| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2021 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Two polynomials, shared factor or separate conditions |
| Difficulty | Standard +0.3 Parts (a) and (b) are routine applications of factor and remainder theorems requiring simple substitution. Part (c) involves standard polynomial subtraction and factorization. Part (d) adds a trigonometric twist but follows directly from (c) by solving cosec θ = root values, requiring knowledge of the cosecant function range and basic inverse trig. This is a well-structured multi-part question slightly easier than average due to its guided nature and standard techniques throughout. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Substitute \(x = -3\), equate to zero and attempt solution for \(a\) | M1 | Allow attempt at synthetic division (complete method); allow one sign error carried through; allow attempt at algebraic long division complete with remainder equated to zero |
| Obtain \(a = 13\) | A1 | |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Substitute \(x = 2\), equate to \(40\) and attempt solution for \(b\) | M1 | |
| Obtain \(b = 9\) | A1 | |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Identify \(x+3\) as factor of \(f(x) - g(x)\) | B1 | May be implied by synthetic division. If working backwards from solutions from a calculator then B0 M0 |
| Attempt, by division or equivalent, to find quadratic factor | M1 | \(k(x+3)(2x+1)(2x-1)\) where \(k \neq 1\) gets B1 M1 |
| Obtain \((x+3)(2x-1)(2x+1)\) | A1 | |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Attempt correct process to find at least 1 value from \(\cosec\theta = k\) where \(k < -1\) | M1 | Allow for \(199.5°\) or \(-19.5°\) |
| Obtain \(3.48\) or \(5.94\) | A1 | |
| Obtain a second correct solution | A1 | And no others within the range |
| 3 |
## Question 6(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Substitute $x = -3$, equate to zero and attempt solution for $a$ | M1 | Allow attempt at synthetic division (complete method); allow one sign error carried through; allow attempt at algebraic long division complete with remainder equated to zero |
| Obtain $a = 13$ | A1 | |
| | **2** | |
---
## Question 6(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Substitute $x = 2$, equate to $40$ and attempt solution for $b$ | M1 | |
| Obtain $b = 9$ | A1 | |
| | **2** | |
---
## Question 6(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| Identify $x+3$ as factor of $f(x) - g(x)$ | B1 | May be implied by synthetic division. If working backwards from solutions from a calculator then B0 M0 |
| Attempt, by division or equivalent, to find quadratic factor | M1 | $k(x+3)(2x+1)(2x-1)$ where $k \neq 1$ gets B1 M1 |
| Obtain $(x+3)(2x-1)(2x+1)$ | A1 | |
| | **3** | |
---
## Question 6(d):
| Answer | Mark | Guidance |
|--------|------|----------|
| Attempt correct process to find at least 1 value from $\cosec\theta = k$ where $k < -1$ | M1 | Allow for $199.5°$ or $-19.5°$ |
| Obtain $3.48$ or $5.94$ | A1 | |
| Obtain a second correct solution | A1 | And no others within the range |
| | **3** | |
---6 The polynomials $\mathrm { f } ( x )$ and $\mathrm { g } ( x )$ are defined by
$$\mathrm { f } ( x ) = 4 x ^ { 3 } + a x ^ { 2 } + 8 x + 15 \quad \text { and } \quad \mathrm { g } ( x ) = x ^ { 2 } + b x + 18$$
where $a$ and $b$ are constants.
\begin{enumerate}[label=(\alph*)]
\item Given that $( x + 3 )$ is a factor of $\mathrm { f } ( x )$, find the value of $a$.
\item Given that the remainder is 40 when $\mathrm { g } ( x )$ is divided by $( x - 2 )$, find the value of $b$.
\item When $a$ and $b$ have these values, factorise $\mathrm { f } ( x ) - \mathrm { g } ( x )$ completely.
\item Hence solve the equation $\mathrm { f } ( \operatorname { cosec } \theta ) - \mathrm { g } ( \operatorname { cosec } \theta ) = 0$ for $0 < \theta < 2 \pi$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2021 Q6 [10]}}