SPM 2023 Add Maths Paper 2, Question 7Circular Measure

Solved on video in English and Bahasa Melayu · Form 5, Circular Measure · 6:35 video

The question

(a)

Diagram 3 shows a shape formed by a combination of four sectors and a square ABCD. A, B, C and D are the centres of sectors BAD, PBQ, BCD and PDQ respectively. It is given that AB = 5 cm and the length of the straight line PQ, which passes through points A and C, is 23 cm. [Use π = 3.142]

(a) Find ∠PBQ in radians, correct to two decimal places.

(b)(i)

Diagram 3 shows a shape formed by four sectors and a square ABCD, with AB = 5 cm and PQ = 23 cm; ∠PBQ = 2.55 rad and the sector radius BP = 12.0312 cm. [Use π = 3.142]

(b)(i) Hence, find the outside perimeter, in cm, of the shape.

(b)(ii)

Diagram 3 shows a shape formed by four sectors and a square ABCD, with AB = 5 cm and PQ = 23 cm; ∠PBQ = 2.55 rad and the sector radius BP = 12.0312 cm. [Use π = 3.142]

(b)(ii) Find the area, in cm², of the shaded region.

SPM 2023 Add Maths Paper 2, Question 7: the question as drawn in Eduly's video6:35

Part (a)

Step 1

The square ABCD has side AB = 5, so the diagonal AC = 5√2 ≈ 7.0711 cm, and the square's centre O is the midpoint of AC.

5 more steps, each with the reason for it, in the full solution

Final answer

∠PBQ = 2.55 rad

Where students lose marks

Watch out: inverse cosine gives degrees. Convert to radians with π/180 (π = 3.142) and round to 2 d.p.

Part (b)(i)

3 more steps, each with the reason for it, in the full solution

Final answer

61.35 cm

Where students lose marks

Watch out: the outside edge is two arcs PQ (from B and D), so double the arc length. One arc is not enough.

Part (b)(ii)

7 more steps, each with the reason for it, in the full solution

Final answer

144.89 cm²

Where students lose marks

Watch out: subtract the triangle, half the square and both 135° inner sectors from the sector. Subtracting the wrong regions gives a different area.

Watch every step of this question solved on video (6:35), with the reason behind each step and a box that checks your own answer.

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