| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2012 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Roots of polynomials |
| Type | Complex roots with real coefficients |
| Difficulty | Standard +0.3 This is a straightforward application of the complex conjugate root theorem and polynomial factorization. Part (i) requires routine substitution to verify u is a root and recalling that complex roots come in conjugate pairs. Part (ii) involves dividing the quartic by a quadratic factor (x-u)(x-ū) and solving the resulting quadratic—all standard techniques with no novel insight required. Slightly easier than average due to being highly procedural. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02b Express complex numbers: cartesian and modulus-argument forms4.02g Conjugate pairs: real coefficient polynomials4.02i Quadratic equations: with complex roots4.02j Cubic/quartic equations: conjugate pairs and factor theorem |
| Answer | Marks | Guidance |
|---|---|---|
| (i) | ||
| EITHER Substitute \(x = 1 + \sqrt{2}i\) and attempt the expansions of the \(x^2\) and \(x^4\) terms | M1 | |
| Use \(i^2 = -1\) correctly at least once | B1 | |
| Complete the verification | A1 | |
| State second root \(1 - \sqrt{2}i\) | B1 | |
| OR 1 | ||
| State second root \(1 - \sqrt{2}i\) | B1 | |
| Carry out a complete method for finding a quadratic factor with zeros \(1 \pm \sqrt{2}i\) | M1 | |
| Obtain \(x^2 - 2x + 3\), or equivalent | A1 | |
| Show that the division of \(p(x)\) by \(x^2 - 2x + 3\) gives zero remainder and complete the verification | A1 | |
| OR 2 | ||
| Substitute \(x = 1 + \sqrt{2}i\) and use correct method to express \(x^2\) and \(x^4\) in polar form | M1 | |
| Obtain \(x^4\) and \(x^4\) in any correct polar form (allow decimals here) | B1 | |
| Complete an exact verification | A1 | |
| State second root \(1 - \sqrt{2}i\), or its polar equivalent (allow decimals here) | B1 | [4] |
| (ii) | ||
| Carry out a complete method for finding a quadratic factor with zeros \(1 \pm \sqrt{2}i\) | M1* | |
| Obtain \(x^2 - 2x + 3\), or equivalent | A1 | |
| Attempt division of \(p(x)\) by \(x^2 - 2x + 3\) reaching a partial quotient \(x^2 + kx\), or equivalent | M1(dep*) | |
| Obtain quadratic factor \(x^2 - 2x + 2\) | A1 | |
| Find the zeros of the second quadratic factor, using \(i^2 = -1\) | M1(dep*) | |
| Obtain roots \(-1 + i\) and \(-1 - i\) | A1 | [6] |
**(i)** | |
**EITHER** Substitute $x = 1 + \sqrt{2}i$ and attempt the expansions of the $x^2$ and $x^4$ terms | M1 |
Use $i^2 = -1$ correctly at least once | B1 |
Complete the verification | A1 |
State second root $1 - \sqrt{2}i$ | B1 |
**OR 1** | |
State second root $1 - \sqrt{2}i$ | B1 |
Carry out a complete method for finding a quadratic factor with zeros $1 \pm \sqrt{2}i$ | M1 |
Obtain $x^2 - 2x + 3$, or equivalent | A1 |
Show that the division of $p(x)$ by $x^2 - 2x + 3$ gives zero remainder and complete the verification | A1 |
**OR 2** | |
Substitute $x = 1 + \sqrt{2}i$ and use correct method to express $x^2$ and $x^4$ in polar form | M1 |
Obtain $x^4$ and $x^4$ in any correct polar form (allow decimals here) | B1 |
Complete an exact verification | A1 |
State second root $1 - \sqrt{2}i$, or its polar equivalent (allow decimals here) | B1 | [4]
**(ii)** | |
Carry out a complete method for finding a quadratic factor with zeros $1 \pm \sqrt{2}i$ | M1* |
Obtain $x^2 - 2x + 3$, or equivalent | A1 |
Attempt division of $p(x)$ by $x^2 - 2x + 3$ reaching a partial quotient $x^2 + kx$, or equivalent | M1(dep*) |
Obtain quadratic factor $x^2 - 2x + 2$ | A1 |
Find the zeros of the second quadratic factor, using $i^2 = -1$ | M1(dep*) |
Obtain roots $-1 + i$ and $-1 - i$ | A1 | [6]
[The second M1 is earned if inspection reaches an unknown factor $x^2 + Bx + C$ and an equation in $B$ and/or $C$, or an unknown factor $Ax^2 + Bx + (6/3)$ and an equation in $A$ and/or $B$]
[If part (i) is attempted by the OR 1 method, then an attempt at part (ii) which uses or quotes relevant working or results obtained in part (i) should be marked using the scheme for part (ii)]9 The complex number $1 + ( \sqrt { } 2 ) \mathrm { i }$ is denoted by $u$. The polynomial $x ^ { 4 } + x ^ { 2 } + 2 x + 6$ is denoted by $\mathrm { p } ( x )$.\\
(i) Showing your working, verify that $u$ is a root of the equation $\mathrm { p } ( x ) = 0$, and write down a second complex root of the equation.\\
(ii) Find the other two roots of the equation $\mathrm { p } ( x ) = 0$.
\hfill \mbox{\textit{CAIE P3 2012 Q9 [10]}}