plc papers - king's lynn academykingslynnacademy.co.uk/.../2015/10/grade-6-geometry.pdf ·...

PLC Papers

Created For:

PiXL PLC 2017 Certification

Arcs and Sectors 2 Grade 6

Objective: Calculate arc lengths and angles, and areas of sectors

Question 1.

PQ is an arc of a circle with radius 9.4cm.

Angle POQ = 143o

(a) Calculate the perimeter of sector POQ.

Give your answer to three significant figures.

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

(3)

(b) Calculate the area of sector POQ.


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

(2)

(Total 5 marks)


Question 2.

A sector has a radius of 8cm and an arc length of 5cm.

(a) What is the angle of the sector? Give your answer correct to 3 significant figures.

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

(3)

(b) Calculate the area of the sector. Give your answer to three significant figures.

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

(2)

(Total 5 marks)

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


Derive triangle results 2 Grade 5

Objective: Derive results about triangle angles and sides using known angle facts, triangle congruence, similarity and properties of quadrilaterals, and use known results to obtain simple proofs. Include the fact that the base angles of an isosceles triangle are equal, and include derivation of Pythagoras' theorem.

Question 1

The diagram below shows two parallel lines, ABC and DEF cut by the line GH.

H

A B C

D E F

G

Show that corresponding angles are equal

(Total 4 marks)


Question 2

Show that the area of a triangle is equal to half the base area multiplied by the perpendicular height.

A

B C

(Total 6 marks)

Total /10


Enlargements and negative scale factors 2 Grade 5

Objective: Identify and construct enlargements including using negative scale factors

Question 1

Enlarge the triangle on the grid by a scale factor of ¼ using the origin as the centre of enlargement.

(3)

Question 2

Enlarge this shape by a scale factor of 3 from the point ( – 5 , – 2 )

(3)

– 5

10

5

– 5 10 5 15 x

y

– 5

10

5

– 5 10 5 15 x

y


Question 3

a) Enlarge this shape by a scale factor of -2 from the point ( 2 , 1 )

b) What is the image of the point ( 5 , 2 ) under this enlargement?

(4)

Total / 10

Total marks / 10

– 5

10

5

– 5 10 5

15 x

y


Loci 2 Grade 5

Objective: Use constructions to solve loci problems

Question 1.

Draw the locus of the points 30km from the line AB

Scale: 1 cm represents 10 km

(Total 2 marks)

Question 2. On the diagram, draw the locus of the points, outside the rectangle, that are 3cm from the edges of this rectangle.

(Total 3 marks)

B

A


Question 3. There is a lighthouse at A and B. To avoid the rocks, a ship must sail equidistant to both A and B. Draw a line that is equidistant to A and B on the diagram.

(Total 2 marks)

Question 4. The diagram represents a triangular garden ABC

The scale of the diagram is 1 cm represents 1 m.

A tree is to be planted in the garden so that it is Nearer to AB than to AC, Within 5m of point A. On the diagram, shade the region where the tree may be planted.

(Total 3 marks)

Total /10

A

B


Pythagoras 2 Grade 5

Objective: Know and use Pythagoras's theorem for right-angled triangles

Question 1

ABC is a right angled triangle. AB = 8 m, BC = 14 m Calculate the length of AC. Give your answer correct to 1 decimal place.

………………………. (3) (Total 3 marks) Question 2

ABC is a right angled triangle. AB = 10 cm, AC = 21 cm Calculate the length of BC. Give your answer correct to 1 decimal place.

………………………. (Total 3 marks) (3)


Question 3

ABCD is a rectangle. AB = 23 m, AD = 12 m Work out the length of the diagonal BD. Give your answer correct to 3 significant figures.

………………………. (4) (Total 4 marks)

Total marks / 10


Similarity and Congruence 2 Grade 5

Objective: Apply the concepts of congruence and similarity

Question 1

Which of the two shapes above are congruent?

……………………… (2) (Total 2 marks) Question 2

In this diagram AB and DE are parallel. Find the lengths of DE and BC.

…………………. (Total 4 marks) (4)

CA

FED

B


Question 3

The diagram shows a model of a post. It has a surface area of 290 cm2. The scale of the model is 1:30. What is the surface area of the real post?

