Standard +0.3 This is a straightforward calculus question requiring differentiation, factorization of a cubic, and finding stationary points. While it involves a quintic and requires showing uniqueness (which adds slight complexity beyond routine exercises), the algebraic manipulation is manageable and the overall approach is standard for AS-level, making it slightly easier than average.
Question 8:
8 | Selects differentiation as the first
step. At least one term correct | 1.1a | M1 | y = 2x5 + 5x4 + 10x3 – 8
𝑑𝑑𝑦𝑦 4 3 2
= 10𝑑𝑑 +20𝑑𝑑 +30𝑑𝑑
𝑑𝑑𝑑𝑑
= 0
2 2
10𝑑𝑑 (𝑑𝑑 +2 𝑑𝑑+3)
x = 0 or
2
𝑑𝑑 + 2𝑑𝑑+3 = 0
discriminant = 4 – 12
2
=𝑏𝑏 –8− 4𝑎𝑎𝑎𝑎 =
negative so no real solutions
Only stationary point at (0, –8)
Differentiates fully correctly | 1.1b | A1
Equates their derivative to zero | 1.1a | M1
States x = 0 is one solution
or
verifies x = 0 is a solution | 1.1b | A1
Deduces the quadratic factor has no
real roots using discriminant,
completing the square, using
formula
Or uses a sketch from their
calculator
Or finds roots of quartic but
discounts non-real roots (only real
root is x = 0) | 2.2a | M1
Deduces that there are no further
stationary points and concludes that
(0, –8) is the only one. | 2.1 | R1
Total | 6
Q | Marking Instructions | AO | Marks | Typical Solution
Prove that the curve with equation
$$y = 2x^5 + 5x^4 + 10x^3 - 8$$
has only one stationary point, stating its coordinates.
[6 marks]
\hfill \mbox{\textit{AQA AS Paper 1 2019 Q8 [6]}}