Edexcel IAL P2 October 2020 question paper and mark scheme (WMA12/01)

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1. Further Trigonometry · 5 marks

   Solve $2\sin^2 x + 3\cos x - 3 = 0$ for $0^\circ \le x \le 360^\circ$.

   Worked answer

   1. M1 for substituting $\sin^2 x$ with $1 - \cos^2 x$ to get $2(1 - \cos^2 x) + 3\cos x - 3 = 0$.
   2. M1 for expanding and rearranging to form the quadratic $2\cos^2 x - 3\cos x + 1 = 0$.
   3. M1 for factorising the quadratic as $(2\cos x - 1)(\cos x - 1) = 0$.
   4. A1 for finding the solutions from $\cos x = 0.5$ giving $x = 60^\circ$ and $300^\circ$.
   5. A1 for finding the solutions from $\cos x = 1$ giving $x = 0^\circ$ and $360^\circ$.

2. Proof · 6 marks

   Prove that \(n^4-n^2\) is a multiple of 12 for every integer \(n\).

   Worked answer

   Factorises: \(n^4-n^2=n^2(n-1)(n+1)\) M1

   \((n-1)n(n+1)\) is a product of three consecutive integers, so one factor is a multiple of 3 B1

   Considers \(n\) even and \(n\) odd M1

   \(n\) even: \(n=2k\), so \(n^2=4k^2\) is a multiple of 4 A1

   \(n\) odd: \(n-1\) and \(n+1\) are both even, so \((n-1)(n+1)\) is a multiple of 4 A1

   The number is a multiple of 3 and of 4, which have no common factor, so it is a multiple of 12 A1 (cso)

3. The Binomial Expansion · 5 marks

   The binomial expansion of $(1+ax)^b$, where $b$ is a positive integer, starts with $1 - 15x + 90x^2$. Find the values of $a$ and $b$.

   Worked answer

   1. M1 for equating the $x$ coefficient: $ab = -15$.
   2. M1 for equating the $x^2$ coefficient: $\frac{b(b-1)}{2} a^2 = 90 \implies b(b-1)a^2 = 180$.
   3. M1 for substituting $(ab) = -15$ or expanding to $a^2b^2 - a^2b = 180$ and substituting $a^2b^2 = 225$.
   4. M1 for solving for $a$: $225 - a(-15) = 180 \implies 15a = -45 \implies a = -3$.
   5. A1 for finding the corresponding value $b = 5$.

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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Formula booklet

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