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PLC Papers

Created For:

PiXL PLC 2017 Certification

Area of a Triangle 2 Grade 7

Objective: Know and apply the formula A = ½absinC to calculate the area, sides or angles of a triangle

Question 1.

AB = 8cm

BC = 14cm

Angle ABC = 106o

Calculate the area of triangle ABC.

Give your answer correct to 3 significant figures

(3 marks)

Question 2.

ABC is a triangle with area 27cm2

AC = 14cm

Angle BAC = 115o

Calculate the length of AB. Give your answer correct to two decimal places.

(3 marks)

14cm 115o

Diagram not drawn

accurately


Question 3.

ABC is a triangle

AB = 5cm

BC = 7cm

Angle ABC = 38o

Calculate the area of triangle ABC. Give your answer to 1 decimal place.

(2 marks)

Question 4.

RST is a triangle

RS = 7m

ST = 3m

Angle RST = 35o

Calculate the area of triangle RST.

Give your answer to 2 decimal places.

(2 marks)

7cm

38o

Diagram not drawn

accurately 5cm

3m 7m

35o

R

S

T

Diagram not drawn

accurately 3m


Total /10


Combined transformations 2 Grade 6

Objective; Describe the effects of combinations of rotations, reflections and translations (using column vector notation for translations) Question 1

a) Reflect shape A in the y axis. Label the reflection with the letter B

b) Translate shape B through the vector Label the translation with the letter C

c) Describe fully the single transformation that will transform shape C onto shape A.

x

y

– 5

10

5

– 10

15 10 5 – 5 – 15 – 10

A

8

0


(4) Question 2

a) Rotate shape A through 900 clockwise about the origin Label the rotated shape with the letter B

b) Translate shape B by the vector . Label the translated shape with the letter C

c) Describe fully the transformation that will transform shape C onto shape A

(6)

Total / 10

5

– 1

x

y

– 5

10

5

– 10

15 10 5 – 5 – 15 – 10

A


Congruence and Similarity 2 Grade 6

Objective: Apply the concepts of congruence and similarity, including the relationships between lengths, areas and volumes in similar figures

Question 1.

Two similar cylinders have heights 6cm and 15cm

(a) If the smaller cylinder has a volume of 100cm3, find the volume of the larger cylinder.

(3 marks) (b) If the curved surface area of the larger cylinder is 175cm2, find the curved surface area of the smaller cylinder.

(3 marks)

6cm

15cm

Diagram not drawn

accurately


Question 2.

AB = 6.3cm

DE = 2.1cm

BC = 15.6cm

Calculate the length of EC.

(2 marks)

Question 3.

Two similar regular hexagons have an area of 24cm2 and 84cm2.

The side lengths of the smaller hexagon are 4cm.

How long are the sides of the larger hexagon?

Give your answer correct to two decimal places.

(2 marks)

Total /10

Diagram not drawn

accurately

6.3 cm 2.1cm

15.6cm

A

B C C

D

E


Cosine Rule 2 Grade 7

Objective: Know and apply the Cosine rule to find unknown lengths and angles

Question 1.

ABC is a triangle.

AB = 8cm

BC = 14cm

Angle ABC = 106o

Calculate the length AC. Give your answer correct to one decimal place.

................................

(3 marks)


Question 2.

ABC is a triangle.

AB = 7cm

AC = 5cm

BC = 8cm

Calculate the size of angle BAC. Give your answer correct to one decimal place.

................................°

(4 marks)

8cm C

7cm

5cm


Question 3.

ABC is a triangle.

AC = 7cm

BC = 3cm

Angle ACB = 35o

Calculate the length AB. Give your answer correct to one decimal place.

................................

(3 marks)

Total /10

7cm

3cm

35o


Pythagoras’ and Trigonometry 2D and 3D 2 Grade 7

Objective: Solve problems using Pythagoras's theorem and trigonometry in general 2-D triangles and 3-D figures

Question 1.

The diagram represents a cuboid ABCDEFGH.

Its height is 2.5metres and its width is 4 metres.

Angle GHF = 62o

(a) Calculate the length of the diagonal HF. Give your answer to one decimal place.

