9709 Pure Mathematics 1 Oct/Nov 2022 Paper 11

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

1. Question 1

   3 marks · 3/75

   Question paper p.2 · Mark scheme p.6

2. Question 2 (a)

   2 marks · 5/75

   Finding the tangent to a curve from its derivative and a point on the curve.

   Examiner report: A recurring issue was treating the derivative as something to integrate, or using its reciprocal and thereby finding a normal rather than a tangent.

   Question paper p.3 · Mark scheme p.6

3. Question 2 (b)

   4 marks · 9/75

   Recovering a curve equation by integrating its gradient and using a point condition.

   Examiner report: Errors commonly arose from incorrect powers and from omitting the factor required by the inner linear expression. Some candidates also interchanged the coordinates when applying the point condition.

   Question paper p.3 · Mark scheme p.7

4. Question 3

   5 marks · 14/75

   Using the stationary-point condition to determine a parameter and then find a coordinate.

   Examiner report: Although differentiation was usually attempted, mistakes in fractional powers and coefficients were frequent. Some candidates substituted their parameter into the derivative instead of returning to the original equation.

   Question paper p.4 · Mark scheme p.7

5. Question 4

   6 marks · 20/75

   Using selected binomial coefficients to determine a parameter from the coefficient of a specified power.

   Examiner report: Expanding every term was unnecessary and sometimes led to errors. Common slips included failing to square the factor involving p and omitting one sign when taking square roots.

   Question paper p.5 · Mark scheme p.8

6. Question 5 (a)

   4 marks · 24/75

   Finding the perimeter of a circular segment using sector geometry and a chord length.

   Examiner report: A frequent misconception was to treat the two radii and chord as an equilateral triangle. Candidates also sometimes confused radian measures with degree measures when finding the chord.

   Question paper p.6 · Mark scheme p.9

7. Question 5 (b)

   2 marks · 26/75

   Finding the area of a minor circular segment.

   Examiner report: The required decomposition into sector area minus triangle area was sometimes missed. Incorrect angle units, or using the full central angle in a half-triangle calculation, caused further errors.

   Question paper p.7 · Mark scheme p.9

8. Question 6 (a)

   3 marks · 29/75

   Algebraic manipulation of a trigonometric equation into a quadratic in sine.

   Examiner report: Incomplete working was a problem in this show-that part. Candidates often struggled with the common denominator or applied the squared-sine and squared-cosine identity incorrectly.

   Question paper p.8 · Mark scheme p.10

9. Question 6 (b)

   3 marks · 32/75

   Solving a trigonometric equation by first solving a quadratic in sin θ and then locating all angles in an interval.

   Examiner report: Some solutions contained only one of the valid angles because the later quadrants were overlooked. Rounding to the specified degree accuracy was also inconsistent.

   Question paper p.9 · Mark scheme p.10

10. Question 7 (a)

    3 marks · 35/75

    Using the terms of a geometric progression to verify when a quantity first falls below a threshold.

    Examiner report: Several candidates treated the data as an arithmetic rather than geometric progression, or inverted the common ratio. A conclusion identifying the first qualifying impact was required, not just two numerical comparisons.

    Question paper p.10 · Mark scheme p.11

11. Question 7 (b)

    2 marks · 37/75

    Finding the sum of a fixed number of terms of a geometric progression and applying a rounding instruction.

    Examiner report: The series formula was often selected correctly, but some candidates failed to apply the specified rounding. Directly adding rounded individual terms also led to avoidable inaccuracies.

    Question paper p.11 · Mark scheme p.11

12. Question 7 (c)

    2 marks · 39/75

    Finding the limiting total of a convergent geometric progression.

    Examiner report: This part was generally handled more successfully with the standard infinite-series formula. Summing terms individually was less reliable and often introduced rounding or arithmetic errors.

    Question paper p.11 · Mark scheme p.11

13. Question 8 (a)

    3 marks · 42/75

    Differentiating a reciprocal linear function and using the derivative to determine monotonicity.

    Examiner report: Some candidates confused derivative notation with inverse-function notation. Other errors involved the chain-rule multiplier, the power in the denominator, or failing to interpret the sign of the derivative.

    Question paper p.12 · Mark scheme p.12

14. Question 8 (b)

    4 marks · 46/75

    Finding the inverse of a rational function and expressing it in a prescribed form.

    Examiner report: The general inverse-function approach was understood by many, although sign and algebraic rearrangement errors remained common.

    Question paper p.13 · Mark scheme p.12

15. Question 8 (c)

    1 mark · 47/75

    Question paper p.13 · Mark scheme p.12

16. Question 9 (a)

    1 mark · 48/75

    Question paper p.14 · Mark scheme p.13

17. Question 9 (b)

    2 marks · 50/75

    Question paper p.14 · Mark scheme p.13

18. Question 9 (c)

    1 mark · 51/75

    Relating two quadratic functions through a shifted input and a vertical scale factor.

    Examiner report: Many candidates found the mapping notation difficult to interpret and did not successfully connect the two completed-square representations.

    Question paper p.15 · Mark scheme p.13

19. Question 9 (d)

    4 marks · 55/75

    Describing the separate graph transformations represented by a function relation.

    Examiner report: Some responses identified one transformation but omitted or misdescribed the other. Precise directional language and a clear scale description were needed for full credit.

    Question paper p.15 · Mark scheme p.13

20. Question 10 (a)

    5 marks · 60/75

    Finding the area enclosed between two intersecting curves by integration.

    Examiner report: Typical mistakes included mishandling fractional exponents or coefficients during integration and reversing the limits without compensating for the sign.

    Question paper p.16 · Mark scheme p.14

21. Question 10 (b)

    5 marks · 65/75

    Finding the acute angle between two tangents from their gradients.

    Examiner report: This was a demanding problem-solving part. Successful approaches began with both gradients and then used inverse tangent, tangent equations with geometry, or a right-triangle construction.

    Question paper p.17 · Mark scheme p.14

22. Question 11 (a)

    4 marks · 69/75

    Finding tangent gradients to a circle by imposing a repeated-intersection condition.

    Examiner report: Arithmetic and algebraic slips in forming the quadratic were common. The tangency condition had to be expressed through a zero discriminant rather than merely finding ordinary intersections.

    Question paper p.18 · Mark scheme p.15

23. Question 11 (b)

    3 marks · 72/75

    Finding the coordinates of points of contact between tangents and a circle.

    Examiner report: Candidates used either substitution into the quadratic or the perpendicular tangent-normal method. Some found one point but did not complete the symmetric pair or paired coordinates incorrectly.

    Question paper p.19 · Mark scheme p.16

24. Question 11 (c)

    3 marks · 75/75

    Using circle geometry and trigonometry to determine an angle subtended by two contact points.

    Examiner report: This was difficult for many candidates. The most effective solutions used either the cosine rule, a central-angle argument, or complementary angles formed with radii and vertical lines.

    Question paper p.19 · Mark scheme p.17