| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2019 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Solve p(exponential) = 0 |
| Difficulty | Standard +0.3 Part (i) is routine application of factor theorem to find a constant. Part (ii) is standard factorisation after finding one factor. Part (iii) requires substitution x = e^(√y) and solving exponential equations, which is slightly above average but still a standard technique for P2 level. The multi-step nature and exponential solving elevate it slightly above average difficulty. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.06f Laws of logarithms: addition, subtraction, power rules |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Substitute \(x=2\), equate to zero and attempt solution | M1 | |
| Obtain \(a=4\) | A1 | |
| Total | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Divide by \(x-2\) at least as far as the \(x\) term | M1 | By inspection or use of identity |
| Obtain \(4x^2 + 12x + 9\) | A1 | |
| Conclude \((x-2)(2x+3)^2\) | A1 | Each factor must be simplified to integer form |
| Total | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Attempt correct process to solve \(e^{\sqrt{y}} = k\) where \(k > 0\) | M1 | For \(y = (\ln k)^2\) |
| Obtain \(0.48\) and no others | A1 | AWRT |
| 2 |
## Question 4(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Substitute $x=2$, equate to zero and attempt solution | M1 | |
| Obtain $a=4$ | A1 | |
| **Total** | **2** | |
## Question 4(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Divide by $x-2$ at least as far as the $x$ term | M1 | By inspection or use of identity |
| Obtain $4x^2 + 12x + 9$ | A1 | |
| Conclude $(x-2)(2x+3)^2$ | A1 | Each factor must be simplified to integer form |
| **Total** | **3** | |
## Question 4(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Attempt correct process to solve $e^{\sqrt{y}} = k$ where $k > 0$ | **M1** | For $y = (\ln k)^2$ |
| Obtain $0.48$ and no others | **A1** | AWRT |
| | **2** | |
---4 The polynomial $\mathrm { p } ( x )$ is defined by
$$\mathrm { p } ( x ) = a x ^ { 3 } + a x ^ { 2 } - 15 x - 18$$
where $a$ is a constant. It is given that ( $x - 2$ ) is a factor of $\mathrm { p } ( x )$.\\
(i) Find the value of $a$.\\
(ii) Using this value of $a$, factorise $\mathrm { p } ( x )$ completely.\\
(iii) Hence solve the equation $\mathrm { p } \left( \mathrm { e } ^ { \sqrt { } y } \right) = 0$, giving the answer correct to 2 significant figures.\\
\hfill \mbox{\textit{CAIE P2 2019 Q4 [7]}}