Hard

Symmetry — Hard

15 questions 18 min PYQ-grade reasoning

Q1. About the four-puzzle-piece figure with a dotted middle line, consider: I. The line splits the figure into two equal-area parts. II. Folding along the line, the two parts overlap exactly. III. The line is a line of symmetry. Which is/are correct?
- A. I and II only
- B. II and III only
- C. I and III only
- D. I only
Q2. Assertion (A): A diagonal of a square is a line of symmetry of the square. Reason (R): A diagonal of any rectangle is a line of symmetry of the rectangle. Choose the correct option.
- A. Both A and R are true, and R is the correct explanation of A.
- B. Both A and R are true, but R is NOT the correct explanation of A.
- C. A is true, R is false.
- D. A is false, R is true.
Q3. A regular six-pointed star (like the one shown for Q3(c)) is rotated about its centre. Its angles of symmetry are
- A. 60 deg, 120 deg, 180 deg, 240 deg, 300 deg, 360 deg
- B. Only 60 deg and 360 deg
- C. 30 deg, 60 deg, 90 deg, 360 deg
- D. Only 180 deg and 360 deg
Q4. A figure with 5 equal radial arms is to have rotational symmetry. The angle between two adjacent arms must be
- A. 60 deg
- B. 72 deg
- C. 90 deg
- D. 108 deg
Q5. Can a figure have rotational symmetry whose smallest angle of symmetry is 17 deg?
- A. Yes, because 17 is a whole number of degrees
- B. Yes, because 360 deg is always an angle of symmetry
- C. No, because 17 deg is not a factor of 360 deg
- D. No, because the smallest angle must be greater than 30 deg
Q6. A square has corners labelled A, B (top) and D, C (bottom). After a 90 deg clockwise rotation about its centre, where does corner A go?
- A. The position earlier occupied by B
- B. The position earlier occupied by C
- C. The position earlier occupied by D
- D. Its own original position
Q7. A square ABCD has corners A (top-left), B (top-right), C (bottom-right), D (bottom-left). When it is reflected along the diagonal from A to C, the points B and D respectively go to the positions occupied earlier by
- A. A and C
- B. D and B
- C. C and A
- D. B and D themselves
Q8. The outer boundary of the new Parliament Building (a regular triangular shape) shown in the chapter has
- A. 2 lines of symmetry and 2 angles of symmetry
- B. 3 lines of symmetry and 3 angles of symmetry
- C. 4 lines of symmetry and 4 angles of symmetry
- D. 6 lines of symmetry and 3 angles of symmetry
Q9. A circle is divided into 12 equal sectors that can be coloured in different ways. By choosing different colourings, the possible numbers of angles of rotational symmetry one can obtain are exactly the
- A. Multiples of 12 up to 360
- B. Divisors (factors) of 12
- C. Multiples of 30 up to 360
- D. All numbers from 1 to 12
Q10. Consider the following statements about a kite-shaped quadrilateral (two pairs of equal adjacent sides): I. It has exactly one line of symmetry. II. The line of symmetry joins the two vertices between unequal sides. III. Its diagonals are perpendicular and both are lines of symmetry. Which statements are correct?
- A. I and II only
- B. II and III only
- C. I and III only
- D. All three
Q11. A figure with radial arms has exactly 7 angles of symmetry. Its smallest angle of symmetry, expressed as a mixed number, is
- A. 50 deg
- B. 51 and 1/7 deg
- C. 51 and 3/7 deg
- D. 52 deg
Q12. Assertion (A): Every figure with rotational symmetry also has at least one line of symmetry. Reason (R): A paper windmill has rotational symmetry of order 4 but has no line of symmetry. Choose the correct option.
- A. Both A and R are true, and R is the correct explanation of A.
- B. Both A and R are true, but R is NOT the correct explanation of A.
- C. A is true, R is false.
- D. A is false, R is true.
Q13. An isosceles triangle has its vertical line of symmetry marked. A small hole is shown on the left side of this line, halfway down. To make the figure symmetric, where must a second hole be placed?
- A. At the same height on the right side of the line, at equal distance from the line
- B. At the top vertex of the triangle
- C. At the bottom-left vertex
- D. On the line itself, halfway down
Q14. What is the order of rotational symmetry of a regular pentagon?
- A. 3
- B. 4
- C. 5
- D. 10
Q15. In a certain figure, 60 deg is the smallest angle of symmetry. The other angles of symmetry of this figure are
- A. Only 120 deg and 360 deg
- B. 120 deg, 180 deg, 240 deg, 300 deg, 360 deg
- C. 60 deg, 90 deg, 120 deg, 360 deg
- D. Only 180 deg and 360 deg

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