…………..…………. (4) (Total 4 marks) Total /10

6 cm

10 cm

5 cm

5 cm

Diagram accurately drawn

NOT


Standard constructions 2 Grade 5

Objective: Use the standard ruler and compass constructions to construct a 60° angle, a perpendicular bisector of a line segment, a perpendicular to a given line from/at a given point, and an angle bisector

Question 1.

Bisect this angle

(Total 3 marks)

Question 2.

Construct an isosceles triangle with side length 5cm, 4cm, 4cm.

(Total 4 marks)


Question 3

Construct a perpendicular bisector on the line AB below

(Total 3 marks)

Total /10

B

A


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


Surface Area 2 Grade 5

Objective: Calculate the surface area of spheres, pyramids, cones and composite solids

Question 1

Find the surface area of this cone, including the base. Give you answer to the nearest square centimetre.

…………………. (3) (Total 3 marks) Question 2 A pyramid has a square base of side 8 cm. Each of the triangular faces are isosceles

triangles with a height of 10 cm from the apex to the middle of the base. Find the surface area of the pyramid in square centimetres.

…… ……………. (Total 3 marks) (3)


Question 3 Find the total surface area of

this box, which is a prism with an L shaped face.

…………….…. (4) (Total 4 marks) Total /10


Trigonometry 2 Grade 5

Objective: Know and use the trigonometric ratios for right-angled triangles

Question 1

ABC is a right angled triangle. AC = 11 cm and the angle ACB is 67o Calculate the length of AB. Give your answer to 1 decimal place.

…….………………. (3) (Total 3 marks) Question 2 ABC is a right angled triangle.

AB = 9 cm, BC = 12 cm Calculate the angle ACB. Give your answer correct to the nearest degree.

…………………. (Total 3 marks) (3)


Question 3 ABCD is a rectangle.

AD = 13 m and the diagonal BD makes an angle of 74o with BC. Work out the length of the diagonal BD. Give your answer correct to 3 significant figures

………………. (4) (Total 4 marks) Total /10


Volume 2 Grade 5

Objective: Calculate the volume of spheres, pyramids, cones and composite solids.

Question 1

Find the volume of this cone. Give you answer to the nearest cubic centimetre.

….………………. (3) (Total 3 marks) Question 2 A pyramid has a square base of side 8 cm. The height of the pyramid (measured

perpendicular to the base) is 9.2 cm. Find the volume of the pyramid to the nearest cubic centimetre.

…. ……………. (Total 3 marks) (3)


Question 3 Find the volume of this box,

which is a prism with an L shaped face.

………….…. (4) (Total 4 marks) Total /10

PLC Papers

Created For:


Arcs and Sectors 2 Grade 6 Solutions

Objective: Calculate arc lengths and angles, and areas of sectors

Question 1.

PQ is an arc of a circle with radius 9.4cm.

Angle POQ = 143o

(a) Calculate the perimeter of sector POQ.


Arc length = 143360 x π x 2 x 9.4 (=23.4607.) (M1)

Perimeter = Arc length + 9.4 x 2 (M1)

Perimeter = 42.3cm (A1)

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

(3)

(b) Calculate the area of sector POQ.


Area = 143360 x π x 9.42 (M1)

Area = 110cm2 (A1)

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

(2)

(Total 5 marks)


Question 2.

A sector has a radius of 8cm and an arc length of 5cm.

(a) What is the angle of the sector? Give your answer correct to 3 significant figures.

Arc length = 5 = �360 x π x 2 x 8 (B1)

θ = 5 × 3602 × � ×8 (M1)

θ = 35.8o (A1)

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

(3)

(b) Calculate the area of the sector. Give your answer to three significant figures.

Area = 35.8360 x π x 82 (M1 ft their angle)

Area = 20.0cm2 (A1)

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

(2)

(Total 5 marks)

Total /10


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


Derive triangle results 2 Grade 5 Solutions

Objective: Derive results about triangle angles and sides using known angle facts, triangle congruence, similarity and properties of quadrilaterals, and use known results to obtain simple proofs. Include the fact that the base angles of an isosceles triangle are equal, and include derivation of Pythagoras' theorem.

