Cambridge Mathematics 7 — Units 8 and 15 Comprehensive Assessment — Paper B

INSTRUCTIONS

Answer all 30 questions.

Section I contains 10 multiple-choice questions worth 20 marks.
Section II contains 10 written-response questions worth 40 marks.
Section III contains 5 explanation questions worth 15 marks.
Section IV contains 5 reasoning questions worth 25 marks.

Choose one answer for each multiple-choice question.

For all other questions, show your calculations and explain your reasoning when requested.

Include appropriate units in every measurement answer.

A ruler, a pencil and a calculator may be used. Use squared paper for nets and views where appropriate.

Diagrams are not necessarily drawn to scale. When considering symmetry, ignore vertex labels and dimension labels.

The order of rotational symmetry is the number of times a shape fits exactly onto itself during one complete turn, including its final position.

Illustrations for the written-response questions are provided in the illustrated solutions after the assessment.

SECTION I. MULTIPLE-CHOICE QUESTIONS — 20 MARKS

Choose the correct answer. Each question is worth 2 marks.

Question 1 (2 marks). The outline below is a rectangle that is not a square. Which statement describes its symmetry?

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Question 2 (2 marks). The circle has centre $O$. Points $A$, $B$, $C$ and $D$ lie on the circle, and $AB$ passes through $O$. Which labelled segment is a chord but not a diameter?

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Question 3 (2 marks). The outline below is a regular hexagon. What is the smallest positive angle through which it can be rotated about its centre to fit exactly onto itself?

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Question 4 (2 marks). Triangles $ABC$ and $PQR$ are congruent, with corresponding vertices in the order shown by $\triangle ABC \cong \triangle PQR$. Given that $AB=6$ cm, $BC=8$ cm and $AC=10$ cm, what is the length of $PR$?

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Question 5 (2 marks). The solid shown is a tetrahedron. How many faces, vertices and edges does it have?

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Question 6 (2 marks). A cylinder stands upright on a horizontal table. What shape is its top view?

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Question 7 (2 marks). Square $ABCD$ has side length $0.6$ m. What is its area in square centimetres?

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Question 8 (2 marks). In triangle $ABC$, $BC=12$ cm and the perpendicular height $AH=5$ cm. What is the area of the triangle?

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Question 9 (2 marks). A cuboid has length $10$ cm, width $4$ cm and height $3$ cm. What is its volume?

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Question 10 (2 marks). A cube has side length $5$ cm. What is its total surface area?

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SECTION II. WRITTEN-RESPONSE AND PRACTICAL QUESTIONS — 40 MARKS

Show your working. Each question is worth 4 marks.

Question 1 (4 marks). Draw a square $ABCD$ with side length $4$ cm.
a) Draw all its lines of symmetry. (2 marks)
b) State its order of rotational symmetry and its smallest positive angle of rotational symmetry. (2 marks)

Question 2 (4 marks). Draw a circle with centre $O$ and radius $3$ cm. Draw a diameter $AB$ and a chord $CD$ that does not pass through $O$.
a) State the length of $AB$. (1 mark)
b) Explain why $CD$ is a chord but not a diameter. (1 mark)
c) Draw a tangent at $A$. Mark the angle between the tangent and $OA$, and state its size. (2 marks)

Question 3 (4 marks). Sketch a regular octagon.
a) State how many equal sides and equal angles it has. (1 mark)
b) State its number of lines of symmetry. (1 mark)
c) State its order of rotational symmetry. (1 mark)
d) Explain what makes it a regular polygon, rather than simply an eight-sided polygon. (1 mark)

Question 4 (4 marks). Triangles $ABC$ and $DEF$ are congruent, with corresponding vertices given by $\triangle ABC \cong \triangle DEF$. Triangle $ABC$ is right-angled at $B$, and $AB=6$ cm, $BC=8$ cm and $AC=10$ cm.
a) State the lengths of $DE$, $EF$ and $DF$. (1 mark)
b) State the size of $\widehat{DEF}$. (1 mark)
c) Calculate the perimeter of triangle $DEF$. (1 mark)
d) Calculate the area of triangle $DEF$. (1 mark)

Question 5 (4 marks). A cuboid rests on a horizontal table. Its front face is $8$ cm wide and $3$ cm high. The cuboid extends $5$ cm from front to back.
a) Draw and label its front view. (1 mark)
b) Draw and label its top view. (1 mark)
c) Draw and label its right-side view. (1 mark)
d) State the number of faces, vertices and edges of the cuboid. (1 mark)

