| Exam Board | Edexcel |
|---|---|
| Module | PMT Mocks (PMT Mocks) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Factor & Remainder Theorem |
| Type | Multiple unknowns with derivative condition |
| Difficulty | Standard +0.3 This is a straightforward integration and simultaneous equations problem. Students integrate f'(x), apply two given conditions to form equations, solve the linear system, then verify a factor—all standard AS-level techniques with no novel insight required. Slightly easier than average due to clear scaffolding. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.08a Fundamental theorem of calculus: integration as reverse of differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| Show that \(a - b = 99\) for curve \(y = f(x)\) with given conditions | (4) | M1 Integrates \(f'(x) = 18x^2 + 2ax + b\); A1 Fully correct integration; B1 Deduces constant term is \(-48\); B1 Uses \(f(-1) = 45\) and obtains linear equation in terms of \(a\) and \(b\) |
| Answer | Marks | Guidance |
|---|---|---|
| Find values of \(a\) and \(b\) where tangent at \(A(-1, 45)\) has gradient \(-84\) | (3) | B1 Uses \(f'(-1) = -84\) and obtains linear equation in terms of \(a\) and \(b\); M1 Attempts to solve simultaneously to get values for both \(a\) and \(b\); A1 \(a = 3\), \(b = -96\) |
| Answer | Marks | Guidance |
|---|---|---|
| Show that \((2x + 1)\) is a factor of \(f(x)\) | (2) | M1 Attempts \(f(-\frac{1}{2})\) or divides by \((2x + 1)\). Look for constant remainder in long division; A1 Obtains remainder zero and makes conclusion, e.g. remainder = 0, hence \((2x + 1)\) is a factor |
**Part (a):**
Show that $a - b = 99$ for curve $y = f(x)$ with given conditions | (4) | M1 Integrates $f'(x) = 18x^2 + 2ax + b$; A1 Fully correct integration; B1 Deduces constant term is $-48$; B1 Uses $f(-1) = 45$ and obtains linear equation in terms of $a$ and $b$
**Part (b):**
Find values of $a$ and $b$ where tangent at $A(-1, 45)$ has gradient $-84$ | (3) | B1 Uses $f'(-1) = -84$ and obtains linear equation in terms of $a$ and $b$; M1 Attempts to solve simultaneously to get values for both $a$ and $b$; A1 $a = 3$, $b = -96$
**Part (c):**
Show that $(2x + 1)$ is a factor of $f(x)$ | (2) | M1 Attempts $f(-\frac{1}{2})$ or divides by $(2x + 1)$. Look for constant remainder in long division; A1 Obtains remainder zero and makes conclusion, e.g. remainder = 0, hence $(2x + 1)$ is a factor
---7. A curve $C$ has equation $y = \mathrm { f } ( x )$.
Given that
\begin{itemize}
\item $\mathrm { f } ^ { \prime } ( x ) = 18 x ^ { 2 } + 2 a x + b$
\item the $y$-intercept of $C$ is - 48
\item the point $A$, with coordinates $( - 1,45 )$ lies on $C$\\
a. Show that $a - b = 99$\\
b. Find the value of $a$ and the value of $b$.\\
c. Show that $( 2 x + 1 )$ is a factor of $\mathrm { f } ( x )$.
\end{itemize}
\hfill \mbox{\textit{Edexcel PMT Mocks Q7 [9]}}