| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2021 |
| Session | November |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Complex roots with real coefficients |
| Difficulty | Standard +0.8 This is a multi-part question combining standard complex roots theory (parts a-c are routine: using conjugate root theorem and substitution) with a more challenging Argand diagram optimization problem (part d(ii) requires finding the minimum imaginary part of the intersection of a disk and half-plane, involving geometric insight and coordinate geometry). The first three parts are straightforward A-level fare, but part d(ii) elevates this to moderately above average difficulty. |
| Spec | 4.02f Convert between forms: cartesian and modulus-argument4.02g Conjugate pairs: real coefficient polynomials4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| Substitute \(1+2i\) in the polynomial and attempt expansions of \(x^2\) and \(x^3\) | M1 | \(u^2 = -3+4i\), \(u^3 = -11-2i\); full substitution but need not simplify. |
| Equate real and/or imaginary parts to zero | M1 | \(-18-3a+b=0\), \(4+4a=0\) |
| Obtain \(a = -1\) | A1 | |
| Obtain \(b = 15\) | A1 | |
| Total: 4 |
| Answer | Marks |
|---|---|
| State second root \(1-2i\) | B1 |
| Total: 1 |
| Answer | Marks |
|---|---|
| State the quadratic factor \(x^2-2x+5\) | B1 |
| State the linear factor \(2x+3\) | B1 |
| Total: 2 |
| Answer | Marks |
|---|---|
| Show a circle with centre \(1+2i\) | B1 |
| Show circle passing through the origin | B1 |
| Show the half line \(y=x\) in the first quadrant (accept chord of circle) | B1 |
| Shade the correct region on a correct diagram | B1 |
| Total: 4 |
| Answer | Marks |
|---|---|
| State answer \(2-\sqrt{5}\) | B1 |
| Total: 1 |
## Question 10(a):
| Substitute $1+2i$ in the polynomial and attempt expansions of $x^2$ and $x^3$ | M1 | $u^2 = -3+4i$, $u^3 = -11-2i$; full substitution but need not simplify. |
| Equate real and/or imaginary parts to zero | M1 | $-18-3a+b=0$, $4+4a=0$ |
| Obtain $a = -1$ | A1 | |
| Obtain $b = 15$ | A1 | |
| **Total: 4** | | |
## Question 10(b):
| State second root $1-2i$ | B1 | |
| **Total: 1** | | |
## Question 10(c):
| State the quadratic factor $x^2-2x+5$ | B1 | |
| State the linear factor $2x+3$ | B1 | |
| **Total: 2** | | |
## Question 10(d)(i):
| Show a circle with centre $1+2i$ | B1 | |
| Show circle passing through the origin | B1 | |
| Show the half line $y=x$ in the first quadrant (accept chord of circle) | B1 | |
| Shade the correct region on a correct diagram | B1 | |
| **Total: 4** | | |
## Question 10(d)(ii):
| State answer $2-\sqrt{5}$ | B1 | |
| **Total: 1** | | |10 The complex number $1 + 2 \mathrm { i }$ is denoted by $u$. The polynomial $2 x ^ { 3 } + a x ^ { 2 } + 4 x + b$, where $a$ and $b$ are real constants, is denoted by $\mathrm { p } ( x )$. It is given that $u$ is a root of the equation $\mathrm { p } ( x ) = 0$.
\begin{enumerate}[label=(\alph*)]
\item Find the values of $a$ and $b$.
\item State a second complex root of this equation.
\item Find the real factors of $\mathrm { p } ( x )$.
\item \begin{enumerate}[label=(\roman*)]
\item On a sketch of an Argand diagram, shade the region whose points represent complex numbers $z$ satisfying the inequalities $| z - u | \leqslant \sqrt { 5 }$ and $\arg z \leqslant \frac { 1 } { 4 } \pi$.
\item Find the least value of $\operatorname { Im } z$ for points in the shaded region. Give your answer in an exact form.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2021 Q10 [12]}}