9709 Pure Mathematics 1 Feb/Mar 2024 Paper 12

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

1. Question 1

   3 marks · 3/75

   Evaluating an improper definite integral involving a negative power of the variable.

   Examiner report: Most candidates could find an antiderivative, but errors arose from reversing the limits, mishandling the behaviour at infinity, and losing a minus sign.

   Question paper p.2 · Mark scheme p.6

2. Question 2 (a)

   1 mark · 4/75

   Question paper p.3 · Mark scheme p.6

3. Question 2 (b)

   3 marks · 7/75

   Applying successive translations and reflections to a trigonometric graph, including tracking a corresponding point.

   Examiner report: A substantial number gave only a transformed equation and forgot the point. Other common mistakes treated the translation as horizontal or reflected in the wrong axis.

   Question paper p.3 · Mark scheme p.6

4. Question 3

   5 marks · 12/75

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

   Examiner report: The calculus method was generally secure; mistakes were mainly numerical, especially when finding the constant or carrying out the final substitution.

   Question paper p.4 · Mark scheme p.7

5. Question 4 (a)

   3 marks · 15/75

   Proving a trigonometric identity by expansion and use of the Pythagorean identity.

   Examiner report: Many candidates had the right overall idea, but incomplete expansion, sign errors, poor fraction notation, and cancellation mistakes prevented some proofs from being complete.

   Question paper p.5 · Mark scheme p.7

6. Question 4 (b)

   3 marks · 18/75

   Solving a trigonometric equation after reducing it to an equation in tangent, within a specified degree interval.

   Examiner report: Using the preceding identity was common, but calculator-only solutions did not show enough method. Several candidates lost a solution by overlooking the negative tangent case.

   Question paper p.5 · Mark scheme p.8

7. Question 5

   6 marks · 24/75

   Finding the equation of a normal to a rational curve using differentiation and the reciprocal-gradient relationship.

   Examiner report: The procedure was well understood and chain-rule differentiation was often used successfully. Errors usually involved evaluating the gradient at the given x-value or rearranging the final line.

   Question paper p.6 · Mark scheme p.8

8. Question 6

   5 marks · 29/75

   Extracting a specified coefficient from a product using a binomial expansion.

   Examiner report: This was generally answered well. Most efficient solutions used only the relevant terms; a few candidates selected an inappropriate root when solving the resulting cubic relation.

   Question paper p.7 · Mark scheme p.9

9. Question 7 (a)

   4 marks · 33/75

   Using the repeated-intersection condition to find the parameter for which a line is tangent to a quadratic curve.

   Examiner report: Elimination was the normal successful route. Attempts based on comparing gradients were uncommon, while algebraic and sign slips in the discriminant calculation caused avoidable losses.

   Question paper p.8 · Mark scheme p.9

10. Question 7 (b)

    2 marks · 35/75

    Finding the unique intersection point after determining the parameter of a tangent line.

    Examiner report: Most candidates understood that the parameter from part (a) had to be reused. Where the quadratic solution was merely asserted, the method mark was often unavailable.

    Question paper p.8 · Mark scheme p.9

11. Question 8 (a)

    4 marks · 39/75

    Determining an arithmetic progression's common difference and summing a finite number of terms.

    Examiner report: This part was handled confidently by most candidates. The main errors involved using the wrong term number in the sum formula or making a small arithmetic slip.

    Question paper p.9 · Mark scheme p.10

12. Question 8 (b)

    4 marks · 43/75

    Finding infinite sums for a geometric progression and for its even-indexed subsequence.

    Examiner report: The original infinite sum was usually found correctly. Some candidates treated the even terms as though they began with the original first term or simply divided the total by two.

    Question paper p.9 · Mark scheme p.10

13. Question 9 (a)

    4 marks · 47/75

    Using composition and the minimum of a quadratic expression to determine a parameter from a function range.

    Examiner report: The composite was usually formed correctly, though the parameter was sometimes not multiplied through. Recognising the square's minimum gave a much shorter route than using a discriminant, and some candidates gave an interval of possible constants instead of one value.

    Question paper p.10 · Mark scheme p.11

14. Question 9 (b)

    2 marks · 49/75

    Finding the range of a composite function by analysing a completed-square-type expression.

    Examiner report: The composition was often obtained, but many solutions did not exploit the immediate minimum supplied by the squared term and consequently failed to state the range correctly.

    Question paper p.10 · Mark scheme p.11

15. Question 9 (c)

    3 marks · 52/75

    Finding the inverse of a linear function and using it to recover a function from a composition.

    Examiner report: This was among the better-answered parts. Most candidates could invert the linear rule and then obtain the unknown function, sometimes by direct inspection rather than formal composition.

    Question paper p.11 · Mark scheme p.11

16. Question 10 (a)

    3 marks · 55/75

    Finding a tangent line to a circle from the perpendicular radius at a specified point.

    Examiner report: The radius-and-perpendicular method was most common and generally successful. A minority first found the circle equation and differentiated it, which was also valid when carried out correctly.

    Question paper p.12 · Mark scheme p.12

17. Question 10 (b)

    2 marks · 57/75

    Writing a circle equation from its centre and radius and expanding it into the requested general form.

    Examiner report: This was well understood, with many correct equations. Both the centre-radius approach and the standard general-circle form were accepted when used consistently.

    Question paper p.12 · Mark scheme p.12

18. Question 10 (c)

    3 marks · 60/75

    Finding a central angle in radians from a circle's intersections with the y-axis.

    Examiner report: Several valid approaches were seen, although many candidates used a longer route through the intersection coordinates. Giving the final angle in degrees rather than radians prevented full credit, as did insufficient numerical accuracy.

    Question paper p.13 · Mark scheme p.12

19. Question 10 (d)

    4 marks · 64/75

    Calculating the perimeter and area of a circular segment from its radius, chord, and central angle.

    Examiner report: Candidates who had already found the angle generally used the arc and sector formulae effectively. The main issue was confusing the sector with the segment or using a half-angle without making the necessary adjustment.

    Question paper p.13 · Mark scheme p.13

20. Question 11 (a)

    4 marks · 68/75

    Locating the minimum point of a curve involving negative fractional powers.

    Examiner report: Both calculus and quadratic approaches were used. Handling the negative fractional indices was difficult for some, and a clear derivation was needed because an unsupported result could arise from an invalid cubing step.

    Question paper p.14 · Mark scheme p.13

21. Question 11 (b)

    7 marks · 75/75

    Finding the area between a power-function curve and the x-axis by locating its intercepts and integrating.

    Examiner report: Successful solutions usually factored the transformed quadratic rather than relying on a calculator. Some candidates found the integral but left a signed result, without converting it into the positive area of the enclosed region.

    Question paper p.15 · Mark scheme p.14