Year 11 Paper 5: Quadratics and Trigonometry
Covers number and calculation, ratio and proportion, indices and standard form, quadratics, angles and geometrical reasoning, area, volume and measures, and Pythagoras' theorem and trigonometry.
Year 11 here means a typical teaching order, not a syllabus rule. No exam board defines what belongs to Year 11, and schools sequence the course differently. Check it against your own scheme of work before using it to decide what a class has covered.
Questions
Question 1 [1 marks]
Angles and Geometrical Reasoning
Point Q has coordinates (-2, 4). Write down the coordinates of the image of Q after a reflection in the y-axis.
Question 2 [1 marks]
Indices and Standard Form
Write the number 0.0032 in standard form.
Question 3 [1 marks]
Ratio and Proportion
The ratio of boys to girls in a class is 5:7. Write the number of boys as a fraction of the total number of pupils in the class.
Question 4 [1 marks]
Area, Volume and Measures
Rectangle A measures 6 cm by 9 cm.
Which of these rectangles is mathematically similar to rectangle A?
Question 5 [1 marks]
Pythagoras and Trigonometry
The diagram shows a right-angled triangle with the right angle marked. The sides are labelled p, q and r, where r is the side opposite the right angle. Diagram: right-angled triangle, right angle shown at the bottom-left corner, side p along the base, side q up the vertical side, side r as the slanted side joining the two ends.
Question 6 [2 marks]
Quadratics
Factorise 2x^2 - 7x + 3
Question 7 [1 marks]
Number and Calculation
Write down 5/6 - 1/4 as a single fraction.
Question 8 [2 marks]
Indices and Standard Form
Simplify sqrt(27) - sqrt(3) fully.
Question 9 [2 marks]
Area, Volume and Measures
Two mathematically similar shapes have areas of 20 cm^2 and 45 cm^2. Find the linear scale factor from the smaller shape to the larger shape.
Question 10 [3 marks]
Number and Calculation
A charity fun run raises £3,460, to be shared equally between 7 local charities. Work out the exact amount each charity receives, then round your answer to the nearest penny to state the amount that will actually be paid to each charity.
Question 11 [2 marks]
Pythagoras and Trigonometry
A right-angled triangle has hypotenuse length 10 cm and one acute angle 30 degrees. Work out the exact length of the side opposite the 30 degree angle.
Question 12 [3 marks]
Number and Calculation
Work out 4 1/3 - 1 3/4, giving your answer as a mixed number. You may use the fraction key on your calculator.
Question 13 [3 marks]
Ratio and Proportion
Ravi has £3.60 and his sister has 90p. Write the ratio of Ravi's money to his sister's money in its simplest form.
Question 14 [4 marks]
Number and Calculation
a = 2^3 x 5 and b = 2^2 x 5^2.
Find the HCF of a and b.
Find the LCM of a and b.
Question 15 [5 marks]
Ratio and Proportion
A cake recipe for 8 people uses 240 g of caster sugar and 200 g of butter. The amounts of each ingredient needed are directly proportional to the number of people.
Work out the mass of caster sugar needed to make the recipe for 6 people.
Nadia has 620 g of butter. Work out the maximum number of people she can make the recipe for.
Question 16 [5 marks]
Quadratics
Solve the quadratic equation by factorising: x^2 - x - 30 = 0. Give answers in exact form.
Question 17 [8 marks]
Ratio and Proportion
A town council divides its budget between parks and libraries in the ratio 5:(t + 3). The total budget is 1600 units. The council then changes the split so that the ratio of parks to libraries becomes 3:2, while keeping the total budget at 1600 units and keeping the actual amount allocated to parks exactly the same as before. Find t.
Question 18 [4 marks]
Angles and Geometrical Reasoning
ABCDEF is a regular hexagon with centre O; the vertices are labelled in order around the hexagon. OA = a and OB = b.
Find AB in terms of a and b.
D is the vertex opposite A. Find OD in terms of a.
E is the vertex opposite B. Find OE in terms of b.
Find EB in terms of b, giving your answer in its simplest form.
Question 19 [4 marks]
Area, Volume and Measures
A solid sphere has volume 500 cm^3. Calculate the surface area of the sphere. Give your answer correct to 3 significant figures.
Question 20 [7 marks]
Pythagoras and Trigonometry
A tall building BC stands on horizontal ground, and a vertical flagpole CD stands on top of the building at C. From a point A on the ground, 40 m from the base B of the building, the angle of elevation to the top of the building, C, is 28 degrees, and the angle of elevation to the top of the flagpole, D, is 34 degrees.
Calculate the height BC of the building. Give your answer correct to 3 significant figures.
Calculate the height CD of the flagpole. Give your answer correct to 3 significant figures.
Calculate the distance AD, from A to the top of the flagpole. Give your answer correct to 3 significant figures.
