Edexcel IAL P1 January 2025 question paper and mark scheme (WMA11/01)

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1. Algebra · 7 marks

   Let $f(x) = 5 - 4x - 2x^2$.
   
   (a) Express $f(x)$ in the form $A - B(x+C)^2$, where $A$, $B$, and $C$ are integers.
   (b) State the maximum value of $f(x)$ and the value of $x$ at which it occurs.
   (c) Solve $f(x) = 0$, giving your answers in exact form.

   Worked answer

   (a) M1 Factors out $-2$ from the $x$ terms: $5 - 2(x^2 + 2x)$.

   1. M1 Completes the square inside the bracket: $5 - 2((x+1)^2 - 1)$.
   2. A1 Expands and simplifies to obtain $7 - 2(x+1)^2$.

   (b) B1 States the maximum value is $7$.

   1. B1 States this occurs at $x = -1$.

   (c) M1 Sets $7 - 2(x+1)^2 = 0$ and rearranges to $(x+1)^2 = \frac{7}{2}$.

   1. A1 Solves for $x$ by square rooting, yielding $x = -1 \pm \sqrt{\frac{7}{2}}$ (or equivalent exact surd).

2. Integration · 6 marks

   Given that $f''(x) = 12x - 4$, $f'(1) = 2$ and $f(-1) = 5$, find $f(x)$.

   Worked answer

   1. M1 for integrating $f''(x)$ to find $f'(x) = 6x^2 - 4x + c_1$.
   2. M1 for substituting $f'(1) = 2$ to find $c_1$: $2 = 6(1)^2 - 4(1) + c_1 \implies c_1 = 0$.
   3. M1 for substituting $c_1 = 0$ and integrating again to find $f(x) = 2x^3 - 2x^2 + c_2$.
   4. M1 for substituting $f(-1) = 5$: $5 = 2(-1)^3 - 2(-1)^2 + c_2$.
   5. M1 for solving for $c_2$: $5 = -2 - 2 + c_2 \implies c_2 = 9$.
   6. A1 for the final correct function $f(x) = 2x^3 - 2x^2 + 9$.

3. Graphs and Transformations · 4 marks

   A cubic curve crosses the x-axis at $x = -2$, $x = 1$, and $x = 4$. It crosses the y-axis at the point $(0, 16)$. Find the equation of the curve in the form $y = ax^3 + bx^2 + cx + d$.

   Worked answer

   1. M1 for writing the equation in the factored form $y = k(x+2)(x-1)(x-4)$.
   2. M1 for substituting $(0, 16)$ to find $k$: $16 = k(2)(-1)(-4) \implies 16 = 8k$.
   3. A1 for finding $k = 2$.
   4. A1 for expanding to get $y = 2(x^3 - 3x^2 - 6x + 8) = 2x^3 - 6x^2 - 12x + 16$.

More on these topics: Algebra · Integration · Graphs and Transformations

P1 topics this paper can test

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• Algebra
• Coordinate Geometry
• Trigonometry
• Differentiation
• Integration
• Graphs and Transformations

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What the examiners said about this paper

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

• WMA11/01: Calculator warning ignored (Q2, Q5(b)); Indices (Q4); Radians (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)