| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2011 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | One factor, one non-zero remainder |
| Difficulty | Moderate -0.3 This is a standard application of the Factor and Remainder Theorems requiring students to set up two simultaneous equations from the given conditions (p(1/2)=0 and p(2)=12), solve for a and b, then perform polynomial division. While it involves multiple steps, the techniques are routine and commonly practiced at A-level with no novel insight required. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
| Answer | Marks | Guidance |
|---|---|---|
| Substitute \(x = \frac{1}{2}\) and equate to zero, or divide, and obtain a correct equation, e.g. \(\frac{1}{8}a + \frac{1}{4}b + \frac{5}{2} - 2 = 0\) | B1 | |
| Substitute \(x = 2\) and equate result to 12, or divide and equate constant remainder to 12 | M1 | |
| Obtain a correct equation, e.g. \(8a + 4b + 10 - 2 = 12\) | A1 | |
| Solve for \(a\) or for \(b\) | M1 | |
| Obtain \(a = 2\) and \(b = -3\) | A1 | [5] |
| Answer | Marks | Guidance |
|---|---|---|
| Attempt division by \(2x - 1\) reaching a partial quotient \(\frac{1}{2}ax^2 + kx\) | M1 | |
| Obtain quadratic factor \(x^2 - x + 2\) | A1 | [2] |
| [The M1 is earned if inspection has an unknown factor \(Ax^2 + Bx + 2\) and an equation in \(A\) and/or \(B\), or an unknown factor of \(\frac{1}{2}ax^2 + Bx + C\) and an equation in \(B\) and/or \(C\).] |
**(i)**
Substitute $x = \frac{1}{2}$ and equate to zero, or divide, and obtain a correct equation, e.g. $\frac{1}{8}a + \frac{1}{4}b + \frac{5}{2} - 2 = 0$ | B1 |
Substitute $x = 2$ and equate result to 12, or divide and equate constant remainder to 12 | M1 |
Obtain a correct equation, e.g. $8a + 4b + 10 - 2 = 12$ | A1 |
Solve for $a$ or for $b$ | M1 |
Obtain $a = 2$ and $b = -3$ | A1 | [5]
**(ii)**
Attempt division by $2x - 1$ reaching a partial quotient $\frac{1}{2}ax^2 + kx$ | M1 |
Obtain quadratic factor $x^2 - x + 2$ | A1 | [2]
[The M1 is earned if inspection has an unknown factor $Ax^2 + Bx + 2$ and an equation in $A$ and/or $B$, or an unknown factor of $\frac{1}{2}ax^2 + Bx + C$ and an equation in $B$ and/or $C$.] |5 The polynomial $a x ^ { 3 } + b x ^ { 2 } + 5 x - 2$, where $a$ and $b$ are constants, is denoted by $\mathrm { p } ( x )$. It is given that $( 2 x - 1 )$ is a factor of $\mathrm { p } ( x )$ and that when $\mathrm { p } ( x )$ is divided by $( x - 2 )$ the remainder is 12 .\\
(i) Find the values of $a$ and $b$.\\
(ii) When $a$ and $b$ have these values, find the quadratic factor of $\mathrm { p } ( x )$.
\hfill \mbox{\textit{CAIE P3 2011 Q5 [7]}}