A-LevelMathematicsIntegration9709/12 · Oct/Nov 2022 · Q1111 Marks

Find the coordinates of the minimum point of the curve y = 9/4 x² – 12x + 18.

(a)Find the coordinates of the minimum point of the curve y = 9/4 x² – 12x + 18.[3]
(b)The diagram shows the curves with equations y = 9/4 x² – 12x + 18 and y = 18 – 3/8 x². The curves intersect at the points (0, 18) and (4, 6). Find the area of the shaded region.[5]
(c)A point P is moving along the curve y = 18 – 3/8 x² in such a way that the x-coordinate of P is increasing at a constant rate of 2 units per second. Find the rate at which the y-coordinate of P is changing when x = 4.[3]

This question uses a diagram from the original paper, which isn't shown here. Open the full paper from the past papers archive to see it.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2022). The mark or mark type is shown in brackets.

B1 OE Either dy/dx or a correct expression in completed square form. Allow unsimplified. dy/dx = 9/2 x - 12 [= 0]
B1 OE Condone 2.67 AWRT. x = 24/9 or 8/3
B1 CAO. y = 2. Note: x = -b/2a = -(-12)/(2*9/4) = 12/(9/2) = 24/9 = 8/3 B1; substitute 8/3 for x in y = B1; y=2 B1.
M1 Intention to integrate and subtract areas (either way around). Can be two separate functions or combined. Using y² scores 0/5 but condone inclusion of π except for the final mark. ∫( (18 - 3/8 x²) - (9/4 x² - 12x + 18) ) dx
B1,B1 One mark for correct integration of each curve, allow unsimplified. [18x - 3/8 * x³/3] or [9/4 * x³/3 - 12x²/2 + 18x]
B1,B1 Or combined [18x - 3/8 x³/3 - (9/4 x³/3 - 12x²/2 + 18x)] BUT condone sign errors that are only due to missing brackets. [18x - 1/8 x³ - (3/4 x³ - 6x² + 18x)]
M1 Clear substitution of 4 into at least one integrated expression (defined by at least one correct power) which can be unsimplified. [ (18(4) - 1/8(4)³) - (3/4(4)³ - 6(4)² + 18(4)) ] - [0]
A1 SC: If all marks awarded except the final M1, SCB1 is available for the correct final answer. 240/7 or 34.3 AWRT
B1 Allow unsimplified. dy/dx = -3/8 * 2x = -3/4 x
M1 Substitute x=4 into their dy/dx and multiply by 2. dy/dt = dy/dx * dx/dt = (-3/4 * 4) * 2
A1 Accept decreasing [at/by] 15. -15
Note: If incorrect curve used, this is not a MR and only M1 mark is available. Expect (9/2 (4) - 12) * 2 [= 12]

On Oracle Prep you can write or photograph your own answer and have it marked by AI against this mark scheme, with feedback on what you missed.