AQA AS Level Mathematics Paper 2, June 2025: Question 11

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Q Marking instructions AO Marks Typical solution

11 Obtains (2k − 1)2 − 4 1( )(4 − 2k ) 3.1a B1 (2k −1)2 − 4 1( )(4 − 2k ) > 0

Compares their b2 − 4ac with 0 1.1a M1 k 2 − k + − + k >

44 1 16 8 0

and obtains two critical values 2

4k + 4k −15 > 0

53 1.1b A1

Obtains − , 5 3

22 k = − ,k =

53 1.1b A1 2 2

Obtains k < − k > 5 3

22 k < − or k >

ACF 2 2

Question 11 Total 4

How to answer it

Finding the Range of Values of a Parameter Using the Discriminant

📌 What this question tests

This question evaluates your ability to apply the quadratic discriminant to deduce properties about the number of real roots, and solve the resulting quadratic inequality:

Identifying coefficients a , b , and c from a non-trivial quadratic in x .
Applying the condition for two distinct real roots: b² − 4ac > 0 .
Expanding algebraic expressions and collecting terms accurately without sign errors.
Finding critical values and correctly identifying outside regions for a strict inequality ( > 0 ).

Question 11 Walkthrough

The equation x² + (2k − 1)x + 4 − 2k = 0 has two distinct real roots. Find the range of possible values for k . [4 marks]

📐 Step-by-Step Calculation

Step 1: Identify coefficients and form the discriminant
For ax² + bx + c = 0 :
• a = 1
• b = 2k − 1
• c = 4 − 2k
Condition for two distinct real roots: b² − 4ac > 0
Discriminant expression: (2k − 1)² − 4(1)(4 − 2k)

awarded B1 for obtaining the correct expression for b² − 4ac

Step 2: Expand and simplify the quadratic inequality
Expand (2k − 1)² : 4k² − 4k + 1
Expand −4(1)(4 − 2k) : −16 + 8k
Combine terms:
(4k² − 4k + 1) + (−16 + 8k) > 0
4k² + 4k − 15 > 0

Step 3: Find the critical values
Set the quadratic equal to zero: 4k² + 4k − 15 = 0
Factorise (or use quadratic formula):
Looking for factors of 4 × (−15) = −60 that add to +4 → +10 and −6 :
4k² + 10k − 6k − 15 = 0
2k(2k + 5) − 3(2k + 5) = 0
(2k + 5)(2k − 3) = 0
Critical values: k = −5/2 and k = 3/2

awarded M1 for comparing b² − 4ac to 0 to find two critical values, and A1 for obtaining −5/2 and 3/2

Step 4: Determine the inequality region
Since the coefficient of k² is positive ( 4 > 0 ), the parabola opens upwards (U-shape).
We require the expression to be strictly greater than zero ( > 0 ), so we take the region above the horizontal axis (outside the roots):
k < −5/2 or k > 3/2

awarded A1 for the correct final inequality in any equivalent form (ACF)

✅ Final Correct Answer

The range of possible values for k is:

k < −5/2 or k > 3/2

Acceptable alternative forms (ACF) include decimal notation ( k < −2.5 or k > 1.5 ) or set notation: {k : k < −5/2} ∪ {k : k > 3/2} or (−∞, −5/2) ∪ (3/2, ∞) .

💡 Key Knowledge: The Discriminant

Distinct real roots: b² − 4ac > 0
Equal/repeated roots: b² − 4ac = 0
Real roots (at least one): b² − 4ac ≥ 0
No real roots: b² − 4ac < 0
For an inequality A(k − α)(k − β) > 0 with A > 0 and α < β , the solution is always the two separated intervals: k < α or k > β .

🧠 Exam Technique & Examiner Insights

Draw a sketch: Always sketch a quick U-shaped parabola marking −5/2 and 3/2 on the horizontal axis. It takes 5 seconds and prevents writing the inside region by mistake.
Watch the language: The question asks for "two distinct" roots. Using ≥ 0 instead of > 0 loses the final accuracy mark!
Check bracket expansion carefully: Distributing −4(4 − 2k) gives −16 + 8k . Forgetting that a negative times a negative equals a positive is the single biggest cause of dropped marks here.

❌ Common Traps & Student Misconceptions

Writing as a single double inequality: Writing 3/2 < k < −5/2 is mathematically impossible and will be penalised severely by examiners. Separate ranges must be joined by the word "or", or written as two distinct inequalities separated by a comma.
Including equality signs: Writing k ≤ −5/2 or k ≥ 3/2 . The question specifies distinct roots, which demands a strict inequality ( > 0 ).
Sign errors when expanding: −4(1)(4 − 2k) = −16 − 8k instead of −16 + 8k leads to an incorrect linear term, throwing off the critical values completely.

Topics

Pure Mathematics · B: Algebra and functions