..............................................

(2)

(b) Calculate the angle CHF. Give your answer to one decimal place

..............................................

(2)

(Total 4 marks)

Question 2.

ABC is an isosceles triangle.

AC = 18cm

Vertical height = 14cm

Calculate angle BCA to 1dp.

..............................................

(2 marks)

14cm

18cm A C

B

Diagram NOT drawn accurately

Diagram NOT drawn

accurately


Question 3.

ABCDE is a square based pyramid.

The base has sides 9cm.

The vertical height of the pyramid is 8cm.

(a) Calculate the length of AC. Give your answer correct to one decimal place.

..............................................

(1)

(b) Calculate the length of AE. Give your answer correct to one decimal place.

..............................................

(1)

(c) Calculate the size of angle EAC.

..............................................

(2)

Total /10

Diagram NOT drawn

accurately


Sine Rule 2 Grade 7

Objective: Know and apply the Sine rule to find unknown lengths and angles

Question 1.

ABC is a triangle AB = 6cm AC = 7cm Angle ACB = 40o Calculate the size of angle ABC. Give your answer correct to one decimal place.

................................

(4 marks)

400

6cm 7cm


Question 2.

Total /10

ABC is a triangle AB = 12m Angle ACB = 80o

Angle ABC = 40o Calculate the length of AC. Give your answer correct to 1 decimal place.

................................

(3 marks)

400

800

12m


Question 3.

Total /10

ABC is a triangle BC = 7cm Angle CAB = 60o

Angle ACB = 80o Calculate the length of AB. Give your answer correct to 3 significant figures.

................................

(3 marks)

Total /10

600

7cm

800


Standard trigonometric ratios 2 Grade 7

Objective: Know and derive the exact values for Sin and Cos 0, 30, 45, 60 and 90 and Tan 0, 30, 45 and 60 degrees.

Question 1.

A right angled triangle has the dimensions as shown in the diagram.

Using the diagram, or otherwise, state the exact values of:

(a) Sin y

(b) Cos y

(c) Tan y

(d) Sin x

(e) Cos x

(f) Tan x

(Total 6 marks)

Question 2.

State the values of:

(a) Tan 0

(b) Cos 90

(Total 2 marks)

Question 3.

The relationship Sin 30 = Cos 60, can be found for other values of sin and cos. What must the angles add up to for this relationship to work?

(Total 2 marks)

Total /10

4

x

3 5

y

PLC Papers

Created For:


Area of a Triangle 2 Grade 7 Solutions

Objective: Know and apply the formula A = ½absinC to calculate the area, sides or angles of a triangle

Question 1.

AB = 8cm

BC = 14cm

Angle ABC = 106o

Calculate the area of triangle ABC.

Give your answer correct to 3 significant figures

0.5 x a x b x SinC

0.5 x 8 x 14 x Sin106 (M1)

53.83065497 (A1)

53.8cm2 (A1 ft)

(3 marks)

Question 2.

ABC is a triangle with area 27cm2

AC = 14cm

Angle BAC = 115o

Calculate the length of AB. Give your answer correct to two decimal places.

0.5 x a x b x SinC

0.5 x 14 x BA x Sin115 = 27 (M1)

BA = 27 ÷ (0.5 x 14 x Sin115) (M1)

4.26cm (A1)

(3 marks)

14cm 115o

Diagram not drawn

accurately


Question 3.

ABC is a triangle

AB = 5cm

BC = 7cm

Angle ABC = 38o

Calculate the area of triangle ABC. Give your answer to 1 decimal place.

0.5 x 5 x 7 x Sin38 (M1)

10.8cm2 (A1)

(2 marks)

Question 4.

RST is a triangle

RS = 7m

ST = 3m

Angle RST = 35o

Calculate the area of triangle RST.

Give your answer to 2 decimal places.

