9709 Pure Mathematics 1 Oct/Nov 2020 Paper 12
9709 Pure Mathematics 1 Oct/Nov 2020 Paper 12: all 25 questions and parts with their marks, taken from the published mark scheme rather than estimated.
Every question in 9709 Pure Mathematics 1 Oct/Nov 2020 Paper 12
25 questions and parts, 75 marks in total. Marks come from the published mark scheme.
Question 1
4 marks · 4/75This assesses extraction of selected terms from a binomial expansion and use of a coefficient condition to determine an unknown constant.
Examiner report: Most candidates found this an accessible opening question; typical errors involved mishandling powers or signs, selecting only one contribution, or failing to equate the combined coefficient to the stated value.
Question 2
5 marks · 9/75This assesses finding the parameters of a geometric progression from consecutive terms and then applying the convergent infinite-sum formula.
Examiner report: Many candidates obtained a ratio in terms of the parameter but did not continue to determine that parameter; others used a non-convergent ratio without recognising the resulting error.
Question 3
5 marks · 14/75This assesses proving that a line meets a parameterised quadratic curve twice by analysing the discriminant of their intersection equation.
Examiner report: Candidates commonly formed the correct quadratic and discriminant but stopped before proving its sign for every parameter value, or incorrectly tried to assign a particular value to the parameter.
Question 4
5 marks · 19/75This assesses recovering the first term and common difference of an arithmetic progression from its sum formula, then applying an nth-term inequality.
Examiner report: Some candidates subtracted terms incorrectly, used the sum rather than the individual term, or stopped at a non-integer threshold instead of choosing the required integer.
Question 5 (a)
1 mark · 20/75Question 5 (b)
5 marks · 25/75This assesses finding inverse functions, equating them, and solving the resulting quadratic equation.
Examiner report: Most candidates found and equated the inverses successfully; lost credit mainly arose from algebraic simplification errors or giving quadratic roots without showing how they were obtained.
Question 6 (a)
4 marks · 29/75This assesses proving a trigonometric identity through substitution of the tangent definition, algebraic fraction manipulation, and the Pythagorean identity.
Examiner report: Although many solutions were complete, errors in combining algebraic fractions prevented some candidates from using the trigonometric identities effectively.
Question 6 (b)
2 marks · 31/75This assesses using a previously proved identity to reduce and solve a trigonometric equation over a specified degree interval.
Examiner report: Using the earlier identity usually led to a successful solution; common problems were failing to take the required root and retaining an additional angle outside the permitted range.
Question 7 (a)
3 marks · 34/75This assesses a related-rates calculation using the chain rule to connect the rates of change of x and y.
Examiner report: Most candidates used the supplied derivative correctly, although some differentiated again or applied the chain rule to the wrong quantities.
Question 7 (b)
4 marks · 38/75This assesses recovering a curve equation by integrating a derivative and using a point on the curve to determine the constant.
Examiner report: Integration was generally handled well, but omissions of the integration constant, errors with fractional powers, and failure to write an explicit y-equation cost marks.
Question 8 (a)
3 marks · 41/75This assesses finding a shaded area by subtracting a sector area from the area of an isosceles triangle using radians.
Examiner report: Many candidates knew the relevant area formulae, but some selected the wrong angle for the triangle or failed to account for both halves when splitting the isosceles triangle.
Question 8 (b)
4 marks · 45/75This assesses finding a composite perimeter involving a circular arc and straight sides of the isosceles-triangle construction.
Examiner report: The arc was usually found readily; determining the remaining straight segment was less secure, and early rounding sometimes affected the final accuracy.
Question 9 (a)
3 marks · 48/75This assesses forming the equation of a circle from its centre and a point on its circumference.
Examiner report: Most candidates found the radius and used the centre-radius form correctly, though some treated the required locus as a straight line or forgot that the radius must be squared in the equation.
Question 9 (b)
4 marks · 52/75This assesses proving tangency by locating the diameter endpoint and showing that the line from it to the external point is perpendicular to the corresponding radius.
Examiner report: Several valid approaches were seen, but candidates often lost the concluding mark by showing perpendicular gradients without explicitly linking that fact to tangency.
Question 9 (c)
2 marks · 54/75This assesses locating the second point of contact by exploiting the symmetry of the circle and the two tangents from the external point.
Examiner report: This part was most often completed by candidates who recognised the reflection relationship; lengthy coordinate calculations were generally unnecessary.
Question 10 (a)
6 marks · 60/75This assesses differentiating a rational power expression twice and integrating the given function term by term.
Examiner report: Most candidates were comfortable with the calculus, but some copied the function incorrectly by omitting its linear term or mishandled the factors arising from the inner linear expression.
Question 10 (b)
2 marks · 62/75This assesses finding the x-coordinate of a minimum by solving the stationary-point condition.
Examiner report: Setting the derivative to zero was generally the clearest route; solutions based on setting y equal to zero were less efficient and required fully demonstrated algebra rather than an unsupported numerical output.
Question 10 (c)
2 marks · 64/75This assesses evaluating the area bounded by a curve and the coordinate axes using a definite integral.
Examiner report: Some candidates left this part blank after difficulties in finding the minimum, while others failed to show both substitutions in the definite integral.
Question 11 (a)
2 marks · 66/75Question 11 (b)
2 marks · 68/75This assesses sketching one complete cycle of a cosine graph over the specified interval with the correct amplitude, midline, period, and endpoint behaviour.
Examiner report: Some sketches did not agree with the stated range, and calculator settings or straight-line approximations led to unsuitable graphs.
Question 11 (c)(i)
1 mark · 69/75Question 11 (c)(ii)
1 mark · 70/75Question 11 (c)(iii)
1 mark · 71/75Question 11 (d)
2 marks · 73/75This assesses describing the horizontal scaling and vertical translation needed to transform the graph of f into the graph of g.
Examiner report: Transformation terminology was a frequent weakness: candidates needed to distinguish a stretch from a translation and specify the scale factor and direction precisely.
Question 11 (e)
2 marks · 75/75This assesses describing a horizontal translation followed by a vertical stretch for the graph of h.
Examiner report: Many candidates did not express these changes using precise transformation language or failed to state the directions and scale factor unambiguously.
Grade thresholds for this paper
| Session | Paper | A | B | C | D | E |
|---|---|---|---|---|---|---|
| 9709 Pure Mathematics 1 Oct/Nov 2020 Paper 12 | 12 | 54 | 45 | 34 | 23 | 12 |