CAMBRIDGE IGCSE MATHS (0580)

Question 1:


The diagram represents a boat race with three buoys K, L, and M.
MK = 4 km, KL = 9 km, and angle MKL = 108°.

(a) Calculate the distance ML.
Answer(a) ML = ............................................... km [4]

(b) The bearing of L from K is 125°.
(i) Calculate how far L is south of K.
Answer(b)(i) ............................................... km [3]

(ii) Find the three-figure bearing of K from M.
Answer(b)(ii) ............................................... [2]

Question 2:


The frame of a child’s bicycle is made from metal rods. ABC is an isosceles triangle with base 22 cm and base angles 50°.
Angle ACD = 100° and CD = 31 cm.

Calculate the length AD.
Answer(c) AD = ............................................... cm [6]

Question 3:


The diagram shows straight roads connecting the towns A, B, C, and D.
AB = 17 km, AC = 12 km, and CD = 10 km.
Angle BAC = 30° and angle ADC = 95°.

(a) Calculate angle CAD.
Answer(a) Angle CAD = ............................................... [3]

(b) Calculate the distance BC.
Answer(b) BC = ............................................... km [4]

(c) The bearing of D from A is 040°.
Find the bearing of
(i) B from A,
Answer(c)(i) ............................................... [1]
(ii) A from B.
Answer(c)(ii) ............................................... [1]

(d) Angle ACB is obtuse.
Calculate angle BCD.
Answer(d) Angle BCD = ............................................... [4]

Question 4:


In the diagram, BCD is a straight line and ABDE is a quadrilateral.
Angle BAC = 90°, angle ABC = 30° and angle CAE = 52°.
AC = 15.7cm, CE = 16.5cm and CD = 23.4cm.

(a) Calculate BC.
Answer(a) BC = ........................................... cm [3]

(b) Use the sine rule to calculate angle AEC.
Show that it rounds to 48.57°, correct to 2 decimal places.
Answer(b) ........................................... [3]

(c) (i) Show that angle ECD = 40.6°, correct to 1 decimal place.
Answer(c)(i) ........................................... [2]

(ii) Calculate DE.
Answer(c)(ii) DE = ........................................... cm [4]

(d) Calculate the area of the quadrilateral ABDE.
Answer(d) ........................................ cm² [4]

Question 5:


(a) The diagram shows triangle LMN with:

• LM = 12 cm
• LN = 15 cm
• MN = 21 cm

(i) Calculate angle LMN. Show that this rounds to 44.4°, correct to 1 decimal place.
Answer(a)(i): ........................................ [4]
(ii) Calculate the area of triangle LMN.
Answer(a)(ii): ........................................ cm² [2]

(b) The diagram shows triangle PQR with:

• PQ = 6.4 cm
• Angle PQR = 82°
• Angle QPR = 43°

Calculate the length of PR.
Answer(b): PR = ......................................... cm [4]

Question 6:
(a)

The diagram shows a circle with center O and points A, B, C, D, and E on the circle. Given:

• Angle ABD = 27°.

Find:

1. Angle ACD
   Answer(a)(i): Angle ACD = ............................................... [1]
2. Angle AOD
   Answer(a)(ii): Angle AOD = ............................................... [1]
3. Angle AED
   Answer(a)(iii): Angle AED = ............................................... [1]

(b) The diagram shows quadrilateral KLMN with:

• KL = 45 cm
• LN = 32 cm
• Angle KLN = 100°
• Angle NLM = 67°

1. Calculate the length KN.
   Answer(b)(i): KN = ......................................... cm [4]
2. The area of triangle LMN is 324 cm². Calculate the length LM.
   Answer(b)(ii): LM = ......................................... cm [3]



3. Another triangle XYZ is mathematically similar to triangle LMN. Given:
   • XZ = 16 cm
   • The area of triangle LMN is 324 cm².
   Calculate the area of triangle XYZ.
   Answer(b)(iii): ........................................ cm² [2]

Question 7:


The diagram shows a quadrilateral ABCD where:

• Angle BAD = 49°
• Angle ABD = 55°
• BD = 80 m, BC = 95 m, CD = 90 m

Solve the following:

1. Use the sine rule to calculate the length of AD.
   Answer(a): AD = ............................................ m [3]
2. Use the cosine rule to calculate angle BCD.
   Answer(b): Angle BCD = ................................................ [4]
3. Calculate the area of the quadrilateral ABCD.
   Answer(c): ........................................... m² [3]
4. The quadrilateral represents a field. Corn seeds are sown across the entire field at a cost of $3250 per hectare.
   1 hectare = 10,000 m².
   Calculate the cost of the corn seeds used.
   Answer(d): $ ................................................ [3]

Question 8:

A plane flies from A to C and then from C to B.
AC = 510 km, CB = 720 km.
The bearing of C from A is 135° and angle ACB = 40°.

1. Find the bearing of:
   1. B from C:
      Answer: ................................................... [2]
   2. C from B:
      Answer: ................................................... [2]
2. Calculate AB and show that it rounds to 464.7 km, correct to 1 decimal place.
   Answer: ................................................... [4]
3. Calculate angle ABC.
   Answer: Angle ABC = .................................................. [3]

Question 9:

The diagram shows a field ABCDE:

1. Calculate the perimeter of the field ABCDE.
   Answer: ................................................ m [4]
2. Calculate angle ABD.
   Answer: Angle ABD = .......................................................... [4]
3. (i) Calculate angle CBD.
   Answer: Angle CBD = .................................................... [2]
   (ii) Find the bearing of D from B.
   Answer: .................................................... [2]
4. Calculate the area of the field ABCDE.
   Give your answer in hectares (1 hectare = 10,000 m²).
   Answer: ...................................... hectares [4]

Question 10:

1. (a) Show that each interior angle of a regular pentagon is 108°.
   Answer: ........................................ [2]



2. The diagram shows a regular pentagon ABCDE where:
   • The vertices of the pentagon lie on a circle, center \( O \), radius \( 12 \, \text{cm} \).
   • \( M \) is the midpoint of \( BC \).
   1. Find \( BM \).
      Answer: \( BM = \) ........................................ cm [3]
   2. \( OMX \) and \( ABX \) are straight lines:
      • (a) Find \( BX \).
      • Answer: \( BX = \) ........................................ cm [3]
      • (b) Calculate the area of triangle \( AOX \).
      • Answer: ........................................ cm² [3]

Question 11:

The diagram shows two triangles \( \triangle ABD \) and \( \triangle BCD \):

• \( AD = 16.5 \, \text{cm} \)
• \( BD = 12.4 \, \text{cm} \)
• \( \angle ADB = 64^\circ \), \( \angle BDC = 53^\circ \), \( \angle DBC = 95^\circ \)

1. (i) Find \( AB \).
   Answer: \( AB = \) ........................................ cm [4]
2. (ii) Find \( BC \).
   Answer: \( BC = \) ........................................ cm [4]