25

Junior Certifi cate Ordinary Level Maths Solutions

Sample Paper 2, no. 5

1. 15 × 15 = 225 × 6 = 1,350 cm2

2. (a)

70°

(b)

(c) To bisect means to divide an angle into two equal parts.

3. (a)

k

Object

Image

(b) No, the length remains 8 units and the width remains 4 units.

4. (a) 16

___

40 =

2 __

5

(b) 12

___

40 =

3 ___

10

(c) 4 ___

40 =

1 ___

10

(d) 8 ___

40 =

1 __

5

(e) Agree. All the coins are not the same size; for example, a fi fty-cent coin is larger than a twenty-cent coin.

26

5. (a) [YZ] Reason: Opposite sides of a parallelogram are equal in measure.

(b) [XY] Reason: Opposite sides are equal in measure.

(c) |∠XYZ| Reason: Opposite angles are equal in measure.

(d) 104 cm2 = 52 × 2. Two triangles make up the parallelogram.

6. (a) 5 + 5 + 8 + 8 = 26 cm

(b) 30 × 18 × 16 = 8,640 cm3

(c) 15 × 7 × 8 = 840

8640 − 840 = 7,800 cm3

7. Answer: 35°

Explanation: |∠xyw| = 180° − 115° = 65°

|∠xwy| = 180° − 100° = 80°

The 3 angles add up to 180° in a triangle so

|∠yxw| = 180° − 65° − 80°

B = 35°

8. (a) Marks scored 10 20 30 40 50

Number of students 4 3 6 2 5

(b)

10 20 30Marks

40 50

9. (a) Primary data is data that you collect yourself.

(b) Secondary data is data obtained from a book or the internet or gathered by alternative surveyors.

(c) Primary, Secondary, Secondary, Secondary, Secondary

27

10. (a)

x

y

1

–4 –3 –2 –1 100

2 3 4

2

3

4

–4

–3

–2

–1

C(4,–4)F(0,–4)D(–4,–4)

B(4,4)E(0,4)A(–4,4)

(b) (−4,−4)

(c) |AB| = 8, |BC| = 8, |CD| = 8, |DA| = 8

(d) E = (0,4)

(e) F = (0,−4)

(f) See diagram in part (a)

(g) Multiple possible answers: (0,0), (0,1), (0,2), … (0,−1), (0,−2)...

11. (a) 80 × 2 = 160 km

(b) 150

____

100 = 1.5 hours

(c) 2 + 1.5 = 3.5 hours

12. (a) King Queen Jack AceKing (K,K) (Q,K) (J,K) (A,K)

Queen (K,Q) (Q,Q) (J,Q) (A,Q)

Jack (K,J) (Q,J) (J,J) (A,J)

Ace (K,A) (Q,A) (J,A) (A,A)

(b) 1 ___

16

(c) 7 ___

16

28

(d) 4 ___

16 =

1 __

4

(e) 9 ___

16

(f) 9 ___

16

13. (a) People would be more likely to answer Yes in order to seem ‘normal’. Alternative question: ‘Do you like movies?’

(b) Unclear. Alternative question: ‘How many times do you go to the cinema in a year?’

Never, 0–5 times, 6–10 times, More often

(c) The question is too personal. Alternative question: ‘What is your monthly income?’

0–500 euro, 501–1,000 euro, more than 1,001 euro

(d) It is a leading question.

Alternative question: ‘What is your opinion about actors’ pay?’

14. (a) A = 7°

(b) A = 70°

(c) A = 37°

15. (a)

14 m

40°

x

(b) sin 40° = 14

___ x

x sin 40° = 14

x = 14 ______

sin 40°

x = 21.78

x = 22 m

16. Line a contains points (0,2) and (−3,0).

Its slope is 0 − 2

______

−8 − 0 =

2 __

3 .

29

Its equation is y − 0 = 2 __

3 (x + 3)

3y = 2x + 6 or 3x − 2y = −6

Line b contains (−1,0) and (0,−2).

Its slope is −2

___

1 = −2.

Its equation is y − 0 = −2(x + 1)

y = −2x − 2

2x + y = −2

Line c is the line y = −2 or y + 2 = 0.

