9709 Pure Mathematics 1 May/June 2022 Paper 12

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

1. Question 1

   4 marks · 4/75

   This assesses the extraction of specified coefficients from binomial expansions and the formation of an equation involving an unknown parameter.

   Examiner report: Some candidates worked with whole terms rather than coefficients, leaving unnecessary powers of x and becoming unsure how to proceed.

   Question paper p.2 · Mark scheme p.6

2. Question 2

   4 marks · 8/75

   This assesses the use of consecutive terms in a geometric progression to determine its sum to infinity.

   Examiner report: A small number treated the sequence as arithmetic, while others inverted the ratio and consequently used a value for which an infinite sum would not converge.

   Question paper p.3 · Mark scheme p.6

3. Question 3

   4 marks · 12/75

   This assesses integration of fractional powers and the use of a point on a curve to determine the integration constant.

   Examiner report: Integration was generally recognised, but errors commonly arose from omitting a factor in the first antiderivative, leaving out the constant, or failing to state the resulting curve as y equal to an expression.

   Question paper p.4 · Mark scheme p.7

4. Question 4 (a)

   2 marks · 14/75

   This assesses the defining equal-difference property of an arithmetic progression.

   Examiner report: Most responses were successful, although some candidates mistakenly applied geometric-progression methods.

   Question paper p.5 · Mark scheme p.7

5. Question 4 (b)

   3 marks · 17/75

   This assesses calculation of a finite arithmetic-series sum using information obtained about the progression.

   Examiner report: The main reported difficulty was confusion between arithmetic and geometric progressions, mirroring the issue seen in the preceding part.

   Question paper p.5 · Mark scheme p.8

6. Question 5 (a)

   4 marks · 21/75

   This assesses the use of two specified intersection abscissae to determine two constants in the equations of a curve and a line.

   Examiner report: Many candidates recognised that the equations had to be equated but then used a repeated-root argument instead of the two supplied x-values, confused an x-coordinate with a y-coordinate, or produced only one equation for the two unknowns.

   Question paper p.6 · Mark scheme p.9

7. Question 5 (b)

   5 marks · 26/75

   This assesses the condition for tangency between a line and a quadratic curve when a parameter is present.

   Examiner report: Weaker solutions reused the earlier values despite the changed information. Although many formed the tangency condition correctly, final parameter values were often obtained from a calculator without showing an acceptable algebraic solution.

   Question paper p.7 · Mark scheme p.10

8. Question 6

   5 marks · 31/75

   This assesses finding curve-line intersections and evaluating the exact area between a square-root curve and a straight line.

   Examiner report: Candidates often identified integration but struggled with the square-root equation or confused solutions for the root of x with solutions for x itself. Limits were also sometimes omitted from the displayed substitution, and exact answers were unnecessarily rounded.

   Question paper p.8 · Mark scheme p.11

9. Question 7 (a)

   2 marks · 33/75

   This assesses conversion between arc length and central angle, followed by angle subtraction in a sector.

   Examiner report: Some candidates treated the arc length as though it were the chord AB, while others stopped at the intermediate angle or simplified the sum of angles incorrectly.

   Question paper p.10 · Mark scheme p.13

10. Question 7 (b)

    4 marks · 37/75

    This assesses the decomposition of a circular segment-like region into a sector and a right-triangle area.

    Examiner report: The sector calculation was usually familiar, but finding the triangular contribution caused more difficulty. A noticeable number also failed to apply the requested three-significant-figure presentation.

    Question paper p.11 · Mark scheme p.14

11. Question 8 (a)

    4 marks · 41/75

    This assesses substitution of points into a circle equation and extraction of the centre from the general circle form.

    Examiner report: Responses varied considerably: some used only one point or incorrectly assumed that the two points formed a diameter, rather than determining the coefficients from both conditions.

    Question paper p.12 · Mark scheme p.15

12. Question 8 (b)

    4 marks · 45/75

    This assesses finding a tangent to a circle by using the radius as a normal to the tangent.

    Examiner report: The usual radius-and-perpendicular method was widely attempted, though differentiation was sometimes mishandled and some final equations were not converted to integer coefficients.

    Question paper p.13 · Mark scheme p.16

13. Question 9 (a)

    3 marks · 48/75

    This assesses first and second differentiation of a composite expression involving a square root.

    Examiner report: This part was generally answered well, although algebraic simplification occasionally introduced errors after the differentiations had been performed correctly.

    Question paper p.14 · Mark scheme p.17

14. Question 9 (b)

    4 marks · 52/75

    This assesses locating a stationary point and classifying it with the second derivative.

    Examiner report: A minority set the second derivative equal to zero instead of the first, and some omitted the y-coordinate. Others evaluated the second derivative correctly but assigned the wrong classification.

    Question paper p.15 · Mark scheme p.17

15. Question 10 (a)

    1 mark · 53/75

    This assesses identifying the domain of an inverse function from the range of the original function.

    Examiner report: This was one of the least successfully attempted parts, with many candidates leaving it blank.

    Question paper p.16 · Mark scheme p.18

16. Question 10 (b)

    3 marks · 56/75

    This assesses algebraic inversion of a rational function.

    Examiner report: Candidates who reached this part were often comfortable with the standard rearrangement, although difficulty with the preceding domain question led some to abandon the whole question.

    Question paper p.16 · Mark scheme p.18

17. Question 10 (c)

    2 marks · 58/75

    This assesses evaluation of a composition involving an inverse function.

    Examiner report: Success was greater when candidates evaluated the inverse value before applying g; attempts that combined the functions prematurely or multiplied their expressions were less successful.

    Question paper p.16 · Mark scheme p.18

18. Question 10 (d)

    1 mark · 59/75

    This assesses the reason a quadratic function defined on all real numbers has no inverse function on that full domain.

    Examiner report: Several candidates omitted this part, while others blamed the issue on taking a square root of a negative number rather than recognising the failure of one-to-one behaviour.

    Question paper p.17 · Mark scheme p.18

19. Question 10 (e)

    6 marks · 65/75

    This assesses algebraic rearrangement, differentiation of the resulting rational form, construction of a tangent, and calculation of an intercept triangle.

    Examiner report: A substantial number stopped after the preliminary identity. Common errors among attempts included finding area under the curve instead of under the tangent and differentiating the original quotient unnecessarily rather than exploiting the rearranged form.

    Question paper p.17 · Mark scheme p.19

20. Question 11 (a)

    5 marks · 70/75

    This assesses reduction of a trigonometric quartic to a quadratic in cos²x and the listing of every solution in a specified interval.

    Examiner report: Many complete solutions were seen, but some candidates did not show enough work to justify solving the auxiliary quadratic or failed to include the full set of four angles.

    Question paper p.18 · Mark scheme p.20

21. Question 11 (b)

    5 marks · 75/75

    This assesses parameter-dependent use of the quadratic formula and comparison of the resulting roots with the possible values of cos²x.

    Examiner report: This was the most difficult part for many candidates. Frequent shortcomings were ignoring the instruction to retain the parameter, testing only one particular value above five, or writing the root formula without substituting the coefficients into it.

    Question paper p.19 · Mark scheme p.21