9709 Pure Mathematics 1 Oct/Nov 2023 Paper 12

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

1. Question 1

   4 marks · 4/75

   This assesses extraction of selected binomial coefficients and use of a coefficient relationship to determine a constant.

   Examiner report: Most candidates handled this successfully, but some included powers of x in the coefficient equation or applied the multiplier to the wrong coefficient.

   Question paper p.2 · Mark scheme p.6

2. Question 2

   2 marks · 6/75

   This assesses exact manipulation of inverse trigonometric expressions in radians.

   Examiner report: A frequent misunderstanding was treating inverse cosine as a reciprocal. Other errors came from mixing degree and radian modes, combining the angles incorrectly, losing a minus sign, or giving a decimal instead of an exact result.

   Question paper p.3 · Mark scheme p.6

3. Question 3 (a)

   2 marks · 8/75

   This assesses finding a normal line from a derivative value at a specified point.

   Examiner report: Some weaker responses used the integration work intended for the next part or took the negative reciprocal of the wrong quantity when finding the normal gradient.

   Question paper p.4 · Mark scheme p.7

4. Question 3 (b)

   4 marks · 12/75

   This assesses integrating a derivative and using a point condition to determine the integration constant.

   Examiner report: Most candidates recognised that integration was needed; the main slips involved the sign of the term with a negative power and failing to simplify its coefficient fully.

   Question paper p.4 · Mark scheme p.7

5. Question 4 (a)

   2 marks · 14/75

   This assesses the use of radian arc-length formulae in a composite geometric boundary.

   Examiner report: Most candidates identified the relevant angle and applied the arc formula, although some inserted a degree measure into a formula that requires radians.

   Question paper p.5 · Mark scheme p.7

6. Question 4 (b)

   4 marks · 18/75

   This assesses finding an area made from circular sectors, circular segments, and an equilateral triangle.

   Examiner report: Successful candidates used a valid sector-and-triangle decomposition. Less secure attempts often selected an incorrect combination of sectors, segments, and triangle areas, or used degrees in the sector formula.

   Question paper p.5 · Mark scheme p.8

7. Question 5 (a)

   3 marks · 21/75

   This assesses using the common-ratio property of a geometric progression together with a trigonometric identity.

   Examiner report: The direct relation between the three terms was generally the most effective route. Some responses reported the angle in degrees, while a smaller number used arithmetic-progression methods instead.

   Question paper p.6 · Mark scheme p.8

8. Question 5 (b)

   3 marks · 24/75

   This assesses summing a finite geometric progression after determining its first term and ratio.

   Examiner report: Most candidates used the sum formula correctly, but some used a formula for an individual term, took the number of terms to be nine, or stopped before reaching the requested form; this part was also omitted by a noticeable minority.

   Question paper p.7 · Mark scheme p.9

9. Question 6 (a)

   2 marks · 26/75

   This assesses locating the vertex of a quadratic function.

   Examiner report: This was usually answered well. Common weaker responses set the quadratic itself equal to zero or supplied only the horizontal coordinate.

   Question paper p.8 · Mark scheme p.10

10. Question 6 (b)

    2 marks · 28/75

    This assesses tracking the coordinates of a vertex under a vertical stretch followed by a translation.

    Examiner report: Many candidates tried to derive a whole transformed equation instead of moving the point. Confusion about the order and effects of the two transformations was common; working in two stages tended to avoid these errors.

    Question paper p.8 · Mark scheme p.10

11. Question 6 (c)

    4 marks · 32/75

    This assesses forming and expanding the equation produced by combined graph transformations.

    Examiner report: A substantial number of candidates left this part unanswered, and some gave only an expression rather than the requested equation.

    Question paper p.9 · Mark scheme p.11

12. Question 7 (a)

    1 mark · 33/75

    This assesses verification of an algebraic identity by direct expansion and collection of terms.

    Examiner report: The main issue was that some candidates did not show all the products and instead checked the relation at one or two numerical values, which does not verify an identity.

    Question paper p.10 · Mark scheme p.11

13. Question 7 (b)

    3 marks · 36/75

    This assesses proving a trigonometric identity by rewriting tangent in terms of sine and cosine and simplifying an algebraic fraction.