Question 6 (4 marks). A rectangular card is $12$ cm long and $8$ mm wide.
a) Calculate its area in square millimetres. (1 mark)
b) Convert this area to square centimetres. (1 mark)
c) Convert $2.75$ m² to square centimetres. (1 mark)
d) Convert $650$ mm² to square centimetres. (1 mark)

Question 7 (4 marks). A rectangular field is $600$ m long and $250$ m wide.
a) Calculate its area in square metres. (1 mark)
b) Express its area in hectares. (1 mark)
c) The field is divided into two parts in the ratio $2:3$. Calculate the area of the larger part in hectares and in square metres. (2 marks)

Question 8 (4 marks). A sign is made from a rectangle with a triangle attached along its entire top edge. The rectangle is $12$ cm wide and $5$ cm high. The triangle has base $12$ cm and perpendicular height $4$ cm. A rectangular opening measuring $2$ cm by $1$ cm is cut entirely inside the rectangular part.
a) Calculate the area of the rectangular part before the opening is cut. (1 mark)
b) Calculate the area of the triangular part. (1 mark)
c) Calculate the remaining area of the sign after the opening is cut. (2 marks)

Question 9 (4 marks). A closed cuboid-shaped box has length $12$ cm, width $8$ cm and height $25$ mm.
a) Convert the height to centimetres. (1 mark)
b) Calculate the volume of the box in cubic centimetres. (1 mark)
c) Calculate its total surface area in square centimetres. (2 marks)

Question 10 (4 marks). A cube is made by folding a net of six congruent squares. Each square has side length $4$ cm.
a) Draw one possible net of the cube. (2 marks)
b) Calculate the total surface area of the cube. (1 mark)
c) Calculate the volume of the cube. (1 mark)

SECTION III. EXPLANATION AND JUSTIFICATION — 15 MARKS

Give clear mathematical explanations. Each question is worth 3 marks.

Question 1 (3 marks). Two equilateral triangles have side lengths $4$ cm and $6$ cm respectively.

A student says, “Both triangles have three angles of $60^\circ$, so they must be congruent.”

Is the student correct? Explain what congruence requires and use the given measurements to support your answer.

Question 2 (3 marks). Rectangle $A$ measures $6$ cm by $4$ cm. Rectangle $B$ measures $8$ cm by $2$ cm.

A student says, “The rectangles have the same perimeter, so they must be congruent and have the same area.”

Calculate their perimeters and areas. Explain why the conclusion is incorrect.

Question 3 (3 marks). In triangle $ABC$, $BC=14$ cm, $AC=10$ cm and the perpendicular height from $A$ to $BC$ is $AH=6$ cm.

A student writes:

$$\text{Area}=\dfrac12\times14\times10=70.$$

Identify the mistake, explain which measurement should be used, and calculate the correct area with its unit.

Question 4 (3 marks). A square panel has side length $0.6$ m, so its area is $0.36$ m².

A student says, “To change square metres to square centimetres, I multiply by $100$. Therefore, the area is $36$ cm².”

Explain why the conversion factor is incorrect and find the correct area in square centimetres.

Question 5 (3 marks). Cube $A$ has side length $3$ cm. Cube $B$ has side length $6$ cm.

A student says, “The side length has doubled, so both the volume and the total surface area have doubled.”

Calculate the volume and total surface area of each cube. Explain the actual multiplication factor for each quantity.

SECTION IV. REASONING AND DEDUCTION — 25 MARKS

Show a complete chain of reasoning. Each question is worth 5 marks.

Question 1 (5 marks). Nine congruent square tiles form a $3\times3$ square. Five tiles are dark and four are white.

The arrangement must have rotational symmetry of order 4, and tile colours must match after rotation.
a) Explain why the centre tile must be dark. (1 mark)
b) Draw the two possible colour arrangements. (2 marks)
c) State the number of lines of symmetry of each arrangement. (1 mark)
d) Explain why there are no other possible arrangements. (1 mark)

Question 2 (5 marks). A right prism has two congruent regular polygonal bases and rectangular side faces. It has 18 edges.
a) Determine the number of sides of each base. Explain how you use the edge count. (2 marks)
b) Name the prism and state its number of faces and vertices. (1 mark)
c) State the number of lines of symmetry of one base and its smallest positive angle of rotational symmetry. (2 marks)

Question 3 (5 marks). A rectangular site measures $800$ m by $600$ m. A rectangle measuring $300$ m by $200$ m is removed from its top-right corner, leaving an L-shaped site.

