Write your name, register number and class on all the work you hand in.
Write in dark blue or black ink.
You may use an HB pencil for any diagrams or graphs.
Do not use staples, paper clips, glue or correction tape/fluid.
Write your answers and working in the blank spaces provided.
Answer all questions.
Omission of essential working will result in loss of marks.
The use of an approved scientific calculator is expected, where appropriate.
If the degree of accuracy is not specified in a question and the answer is not exact, give the answer to three significant figures. For π, use either your calculator value or 3.142.
The number of marks is given in brackets [ ] at the end of each question or part question.
The total number of marks for this paper is 50.
INFORMATION
This paper contains 10 questions and is worth 50 marks. A calculator is allowed.
Every topic family that appeared in at least two of the 2023, 2024 and 2025 EOY Paper 2 papers is covered, with new numbers in every question.
Topics: equations in indices; simultaneous equations including the no-solution case; similar solids; similar triangles with a proof and an area ratio; trigonometry with angles of elevation; sketching straight lines and parabolas on given axes; reading a graph in context and forming its equation; completing the square; statistics from a cumulative frequency curve; and forming a quadratic from a word problem.
Suggested pace: about 8 minutes per question, leaving 10 minutes at the end to check your work.
Questions
Q1.(a) Solve the equation 93x−1 = 27x+2. [2]
(b) Simplify (8a−3b6)2/3 × (4a4b−2)1/2, leaving your answer in positive index form. [3]
Q2.(a) The pair of simultaneous equations
x + 2y = 7 and 3x + ky = 12
has no solution. Find the value of k. [2]
(b) Solve the simultaneous equations y = x − 3 and x2 + y2 = 17. [3]
Q3.Two similar cylindrical containers, P and Q, have heights 6 cm and 9 cm respectively. The volume of container P is 240 cm3.
(a) Calculate the volume of container Q. [2]
(b) The curved outside surface of both containers is painted. Find the area painted on container Q as a percentage of the area painted on container P. [2]
(c) Container Q is filled completely with water and all of the water is poured into container P, which is empty. Explain whether container P overflows. [1]
Q4.In triangle ABC, D lies on AB and E lies on AC such that DE is parallel to BC. AD = 6 cm, DB = 4 cm and DE = 9 cm.
(a) Prove that triangle ADE is similar to triangle ABC. [2]
(b) Find the length of BC. [2]
(c) Given that the area of triangle ADE is 27 cm2, find the area of triangle ABC. [1]
Q5.A vertical mast ST stands on level ground. The foot of the mast is T and P is a point on the ground 45 m from T. The angle of elevation of the top of the mast, S, from P is 32°.
(a) Calculate the height of the mast, ST. [2]
(b) A small flag is fixed to the mast at a point F, 5 m vertically below S. Calculate the angle of elevation of F from P. [2]
(c) Calculate the distance PS. [1]
Q6.The axes below show the graph of y = x. On the same axes, sketch and label each of the following.
(a) y = −2x + 4, indicating the coordinates of both intercepts, [2]
(b) y = 3, [1]
(c) y = x2 − 4, indicating the coordinates of both intercepts and of the turning point. [2]
Q7.A taxi company charges a fixed booking fee plus a charge for every kilometre travelled. The graph shows the fare, $C, against the distance travelled, d km, for one journey.
(a) State the fare when the distance travelled is 0 km, and explain what it represents. [1]
(b) Find the gradient of the graph and explain its significance. [2]
(c) Form an equation connecting C and d. [1]
(d) Calculate the fare for a journey of 15 km. [1]
Q8.(a) Express x2 − 6x + 2 in the form (x + p)2 + q. [2]
(b) Hence state the coordinates of the turning point of the graph of y = x2 − 6x + 2. [1]
(c) The line y = 1 meets the curve y = x2 − 6x + 2 at two points. Find the x-coordinates of both points, giving your answers correct to 2 decimal places. [2]
Q9.A survey recorded the monthly electricity usage of 120 households. The cumulative frequency curve shows the results.