    Examiner report: Most candidates began appropriately, but many lost marks while manipulating the resulting algebraic fraction, particularly when transforming the denominator.

    Question paper p.10 · Mark scheme p.12

14. Question 7 (c)

    5 marks · 41/75

    This assesses solving a trigonometric equation by combining a proved identity with algebraic factorisation over a specified interval.

    Examiner report: Although many used the identity from the preceding part, most did not exploit the earlier factorisation and consequently struggled with the cubic or relied on an unsupported calculator result.

    Question paper p.11 · Mark scheme p.14

15. Question 8 (a)

    3 marks · 44/75

    This assesses finding the inverse of a restricted quadratic function and selecting the appropriate branch.

    Examiner report: Keeping the original completed-square structure helped candidates make progress. The most frequent mistake was retaining both square-root branches instead of selecting the one allowed by the restriction; expansion also caused difficulties for weaker candidates.

    Question paper p.12 · Mark scheme p.15

16. Question 8 (b)(i)

    1 mark · 45/75

    Question paper p.12 · Mark scheme p.15

17. Question 8 (b)(ii)

    1 mark · 46/75

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

    Examiner report: Candidates commonly confused the roles of domain and range, despite the inverse relationship between them.

    Question paper p.12 · Mark scheme p.15

18. Question 8 (c)

    3 marks · 49/75

    This assesses evaluating a composite function, solving the resulting quadratic, and enforcing a domain restriction.

    Examiner report: Most candidates understood how to compose the functions, but many used a calculator without showing the quadratic solution. Only the strongest responses consistently rejected the root that violated the domain restriction.

    Question paper p.13 · Mark scheme p.16

19. Question 9 (a)

    4 marks · 53/75

    This assesses finding intersections of curves containing positive and negative half-powers of x.

    Examiner report: Many candidates struggled to manipulate the negative half-power correctly, sometimes confusing it with the positive half-power or with a square of y. Writing the negative power as a reciprocal generally made the algebra more manageable.

    Question paper p.14 · Mark scheme p.17

20. Question 9 (b)

    5 marks · 58/75

    This assesses finding the area between two intersecting curves using definite integration.

    Examiner report: A considerable number did not attempt this after failing to find the intersections. Those who integrated often earned method credit, though errors with fractional powers and mishandling the coefficient of the x-term were seen.

    Question paper p.15 · Mark scheme p.18

21. Question 10 (a)

    6 marks · 64/75

    This assesses locating stationary points of a composite power function and classifying their nature.

    Examiner report: Differentiation and the chain rule were generally handled well, but many candidates found only one stationary value despite the wording requiring more than one. The second-derivative classification was less reliable.

    Question paper p.16 · Mark scheme p.20

22. Question 10 (b)

    1 mark · 65/75

    This assesses identifying intervals of increase from the stationary-point analysis.

    Examiner report: Many candidates omitted this one-mark response or attempted an unnecessary inequality. Some did not recognise that the earlier stationary-point information already supplied the required conclusion.

    Question paper p.17 · Mark scheme p.21

23. Question 11 (a)

    4 marks · 69/75

    This assesses using perpendicular gradients or a scalar product to determine an unknown coordinate parameter.

    Examiner report: A sketch often helped. Multiplying the two gradients was usually more successful than manipulating a negative reciprocal, while weaker attempts frequently confused multiplication and addition of algebraic fractions.

    Question paper p.18 · Mark scheme p.22

24. Question 11 (b)

    3 marks · 72/75

    This assesses finding the equation of a circle through three points when a right angle identifies its diameter.

    Examiner report: There was considerable uncertainty about which segment formed the diameter, and some candidates incorrectly chose a given point or the midpoint of another side as the centre. A diagram would often have clarified the geometry.

    Question paper p.19 · Mark scheme p.23

25. Question 11 (c)

    3 marks · 75/75

    This assesses finding a tangent line to a circle at a specified point.

    Examiner report: Some candidates used the gradient of the wrong chord, particularly the line joining the other two relevant points. Sketching the circle and identifying the radius direction would have reduced this confusion.

    Question paper p.19 · Mark scheme p.23