| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2015 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polynomial Division & Manipulation |
| Type | Division then Show Number of Real Roots |
| Difficulty | Standard +0.3 This is a straightforward two-part question requiring polynomial division (a standard P2 technique) followed by a simple application to show a cubic has one real root. Part (i) is routine calculation, and part (ii) requires recognizing that subtracting 39 from the remainder gives a factorization, then analyzing the quadratic factor's discriminant—all standard procedures with no novel insight needed. Slightly above average difficulty due to the 'hence' connection requiring some thought. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02p Interpret algebraic solutions: graphically |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Attempt division, or equivalent, at least as far as quotient \(3x^2+kx\) | M1 | |
| Obtain partial quotient \(3x^2+11x\) | A1 | |
| Obtain complete quotient \(3x^2+11x+20\) with no errors seen | A1 | |
| Confirm remainder is \(39\) | B1 | [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| State or imply \((x-2)(3x^2+11x+20)=0\) | B1 | |
| Calculate discriminant of quadratic factor or equivalent | M1 | |
| Obtain \(-119\) or equivalent and confirm only one real root | A1 | [3] |
## Question 4:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Attempt division, or equivalent, at least as far as quotient $3x^2+kx$ | M1 | |
| Obtain partial quotient $3x^2+11x$ | A1 | |
| Obtain complete quotient $3x^2+11x+20$ with no errors seen | A1 | |
| Confirm remainder is $39$ | B1 | [4] |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| State or imply $(x-2)(3x^2+11x+20)=0$ | B1 | |
| Calculate discriminant of quadratic factor or equivalent | M1 | |
| Obtain $-119$ or equivalent and confirm only one real root | A1 | [3] |
---4 (i) Find the quotient when $3 x ^ { 3 } + 5 x ^ { 2 } - 2 x - 1$ is divided by ( $x - 2$ ), and show that the remainder is 39 .\\
(ii) Hence show that the equation $3 x ^ { 3 } + 5 x ^ { 2 } - 2 x - 40 = 0$ has exactly one real root.
\hfill \mbox{\textit{CAIE P2 2015 Q4 [7]}}