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
Circle Theorems 2
Grade 8
Objective: Apply and prove the standard circle theorems concerning angles, radii, tangents and
chords, and use them to prove related results
Question 1
A and B are points on the circumference of a circle, centre O.
AC and BC are tangents to the circle.
Angle ACB = 36°.
Find the size of angle OBA.
Give reasons for your answer.
(Total 3 marks)
PiXL PLC 2017 Certification
Question 2
*
A, B, C and D are points on the circumference of a circle, centre O.
Angle AOC = y.
Find the size of angle ABC in terms of y.
Give a reason for each stage of your working.
(Total 4 marks)
PiXL PLC 2017 Certification
Question 3
B, C and D are points on the circumference of a circle, centre O.
ABE and ADF are tangents to the circle.
Angle DAB = 40°
Angle CBE = 75°
Work out the size of angle ODC.
(Total 3 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
Vector Arguments and Proof 2
Grade 8
Objective: Use vectors to construct geometric argument and proof
Question 1.
OPT is a triangle
OT = 3a and TP = 5b
5b
Diagram NOT
drawn
accurately
M is the midpoint of OP
3a
(a) Find OM in terms of a and b
..............................................
(2)
(b) Find TM in terms of a and b
..............................................
(2)
(Total 4 marks)
Question 2.
OAP is a triangle
OA = 2f + g and OB = 3h
2f + g
Diagram NOT
drawn
accurately
P is the point on AB such that AP: PB = 2:1
(a) Find the vector BA in terms of f, g and h.
3h
..............................................
(1)
(b) Find the vector PO in terms of f, g and h
..............................................
(2)
(Total 3 marks)
PiXL PLC 2017 Certification
Question 3.
OABC is a parallelogram.
X is the midpoint of OB
Diagram NOT
drawn
accurately
OA = a and OC = c
(a) Find the vector OX in terms of a and c.
..............................................
(1)
(b) Find the vector XC in terms of a and c.
..............................................
(2)
(Total 3 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
Circle Theorems 2
Grade 8
Solutions
Objective: Apply and prove the standard circle theorems concerning angles, radii, tangents and
chords, and use them to prove related results
Question 1
A and B are points on the circumference of a circle, centre O.
AC and BC are tangents to the circle.
Angle ACB = 36°.
Find the size of angle OBA.
Give reasons for your answer.
OBC = 90° or AOB = 144°
or both angles CAB and CBA = 72°
or ACO and BOC = 18°
or AOC = 72°
Tangent/radius is 90
BOA an isosceles triangle
tangents from a point are equal
for complete correct method 90 one correct reason given
answer 18°
180−36
2
(M1)
(C1)
(A1)
(Total 3 marks)
PiXL PLC 2017 Certification
Question 2
*
A, B, C and D are points on the circumference of a circle, centre O.
Angle AOC = y.
Find the size of angle ABC in terms of y.
Give a reason for each stage of your working.
ADC =
180 -
�
2
�
2
(M1)
(A1)
Two reasons given
Angle at centre is twice the angle at the circumference
(C1)
Opposite angles in cyclic quadrilateral add to 180°
(C1)
OR
Reflex AOC = 360 - y
360−�
2
(M1)
(A1)
Two reasons given
Angles round a point add to 360° (C1)
Angle at centre is twice the angle at the circumference (C1)
(Total 4 marks)
PiXL PLC 2017 Certification
Question 3
B, C and D are points on the circumference of a circle, centre O.
ABE and ADF are tangents to the circle.
Angle DAB = 40°
Angle CBE = 75°
Work out the size of angle ODC.
ABO = 90° or OBC = 15° or ADO = 90° or FDO = 90° or EBO = 90° (M1)
Reflex angle BOD = 360 – (360-90-90-40) = 220°
Or BCD = (360 -90-90-40) = 70
Or DBO = 90 –(180-40)/2 = 20
Or BOC = 180 – (15 + 15) = 150
With valid reasons
(M1)
ODC = 55°
(A1)
(Total 3 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
Vector Arguments and Proof 2
Grade 8
Solutions
Objective: Use vectors to construct geometric argument and proof
Question 1.
OPT is a triangle
OT = 3a and TP = 5b
5b
Diagram NOT
drawn
accurately
M is the midpoint of OP
3a
(a) Find OM in terms of a and b
OM = ½ OP (M1)
OM = ½ (3a + 5b) oe (A1)
.........................................
(2)
(b) Find TM in terms of a and b
TM = TO + OM
TM = -3a + ½ (3a + 5b) (M1) ft from (a)
TM = -1.5a + 2.5b or ½ (5b – 3a) (A1)
..............................................
(2)
(Total 4 marks)
Question 2.
OAP is a triangle
2f + g
OA = 2f + g and OB = 3h
Diagram NOT
drawn
accurately
P is the point on AB such that AP: PB = 2:1
(a) Find the vector BA in terms of f, g and h.
3h
BA = -3h + 2f + g (B1)
..............................................
(1)
(b) Find the vector PO in terms of f, g and h
PO = PA + AO = ½ (-3h + 2f + g) – (2f + g) (M1)
PO = -1.5h – f - 0.5g or -½ (3h + 2f + g) oe simplified expression (A1)
..............................................
(2)
(Total 3 marks)
PiXL PLC 2017 Certification
Question 3.
OABC is a parallelogram.
X is the midpoint of OB
Diagram NOT
drawn
accurately
OA = a and OC = c
(a) Find the vector OX in terms of a and c.
OX = ½ (a + c)
(B1)
..............................................
(1)
(b) Find the vector XC in terms of a and c.
XC = XO + OC = - ½ (a + c) + c (M1)
XC = - ½ a + ½ c or ½ (c – a) (A1)
..............................................
(2)
(Total 3 marks)
Total /10
PiXL PLC 2017 Certification