Equations Sketched line

x = −2y + 2 = 02x − 3y = −62x + y = −2x −3y = 4

cab

30

Junior Certifi cate Ordinary Level Maths Solutions

Sample Paper 2, no. 6

1. Sample answer: He might do a street survey on the main street; he could randomly stop people of all ages and ask them what foods they like.

2. (a) All three angles are 60° and the lengths of the three sides of the triangle are equal.

(b)

6 cm

6 cm

60°

60°

60°

6 cm

3. (a) 80 cm ÷ 2 = 40 cm

(b) pr2h = (3.142)(40)(40)(440)

= 2,211,968 cm3

(c) 2,211,968 × 8 = 17,695,744 cm3

4. Equation of the line Line

x = 2y = –3x = −4y = 2

wzxy

5. (a) 5.2 × 10 = 52 mm

(b) 1250 ÷ 1000 = 1.25 kg

(c) 3.75 × 1000 = 3,750 m

6. (a)

F B

U = 50

30 – x 25 – x

8

x

31

(b) 50 = 30 – x + x + 25 − x + 8

50 = 63 − x

x = 63 − 50

x = 13

(c) 8 ___

50 =

4 ___

25

(d) 17 + 12

_______

50 =

29 ___

50

7. (a)

D A

C B

(b) Spinner

A B C D

Die

1 (A,1) (B,1) (C,1) (D,1)

2 (A,2) (B,2) (C,2) (D,2)

3 (A,3) (B,3) (C,3) (D,3)

4 (A,4) (B,4) (C,4) (D,4)

5 (A,5) (B,5) (C,5) (D,5)

6 (A,6) (B,6) (C,6) (D,6)

(c) 3 ___

24 =

1 __

8

(d) 3 ___

24 =

1 __

8

8. (a) Stem Leaf

2 8

3 6 6 7 7 7 7 7 8 9 9

4 1 2 2 2 2 3 4 7 9

Key: 4|1 = 41

(b) 37 cm

(c) 39 cm

(d) In this case both are ideal, as they are relatively close in value.

(e) 49 − 28 = 21 cm

32

9. (a) From looking at the graph, it seems that double the amount of money was spent in 2014 compared with 2012.

(b) Based on the y-axis, it seems that each year double the amount of money is spent on new buildings, but without a scale we cannot be sure. Also, the bars are increasing in width, suggesting it maybe more than double.

10. (a) Use the midpoint formula:

( 2 + 8 _____

2 ,

3 + 11 ______

2 ) = ( 10

___

2 ,

14 ___

2 ) = (5,7)

(b) Use (0,0) and (2,3)

√_______________

(2 − 0)2 + (3 − 0)2

= √_____

4 + 9 = √___

13 = 3.6 units

11. (a) 1st line: Positive; 2nd line: Negative

(b) Slope

Stage A 0

Stage B 5 __

2

Stage C 1 __

4

12. (a) tan 30° = 0.5773502692 = 0.577

(b), (c), (d)

30°

Adjacent100 m

Opp

osite

Hypotenuse

(e) tan 30° = height

______

100

height = 100 tan 30°

height = 57.73502692

height = 57.7 m

13. The diagonal bar:

(diagonal)2 = (1.8)2 + (2.5)2

(diagonal)2 = 3.24 + 6.25

33

(diagonal)2 = 9.49

diagonal = √____

9.49

= 3.1 m

Total number needed 4(2.5) + 2(1.8) + 3.1 = 16.7 metres

14. Transformation Co-ordinates of vertices

Axial symmetry in the y-axis (3,2) (7,2) (7,6) (3,6)

Central symmetry in the point (0,0) (3,−2) (7,−2) (7,−6) (3,−6)

Axial symmetry in the x-axis (−7,−2) (−3,−2) (−3,−6) (−7,−6)

15. The mode, as it is the county that appears most often, in this case Kerry.

16. (a)

4 axes of symmetry

No axis of symmetry

5 axes of symmetry

(b) A = 60°

In a straight line, the angles add up to 180°.

A + 120 = 180

A = 180 − 120

A = 60°

(c) B = 50°

The three angles in a triangle add up to 180°.

