| Exam Board | Edexcel |
|---|---|
| Module | CP AS (Core Pure AS) |
| Year | 2018 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Geometric properties using complex numbers |
| Difficulty | Challenging +1.8 This question requires multiple sophisticated steps: using the factor theorem with a real root to reduce the cubic, applying the conjugate root theorem for complex coefficients, using Vieta's formulas to relate roots to coefficients, and crucially, applying the geometric formula for triangle area on an Argand diagram (½|Im(z₂ - z₃)| × base). While each technique is A-level standard, the combination—especially the geometric constraint with complex numbers—requires strong problem-solving and is well above typical textbook exercises. |
| Spec | 4.02g Conjugate pairs: real coefficient polynomials4.02k Argand diagrams: geometric interpretation4.05a Roots and coefficients: symmetric functions |
| VILU SIHI NI IIIUM ION OC | VGHV SIHILNI IMAM ION OO | VJYV SIHI NI JIIYM ION OC |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Complex roots are \(\alpha \pm \beta i\), or \((z^3+z^2+pz+q)\div(z-3) = z^2+4z+p+12\), or \(f(3)=0 \Rightarrow 3^3+3^2+3p+q=0\), or one of: \(3+z_2+z_3=-1\), \(3z_2z_3=-q\), \(3z_2+3z_3+z_2z_3=p\) | B1 | Recognises other roots form a conjugate pair or obtains \(z^2+4z+p+12\) as quadratic factor or writes correct equation for \(p\) and \(q\) or writes correct equation involving \(z_2\) and \(z_3\) |
| Sum of roots: \(\alpha+\beta i + \alpha - \beta i + 3 = -1 \Rightarrow \alpha = \ldots\), or \(\alpha+\beta i+\alpha-\beta i = -4 \Rightarrow \alpha = \ldots\) | M1 | Uses sum of roots of cubic or sum of roots of quadratic to find \(\alpha\) |
| \(\alpha = -2\) | A1 | Correct value for \(\alpha\) |
| \(\frac{1}{2}\times 2\beta \times 5 = 35 \Rightarrow \beta = 7\) | M1 | Uses value of \(\alpha\) and given area to find \(\beta\); must use area and triangle dimensions correctly |
| \(q = -3(-2+7i)(-2-7i) = \ldots\), or \(p = 3(-2+7i)+3(-2-7i)+(-2+7i)(-2-7i)\), or \((z-3)(z-(-2+7i))(z-(-2-7i)) = \ldots\) | M1 | Uses appropriate method to find \(p\) or \(q\) |
| \(q = -159\) or \(p = 41\) | A1 | A correct value for \(p\) or \(q\) |
| \(3p+q = -36 \Rightarrow p = \frac{-36-q}{3} = 41\) and \(q = -159\) | A1 | Correct values for both \(p\) and \(q\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((z^3+z^2+pz+q)\div(z-3) = z^2+4z+p+12\) | B1 | Obtains \(z^2+4z+p+12\) as the quadratic factor |
| \(z^2+4z+p+12=0 \Rightarrow z = \frac{-4\pm\sqrt{4^2-4(p+12)}}{2}\left(= -2\pm i\sqrt{p+8}\right)\) | M1 | Solves quadratic factor by completing the square or quadratic formula |
| \(\alpha = -2\) | A1 | Correct value for \(\alpha\) |
| \(\beta = \sqrt{p+8}\) | M1 | Uses imaginary part to find \(\beta\) in terms of \(p\) |
| \(\frac{1}{2}\times(3+2)\times 2\sqrt{p+8} = 35 \Rightarrow p = \ldots\) | M1 | Draws together imaginary parts of complex conjugate pair and real root to form triangle sides, sets area equal to 35 and solves for \(p\) |
| \(p = 41\) | A1 | Correct value for \(p\) or \(q\) |
| \(3p+q=-36 \Rightarrow q=-159\) | A1 | Correct values for both \(p\) and \(q\) |
## Question 7:
### Main Method:
| Answer/Working | Mark | Guidance |
|---|---|---|
| Complex roots are $\alpha \pm \beta i$, or $(z^3+z^2+pz+q)\div(z-3) = z^2+4z+p+12$, or $f(3)=0 \Rightarrow 3^3+3^2+3p+q=0$, or one of: $3+z_2+z_3=-1$, $3z_2z_3=-q$, $3z_2+3z_3+z_2z_3=p$ | B1 | Recognises other roots form a conjugate pair **or** obtains $z^2+4z+p+12$ as quadratic factor **or** writes correct equation for $p$ and $q$ **or** writes correct equation involving $z_2$ and $z_3$ |
| Sum of roots: $\alpha+\beta i + \alpha - \beta i + 3 = -1 \Rightarrow \alpha = \ldots$, or $\alpha+\beta i+\alpha-\beta i = -4 \Rightarrow \alpha = \ldots$ | M1 | Uses sum of roots of cubic or sum of roots of quadratic to find $\alpha$ |
| $\alpha = -2$ | A1 | Correct value for $\alpha$ |
| $\frac{1}{2}\times 2\beta \times 5 = 35 \Rightarrow \beta = 7$ | M1 | Uses value of $\alpha$ and given area to find $\beta$; must use area and triangle dimensions correctly |
| $q = -3(-2+7i)(-2-7i) = \ldots$, or $p = 3(-2+7i)+3(-2-7i)+(-2+7i)(-2-7i)$, or $(z-3)(z-(-2+7i))(z-(-2-7i)) = \ldots$ | M1 | Uses appropriate method to find $p$ or $q$ |
| $q = -159$ or $p = 41$ | A1 | A correct value for $p$ or $q$ |
| $3p+q = -36 \Rightarrow p = \frac{-36-q}{3} = 41$ and $q = -159$ | A1 | Correct values for both $p$ and $q$ |
### Alternative Method:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(z^3+z^2+pz+q)\div(z-3) = z^2+4z+p+12$ | B1 | Obtains $z^2+4z+p+12$ as the quadratic factor |
| $z^2+4z+p+12=0 \Rightarrow z = \frac{-4\pm\sqrt{4^2-4(p+12)}}{2}\left(= -2\pm i\sqrt{p+8}\right)$ | M1 | Solves quadratic factor by completing the square or quadratic formula |
| $\alpha = -2$ | A1 | Correct value for $\alpha$ |
| $\beta = \sqrt{p+8}$ | M1 | Uses imaginary part to find $\beta$ in terms of $p$ |
| $\frac{1}{2}\times(3+2)\times 2\sqrt{p+8} = 35 \Rightarrow p = \ldots$ | M1 | Draws together imaginary parts of complex conjugate pair and real root to form triangle sides, sets area equal to 35 and solves for $p$ |
| $p = 41$ | A1 | Correct value for $p$ or $q$ |
| $3p+q=-36 \Rightarrow q=-159$ | A1 | Correct values for both $p$ and $q$ |7.
$$f ( z ) = z ^ { 3 } + z ^ { 2 } + p z + q$$
where $p$ and $q$ are real constants.\\
The equation $f ( z ) = 0$ has roots $z _ { 1 } , z _ { 2 }$ and $z _ { 3 }$\\
When plotted on an Argand diagram, the points representing $z _ { 1 } , z _ { 2 }$ and $z _ { 3 }$ form the vertices of a triangle of area 35
Given that $z _ { 1 } = 3$, find the values of $p$ and $q$.
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VILU SIHI NI IIIUM ION OC & VGHV SIHILNI IMAM ION OO & VJYV SIHI NI JIIYM ION OC \\
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\hfill \mbox{\textit{Edexcel CP AS 2018 Q7 [7]}}