9709 Pure Mathematics 1 Feb/Mar 2022 Paper 12

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

1. Question 1

   4 marks · 4/75

   This assesses integration of a derivative involving fractional powers, followed by use of a specified point to determine the constant.

   Examiner report: Most candidates understood that the derivative had to be integrated and usually found the constant successfully. Final forms should be simplified so that fractional powers are not left in denominator notation.

   Question paper p.3 · Mark scheme p.6

2. Question 2

   5 marks · 9/75

   This tests finding the condition for two distinct intersections by forming a quadratic and applying the discriminant criterion.

   Examiner report: Eliminating y and using the discriminant was generally handled well. Selecting the correct parameter intervals from the inequality was less secure, although a sketch often helped candidates choose the regions.

   Question paper p.4 · Mark scheme p.6

3. Question 3 (a)

   3 marks · 12/75

   This assesses selection and evaluation of the constant term in a binomial expansion with positive and negative powers of x.

   Examiner report: Most candidates identified and calculated the relevant term. Errors sometimes arose from handling the powers inside the brackets, and expanding every term was an unnecessary source of risk.

   Question paper p.5 · Mark scheme p.7

4. Question 3 (b)

   3 marks · 15/75

   This tests how to obtain a constant term from a product by combining suitable terms from the binomial expansion.

   Examiner report: Many candidates recognised that two contributions had to be considered. Sign mistakes were the main difficulty for those who did not reach the required result.

   Question paper p.5 · Mark scheme p.7

5. Question 4

   6 marks · 21/75

   This assesses modelling relationships between corresponding terms of geometric and arithmetic progressions and then summing an arithmetic progression.

   Examiner report: The two initial term equations were usually set up correctly. Algebraic and sign errors during elimination often prevented candidates from using the arithmetic-series formula successfully.

   Question paper p.6 · Mark scheme p.8

6. Question 5 (a)

   2 marks · 23/75

   This tests completing the square and matching the result to a specified form.

   Examiner report: The target structure was commonly recognised, though some candidates placed the constant outside the square-bracket expression or used an incorrect version of the completed-square form.

   Question paper p.7 · Mark scheme p.8

7. Question 5 (b)

   4 marks · 27/75

   This assesses describing a composite graph transformation, including horizontal translation, vertical scaling and vertical translation in the required order.

   Examiner report: Most successful responses gave a clear ordered description. Translations were often expressed effectively with vectors, while stretches needed both a scale factor and a direction.

   Question paper p.7 · Mark scheme p.8

8. Question 6 (a)

   4 marks · 31/75

   This tests finding the intersection of a line and a circle by substitution and solving the resulting quadratic.

   Examiner report: Substitution and formation of the quadratic were generally successful. A complete solution required visible working for solving that quadratic rather than only unsupported calculator coordinates.

   Question paper p.8 · Mark scheme p.9

9. Question 6 (b)

   4 marks · 35/75

   This assesses finding a circle with a fixed centre that is tangent to a given line.

   Examiner report: Several valid approaches were used, including midpoint-distance reasoning and a repeated-root discriminant method. Sign errors in the distance calculation were a frequent source of lost marks.

   Question paper p.9 · Mark scheme p.9

10. Question 7 (a)

    4 marks · 39/75

    Question paper p.10 · Mark scheme p.10

11. Question 7 (b)

    3 marks · 42/75

    This assesses using the identity from part (a) to solve a trigonometric equation over a specified degree interval.

    Examiner report: Most candidates obtained the correct ratio from the identity, but some reported only one of the two angles. The answer units needed to match the degree interval in the question.

    Question paper p.11 · Mark scheme p.10

12. Question 8 (a)

    3 marks · 45/75

    This tests locating the endpoints of a horizontal chord using the circle equation and the stated axis condition.

    Examiner report: The algebra was usually handled correctly. A negative square root or swapping the coordinate roles led to avoidable errors; the diagram clarified which endpoint lay on the positive y-axis.

    Question paper p.12 · Mark scheme p.11

13. Question 8 (b)

    5 marks · 50/75

    This assesses finding a volume of revolution for a circular segment by subtracting the cylinder beneath the chord from the rotated circular region.

    Examiner report: Most candidates set up the integral correctly and integrated it successfully. Full credit required the limits from part (a) to be visibly used, as well as inclusion of the cylindrical contribution where appropriate.

    Question paper p.13 · Mark scheme p.11

14. Question 9 (a)

    4 marks · 54/75

    This tests solving an equation containing a square root and expressing the resulting values in the prescribed surd form.

    Examiner report: Clear algebra enabled many candidates to gain full credit by either squaring directly or treating the square root as a temporary variable. Some overlooked that both resulting values were permitted by the domain.

    Question paper p.14 · Mark scheme p.12

15. Question 9 (b)

    4 marks · 58/75

    Question paper p.15 · Mark scheme p.12

16. Question 10 (a)

    5 marks · 63/75

    This tests geometric use of right-triangle trigonometry, central angles and arc length to find the perimeter of a composite circular region.

    Examiner report: The required trigonometric relationships were generally familiar, and many candidates recognised the right-triangle structure. Carrying extra accuracy in intermediate angles was important when obtaining the final perimeter.

    Question paper p.16 · Mark scheme p.13

17. Question 10 (b)

    3 marks · 66/75

    This assesses finding the area of a region made from a triangle with circular sectors removed.

    Examiner report: This part was accessible when the angles from the perimeter calculation were correct. Rounding those angles too early sometimes moved the final area outside the accepted range.

    Question paper p.17 · Mark scheme p.13

18. Question 11 (a)

    4 marks · 70/75

    This tests differentiating a rational expression, setting the derivative to zero and solving for all stationary-point x-values in terms of the parameter.

    Examiner report: Differentiation was generally understood, but algebraic slips after setting the derivative to zero were common. Some candidates also lost a solution by failing to consider the negative square-root branch.

    Question paper p.18 · Mark scheme p.14

19. Question 11 (b)

    3 marks · 73/75

    This assesses locating a stationary point for a particular function and classifying its stationary value using derivative information.

    Examiner report: Most candidates differentiated the particular function and solved for its stationary point. A clear, correct second-derivative argument was less consistently shown, so explanations of the classification needed greater care.

    Question paper p.19 · Mark scheme p.14

20. Question 11 (c)

    2 marks · 75/75

    This tests determining monotonicity by analysing the sign of the derivative throughout the function's domain.

    Examiner report: Candidates generally knew that the derivative was the appropriate tool. The justification had to establish positivity for the whole domain, rather than merely evaluating the derivative at selected points.

    Question paper p.19 · Mark scheme p.15