has concave or convex faces
so, options A & B are also wrong and 4. (a)
When this 3D object is viewed in the direction of arro w,
option C is correct then the drawing obt ained is.

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(b) Scan for Video solution

Shadow Qis not possible
Depen ding upon the angle of incidence of the light, the
shadows P, R and S can be oblained.
5. b)

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Normal Output (2)

3. (a)
Even after both the operations, the centre of the new
shape will also lie on the origin. When operation 1 is
performed then we can get an ellipse and when Output()
operation 2is performed then a tiled ellipse with centre
at origin can be achieved, hencec ion A iis correct.
eoption

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(b)

|TRIANGL
LKIVICE LE Miror

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Output (3) 9. (b)

So, output (1) and (3) are equivalent. PHYLAXIS

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BHATVXI?
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6 a

10. (a)
Reteetion
alons

a horizontal

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7. (b)
11. (b, c)
(b) and (c) both are possible

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-
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12. (b)
The boxes to be coloured black to achieve two lines o
symmetry are c, d and i. 6,4

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6,2
6

13. (MTA)
The bluck shade should be in the 6, 1
direction of line and
the grey surfce should be in the
direction
perpendiculur to the line.
One of the possible unfolding of cube is
6, 0.5
0.s
shown below.
Perimeter =2(a +b)
Sheet M's perimeter =221.5 + 2) =7
Sheet N's perimeter = 2(6+0.5) = 13
ratio of sheet N perimeter
ratio
of sheet M perimeter 7
No option mutches with the unfolding.
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15. (a)
14. (a)
The piece in option A when
M form a rectangle on flipped and rotated, will
assembling with first part.
6,4

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solution
3,4

16
3.2
For lower part of the
we have to add l
figure to be symnetric about
square on the left side and 3 PQ,
1.5, 2 on the right side as squares
15 For upper part, we shown in figure.
have to
only as shown in figure. add 2 squares on the left side
So, total number of
minimum squares needed is
=1+3+2=6

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19. (a)

Scan for Video solution The unfolding of the punched paper is shown below.

17. (c)
As the paper is folded with respect to dotted line, it will
form two triangles vertically one above other. Which is
available in option (c). unfolded
From the above diagram it is clear that the
So, option C' is correct. paper will be like in option A

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18. (a)
20. (a)
if line from 1 to5
In polygon as shown in figure below, The piece in option A if flipped and
rotated by 90 can
is drawn then it lies outside the polygon
hence it is not
fit in the piece in question.
a convex polygon
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