AQA A-Level Mathematics Paper 1, June 2025: Question 11

Mark scheme

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Q Marking instructions AO Marks Typical solution

11(a) Uses implicit differentiation with x y2 + 4 y3 = 8x

2 dy 2 dy 1.1a M1 dy dy

x or Ay seen xy + x2 + y2 =

dx dx 2 12 8

dx dx

Uses the product rule to obtain

dy

2 dy 3.1a M1 = 0

Bxy + x dx

dx

2xy = 8

Obtains

2 dy 2 dy 4

2xy + x +12y = 8 1.1b A1 y =

dx dx x

OE

Completes a reasoned

dy

argument using = 0 and

dx

2 dy 2 dy

2xy + x +12y = 8 OE

dx dx 2.1 R1

with at least one correct line of

intermediate working before

showing that y =

x

Subtotal 4

11(b) 3 3

24 4 2 4 4

Obtains x × + 4 = 8x OE 3.1a B1 x × + 4 = 8x

x x x x

Rearranges their equation to 256

4 −4 2.1 M1 4x + = 8x

obtain ax = k or ax = k OE x3

Deduces x = ± 8 x4 = 64

CAO 2.2a R1

x = ± 8

Subtotal 3

Question 11 Total 7

How to answer it

Implicit Differentiation & Stationary Points

What this question tests

Implicit differentiation: Differentiating terms involving products such as x²y using the product rule.
Chain rule for y-terms: Differentiating powers of y with respect to x (e.g. d/dx(4y³) = 12y² dy/dx).
Stationary points: Applying the condition dy/dx = 0 to establish a relationship between x and y.
Simultaneous non-linear equations: Substituting the derived relationship back into the original curve equation to solve for x in exact surd form.

Question 11 (a)

Implicit Differentiation & Condition for Stationary Points [4 Marks]

📐 Step-by-Step Solution

Step 1: Given equation:
x²y + 4y³ = 8x

Step 2: Differentiate each term with respect to x:

For x²y , apply the product rule:
d/dx(x²) · y + x² · d/dx(y) = 2xy + x²(dy/dx)
For 4y³ , apply the chain rule:
12y²(dy/dx)
For 8x :
8

This gives the differentiated equation:
2xy + x²(dy/dx) + 12y²(dy/dx) = 8

Step 3: Apply the condition for stationary points ( dy/dx = 0 ):

2xy + x²(0) + 12y²(0) = 8
2xy = 8

Step 4: Rearrange to make y the subject:

y = 8 / (2x) ⇒ y = 4 / x (as required)

🧠 Exam Technique & Insight

Why simplify early?

Many students spend unnecessary time rearranging the differentiated equation into the form dy/dx = (8 - 2xy) / (x² + 12y²) before setting dy/dx = 0 .

While mathematically correct, it is much faster and less prone to algebraic slips to substitute dy/dx = 0 directly into the equation as soon as you differentiate!

Mark Breakdown:
• M1 (1.1a): Evidence of implicit differentiation with x²(dy/dx) or Ay²(dy/dx) .
• M1 (3.1a): Correct product rule form Bxy + x²(dy/dx) .
• A1 (1.1b): Completely correct differentiation: 2xy + x²(dy/dx) + 12y²(dy/dx) = 8 .
• R1 (2.1): Clear reasoned argument setting dy/dx = 0 with at least one line of intermediate working (e.g. 2xy = 8 ) leading to y = 4/x .

💡 Key Knowledge

Product Rule: If u = x² and v = y , then d(uv)/dx = u'v + uv' = 2xy + x²(dy/dx) .
Chain Rule for implicit functions: Differentiating g(y) with respect to x gives g'(y) · (dy/dx) .
"Show that" requirements: You must state dy/dx = 0 clearly. Jumping straight from the differentiated expression to the final result will forfeit the final reasoning mark (R1).

❌ Common Errors

Differentiating x²y as 2x(dy/dx): Forgetting the product rule is the single most common mistake in implicit questions.
Omitting dy/dx on y³: Writing the derivative of 4y³ simply as 12y² without multiplying by dy/dx .
Differentiating the RHS incorrectly: Forgetting to differentiate the 8x term (leaving it as 8x instead of 8 ) or turning it into 0.

Question 11 (b)

Finding Stationary Coordinates in Form ±√n [3 Marks]

📐 Step-by-Step Solution

Step 1: Substitute y = 4 / x back into the original curve equation x²y + 4y³ = 8x :

x²(4/x) + 4(4/x)³ = 8x

Step 2: Simplify each term:

4x + 4(64 / x³) = 8x
4x + 256 / x³ = 8x

Step 3: Collect like terms in x:

256 / x³ = 4x

Step 4: Rearrange to obtain a single power of x:

Multiply both sides by x³ :
256 = 4x⁴
x⁴ = 64

Step 5: Solve for x in the form ±√n :

Take the square root of both sides:
x² = √64 = 8 (since x² > 0)
x = ±√8 (where integer n = 8 )

✅ Final Answer

The x-coordinates of the stationary points are:

x = ±√8

Here, the integer is n = 8 .

Mark Breakdown:
• B1 (3.1a): Correct substitution of y = 4/x into the curve equation.
• M1 (2.1): Algebraic manipulation to reach ax⁴ = k (e.g. x⁴ = 64 or 4x⁴ = 256 ).
• R1 (2.2a): Completely correct deduction of x = ±√8 (must include the ± sign).

❌ Common Errors & Pitfalls

Missing the negative root (±): Writing only x = √8 loses the final R1 mark. The question explicitly notes there are two stationary points and specifies the form ±√n .
Simplifying the surd: Writing x = ±2√2 . While mathematically equivalent, the question demands the exact form ±√n where n is an integer. Leave it as ±√8 !
Cubing errors: Calculating 4(4/x)³ incorrectly (e.g. forgetting to cube the 4 inside the brackets, leading to 16/x³ instead of 256/x³ ).

🧠 "Hence" Command Word

The word "Hence" means you must use the result you proved in part (a). If you attempt to find stationary points by another method, you will score 0 marks for this part.

Always double-check that your final answer directly matches the target format: ±√n .

Topics

Pure Mathematics · G: Differentiation