Question 1

The diagram below shows two parallel lines, ABC and DEF cut by the line GH.

H

A B C

D E F

G

Show that corresponding angles are equal

Angle HBC = Angle ABE (vertically opposite angles) M1

Angle ABE = Angle BEF (alternate angles) M1

Hence Angle HBC = Angle BEF (corresponding angles) C1

This argument can be repeated for any pair of corresponding angles C1

(Total 4 marks)


Question 2

Show that the area of a triangle is equal to half the base area multiplied by the perpendicular height.

X A Y

B N C

Add a line perpendicular to BC, from a point N on BC, that goes to A M1

Add two lines, the same length as and parallel to AN, from B to X and from C to Y. M1

Join XY M1

Area of triangle ABN = 1/2 area of rectangle XABN M1

Area of triangle ANC = 1/2 area of rectangle ANCY

Hence area of triangle ABC = 1/2 (area of rectangle XABN + area of rectangle ANCY) M1

= 1/2 area of rectangle XBCY

= 1/2 BC x CY

= 1/2 BC x AN

= 1/2 x base x perpendicular height C1

(Total 6 marks)

Total /10


Enlargements and negative scale factors 2 Grade 5 Solutions

Objective: Identify and construct enlargements including using negative scale factors

Question 1

Enlarge the triangle on the grid by a scale factor of ¼ using the origin as the centre of enlargement.

1M construction lines

(at least one in correct position)

1M using the origin

1M correct size

(3)

Question 2

Enlarge this shape by a scale factor of 3 from the point ( – 5 , – 2 )



1M using the correct point

1M correct size

(3)

– 5

10

5

– 5 10 5 15 x

y

– 5

10

5

– 5 10 5 15 x

y


Question 3



1M using the correct point

1M correct size

a) Enlarge this shape by a scale factor of -2 from the point ( 2 , 1 )

b) What is the image of the point ( 5 , 2 ) under this enlargement?

1M ( -4 , -1 )

(4)

Total / 10

– 5

10

5

– 5 10 5

15 x

y


Loci 2 Grade 5 Solutions

Objective: Use constructions to solve loci problems Question 1.

(Total 2 marks)

Question 2.

(Total 3 marks)


Question 3.

(Total 2 marks)

Question 4.

(Total 3 marks)

Total /10


Pythagoras 2 Grade 5 Solutions

Objective: Know and use Pythagoras's theorem for right-angled triangles

Question 1

ABC is a right angled triangle. AB = 8 m, BC = 14 m Calculate the length of AC. Give your answer correct to 1 decimal place. AC2 = 82 + 142 (M2 square, add)

= 64 + 196 = 260 AC = 16.1 (A1)

16.1 m………………. (3) (Total 3 marks) Question 2

ABC is a right angled triangle. AB = 10 cm, AC = 21 cm Calculate the length of BC. Give your answer correct to 1 decimal place.

BC2 = 212 - 102 (M2 square, subtract)

= 441 - 100 = 341 BC = 18.5 (A1)

18.5 cm……………. (Total 3 marks) (3)


Question 3

ABCD is a rectangle. AB = 23 m, AD = 12 m Work out the length of the diagonal BD. Give your answer correct to 3 significant figures. BD2 = 122 + 232 (M2 square, add)

= 144 + 529 = 673 BD = 25.9 (A2 correct, correct 3sf)

………25.9 m…. (4) (Total 4 marks) Total /10


Similarity and Congruence 2 Grade 5 Solutions

Objective: Apply the concepts of congruence and similarity

Question 1

Which of the two shapes above are congruent?

…A & C (B2)…………. (2) (Total 2 marks) Question 2

In this diagram AB and DE are parallel. Find the lengths of DE and BC.

Scale factor = 3 DE = SF x 2.5 = 7.5 (M1, A1) BC = 11 / SF = 3.7 (M1, A1)

………3.7 cm……. (Total 4 marks) (4)

CA

FED

B


Question 3

The diagram shows a model of a post. It has a surface area of 290 cm2. The scale of the model is 1:30. What is the surface area of the real post? Scale factor = 30 So Area factor = 302 = 900 (M1,A1) Surface area of real post = 900 x 290 = 261 000 (A1, A1)

……261 000 cm2…………. (4) (Total 4 marks) Total /10

6 cm

10 cm

5 cm

5 cm

Diagram accurately drawn

NOT


Standard constructions 2 Grade 5 Solutions

Objective: Use the standard ruler and compass constructions to construct a 60° angle, a perpendicular bisector of a line segment, a perpendicular to a given line from/at a given point, and an angle bisector

Question 1.

