| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2011 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Find constants with divisibility condition |
| Difficulty | Standard +0.3 This is a standard application of the factor theorem requiring polynomial division or comparing coefficients. Students must divide p(x) by the given quadratic factor, set the remainder to zero to find a, then solve the resulting factored equation. While it involves multiple steps and careful algebraic manipulation, it follows a well-practiced procedure with no novel insight required, making it slightly above average difficulty. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem4.02q De Moivre's theorem: multiple angle formulae |
| Answer | Marks |
|---|---|
| Obtain quotient \(x^2 + 4x + 3\) | A1 |
| Equate remainder of form \(kx\) to zero and solve for \(a\), or equivalent | M1 |
| Obtain answer \(a = 1\) | A1 |
| Answer | Marks |
|---|---|
| Obtain a correct equation in \(a\) in any unsimplified form | A1 |
| Expand terms, use \(i^2 = -1\) and solve for \(a\) | M1 |
| Obtain answer \(a = 1\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| State answer, e.g. \(x = -1\) and no others | B1 | [2] |
**(i) EITHER: Attempt division by $x^2 - x + 1$ reaching a partial quotient of $x^2 + kx$**
Obtain quotient $x^2 + 4x + 3$ | A1 |
Equate remainder of form $kx$ to zero and solve for $a$, or equivalent | M1 |
Obtain answer $a = 1$ | A1 |
**OR: Substitute a complex zero of $x^2 - x + 1$ in $p(x)$ and equate to zero**
Obtain a correct equation in $a$ in any unsimplified form | A1 |
Expand terms, use $i^2 = -1$ and solve for $a$ | M1 |
Obtain answer $a = 1$ | A1 |
[SR: The first 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 $Ax^2 + Bx + 3$ and an equation in $A$ and/or $B$. The second M1 is only earned if use of the equation $a = B - C$ is seen or implied.]
**(ii) State answer, e.g. $x = -3$**
State answer, e.g. $x = -1$ and no others | B1 | [2]3 The polynomial $x ^ { 4 } + 3 x ^ { 3 } + a x + 3$ is denoted by $\mathrm { p } ( x )$. It is given that $\mathrm { p } ( x )$ is divisible by $x ^ { 2 } - x + 1$.\\
(i) Find the value of $a$.\\
(ii) When $a$ has this value, find the real roots of the equation $\mathrm { p } ( x ) = 0$.
\hfill \mbox{\textit{CAIE P3 2011 Q3 [6]}}