| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2018 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polynomial Division & Manipulation |
| Type | Division then Show Number of Real Roots |
| Difficulty | Standard +0.3 Part (i) is a straightforward polynomial division exercise requiring systematic application of the division algorithm. Part (ii) follows directly from (i) by recognizing that the equation can be rewritten using the division result, then showing both quotient and remainder are always positive. This is a standard multi-step question requiring technique rather than novel insight, making it slightly easier than average. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.02k Simplify rational expressions: factorising, cancelling, algebraic division |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Carry out division and reach at least partial quotient of form \(x^2 + kx\) | M1 | |
| Obtain quotient \(x^2 - 2x + 2\) | A1 | |
| Obtain remainder 1 | A1 | AG; necessary detail needed and all correct |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State equation as \((x^2+6)(x^2-2x+2)=0\) | B1FT | Following their 3-term quotient from part (i) |
| Calculate discriminant of 3-term quadratic or equivalent | M1 | |
| Obtain \(-4\) and state no root, also referring to no root from \(x^2+6\) factor | A1 | AG; necessary detail needed |
## Question 3(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Carry out division and reach at least partial quotient of form $x^2 + kx$ | M1 | |
| Obtain quotient $x^2 - 2x + 2$ | A1 | |
| Obtain remainder 1 | A1 | AG; necessary detail needed and all correct |
## Question 3(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State equation as $(x^2+6)(x^2-2x+2)=0$ | B1FT | Following their 3-term quotient from part (i) |
| Calculate discriminant of 3-term quadratic or equivalent | M1 | |
| Obtain $-4$ and state no root, also referring to no root from $x^2+6$ factor | A1 | AG; necessary detail needed |3 (i) Find the quotient when
$$x ^ { 4 } - 2 x ^ { 3 } + 8 x ^ { 2 } - 12 x + 13$$
is divided by $x ^ { 2 } + 6$ and show that the remainder is 1 .\\
(ii) Show that the equation
$$x ^ { 4 } - 2 x ^ { 3 } + 8 x ^ { 2 } - 12 x + 12 = 0$$
has no real roots.\\
\hfill \mbox{\textit{CAIE P2 2018 Q3 [6]}}