AQA AS Level Mathematics Paper 2, June 2025: Question 10

Mark scheme

Show the mark scheme

$Mark scheme for Question 10. Part (a) awards M1 for multiplying the numerator and denominator by (sqrt(k) - sqrt(k + 1)) or (sqrt(k + 1) - sqrt(k)), leading to +/-(k - (k + 1)) on the denominator. R1 is awarded for a fully reasoned argument showing the given result with correct simplification of the denominator. Part (b) awards M1 for using part (a) to rewrite the expression with no fractions as (sqrt(2) - sqrt(1)) + (sqrt(3) - sqrt(2)) + (sqrt(4) - sqrt(3)), and A1 for obtaining 1 (evaluating sqrt(4) - sqrt(1) = 2 - 1 = 1; do not accept sqrt(4) - 1). Total: 4 marks.$

Q Marking instructions AO Marks Typical solution

10(a) Uses 1.1a M1

( k + k +1)( k − k +1) 1 k − k +1

×

Or k + k +1 k − k +1

( k + k +1)( k +1− k ) k − k +1

=

PI by ±[k − (k +1)] on k − (k +1)

denominator k − k +1

=

−1

Completes a reasoned 2.1 R1

= k +1− k

argument to show given result

Must see correct simplification

of the denominator

AG

Subtotal 2

10(b) Uses part (a) to rewrite given 3.1a M1 1 1 1

expression with no fractions + +

1+ 2 2 + 3 3 + 4

= ( 2 − 1)+( 3 − 2)+( 4 − 3)

Obtains 1 1.1b A1

Do not accept 4 −1 = 4 − 1

CSO = 2−1

=1

Subtotal 2

Question 10 Total 4

How to answer it

Algebraic Surds & Telescoping Proof

AQA AS Level Mathematics • Pure Core • Surds & Series

What this question tests

Rationalising two-term surd denominators using conjugate pairs, applying the difference of two squares identity to algebraic terms, executing rigorous step-by-step proofs ("Show that"), and recognizing "telescoping" series where intermediate terms cancel out to leave exact simplified values.

Question 10 (a) • 2 Marks

Part (a): Rationalising the General Surd Identity

Show that 1 / (√k + √(k + 1)) ≡ √(k + 1) - √k for k > 0

📐 Step-by-Step Proof

Method 1: Standard Conjugate

1. Multiply numerator & denominator by (√k - √(k + 1)):
= [1 × (√k - √(k + 1))] / [(√k + √(k + 1))(√k - √(k + 1))]

2. Expand denominator using (a+b)(a-b) = a² - b²:
= (√k - √(k + 1)) / [k - (k + 1)]
= (√k - √(k + 1)) / [k - k - 1]
= (√k - √(k + 1)) / (-1)

3. Divide by -1:
= -√k + √(k + 1)
= √(k + 1) - √k (Q.E.D.)

Method 2: Reorder First (Cleaner)

1 / (√(k + 1) + √k) × [√(k + 1) - √k] / [√(k + 1) - √k]
= (√(k + 1) - √k) / [(k + 1) - k]
= (√(k + 1) - √k) / 1
= √(k + 1) - √k

💡 Key Knowledge

Conjugate Pair: The conjugate of (√a + √b) is (√a - √b) .
Difference of Two Squares: (√a + √b)(√a - √b) = a - b eliminates irrational radicals from the denominator.
Sign Rules with Brackets: When squaring √(k + 1) , write -(k + 1) inside brackets to avoid erroneous signs like -k + 1 .

❌ Common Errors & Pitfalls

Bracket omission trap: Writing k - k + 1 = 1 instead of k - (k + 1) = -1 . Fumbling this sign and forcing the answer loses the reasoning mark.
Skipping steps: Going directly from the fraction to the final answer without displaying the simplified denominator ( -1 or 1 ). Because this is an "Answer Given" (AG) question, every step must be explicit.

🧠 Exam Technique & Mark Breakdown

M1 (AO 1.1a): Evidence of multiplying by the conjugate pair (√k - √(k + 1)) or (√(k + 1) - √k) , implied by showing ±[k - (k + 1)] in the denominator.

R1 (AO 2.1): Fully reasoned argument reaching the given result with correct simplification of the denominator seen.

Question 10 (b) • 2 Marks

Part (b): Evaluating the Finite Series

Using part (a), find the exact value of 1/(√1 + √2) + 1/(√2 + √3) + 1/(√3 + √4)

📐 Step-by-Step Calculation

Apply 1/(√k + √(k+1)) = √(k+1) - √k to each fraction:

• For k = 1: 1/(√1 + √2) = √2 - √1
• For k = 2: 1/(√2 + √3) = √3 - √2
• For k = 3: 1/(√3 + √4) = √4 - √3

Sum all three terms:
= (√2 - √1) + (√3 - √2) + (√4 - √3)

Notice the telescoping cancellation:
= -√1 + (√2 - √2) + (√3 - √3) + √4
= -√1 + √4
= -1 + 2
= 1

✅ Correct Answer

1

Must be fully simplified to an integer. The expression √4 - 1 is not accepted for the final accuracy mark.

❌ Common Errors

Incomplete evaluation: Stopping at √4 - √1 or √4 - 1 without evaluating square roots of square numbers.
Calculator without working: Writing down "1" directly from a calculator. The question explicitly states "Using the result from part (a)... Fully justify your answer", which requires the fractionless sum to be shown.
Sign mix-ups: Expanding as √1 - √2 instead of √2 - √1 .

🧠 Examiner Insight & Marks

M1 (AO 3.1a): Uses the result from part (a) to rewrite the entire given expression with no fractions: (√2 - √1) + (√3 - √2) + (√4 - √3) .

A1 (AO 1.1b): Correct solution only (CSO) giving an answer of 1.

Topics

Pure Mathematics · B: Algebra and functions · A: Proof