Bisect this angle

(Total 3 marks)

Question 2.

Construct an isosceles triangle with side length 5cm, 4cm, 4cm.

M1 for 5cm line drawn

B1 B1 for construction arcs drawn from each end

A1 for correct triangle

(Total 4 marks)


Question 3.

Construct a perpendicular bisector on the line AB below

(Total 3 marks)

Total /10


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


Surface Area 2 Grade 5 Solutions

Objective: Calculate the surface area of spheres, pyramids, cones and composite solids

Question 1

Find the surface area of this cone, including the base. Give you answer to the nearest square centimetre.

S. Area = Πrl + Πr2 = Πx4.1x9.3 + Πx4.12 (M2 sloping face, circle) = 119.8 + 52.8 = 172.6 (A1)

173 cm2………………. (3) (Total 3 marks) Question 2 A pyramid has a square base of side 8 cm. Each of the triangular faces are isosceles

triangles with a height of 10 cm from the apex to the middle of the base. Find the surface area of the pyramid in square centimetres.

S. Area = ( 1/2 x 10 x 8)4 + 82 (M2 triangle, square) = 160 + 64 = 224

(A1)

224 cm2 ……………. (Total 3 marks) (3)


Question 3 Find the total surface area of

this box, which is a prism with an L shaped face.

2 L faces = 2(24 + 9) = 66 (M1, A1 three or more faces correct) Back = 84 Base = 72 Steps down = 36+36+36+48 =156 Total = 378 (M1 finding total)

(A1)

………378 cm2…. (4) (Total 4 marks) Total /10


Trigonometry 2 Grade 5 Solutions

Objective: Know and use the trigonometric ratios for right-angled triangles

Question 1

ABC is a right angled triangle. AC = 11 cm and the angle ACB is 67o Calculate the length of AB. Give your answer to 1 decimal place.

sin 67 = AB / 11 (M1) AB = 11 sin 67 (M1) = 10.1 (A1)

10.1 cm………………. (3) (Total 3 marks) Question 2 ABC is a right angled triangle.

AB = 9 cm, BC = 12 cm Calculate the angle ACB. Give your answer correct to the nearest degree.

tan ACB = 9 / 12 (M1) = 0.75 (M1) BAC = 37 (A1)

37o……………. (Total 3 marks) (3)


Question 3 ABCD is a rectangle.

AD = 13 m and the diagonal BD makes an angle of 74o with BC. Work out the length of the diagonal BD. Give your answer correct to 3 significant figures

cos 74 = 13 / BD (M1) BD = 13 /cos 74 (M1) = 47.16 (A1) (A1)

………47.2 m…. (4) (Total 4 marks) Total /10


Volume 2 Grade 5 Solutions

Objective: Calculate the volume of spheres, pyramids, cones and composite solids.

Question 1

Find the volume of this cone. Give you answer to the nearest cubic centimetre.

Volume = 1/3Πr2h = 1/3Π x 4.12 x 8.3 (M2 correct height, subst. into correct formula) = 146.1 (A1)

146 cm3………………. (3) (Total 3 marks) Question 2 A pyramid has a square base of side 8 cm. The height of the pyramid (measured

perpendicular to the base) is 9.2 cm. Find the volume of the pyramid to the nearest cubic centimetre.

Volume = 1/3 Base x height = 1/3 x 82 x 9.2 (M1) = 196.3 (A1)

(A1 rounding)

196 cm3 ……………. (Total 3 marks) (3)


Question 3 Find the volume of this box,

which is a prism with an L shaped face.

Cross section = 24 + 9= 33 (M1, A1) Volume = 33 x 12 (M1)

(A1)

………396 cm3…. (4) (Total 4 marks) Total /10

plc papers - king's lynn academykingslynnacademy.co.uk/.../2015/10/grade-6-geometry.pdf ·...

Documents