9709 Pure Mathematics 1 May/June 2023 Paper 11
9709 Pure Mathematics 1 May/June 2023 Paper 11: all 27 questions and parts with their marks, taken from the published mark scheme rather than estimated.
Every question in 9709 Pure Mathematics 1 May/June 2023 Paper 11
27 questions and parts, 75 marks in total. Marks come from the published mark scheme.
Question 1
3 marks · 3/75Question 2 (a)
2 marks · 5/75This assesses using the binomial theorem to obtain the constant, linear and quadratic terms of a power of a binomial.
Examiner report: The first expansion was generally handled well, although some candidates omitted the constant term or included the wrong terms.
Question 2 (b)
2 marks · 7/75This assesses binomial expansion when the binomial contains a negative term.
Examiner report: Sign mistakes were particularly common in the second and third terms, and some responses retained the wrong powers.
Question 2 (c)
2 marks · 9/75This assesses extracting a specified coefficient from the product of two truncated expansions.
Examiner report: Candidates who matched powers before multiplying often gained both marks, whereas weaker attempts selected only the x² terms or added coefficients without forming the required products.
Question 3
4 marks · 13/75This assesses interpreting graphical changes as a vertical stretch and a translation.
Examiner report: Some candidates described the reverse mapping, while others omitted either the direction or the scale factor. Giving several incompatible transformations also made otherwise correct intentions unclear.
Question 4
4 marks · 17/75This assesses finding a sector angle from its area and using arc and chord lengths to calculate a segment perimeter.
Examiner report: Most successful responses found the angle and arc correctly. Errors included treating the triangle as right-angled, using an unsuitable tangent relation, or calculating the area of the segment instead of its boundary length.
Question 5
5 marks · 22/75This assesses applying the tangency condition to a line and a reciprocal curve.
Examiner report: The usual successful approach was to form a quadratic and impose a zero discriminant, but algebraic slips with the denominator were frequent. Some candidates retained an impossible parameter value or gave unsupported calculator results.
Question 6 (a)
3 marks · 25/75This assesses using equal consecutive differences in an arithmetic progression to determine a parameter.
Examiner report: Many candidates reached the appropriate quadratic, but some kept both roots despite the explicit non-zero-difference condition. Poorly organised algebra and multiple unchecked attempts also led to avoidable errors.
Question 6 (b)
2 marks · 27/75This assesses finding the common ratio and sum to infinity of a geometric progression after using a parameter found earlier.
Examiner report: Successful candidates used the earlier parameter value before forming the ratio and then applied the standard sum formula. Follow-through credit was available when an earlier parameter error still produced a valid ratio.
Question 7 (a)
2 marks · 29/75This assesses finding the range of a vertically shifted and scaled sine function.
Examiner report: A frequent misconception was to use the endpoints of the x-interval as though they determined the range. Some candidates confused domain and range, although correct extrema shown later on a graph could still receive credit.
Question 7 (b)
2 marks · 31/75This assesses sketching a sinusoidal graph with the correct period, amplitude, phase position and endpoint behaviour.
Examiner report: Many sketches had the right general wave shape but an incorrect period or lacked periodicity. A graph could earn follow-through credit when its extrema matched the candidate’s earlier values.
Question 7 (c)
1 mark · 32/75Question 8 (a)
4 marks · 36/75This assesses solving a rational-function condition and a composite-function condition for two constants.
Examiner report: A common algebraic error was failing to multiply the constant term when clearing the denominator. Finding the intermediate value of f first generally helped candidates handle the composite function correctly.
Question 8 (b)
1 mark · 37/75This assesses recognising that the domain of an inverse function is the range of the original function.
Examiner report: Many responses gave the domain of f instead of the range of f, or excluded a single value rather than describing the correct interval restriction. Notation was also sometimes unclear.
Question 8 (c)
3 marks · 40/75This assesses finding the inverse of a restricted rational function by rearranging and interchanging variables.
