9709 Pure Mathematics 1 Feb/Mar 2025 Paper 12

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

1. Question 1

   4 marks · 4/75

   This assesses using the discriminant to determine when a quadratic equation has no real solutions.

   Examiner report: This was generally a good opening question, but some candidates found the critical values without stating the required interval.

   Question paper p.2 · Mark scheme p.9

2. Question 2 (a)

   2 marks · 6/75

   This assesses differentiating a rational-power expression and evaluating the derivative at a specified point to obtain a gradient.

   Examiner report: Most candidates differentiated correctly and then used the given x-coordinate successfully; this usually led to full credit.

   Question paper p.3 · Mark scheme p.9

3. Question 2 (b)

   3 marks · 9/75

   This assesses finding a stationary point by solving a derivative equation and then evaluating the corresponding coordinate numerically.

   Examiner report: A common mistake was to differentiate the already obtained derivative instead of setting it to zero, while early decimal rounding also caused inaccurate coordinates.

   Question paper p.3 · Mark scheme p.10

4. Question 3 (a)

   4 marks · 13/75

   This assesses applying the binomial theorem to expand a fourth power involving positive and negative powers of x.

   Examiner report: Most responses were correct, but some lost signs or applied the fourth power incorrectly; multiplying four separate brackets was generally an inefficient and unsuccessful approach.

   Question paper p.4 · Mark scheme p.10

5. Question 3 (b)

   2 marks · 15/75

   This assesses extracting a particular coefficient from a product of algebraic expressions using a preceding expansion.

   Examiner report: Candidates who had a usable expansion usually selected the necessary terms correctly and gained the available marks.

   Question paper p.4 · Mark scheme p.11

6. Question 4 (a)

   3 marks · 18/75

   This assesses finding a perimeter involving a circular arc and the side of an isosceles triangle, while working in radians.

   Examiner report: The main successful approaches used the arc-length formula in radians together with the cosine rule or a half-angle triangle calculation; converting to degrees often reduced accuracy.

   Question paper p.5 · Mark scheme p.11

7. Question 4 (b)

   3 marks · 21/75

   This assesses finding a composite area by subtracting a sector from a triangle.

   Examiner report: Many candidates used the intended triangle-minus-sector method effectively, whereas degree-based calculations sometimes introduced rounding errors.

   Question paper p.5 · Mark scheme p.11

8. Question 5

   6 marks · 27/75

   This assesses identifying the terms of an arithmetic progression lying strictly within given bounds and summing them.

   Examiner report: A frequent misreading was to calculate sums labelled by the two boundary numbers, even though those numbers were not term positions; finding the actual first and last included terms gave useful progress.

   Question paper p.6 · Mark scheme p.12

9. Question 6 (a)

   1 mark · 28/75

   This assesses translating minimum distances from the coordinate axes into the centre coordinates of a circle in the first quadrant.

   Examiner report: Many candidates found the correct two expressions but placed them in the wrong coordinate order.

   Question paper p.7 · Mark scheme p.13

10. Question 6 (b)

    3 marks · 31/75

    This assesses using the distance formula to determine a circle radius from the distance of its centre from the origin.

    Examiner report: The Pythagorean setup was usually understood, but some calculator-based solutions did not show enough algebra; nearly all candidates correctly recognised that a negative radius was inadmissible.

    Question paper p.7 · Mark scheme p.13

11. Question 6 (c)

    2 marks · 33/75

    This assesses finding a tangent gradient from the perpendicular relationship between a radius and its tangent.

    Examiner report: Most successful solutions used the fact that the tangent is perpendicular to the radius, although some did not make the geometric link explicit.

    Question paper p.7 · Mark scheme p.13

12. Question 7 (a)

    3 marks · 36/75

    Question paper p.8 · Mark scheme p.14

13. Question 7 (b)

    4 marks · 40/75

    This assesses solving a trigonometric equation by applying the identity, treating it as a quadratic in sin squared, and selecting every angle in the specified interval.

    Examiner report: Many candidates found the admissible sine-squared value but overlooked the negative-sine possibility, so they reported too few angles.

    Question paper p.9 · Mark scheme p.14

14. Question 8 (a)

    4 marks · 44/75

    This assesses determining the common ratio of a convergent geometric progression from its second term and infinite sum.

    Examiner report: The equations were generally set up correctly, but a substantial number of candidates rejected the valid negative ratio merely because it was negative rather than checking its magnitude.

    Question paper p.10 · Mark scheme p.15

15. Question 8 (b)

    3 marks · 47/75

    This assesses finding the infinite sum remaining after removing a finite number of initial terms from a geometric progression.

    Examiner report: Both subtracting the first nine-term sum and restarting at the tenth term were valid approaches; missing negative signs and rounding an exact result cost marks.

    Question paper p.10 · Mark scheme p.16

16. Question 9 (a)

    2 marks · 49/75

    This assesses classifying a stationary point using the sign of the second derivative at its given coordinate.

    Examiner report: Most candidates evaluated the second derivative correctly, but some stated the classification without demonstrating why its sign justified that conclusion.

    Question paper p.11 · Mark scheme p.17

17. Question 9 (b)

    7 marks · 56/75

    This assesses integrating a second derivative twice and using a stationary point and a point on the curve to determine both constants.

    Examiner report: Successful responses used the stationary condition after the first integration and the coordinate condition after the second; a common error was treating the given y-coordinate as the first integration constant or omitting a constant altogether.

    Question paper p.11 · Mark scheme p.18

18. Question 10 (a)

    4 marks · 60/75

    This assesses finding the area under a curve involving a square root by definite integration between the intercepts.

    Examiner report: The main algebraic issue was losing the factor arising from integrating a function of 3x plus 4; some candidates also failed to evaluate the non-zero contribution at the lower limit.

    Question paper p.12 · Mark scheme p.19

19. Question 10 (b)

    5 marks · 65/75

    Question paper p.13 · Mark scheme p.20

20. Question 11 (a)

    4 marks · 69/75

    This assesses using an inverse linear function and a given condition to establish a relation between constants before evaluating a composite function.

    Examiner report: The composite-function idea was widely understood, but slips in the constant algebra and insufficient intermediate working were common in this show-that part.

    Question paper p.14 · Mark scheme p.21

21. Question 11 (b)

    3 marks · 72/75

    This assesses applying a translation, vertical stretch, and reflection to the equation of a parabola in the stated order.

    Examiner report: Writing the transformations separately was the safest approach; applying them in an unclear order made it difficult to receive credit for the full sequence.

    Question paper p.15 · Mark scheme p.22

22. Question 11 (c)

    3 marks · 75/75

    This assesses using the maximum value of a transformed quadratic together with a previously obtained relation to determine positive constants.

    Examiner report: This was the most demanding part for many candidates; recognising that the squared term is non-positive makes the maximum immediate, while less suitable forms required more error-prone algebra.

    Question paper p.15 · Mark scheme p.22