B + 70 + 60 = 180

B + 130 = 180

B = 180 − 130

B = 50°

34

Junior Certifi cate Ordinary Level Maths Solutions

Sample Paper 2, no. 7

1. A

P Q

2. (a) [BC] = 6 cm

(b) h = 3.5 cm

(c) Area = Answer from (a) × Answer from (b) = 21 cm2

3. (a) Radius = 40 ÷ 2 = 20 cm

(b) 2(3.14)(20) = 125.6 = 126 cm

(c) Radius = 50 cm

2(3.14)(50) = 314 cm

(d) 314 – 126 = 188 cm

(e) 314 × 150 = 47,100 cm or 471 m

4. Data Type of data

1. The number of horses in a race Numerical

2. Blood types of students in your class Categorical

3. The area of a garden Numerical

4. The score on a die Numerical

5. The colour of a t-shirt Categorical

6. The temperature in a fridge Numerical

5. (a) 3 __

6 =

1 __

2

(b) 1 __

6

(c) 3 __

6 =

1 __

2

6. (a) Letting (x = 0) 2(0) − y = −1

1 = y (0,1)

35

Letting (y = 0) 2x − 0 = −1

2x = −1

x = − 1 __ 2

( − 1 __ 2 ,0 )

(b) Letting x = 0 0 + y = 4 (0,4)

Letting y = 0 x + 0 = 4 (4,0)

(c)

–2

–3

–4

–5

–6

–1

1

2

3

4

5

6

y

–1

x + y = 4

2x – y = –1

–2–3–4–5–6 1 2 3 5 640x

(d) (1,3)

7. (a) 360 − 135 − 45 − 30 − 90 = 60°

(b) Computer games

(c) 90° = 24 students so 360° = 96 students

8. (a) Mode of travel Walk Car Cycle Bus

Number of students 7 11 4 5

36

(b)

Walk Car Cycle Bus

(c) Cycle

(d) It would be typical for a day with good weather, but if it was raining, some students who normally walk or cycle might choose to travel by car instead.

9. (a) Acute: 45° Right-angled: 90° Obtuse: 130° Straight: 180° Refl ex: 270° Null: 0°

(b) x = 130°, as it is vertically opposite 130°

(c) y = 50°; straight line ⇒180 − 130

(d) z = 50°, as it is vertically opposite angle y

(e) Angles Y and Z are equal in measure or vertically opposite or acute angles, but angles X and Y together make a 180° angle.

10. (a) A B C

1 (A,1) (B,1) (C,1)

2 (A,2) (B,2) (C,2)

3 (A,3) (B,3) (C,3)

4 (A,4) (B,4) (C,4)

5 (A,5) (B,5) (C,5)

6 (A,6) (B,6) (C,6)

(b) 18 outcomes

(c) 6 ___

18 =

1 __

3

(d) 3 ___

18 × 100 = 17%

(e) 3 ___

18 =

1 __

6

37

11. (a) 01:25 ≈ 25:25

− 22:40

2:45

2 hours and 45 minutes

(b) 2 3 __

4 × 5 × €1.83 = € 25.16

12. (a)

–2

–3

–4

–5

–6

–1

1

2

3

4

5

6

y

–1

C

BA

–2–3–4–5–6 1 2 3 5 640x

(b) (−3,−3)

(c) ( −3 + 3 ______

2 ,

3 − 3 _____

2 ) = (0,0)

(d) ( 3 − 3 _____

2 ,

3 − 3 _____

2 ) = (0,0)

(e)

CD

BA

(f) Both diagonal intersect at the point (0,0), the midpoint of both diagonals.

38

13. (a)

20°25 m

(b) tan 20° = h ___

25

h = 25 tan 20°

h = 9 metres

14. sin A = 3 __ 5 or cos A = 4 __

5 or tan A = 3 __

4

sin A = 0.6 cos A = 0.8 tan A = 0.75

A = 37°

15. (a) 120 × 80 × 50 = 480,000 cm3

(b) 240,000 ÷ 1,000 = 240 litres

16. Central symmetry in the point O: Δ BGO; axial symmetry in [HF]: Δ CGO; axial symmetry in [KG]: Δ AKO