Examiner report: Candidates who swapped variables and isolated the subject usually completed the task efficiently. Errors commonly arose during rearrangement, and unclear working made it difficult to distinguish the inverse from the original function.
Question 9 (a)
3 marks · 43/75This assesses related rates using the volume-depth relationship and a constant volume inflow.
Examiner report: A large proportion of candidates did not recognise the related-rates structure and either left the question blank or substituted the depth into the volume formula. The essential chain-rule link was often missing.
Question 9 (b)
3 marks · 46/75This assesses using a given depth rate to determine the tank volume at another instant.
Examiner report: Only the strongest responses completed all three stages reliably. A common misconception was to treat the given rate as a depth value rather than using it in the differentiated relationship.
Question 10 (a)
5 marks · 51/75This assesses finding a volume of revolution involving a curved boundary and a horizontal boundary.
Examiner report: Fully justified solutions showed the integration and both limit substitutions. Errors included integrating y rather than y², changing the power in the wrong direction, omitting the chain-rule factor, or forgetting the cylindrical part.
Question 10 (b)
6 marks · 57/75This assesses using differentiation and perpendicular gradients to find the area of a triangle formed by tangent, normal and the x-axis.
Examiner report: A clear plan, often supported by a fresh sketch, helped the stronger candidates. Common difficulties were differentiating without the chain-rule factor, finding the normal gradient, and recognising that the height can be read directly from B.
Question 11 (a)
2 marks · 59/75This assesses using the stationary-point condition to determine the positive parameter.
Examiner report: Successful candidates used dy/dx=0 at the stated x-coordinate. Some incorrectly used the point’s y-coordinate as the derivative value or applied an unnecessarily complicated quadratic method.
Question 11 (b)
2 marks · 61/75This assesses classifying a stationary point with the second-derivative test.
Examiner report: Most candidates could differentiate correctly, but arithmetic and substitution errors sometimes reversed the classification. A conclusion about the type of point was required, not merely the sign of a derivative.
Question 11 (c)
3 marks · 64/75This assesses integrating a derivative and using a known point to determine the constant of integration.
Examiner report: Many candidates integrated successfully and used the given point, but some wrote only an expression in x rather than an equation for y. Careful simplification of the constant term was also important.
Question 11 (d)
2 marks · 66/75This assesses finding and evaluating any stationary point other than the one already supplied.
Examiner report: The required method was to demonstrate the quadratic solution before rejecting the known point. Arithmetic errors in evaluating the remaining ordinate were common, even when the derivative equation had been solved correctly.
Question 12 (a)
1 mark · 67/75This assesses writing the equation of a circle from its centre and radius.
Examiner report: Although many candidates knew the circle form, common errors included using the radius rather than its square, omitting the square on a bracket, interchanging coordinates, and choosing the wrong sign.
Question 12 (b)
2 marks · 69/75This assesses finding a tangent equation using the radius gradient and perpendicularity.
Examiner report: Coordinate-gradient methods were often successful. Other responses inverted the gradient incorrectly, forgot the perpendicular relationship, or made avoidable errors when differentiating the circle implicitly.
Question 12 (c)
3 marks · 72/75This assesses constructing a second circle from the tangent’s y-intercept and verifying a common ordinate of intersection.
Examiner report: Candidates who linked this part to their tangent equation could form Q readily. Many did not perform the requested intersection check; comparing the two equations or testing the stated ordinate provides a concise verification.
Question 12 (d)
3 marks · 75/75This assesses finding exact intersections between a line and a circle and expressing the coordinates in surd form.
Examiner report: Stronger responses completed the substitution and obtained exact roots, but some stopped at decimal approximations or did not understand that the requested surd form required exact radical coordinates.
Grade thresholds for this paper
| Session | Paper | A | B | C | D | E |
|---|---|---|---|---|---|---|
| 9709 Pure Mathematics 1 May/June 2023 Paper 11 | 11 | 50 | 43 | 33 | 23 | 14 |