9709 Pure Mathematics 1 May/June 2023 Paper 13

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

1. Question 1

   4 marks · 4/75

   Interpreting corresponding features of two graphs to identify a horizontal stretch and a translation.

   Examiner report: This newer syllabus area was handled confidently by many candidates, but descriptions were sometimes incomplete. Common faults included reversing the stretch direction or giving the wrong magnitude, especially when the translation was described without a clear vector or equivalent wording.

   Question paper p.2 · Mark scheme p.6

2. Question 2

   4 marks · 8/75

   Using the minimum of a quadratic to determine when a strict inequality holds for every real input.

   Examiner report: Both completing the square and calculus gave reliable routes, while the discriminant method also worked when its inequality was set up correctly. Some candidates used the wrong discriminant direction and therefore did not establish the required condition.

   Question paper p.3 · Mark scheme p.7

3. Question 3 (a)

   2 marks · 10/75

   Applying the binomial theorem to expand a power containing both positive and negative powers of the variable.

   Examiner report: This was one of the strongest parts of the paper. Most candidates used the binomial pattern successfully, with only occasional slips in coefficients or exponents.

   Question paper p.4 · Mark scheme p.8

4. Question 3 (b)

   4 marks · 14/75

   Extracting selected powers from a product of expansions and using their coefficients to form simultaneous equations.

   Examiner report: Candidates who had expanded the preceding expression generally progressed smoothly by selecting the relevant powers. The resulting pair of equations was usually solved correctly once the coefficient contributions had been identified.

   Question paper p.4 · Mark scheme p.8

5. Question 4 (a)

   3 marks · 17/75

   Converting a trigonometric equation into a polynomial equation in cosine squared.

   Examiner report: Several valid algebraic routes appeared, with the most common using the tangent identity followed by elimination of sine. Incorrect brackets and misplaced powers were the main sources of otherwise avoidable errors.

   Question paper p.5 · Mark scheme p.8

6. Question 4 (b)

   4 marks · 21/75

   Solving a trigonometric equation by treating cosine squared as a quadratic and selecting angles in a stated interval.

   Examiner report: Most successful solutions reduced the problem to a quadratic in cosine squared and then considered both signs of cosine. A recurring weakness was writing down a value for cosine squared without showing how it had been obtained.

   Question paper p.5 · Mark scheme p.9

7. Question 5 (a)

   4 marks · 25/75

   Finding intersections of a straight line and a circle by substitution and solving a quadratic.

   Examiner report: Substitution into the circle was generally understood. However, some candidates gave calculator-generated roots without displaying a factorisation, formula substitution, or another acceptable method for solving the quadratic.

   Question paper p.6 · Mark scheme p.9

8. Question 5 (b)

   3 marks · 28/75

   Constructing the equation of the circle whose diameter is the chord joining two known intersection points.

   Examiner report: Most candidates recognised the midpoint as the new centre and used half the chord for the radius. Sign mistakes in the centre were frequent, and the final radius-squared expression was sometimes left unnecessarily unsimplified.

   Question paper p.6 · Mark scheme p.10

9. Question 6 (a)

   5 marks · 33/75

   Using arc length and sector area to determine sector dimensions, then finding the area of a circular segment.

   Examiner report: The sector formulae were used correctly in many scripts, and the sine rule for the triangle often gave a compact calculation. Solutions based on a base-height triangle were more prone to rounding loss, and final numerical accuracy was not always maintained.

   Question paper p.7 · Mark scheme p.10

10. Question 6 (b)

    2 marks · 35/75

    Finding a chord length from the radius and central angle and combining it with the arc to obtain a boundary length.

    Examiner report: A range of sound methods was seen for the chord, and this part generally had fewer rounding problems than the area calculation. In some cases the chord found earlier was correctly reused rather than recalculated.

    Question paper p.7 · Mark scheme p.10

11. Question 7 (a)

    1 mark · 36/75

    Determining the range of a restricted rational function.

    Examiner report: Only about half of candidates expressed the output restriction correctly. Confusing the input restriction with the range, or including the unattainable limiting value, caused many losses.

    Question paper p.8 · Mark scheme p.11

12. Question 7 (b)

    4 marks · 40/75

    Finding an inverse function algebraically and identifying its domain from the original function's range.

    Examiner report: Changing the subject of the formula was the usual and generally effective approach. Algebraic sign errors were uncommon, and many candidates correctly transferred the earlier range restriction to the inverse domain.

    Question paper p.8 · Mark scheme p.11

13. Question 7 (c)

    3 marks · 43/75

    Forming and simplifying a composite of two functions.

    Examiner report: This was answered successfully by nearly all candidates. The composition order and fraction manipulation were usually handled correctly, with very few cases of interchanging the two functions.

    Question paper p.9 · Mark scheme p.11

14. Question 8 (a)

    5 marks · 48/75

    Using the first two terms of a geometric progression and its infinite-sum formula to determine the first term.

    Examiner report: The relationship between consecutive geometric terms was generally clear, and substitution into the infinite-sum formula often led to a correct polynomial. Full credit depended on showing how that polynomial was solved and rejecting the unsuitable root.

    Question paper p.10 · Mark scheme p.12

15. Question 8 (b)

    5 marks · 53/75

    Using an arithmetic-series inequality to find the least integer number of terms meeting a specified bound.

    Examiner report: Most candidates found the common difference and set up the arithmetic sum appropriately, although many replaced the inequality by an equality. The main missing step was often the actual solution of the quadratic and the justification of the first valid integer.

    Question paper p.11 · Mark scheme p.12

16. Question 9 (a)

    4 marks · 57/75

    Integrating a derivative and using a point on the curve to determine the integration constant.

    Examiner report: Integration and determination of the constant were handled well overall. Errors with the negative power or the constant term were relatively infrequent.

    Question paper p.12 · Mark scheme p.13

17. Question 9 (b)

    4 marks · 61/75

    Using a tangent equation and an intersection condition to derive an algebraic equation for a second point.

    Examiner report: Stronger solutions explicitly justified the tangent gradient before writing its equation. Most candidates then equated the line and curve, but some missed a short algebraic route to the requested factorised condition.

    Question paper p.13 · Mark scheme p.13

18. Question 9 (c)

    2 marks · 63/75

    Checking a proposed root of an intersection equation and using it to calculate the corresponding ordinate.

    Examiner report: Calculations were more convincing when both factors in the left-hand side were evaluated rather than when the result was asserted. The tangent provided the simplest way to obtain the second coordinate once the abscissa had been accepted.

    Question paper p.13 · Mark scheme p.13

19. Question 10 (a)

    4 marks · 67/75

    Finding and identifying the maximum point of a curve by differentiation.

    Examiner report: The stationary-point procedure was familiar to most candidates. Sign errors in differentiation and mistakes when squaring were the main difficulties; the ordinate also had to come from the original curve rather than from an incomplete intermediate equation.

    Question paper p.14 · Mark scheme p.14

20. Question 10 (b)

    3 marks · 70/75

    Verifying perpendicularity between a chord and the curve's tangent at a specified point.

    Examiner report: The shortest successful method compared the two gradients and checked their product. Some candidates predicted one gradient from perpendicularity and then verified the other; this was acceptable when the calculation was clearly explained.

    Question paper p.15 · Mark scheme p.14

21. Question 10 (c)

    5 marks · 75/75

    Finding the area between a curve and its normal line over the interval determined by the two points.

    Examiner report: Both the direct difference integral and the curve-minus-trapezium approach were used effectively. Calculator-only definite integrals did not demonstrate the required method; full credit required visible algebra and substitution of the endpoints.

    Question paper p.15 · Mark scheme p.15