9709 Pure Mathematics 1 Oct/Nov 2020 Paper 13

23 questions and parts, 75 marks in total. Marks come from the published mark scheme.

1. Question 1 (a)

   2 marks · 2/75

   Question paper p.2 · Mark scheme p.6

2. Question 1 (b)

   2 marks · 4/75

   Describing a translation from the algebraic form of a transformed quadratic.

   Examiner report: This was noticeably less secure than part (a): candidates often recognised a shift but failed to communicate both its direction and size using acceptable coordinate or axis-based language.

   Question paper p.2 · Mark scheme p.6

3. Question 2 (a)

   3 marks · 7/75

   Evaluating an improper integral involving a rational power function.

   Examiner report: Integration was generally handled well, but a common omission was failing to evaluate what happens to the antiderivative at infinity.

   Question paper p.3 · Mark scheme p.6

4. Question 2 (b)

   2 marks · 9/75

   Finding a curve from its derivative and a specified point.

   Examiner report: Most responses were successful, although some differentiated the given function instead of using it as the derivative to be integrated.

   Question paper p.3 · Mark scheme p.6

5. Question 3

   5 marks · 14/75

   Question paper p.4 · Mark scheme p.7

6. Question 4

   5 marks · 19/75

   Using the discriminant to determine when a line and quadratic meet at two distinct points.

   Examiner report: The standard discriminant approach was widely used, but errors in collecting terms, signs, factorisation, or converting the final result into inequalities were frequent.

   Question paper p.5 · Mark scheme p.7

7. Question 5

   5 marks · 24/75

   Using binomial coefficients and the geometric-progression condition on selected terms.

   Examiner report: The three relevant binomial terms were usually identified; the greater difficulty was forming a useful relation between them and carrying the subsequent algebra through without slips.

   Question paper p.6 · Mark scheme p.8

8. Question 6 (a)

   3 marks · 27/75

   Finding the inverse of a rational function by algebraic rearrangement.

   Examiner report: Clearing the denominator at the outset was generally the most reliable route. Work involving reciprocals or an unclear interchange of the variables tended to lead to mistakes.

   Question paper p.7 · Mark scheme p.8

9. Question 6 (b)

   2 marks · 29/75

   Verifying an equivalent algebraic form of a rational function.

   Examiner report: This identity was answered successfully by most candidates, helped by the expression being supplied for comparison.

   Question paper p.7 · Mark scheme p.8

10. Question 6 (c)

    1 mark · 30/75

    Determining the range of a rational function from its domain and an equivalent form.

    Examiner report: Many candidates did not exploit the rearrangement in part (b), so they struggled to identify the range.

    Question paper p.7 · Mark scheme p.9

11. Question 7 (a)

    4 marks · 34/75

    Deriving the common difference of an arithmetic progression using trigonometric identities.

    Examiner report: Candidates generally knew the intended starting identity, but sign errors during fraction manipulation were common. Since the target was supplied, intermediate algebra needed to be especially clear.

    Question paper p.8 · Mark scheme p.9

12. Question 7 (b)

    3 marks · 37/75

    Finding a specified arithmetic-progression term exactly using trigonometric values.

    Examiner report: The progression formula was usually applied correctly; exact evaluation of the relevant powers of cosine caused difficulty for some candidates.

    Question paper p.9 · Mark scheme p.9

13. Question 8 (a)

    3 marks · 40/75

    Finding first and second derivatives of a function containing a reciprocal linear factor.

    Examiner report: The first derivative was usually obtained accurately, whereas mistakes were more common in the second differentiation.

    Question paper p.10 · Mark scheme p.10

14. Question 8 (b)

    5 marks · 45/75

    Locating and classifying a stationary point using derivatives and the domain restriction.

    Examiner report: A frequent error was retaining a stationary-point candidate excluded by the domain. Other omissions included the y-coordinate or the evidence needed to establish that the point is a minimum.

    Question paper p.11 · Mark scheme p.10

15. Question 9 (a)

    2 marks · 47/75

    Finding an angle in a composite-circle geometry diagram and expressing it in radians.

    Examiner report: A perpendicular construction gave an efficient solution, while less direct cosine-rule approaches sometimes found the wrong angle. Converting from degrees and rounding too early also caused accuracy losses.

    Question paper p.12 · Mark scheme p.10

16. Question 9 (b)

    4 marks · 51/75

    Calculating a shaded area formed by a circular sector and a triangle.

    Examiner report: Most candidates used the intended sector-minus-triangle structure successfully; both the angle and the triangle area needed to be carried accurately into the final subtraction.

    Question paper p.13 · Mark scheme p.11

17. Question 9 (c)

    3 marks · 54/75

    Finding the boundary length of a region involving an arc and straight segments.

    Examiner report: This was generally the most accessible part of the geometry question and was answered well by most candidates.

    Question paper p.13 · Mark scheme p.11

18. Question 10 (a)

    4 marks · 58/75

    Differentiating a function with fractional powers and using a specified gradient to determine a positive parameter.

    Examiner report: The mixed positive and negative powers led to errors for weaker candidates; some also treated the parameter as the differentiation variable or substituted into the wrong expression.

    Question paper p.14 · Mark scheme p.11

19. Question 10 (b)

    5 marks · 63/75

    Evaluating a parameter-dependent definite integral and solving the resulting equation.

    Examiner report: The first two integrations were usually correct. Mistakes commonly arose from the constant term, the lower limit containing the parameter, and failing to justify the factorisation or reject an inadmissible root.

    Question paper p.15 · Mark scheme p.12

20. Question 11 (a)

    3 marks · 66/75

    Showing that a point lies outside a circle using the circle equation or distance from its centre.

    Examiner report: A comparison using inequality was essential; merely showing that the two quantities were unequal only established that the point was not on the circumference.

    Question paper p.16 · Mark scheme p.13

21. Question 11 (b)

    2 marks · 68/75

    Using tangent-radius geometry and an exact right-triangle calculation to find an angle.

    Examiner report: Some candidates tried to use the gradient of CT, while others assumed a tangent length without proving it. The successful method came from the right triangle formed by the centre, contact point, and external point.

    Question paper p.16 · Mark scheme p.13

22. Question 11 (c)

    4 marks · 72/75

    Finding the equation of the chord joining the two points of contact from an external point.

    Examiner report: Many candidates found the relevant gradients but could not locate the line. Recognising the square-like tangent configuration, and hence the midpoint property, was the key additional step.

    Question paper p.17 · Mark scheme p.14

23. Question 11 (d)

    3 marks · 75/75

    Finding the x-coordinates of circle-tangent intersection points.

    Examiner report: Progress in this part usually depended on having obtained the correct line in part (c); once substituted, the resulting quadratic was relatively straightforward.

    Question paper p.17 · Mark scheme p.14