The land costs 2500 dollars per hectare. A fence around the entire boundary of the remaining site costs 8 dollars per metre. A buyer has a total budget of 130 000 dollars.

a) Calculate the area of the remaining site in hectares. (2 marks)
b) Calculate the total length of fence required. (1 mark)
c) Determine whether the budget covers both the land and the fence. State the amount left over or the shortfall. (2 marks)

Question 4 (5 marks). An open-top tank is a cuboid with length $2$ m, width $1.5$ m and height $1$ m. Ignore the thickness of its walls.

Only the outside of its base and four walls will be painted. The tank needs two coats of paint.

One litre of paint covers $4$ m² for one coat. Paint is sold only in $0.75$-litre tins costing 18 dollars each. Assume there is no paint wastage.

a) Calculate the volume of the tank in cubic metres. (1 mark)
b) Calculate the outside area to be painted for one coat. (2 marks)
c) Calculate the minimum number of tins required for two coats and the total cost. (2 marks)

Question 5 (5 marks). A solid metal model consists of a cuboid measuring $12$ cm by $8$ cm by $4$ cm and a cube of side length $4$ cm attached on top.

The cube stands at one corner of the cuboid’s top face. Its entire bottom face is in contact with the cuboid. There are no gaps or overlaps.

a) Calculate the volume of the complete model. (1 mark)
b) Calculate its total exposed surface area. Explain how the contact area is treated. (2 marks)
c) The model is melted down and recast into cubes of side length $2$ cm. Assuming no metal is lost, calculate how many complete cubes can be made. (2 marks)

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Student name: ................................................ Candidate number: ........................

ANSWER KEY AND ILLUSTRATED SOLUTIONS

The illustrations below accompany the written-response questions. Alternative correct drawings, methods and explanations are acceptable.

SECTION I. MULTIPLE-CHOICE ANSWERS

Question 1: B. A non-square rectangle has two lines of symmetry, through the midpoints of opposite sides, and rotational symmetry of order 2.

Question 2: C. Both endpoints of $CD$ lie on the circle, but $CD$ does not pass through the centre.

Question 3: C. The smallest positive rotation is $360^\circ\div6=60^\circ$.

Question 4: C. Side $PR$ corresponds to $AC$, so $PR=10$ cm.

Question 5: A. A tetrahedron has 4 triangular faces, 4 vertices and 6 edges.

Question 6: B. Viewed vertically from above, an upright cylinder has a circular outline.

Question 7: C. The side length is $60$ cm, so the area is $60\times60=3600$ cm².

Question 8: B. The area is $\dfrac12\times12\times5=30$ cm².

Question 9: C. The volume is $10\times4\times3=120$ cm³.

Question 10: D. The total surface area is $6\times5^2=150$ cm².

SECTION II. WRITTEN-RESPONSE AND PRACTICAL SOLUTIONS

Question 1

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a) The four lines of symmetry are the two diagonals and the two lines joining the midpoints of opposite sides.

b) The order of rotational symmetry is 4.

The smallest positive angle of rotational symmetry is:

$$360^\circ\div4=90^\circ.$$

Question 2

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a) The diameter is twice the radius:

$$AB=2\times3=6.$$

Therefore, $AB=6$ cm.

b) Segment $CD$ joins two points on the circle, so it is a chord. It does not pass through $O$, so it is not a diameter.

c) The tangent at $A$ is perpendicular to radius $OA$. The angle is $90^\circ$.

Question 3

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a) It has 8 equal sides and 8 equal interior angles.

b) It has 8 lines of symmetry.

c) Its order of rotational symmetry is 8.

d) A regular polygon has all sides equal and all interior angles equal. Having eight sides alone does not guarantee either property.

Question 4

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a) Corresponding sides are equal:

$$DE=6,\qquad EF=8,\qquad DF=10.$$

All three lengths are in centimetres.

b) Angle $\widehat{DEF}$ corresponds to angle $\widehat{ABC}$, so $\widehat{DEF}=90^\circ$.

c) The perimeter is:

$$6+8+10=24.$$

The perimeter is $24$ cm.

d) The two perpendicular sides are $DE$ and $EF$:

$$\text{Area}=\dfrac12\times6\times8=24.$$

The area is $24$ cm².

