| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2019 |
| Session | March |
| 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 routine polynomial division with verification of remainder—a standard algorithmic procedure. Part (ii) requires recognizing the connection between parts, finding one root by inspection or trial, then using calculus or sign analysis to show uniqueness, which adds modest problem-solving beyond pure recall but remains a familiar exam pattern. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.02j Manipulate polynomials: expanding, factorising, division, factor theorem |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Carry out division at least as far as \(2x^2 + kx\) | M1 | |
| Obtain quotient \(2x^2 + 3x + 4\) | A1 | |
| Confirm remainder is 5 | A1 | Answer given; necessary detail needed |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State or imply equation is \((2x+1)(2x^2+3x+4) = 0\) | B1 | FT their quotient from part (i) |
| Calculate discriminant of 3-term quadratic expression or equivalent | M1 | |
| Obtain \(-23\) or equiv and conclude appropriately | A1 |
## Question 4(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Carry out division at least as far as $2x^2 + kx$ | M1 | |
| Obtain quotient $2x^2 + 3x + 4$ | A1 | |
| Confirm remainder is 5 | A1 | Answer given; necessary detail needed |
## Question 4(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply equation is $(2x+1)(2x^2+3x+4) = 0$ | B1 | FT their quotient from part (i) |
| Calculate discriminant of 3-term quadratic expression or equivalent | M1 | |
| Obtain $-23$ or equiv and conclude appropriately | A1 | |
---4 (i) Find the quotient when $4 x ^ { 3 } + 8 x ^ { 2 } + 11 x + 9$ is divided by ( $2 x + 1$ ), and show that the remainder is 5 .\\
(ii) Show that the equation $4 x ^ { 3 } + 8 x ^ { 2 } + 11 x + 4 = 0$ has exactly one real root.\\
\hfill \mbox{\textit{CAIE P2 2019 Q4 [6]}}