Model solutions
| Question 1[1 mark] | |
|---|---|
| Answer or working | Marks |
| (2, 4) | B1cao |
| Question 2[1 mark] | |
|---|---|
| Answer or working | Marks |
| 3.2 x 10^-3 | B1oe |
| Question 3[1 mark] | |
|---|---|
| Answer or working | Marks |
| 5/12 | B1oe |
| Question 4[1 mark] | |
|---|---|
| Answer or working | Marks |
| B) 12 cm by 18 cm | B1 |
| Final answer: B | |
| Question 5[1 mark] | |
|---|---|
| Answer or working | Marks |
| r stated as the hypotenuse | B1cao |
| Final answer: C) r | |
| Question 6[2 marks] | |
|---|---|
| Answer or working | Marks |
| identifies a pair of numbers with product 6 and sum -7 (-6 and -1), or one correct bracket seen | M1 |
| (2x - 1)(x - 3) | A1cao |
| Question 7[1 mark] | |
|---|---|
| Answer or working | Marks |
| 7/12 | B1cao |
| Question 8[2 marks] | |
|---|---|
| Answer or working | Marks |
| sqrt(27) = 3sqrt(3) identified | M1 |
| 2sqrt(3) oe | A1cao |
| Final answer: 2sqrt(3) | |
| Question 9[2 marks] | |
|---|---|
| Answer or working | Marks |
| area ratio = 45/20 (=2.25) | M1oe |
| 1.5 | A1cao |
| Question 10[3 marks] | |
|---|---|
| Answer or working | Marks |
| 3,460 / 7 = 494.2857... shown (or equivalent method) | M1 |
| 494.29 (pounds) to the nearest penny | A1cao |
| notes that 7 x 494.29 = 3,460.03, which is 3p more than the £3,460 raised, so rounding to the nearest penny does not exactly reconcile with the total | B1oe |
| Final answer: GBP494.29 to the nearest penny; 7 x 494.29 = GBP3,460.03, which is 3p more than the GBP3,460 raised, so the shares do not exactly reconcile with the total once rounded. | |
| Question 11[2 marks] | |
|---|---|
| Answer or working | Marks |
| use sin 30 = opposite/hypotenuse to form opposite = 10 * sin 30 | M1 |
| 5 cm | A1cao |
| Question 12[3 marks] | |
|---|---|
| Answer or working | Marks |
| convert to improper fractions or use fraction key: 13/3 - 7/4 | M1oe |
| common denominator 12 used, or show calculator method to reach 31/12 | M1 |
| 2 7/12 | A1cao |
| Question 13[3 marks] | |
|---|---|
| Answer or working | Marks |
| convert £3.60 to 360p | M1oe |
| form the ratio 360:90 and begin to simplify | M1 |
| 4:1 | A1cao |
| Question 14[4 marks] | |
|---|---|
| Answer or working | Marks |
| correct lowest power of each common prime identified, 2^2 and 5^1 | M1 |
| 20 | A1cao |
| correct highest power of each prime identified, 2^3 and 5^2 | M1 |
| 200 | A1cao |
| Final answer: 20 | 200 | |
| Question 15[5 marks] | |
|---|---|
| Answer or working | Marks |
| finds sugar per person = 240/8 = 30 (g) | M1oe |
| 180 g | A1cao |
| finds butter per person = 200/8 = 25 (g) | M1oe |
| 620/25 (= 24.8) oe, dependent on the first M1 | dM1 |
| 24 cao, rounded down to a whole number of people | A1 |
| Final answer: 180 g | 24 people | |
| Question 16[5 marks] | |
|---|---|
| Answer or working | Marks |
| factorise to (x + 5)(x - 6) or show factors giving product -30 and sum -1 | M1 |
| set each factor to zero: x + 5 = 0 or x - 6 = 0 | M1 |
| x = -5 | A1 |
| x = 6 | A1 |
| list both solutions as x = -5, 6 (exact) | B1cao |
| Final answer: x = -5 or x = 6 (Exact values) | |
| Question 17[8 marks] | |
|---|---|
| Answer or working | Marks |
| express original parts: parks = 5p, libraries = (t+3)p so total = (t+8)p = 1600 | M1 |
| after change ratio becomes 3:2 so parks = 3q, libraries = 2q and total =5q =1600 so q = 320 | M1 |
| parks amount is unchanged: 5p = 3q so p = 3q/5 | M1 |
| substitute q=320 to get p = 3*320/5 = 192 | M1 |
| use total (t+8)p =1600 so (t+8)*192 = 1600 | M1 |
| solve (t+8) = 1600/192 = 25/3? show steps | M1 |
| find t = 1600/192 - 8 = (25/3) - 8 = (25 - 24)/3 = 1/3 | M1 |
| t = 1/3 | A1cao |
| Final answer: t = 1/3. Working check: total 1600 = 5p + (t+3)p = (t+8)p. New ratio 3:2 gives q = 320 and parks 3q = 960 so p = 960/5 = 192. Then t+8 = 1600/192 = 25/3 so t = 25/3 - 8 = 1/3. | |
| Question 18[4 marks] | |
|---|---|
| Answer or working | Marks |
| b - a | B1cao |
| -a cao, using that O is the midpoint of the diagonal AD in a regular hexagon | B1 |
| -b cao, using that O is the midpoint of the diagonal BE | B1 |
| 2b oe, ft their | B1OE |
| Final answer: AB = b - a | OD = -a | OE = -b | EB = 2b | |
| Question 19[4 marks] | |
|---|---|
| Answer or working | Marks |
| rearranges V = (4/3) x pi x r^3 to r^3 = 3V / (4 x pi) | M1oe |
| r = awrt 4.92 cm (unrounded value should be carried forward if possible) | M1 |
| substitutes their r into SA = 4 x pi x r^2 | M1ft |
| awrt 305 cm^2 (cao; accept awrt 304-305 depending on rounding of r) | A1 |
| Final answer: 305 cm^2 (3 s.f.) | |
| Question 20[7 marks] | |
|---|---|
| Answer or working | Marks |
| BC = 40 tan(28) | M1oe |
| awrt 21.3 (m) | A1 |
| BD (total height) = 40 tan(34) | M1oe |
| CD = their BD - their BC | M1ft |
| awrt 5.71 (m) | A1 |
| AD = 40 / cos(34) | M1oe |
| awrt 48.2 (m) | A1 |
| Final answer: BC = 21.3 m (3 s.f.) | CD = 5.71 m (3 s.f.) | AD = 48.2 m (3 s.f.) | |