Question 5

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a) The front view is a rectangle measuring $8$ cm by $3$ cm.

b) The top view is a rectangle measuring $8$ cm by $5$ cm.

c) The right-side view is a rectangle measuring $5$ cm by $3$ cm.

d) A cuboid has 6 faces, 8 vertices and 12 edges.

Question 6

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The card has length $AB=12$ cm and width $BC=8$ mm.

a) Convert the length first: $12$ cm $=120$ mm.

$$\text{Area}=120\times8=960.$$

The area is $960$ mm².

b) Since $1$ cm² $=100$ mm²:

$$960\div100=9.6.$$

The area is $9.6$ cm².

c) Since $1$ m² $=10\,000$ cm²:

$$2.75\times10\,000=27\,500.$$

Therefore, $2.75$ m² $=27\,500$ cm².

d)

$$650\div100=6.5.$$

Therefore, $650$ mm² $=6.5$ cm².

Question 7

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Here $AB=600$ m and $BC=250$ m. The division shown gives areas in the ratio $2:3$.

a)

$$\text{Area}=600\times250=150\,000.$$

The field has area $150\,000$ m².

b)

$$150\,000\div10\,000=15.$$

The area is $15$ ha.

c) There are $2+3=5$ equal ratio parts. The larger portion contains three parts:

$$\dfrac35\times15=9.$$

The larger portion is $9$ ha, or $90\,000$ m².

Question 8

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The rectangle is $12$ cm by $5$ cm, the triangular height is $4$ cm, and the opening is $2$ cm by $1$ cm.

a)

$$\text{Rectangle area}=12\times5=60.$$

The rectangular part has area $60$ cm².

b)

$$\text{Triangle area}=\dfrac12\times12\times4=24.$$

The triangular part has area $24$ cm².

c) The opening has area $2\times1=2$ cm².

$$\text{Remaining area}=60+24-2=82.$$

The remaining area is $82$ cm².

Question 9

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a) The height is $25\div10=2.5$ cm.

b)

$$V=12\times8\times2.5=240.$$

The volume is $240$ cm³.

c)

$$S=2(lw+lh+wh).$$

$$S=2(12\times8+12\times2.5+8\times2.5) =2(96+30+20)=292.$$

The total surface area is $292$ cm².

Question 10

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a) One valid net is shown. It contains six squares that fold to form the six faces without overlapping.

b)

$$S=6\times4^2=96.$$

The total surface area is $96$ cm².

c)

$$V=4^3=64.$$

The volume is $64$ cm³.

SECTION III. EXPLANATION AND JUSTIFICATION SOLUTIONS

Question 1

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The student is incorrect.

Congruent shapes must have the same shape and the same size. Their corresponding sides and angles must be equal.

Although both equilateral triangles have angles of $60^\circ$, their corresponding sides measure $4$ cm and $6$ cm. Their sizes differ, so they are not congruent.

Question 2

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The left rectangle measures $6$ cm by $4$ cm. The right rectangle measures $8$ cm by $2$ cm.

Their perimeters are:

$$P_A=2(6+4)=20,\qquad P_B=2(8+2)=20.$$

Both perimeters are $20$ cm.

Their areas are:

$$A_A=6\times4=24,\qquad A_B=8\times2=16.$$

Their areas are $24$ cm² and $16$ cm².

Equal perimeters do not imply congruence or equal areas. The rectangles have different side lengths and different areas.

Question 3

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The student used the sloping side $AC=10$ cm as the height.

The height corresponding to base $BC$ must be perpendicular to $BC$. The correct height is $AH=6$ cm.

Therefore:

$$\text{Area}=\dfrac12\times14\times6=42.$$

The correct area is $42$ cm².

Question 4

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The side length is $0.6$ m, which is $60$ cm.

A square metre contains $100\times100=10\,000$ square centimetres. The linear conversion factor must be squared:

$$1\text{ m}^2=10\,000\text{ cm}^2.$$

Therefore:

$$0.36\times10\,000=3600.$$

The correct area is $3600$ cm².

This also agrees with $60\times60=3600$ cm².

Question 5

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Cube $A$:

$$V_A=3^3=27,\qquad S_A=6\times3^2=54.$$

Its volume is $27$ cm³ and its total surface area is $54$ cm².