0.5 x 3 x 7 x Sin35 (M1)

6.02cm2 (A1)

(2 marks)

Total /10

7cm

38o

Diagram not drawn

accurately 5cm

3m 7m

35o

R

S

T

Diagram not drawn

accurately 3m


Combined transformations 2 Grade 6 Solutions

Objective; Describe the effects of combinations of rotations, reflections and translations (using column vector notation for translations) Question 1

a) Reflect shape A in the y axis. Label the reflection with the letter B

Shape drawn in position shown on the grid 1M

b) Translate shape B through the vector Label the translation with the letter C

Shape drawn in position shown on the grid 1M

c) Describe fully the single transformation that will transform shape C onto shape A.

Reflection 1M In the line x = 4 1M ( allow reference to a line on the diagram e.g. the dotted blue line)

(4)

x

y

– 5

10

5

– 10

15 10 5 – 5 – 15 – 10

A B C

8

0


Question 2

a) Rotate shape A through 900 clockwise about the origin

Label the rotated shape with the letter B Shape rotated through 900 1M Shape rotated about the correct point 1M

b) Translate shape B by the vector . Label the translated shape with the letter C

Shape translated to position shown in the diagram 1M

c) Describe fully the transformation that will transform shape C onto shape A

Rotation 1M 900 anticlockwise 1M About ( 2 , – 3 ) 1M

(6)

Total / 10

5

– 1

x

y

– 5

10

5

– 10

15 10 5 – 5 – 15 – 10

A

B

C


Congruence and Similarity 2 Grade 6 Solutions

Objective: Apply the concepts of congruence and similarity, including the relationships between lengths, areas and volumes in similar figures

Question 1.

Two similar cylinders have heights 6cm and 15cm

(a) If the smaller cylinder has a volume of 100cm3, find the volume of the larger cylinder.

Length scale factor = 2.5 (B1)

2.53 x 100 (M1)

1562.5cm3 (A1)

(3 marks) (b) If the curved surface area of the larger cylinder is 175cm2, find the curved surface area of the smaller cylinder.

Length scale factor = 2/5 (B1)

(2/5)2 x 175 (M1)

28cm2 (A1)

(3 marks)

6cm

15cm

Diagram not drawn

accurately


Question 2.

AB = 6.3cm

DE = 2.1cm

BC = 15.6cm

Calculate the length of EC.

Scale factor = 1/3 may be implied in working (B1)

EC = 5.2cm (A1)

(2 marks)

Question 3.

Two similar regular hexagons have an area of 24cm2 and 84cm2.

The side lengths of the smaller hexagon are 4cm.

How long are the sides of the larger hexagon?

Give your answer correct to two decimal places.

Scale factor = √3.5 may be seen in working (B1)

Longer sides are 7.48cm (A1)

(2 marks)

Total /10

Diagram not drawn

accurately

6.3 cm 2.1cm

15.6cm

A

B C C

D

E


Cosine Rule 2 Grade 7 Solutions

Objective: Know and apply the Cosine rule to find unknown lengths and angles

Question 1.

ABC is a triangle.

AB = 8cm

BC = 14cm

Angle ABC = 106o

Calculate the length AC. Give your answer correct to one decimal place.

AC2 = 82 + 142 – 2 x 8 x 14 x Cos106 (M1)

AC2 = 260 - 224Cos106

AC2 = 321.74… (M1)

AC = 17.9cm (M1)

................................

(3 marks)


Question 2.

ABC is a triangle.

AB = 7cm

AC = 5cm

BC = 8cm

Calculate the size of angle BAC. Give your answer correct to one decimal place.

82 = 72 + 52 – 2 x 7 x 5 x CosA (M1)

64 = 74 – 70Cos A

70CosA = 10 (M1)

Cos A = 10/70 (M1)

A = Cos-1 (10/70) = 81.8o (A1)

................................°

(4 marks)

8cm C

7cm

5cm


Question 3.

ABC is a triangle.

AC = 7cm

BC = 3cm

Angle ACB = 35o

Calculate the length AB. Give your answer correct to one decimal place.

AB2 = 72 + 32 – 2 x 7 x 3 x Cos35 (M1)

AB2 = 58 - 42Cos35

AB2 = 23,5956 (M1)

AB = 4.86cm (A1)

................................

(3 marks)

Total /10

7cm

3cm

35o


Pythagoras’ and Trigonometry 2D and 3D 2 Grade 7 Solutions

Objective: Solve problems using Pythagoras's theorem and trigonometry in general 2-D triangles and 3-D figures

Question 1.

