PLC Papers
Created For:
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
115o
14cm
Diagram not
drawn
accurately
AC = 14cm
Angle BAC = 115o
Calculate the length of AB. Give your answer correct to two decimal places.
(3 marks)
PiXL PLC 2017 Certification
Diagram not
drawn
accurately
Question 3.
ABC is a triangle
AB = 5cm
5cm
BC = 7cm
38o
Angle ABC = 38o
7cm
Calculate the area of triangle ABC. Give your answer to 1 decimal place.
(2 marks)
S
Question 4.
35o
RST is a triangle
7m
3m
Diagram not
3m drawn
accurately
RS = 7m
T
ST = 3m
Angle RST = 35o
R
Calculate the area of triangle RST.
Give your answer to 2 decimal places.
(2 marks)
PiXL PLC 2017 Certification
Total /10
PiXL PLC 2017 Certification
Combined transformations 2
Objective;
Grade 6
Describe the effects of combinations of rotations, reflections and translations
(using column vector notation for translations)
Question 1
y
10
5
A
– 15
– 10
–5
5
10
–5
– 10
a) Reflect shape A in the y axis.
Label the reflection with the letter B
8
b) Translate shape B through the vector
0
Label the translation with the letter C
c) Describe fully the single transformation that will transform shape C onto shape A.
PiXL PLC 2017 Certification
15 x
(4)
Question 2
y
10
5
A
– 15
– 10
–5
5
10
15 x
–5
– 10
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
5
–1
.
Label the translated shape with the letter C
c) Describe fully the transformation that will transform shape C onto shape A
(6)
Total
PiXL PLC 2017 Certification
/ 10
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
15cm
Diagram not
drawn
accurately
6cm
(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)
PiXL PLC 2017 Certification
Question 2.
AB = 6.3cm
A
DE = 2.1cm
BC = 15.6cm
D
Diagram not
drawn
accurately
Calculate the length of EC.
6.3 cm
B
2.1cm
E
C
15.6cm
(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
PiXL PLC 2017 Certification
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)
PiXL PLC 2017 Certification
Question 2.
7cm
5cm
C
8cm
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)
PiXL PLC 2017 Certification
Question 3.
3cm
35o
7cm
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
PiXL PLC 2017 Certification
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
Diagram NOT
drawn
accurately
(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)
B
Question 2.
Diagram
NOT drawn
accurately
ABC is an isosceles triangle.
AC = 18cm
14cm
Vertical height = 14cm
Calculate angle BCA to 1dp.
A
18cm
C
..............................................
(2 marks)
PiXL PLC 2017 Certification
Question 3.
Diagram NOT
drawn
accurately
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
PiXL PLC 2017 Certification
Sine Rule 2
Grade 7
Objective: Know and apply the Sine rule to find unknown lengths and angles
Question 1.
6cm
7cm
400
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)
PiXL PLC 2017 Certification
Question 2.
800
Total /10
400
12m
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)
PiXL PLC 2017 Certification
Question 3.
800
Total /10
7cm
600
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
PiXL PLC 2017 Certification
Standard trigonometric ratios 2
Objective:
Grade 7
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:
5
y
3
(a) Sin y
x
(b) Cos y
4
(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
PiXL PLC 2017 Certification
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
115o
14cm
Diagram not
drawn
accurately
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)
PiXL PLC 2017 Certification
Diagram not
drawn
accurately
Question 3.
ABC is a triangle
AB = 5cm
5cm
BC = 7cm
38o
Angle ABC = 38o
7cm
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)
S
Question 4.
35o
RST is a triangle
7m
3m
RS = 7m
Diagram not
3m drawn
accurately
T
ST = 3m
Angle RST = 35o
R
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
PiXL PLC 2017 Certification
PiXL PLC 2017 Certification
Combined transformations 2
Objective;
Grade 6
Solutions
Describe the effects of combinations of rotations, reflections and translations
(using column vector notation for translations)
Question 1
y
10
A
– 15
– 10
5
C
B
–5
5
15 x
10
–5
– 10
a) Reflect shape A in the y axis.
Label the reflection with the letter B
Shape drawn in position shown on the grid
b) Translate shape B through the vector
1M
8
0
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
In the line x = 4
( allow reference to a line on the diagram e.g. the dotted blue line)
1M
1M
(4)
PiXL PLC 2017 Certification
Question 2
y
10
5
B
C
A
– 15
– 10
–5
5
15 x
10
–5
– 10
a) Rotate shape A through 900 clockwise about the origin
Label the rotated shape with the letter B
Shape rotated through 900
Shape rotated about the correct point
5
b) Translate shape B by the vector
–1
1M
1M
.
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
900 anticlockwise
About ( 2 , – 3 )
1M
1M
1M
(6)
Total
PiXL PLC 2017 Certification
/ 10
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
15cm
Diagram not
drawn
accurately
6cm
(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)
PiXL PLC 2017 Certification
Question 2.
AB = 6.3cm
A
DE = 2.1cm
BC = 15.6cm
D
Diagram not
drawn
accurately
Calculate the length of EC.
6.3 cm
B
2.1cm
E
C
15.6cm
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
PiXL PLC 2017 Certification
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)
PiXL PLC 2017 Certification
Question 2.
7cm
5cm
C
8cm
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)
PiXL PLC 2017 Certification
Question 3.
3cm
35o
7cm
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
PiXL PLC 2017 Certification
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
Diagram NOT
drawn
accurately
(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)
B
Question 2.
Diagram
NOT drawn
accurately
ABC is an isosceles triangle
AC = 18cm
14cm
Vertical height = 14cm
Calculate the angle BCA to 1dp.
TanBCA = 14 ÷ 9 (M1 use of Tan)
Tan-1 (14 ÷ 9) = 57.3o
A
18cm
C
..........................................
(1 mark)
PiXL PLC 2017 Certification
Question 3.
Diagram NOT
drawn
accurately
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
PiXL PLC 2017 Certification
Sine Rule 2
Grade 7
Solutions
Objective: Know and apply the Sine rule to find unknown lengths and angles
Question 1.
6cm
7cm
400
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.
Sinx
7
Sinx =
Sin40
=
Sin40
6
6
(M1)
x 7 (M1)
Sinx = 0.7499..
x = Sin-1 (0.7499..) (M1)
x = 48.6o (A1)
................................
(4 marks)
PiXL PLC 2017 Certification
Question 2.
800
Total /10
400
12m
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.
12
AC
Sin80
=
=
12
Sin80
AC
Sin40
(M1)
x Sin40 (M1)
AB = 7.8m (A1)
................................
(3 marks)
PiXL PLC 2017 Certification
Question 3.
800
Total /10
7cm
600
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.
7
AB
Sin60
=
7
=
Sin60
AB
Sin80
(M1)
x Sin80 (M1)
AB = 7.96cm (A1)
................................
(3 marks)
Total /10
PiXL PLC 2017 Certification
Standard trigonometric ratios 2
Objective:
Grade 7
Solutions
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��
���
=
���
=0.6
(c) Tan y =
(d) Sin x =
���
ℎ��
(e) Cos x =
(f) Tan x =
4
4
3
���
=0.8
���
= 0.75
ℎ��
���
3
x
=0.6
ℎ��
y
=0.8
���
(b) Cos y =
5
(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
PiXL PLC 2017 Certification