Edexcel IAL P2 January 2025 question paper and mark scheme (WMA12/01)

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1. Further Trigonometry · 5 marks

   Solve the equation $2\cos^2 x - \sin x - 1 = 0$ for $0^\circ \le x \le 360^\circ$.

   Worked answer

   1. M1 for using the identity $\cos^2 x = 1 - \sin^2 x$ to rewrite the equation: $2(1 - \sin^2 x) - \sin x - 1 = 0$.
   2. M1 for simplifying to a quadratic in $\sin x$: $2\sin^2 x + \sin x - 1 = 0$.
   3. M1 for factorising the quadratic to $(2\sin x - 1)(\sin x + 1) = 0$.
   4. A1 for finding the solutions from $\sin x = 0.5$, which are $x = 30^\circ, 150^\circ$.
   5. A1 for finding the solution from $\sin x = -1$, which is $x = 270^\circ$.

2. Proof · 5 marks

   (a) Prove by exhaustion that the square of any integer leaves a remainder of 0, 1 or 4 when it is divided by 5. [4]

   (b) Hence show that the last digit of a square number is never 2, 3, 7 or 8. [1]

   Worked answer

   (a) Considers every case: \(n=5k,\ 5k\pm1,\ 5k\pm2\) (or \(5k,\ 5k+1,\ 5k+2,\ 5k+3,\ 5k+4\)), \(k\) an integer M1

   Squares at least two cases correctly, e.g. \((5k\pm1)^2=25k^2\pm10k+1\) M1

   All cases correct: \((5k)^2=5(5k^2)\); \((5k\pm1)^2=5(5k^2\pm2k)+1\); \((5k\pm2)^2=5(5k^2\pm4k)+4\) A1

   Conclusion: every integer is in one of the cases, so the remainder is always 0, 1 or 4 A1 (cso)

   (b) A whole number ending in 2 or 7 leaves remainder 2, and one ending in 3 or 8 leaves remainder 3, on division by 5; by (a) a square leaves remainder 0, 1 or 4, so it cannot end in 2, 3, 7 or 8 B1

3. The Binomial Expansion · 4 marks

   The first three terms of the expansion $(a+bx)^6$ are $64 - 192x + cx^2$. Find the values of the constants $a$, $b$, and $c$.

   Worked answer

   1. M1 for equating the first term: $a^6 = 64 \implies a=2$.
   2. M1 for equating the $x$ coefficient to find $b$: $\binom{6}{1}a^5 b = -192 \implies 6(32)b = -192 \implies 192b = -192$.
   3. A1 for obtaining $b = -1$.
   4. A1 for finding the $x^2$ coefficient: $c = \binom{6}{2}a^4 b^2 = 15(16)(-1)^2 = 240$.

More on these topics: Further Trigonometry · Proof · The Binomial Expansion

P2 topics this paper can test

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• Sequences and Series
• Logarithms and Exponentials
• Circles
• Further Trigonometry
• Algebraic Methods
• Proof
• The Binomial Expansion
• Differentiation (stationary points)
• Integration (definite integrals and area)

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What the examiners said about this paper

Summarised in our own words from the board's examiner report (Examiner Insights).

• WMA12/01: Trapezium rule (Q7); Factor theorem (Q5); Proof (Q8). Official examiner report ↗

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Formula booklet

Mathematical Formulae and Statistical Tables (International AS/A Level Mathematics) · Pearson Edexcel · For the 2018 IAL specification (current)