The diagram represents a cuboid ABCDEFGH.

Its height is 2.5metres and its width is 4 metres.

Angle GHF = 62o

(a) Calculate the length of the diagonal HF. Give your answer to one decimal place.

cos62 = 4 ÷ HF (M1 using cos62)

HF = 4 ÷ cos62 = 8.5m (A1) ..............................................

(2)

(b) Calculate the angle CHF. Give your answer to one decimal place

tanCHF = 2.5 ÷ 8.5 (M1 Using tanθ)

CHF = tan-1 (2.5 ÷ 8.5) = 16.4m (A1) FT from (a)

..............................................

(2)

(Total 4 marks)

Question 2.

ABC is an isosceles triangle

AC = 18cm

Vertical height = 14cm

Calculate the angle BCA to 1dp.

TanBCA = 14 ÷ 9 (M1 use of Tan)

Tan-1 (14 ÷ 9) = 57.3o ..........................................

(1 mark)

14cm

18cm A C

B

Diagram NOT drawn accurately

Diagram NOT drawn

accurately


Question 3.

ABCDE is a square based pyramid.

The base has sides 9cm.

The vertical height of the pyramid is 8cm.

(a) Calculate the length of AC. Give your answer correct to one decimal place.

AC = √(92 + 92 ) = 12.7cm (B1)

..............................................

(1)

(b) Calculate the length of AE. Give your answer correct to one decimal place.

AE = √ (82 + 6.35 2) = 10.2 cm (B1)

..............................................

(2)

(c) Calculate the size of angle EAC.

CosEAC = AC ÷ AE (M1 use of Cos)

EAC = Cos-1 (AC ÷ AE) = 51.5o (A1)

..............................................

(2)

Total /10

Diagram NOT drawn

accurately


Sine Rule 2 Grade 7 Solutions

Objective: Know and apply the Sine rule to find unknown lengths and angles

Question 1.

ABC is a triangle AB = 6cm AC = 7cm Angle ACB = 40o Calculate the size of angle ABC. Give your answer correct to one decimal place. Sinx7

= Sin406 (M1)

Sinx = Sin406 x 7 (M1)

Sinx = 0.7499..

x = Sin-1 (0.7499..) (M1)

x = 48.6o (A1)

................................

(4 marks)

400

6cm 7cm


Question 2.

Total /10

ABC is a triangle AB = 12m Angle ACB = 80o

Angle ABC = 40o Calculate the length of AC. Give your answer correct to 1 decimal place. 12Sin80

= ACSin40 (M1)

AC = 12Sin80

x Sin40 (M1)

AB = 7.8m (A1)

................................

(3 marks)

400

800

12m


Question 3.

Total /10

ABC is a triangle BC = 7cm Angle CAB = 60o

Angle ACB = 80o Calculate the length of AB. Give your answer correct to 3 significant figures. 7Sin60

= ABSin80 (M1)

AB = 7Sin60

x Sin80 (M1)

AB = 7.96cm (A1)

................................

(3 marks)

Total /10

600

7cm

800


Standard trigonometric ratios 2 Grade 7 Solutions

Objective: Know and derive the exact values for Sin and Cos 0, 30, 45, 60 and 90 and Tan 0, 30, 45 and 60 degrees.

Question 1.

A right angled triangle has the dimensions as shown in the diagram.

Using the diagram, or otherwise, state the exact values of:

(a) Sin y =��ℎ�� =0.8

(b) Cos y =��ℎ�� =0.6

(c) Tan y =�� =

43

(d) Sin x =��ℎ�� =0.6

(e) Cos x =��ℎ�� =0.8

(f) Tan x =�� = 0.75

(Total 6 marks)

Question 2.

State the values of:

(a) Tan 0 = 0

(b) Cos 90 = 0

(Total 2 marks)

Question 3.

The relationship Sin 30 = Cos 60, can be found for other values of sin and cos. What must the angles add up to for this relationship to work? 90

(Total 2 marks)

Total /10

4

x

3 5

y

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