Cube $B$:

$$V_B=6^3=216,\qquad S_B=6\times6^2=216.$$

Its volume is $216$ cm³ and its total surface area is $216$ cm².

The volume multiplication factor is:

$$216\div27=8=2^3.$$

The surface-area multiplication factor is:

$$216\div54=4=2^2.$$

Doubling every length multiplies volume by 8 and surface area by 4, not by 2.

SECTION IV. REASONING AND DEDUCTION SOLUTIONS

Question 1

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a) A quarter-turn leaves the centre tile in the same position.

The four corner positions cycle through one another, so they must all have the same colour. The four edge-middle positions also cycle through one another and must all have the same colour.

Dark tiles outside the centre therefore occur in groups of four. Since there are five dark tiles, the centre must be dark.

b) The other four dark tiles must occupy either all four corners or all four edge-middle positions. These give the two patterns shown.

c) Each arrangement has 4 lines of symmetry: one horizontal, one vertical and two diagonal.

d) Once the centre is dark, exactly four dark tiles remain. Rotational symmetry requires them to fill one complete group of four positions. There are only two such groups, so there are no other arrangements.

Question 2

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a) Let each base have $n$ sides.

The two bases contribute $2n$ edges, and there are $n$ edges joining corresponding vertices. Therefore:

$$3n=18,\qquad n=6.$$

Each base has 6 sides.

b) The solid is a hexagonal prism.

It has $6+2=8$ faces and $2\times6=12$ vertices.

c) A regular hexagonal base has 6 lines of symmetry.

Its smallest positive angle of rotational symmetry is:

$$360^\circ\div6=60^\circ.$$

Question 3

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The original rectangle has dimensions $800$ m by $600$ m. The removed corner $DCGE$ measures $300$ m by $200$ m.

a) The remaining area is:

$$800\times600-300\times200 =480\,000-60\,000 =420\,000.$$

The area is $420\,000$ m².

In hectares:

$$420\,000\div10\,000=42.$$

The remaining site has area $42$ ha.

b) The boundary lengths, in metres, are:

$$800,\ 400,\ 300,\ 200,\ 500,\ 600.$$

Therefore:

$$P=800+400+300+200+500+600=2800.$$

The fence must be $2800$ m long.

Removing the corner does not change the total perimeter: the removed outer lengths are replaced by equal inner lengths.

c) The land cost is:

$$42\times2500=105\,000.$$

The fence cost is:

$$2800\times8=22\,400.$$

The combined cost is:

$$105\,000+22\,400=127\,400.$$

The budget is sufficient, with:

$$130\,000-127\,400=2600.$$

The buyer has 2600 dollars left over.

Question 4

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a)

$$V=2\times1.5\times1=3.$$

The tank volume is $3$ m³.

b) The outside base has area:

$$2\times1.5=3.$$

Its area is $3$ m².

The four outside walls have total area:

$$2(2\times1)+2(1.5\times1)=4+3=7.$$

Their area is $7$ m².

There is no top to paint, so:

$$\text{Area for one coat}=3+7=10.$$

The area is $10$ m².

c) Two coats require coverage of:

$$2\times10=20.$$

This is $20$ m² of single-coat coverage.

The paint required is:

$$20\div4=5.$$

The tank requires $5$ litres.

The number of tins must be a whole number:

$$5\div0.75=6\dfrac23.$$

Therefore, 7 tins are required.

The total cost is:

$$7\times18=126.$$

The paint costs 126 dollars.

Question 5

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a) The cuboid volume is:

$$12\times8\times4=384.$$

The cube volume is:

$$4^3=64.$$

Therefore:

$$V_{\text{total}}=384+64=448.$$

The total volume is $448$ cm³.

b) The full surface area of the cuboid is:

$$S_{\text{cuboid}}=2(12\times8+12\times4+8\times4) =2(96+48+32)=352.$$

This is $352$ cm².

The full surface area of the cube is:

$$S_{\text{cube}}=6\times4^2=96.$$

This is $96$ cm².

The contact area is $4\times4=16$ cm². It is counted once on the cuboid and once on the cube, but neither contact surface is exposed.

Therefore:

$$S_{\text{exposed}}=352+96-2\times16=416.$$

The total exposed surface area is $416$ cm².

c) Each new cube has volume:

$$2^3=8.$$

This is $8$ cm³.

The number of complete cubes is:

$$448\div8=56.$$

Therefore, 56 complete cubes can be made.