GATE General Aptitude: Spatial Aptitude questions with solutions

Q1. An ant walks in a straight line on a plane leaving behind a trace of its movement. The initial position of the ant is at point P facing east. The ant first turns 72° anticlockwise at P, and then does the following two steps in sequence exactly FIVE times before halting. 1. moves forward for 10 cm. 2. turns 144° clockwise. The pattern made by the trace left behind by the ant is

1. PQ = QR = RS = ST = TP = 10 cm

Answer: PQ = QR = RS = ST = TP = 10 cm

The ant consistently moves forward 10 cm after each turn, maintaining equal segment lengths, which results in a regular pattern. The combination of the 72° anticlockwise turn and the 144° clockwise turn creates a geometric shape where all sides are equal, confirming that each segment is indeed 10 cm.

Q2. Looking at the surface of a smooth 3-dimensional object from the outside, which one of the following options is TRUE?

1. The surface of the object must be concave everywhere.
2. The surface of the object must be convex everywhere.
3. The surface of the object may be concave in some places and convex in other places.
4. The object can have edges, but no corners.

Answer: The surface of the object may be concave in some places and convex in other places.

The correct option acknowledges that a smooth 3-dimensional object can have varying surface geometries, allowing for both concave and convex areas, which is common in many natural and manufactured shapes.

Q3. A cube is to be cut into 8 pieces of equal size and shape. Here, each cut should be straight and it should not stop till it reaches the other end of the cube. The minimum number of such cuts required is

1. 3
2. 4
3. 7
4. 8

Answer: 3

Cutting once along each of the three mutually perpendicular axes divides the cube into 2x2x2 = 8 equal pieces. Three cuts suffice, and fewer is impossible, so the minimum is 3.

Q4. A 100 cm × 32 cm rectangular sheet is folded 5 times. Each time the sheet is folded, the long edge aligns with its opposite side. Eventually, the folded sheet is a rectangle of dimensions 100 cm × 1 cm. The total number of creases visible when the sheet is unfolded is _______.

1. 32
2. 5
3. 31
4. 63

Answer: 31

Folding 5 times reduces 32cm to 1cm, doubling layers each time. When unfolded the fold lines number 2^5-1 = 31 creases.

Q5. Which one of the following options has the correct sequence of objects arranged in the increasing number of mirror lines (lines of symmetry)?

1. Circle: Square: Equilateral triangle: Isosceles triangle
2. Isosceles triangle: Equilateral triangle: Square: Circle
3. Equilateral triangle: Isosceles triangle: Square: Circle
4. Isosceles triangle: Square: Equilateral triangle: Circle

Answer: Isosceles triangle: Equilateral triangle: Square: Circle

The isosceles triangle has one line of symmetry, the equilateral triangle has three, the square has four, and the circle has infinite lines of symmetry, making the correct sequence based on the increasing number of mirror lines: Isosceles triangle, Equilateral triangle, Square, Circle.