(a) Use the curve to estimate the median monthly usage. [1]
(b) Find the interquartile range. [2]
(c) The electricity company claims that the median monthly usage of its customers is below 300 kWh. Using your answer to (a), state whether the survey supports this claim, giving a reason. [2]
Q10.A rectangular card is 3 cm longer than it is wide. The area of the card is 150 cm2.
(a) Taking the width of the card as x cm, form an equation in x and show that it reduces to x2 + 3x − 150 = 0. [2]
(b) Solve the equation x2 + 3x − 150 = 0, giving your answers correct to 3 significant figures. [2]
(c) Find the length of the card. [1]
End of Paper 2. Check your work — units on every measurement, angles to 1 d.p. and other answers to 3 s.f. unless the question says otherwise.
Answer Key — S2 EOY Practice, Mock Paper B2
Total: 50 marks · 10 questions. Method marks are awarded for a correct method even where the final answer is wrong. Values read from the graph in Q9 are accepted within about 5 kWh of the value given.
Q1Indices[5]
(a) 93x−1 = 32(3x−1) = 36x−2 and 27x+2 = 33x+6, so 6x − 2 = 3x + 6 and x = 8/3
(a) gradients: −1/2 from the first, −3/k from the second. No solution means they are equal, so −3/k = −1/2 and k = 6
(b) substituting: x2 + (x − 3)2 = 17 gives 2x2 − 6x − 8 = 0, i.e. (x − 4)(x + 1) = 0. x = 4, y = 1 or x = −1, y = −4
Q3Similar solids[5]
(a) length ratio 6 : 9 = 2 : 3, so volume ratio = 8 : 27. Volume of Q = 240 × 27/8 = 810 cm3
(b) surface area ratio = (3/2)2 = 2.25, so the area painted on Q is 225 % of that on P
(c) 810 cm3 of water cannot fit into a 240 cm3 container, so container P overflows
Q4Similar triangles[5]
(a) ∠DAE = ∠BAC (common angle) and ∠ADE = ∠ABC (corresponding angles, DE ∥ BC). Two equal angles, so triangle ADE is similar to triangle ABC (AA similarity test)
(b) AD/AB = 6/10 = 3/5, so BC = 9 ÷ 3/5 = 15 cm
(c) area ratio = (3/5)2 = 9/25, so area of ABC = 27 ÷ 9/25 = 75 cm2
Q5Trigonometry[5]
(a) ST = 45 tan 32° = 28.1 m (3 s.f.)
(b) the flag is 28.119 − 5 = 23.119 m above the ground, so the angle = tan−1(23.119/45) = 27.2° (1 d.p.)
(c) PS = 45 ÷ cos 32° = 53.1 m (3 s.f.)
Q6Graph sketching[5]
(a) y = −2x + 4: a straight line falling from (0, 4) to (2, 0)
(b) y = 3: a horizontal line through (0, 3)
(c) y = x2 − 4: an upward parabola cutting the x-axis at (−2, 0) and (2, 0), the y-axis at (0, −4), with turning point (0, −4)
Q7Graph reading in context[5]
(a) $4. This is the fixed booking fee, charged before any distance is travelled
(b) gradient = (22 − 4)/10 = 1.8, so the fare rises by $1.80 for each kilometre travelled
(c) C = 1.8d + 4
(d) C = 1.8(15) + 4 = $31
Q8Completing the square[5]
(a) x2 − 6x + 2 = (x − 3)2 − 9 + 2 = (x − 3)2 − 7
(b) turning point (3, −7)
(c) (x − 3)2 − 7 = 1 gives (x − 3)2 = 8, so x = 3 ± 2√2 = 0.17 or 5.83 (2 d.p.)
Q9Statistics[5]
(a) the median is the 60th value; read from the curve, ≈ 332 kWh
(b) lower quartile (30th value) ≈ 285 kWh and upper quartile (90th value) ≈ 385 kWh, so the interquartile range ≈ 100 kWh
(c) the survey does not support the claim: the median from the survey (≈ 332 kWh) is higher than 300 kWh
Q10Quadratic word problem[5]
(a) x(x + 3) = 150, which expands to x2 + 3x − 150 = 0
(b) x = [−3 ± √(9 + 600)] / 2 = [−3 ± √609] / 2, so x = 10.8 or −13.8; the negative value is rejected
