| Exam Board | AQA |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2005 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indices and Surds |
| Type | Expand and simplify surd expressions |
| Difficulty | Easy -1.2 This is a straightforward C1 surds question testing standard techniques: (a) uses difference of two squares requiring simple recall, (b) is direct simplification of a surd, and (c) applies rationalizing the denominator—a routine textbook exercise with no problem-solving insight needed. Easier than average A-level questions. |
| Spec | 1.02b Surds: manipulation and rationalising denominators |
| Answer | Marks |
|---|---|
| Total: 7 |
| Answer | Marks | Guidance |
|---|---|---|
| Sides \(24 - 2x, 9 - 2x\) | B1 | Either correct |
| \(V = x(24-2x)(9-2x)\) | M1 | 3 sides involving x multiplied together |
| \(= 4x^3 - 66x^2 + 216x\) | A1 | AG (be convinced) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{dV}{dx} = 12x^2 - 132x + 216\) | M1 | Power decreased by 1 |
| A1 | One term correct | |
| A1 | All correct (no +C etc) |
| Answer | Marks | Guidance |
|---|---|---|
| Putting their \(\frac{dV}{dx} = 0\) (must see this first) | M1 | Or their \(12x^2 - 132x + 216 = 0\) or \(12(x^2 - 11x + 18) = 0\) or statement |
| \(\Rightarrow x^2 - 11x + 18 = 0\) | A1 | AG (be convinced) |
| Answer | Marks | Guidance |
|---|---|---|
| \((x - 2)(x - 9) = 0\) | M1 | Factors, comp sq or formulae used (1 slip) |
| \(\Rightarrow x = 2, x = 9\) | A1 | |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Reject \(x = 9\), since \(9 - 2x < 0\) | E1 | \(x = 2\) is only possible value |
| Total: 1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{d^2V}{dx^2} = 24x - 132\) | M1 | Differentiating their \(\frac{dV}{dx}\) (eg 2x-11) |
| A1 | Correct | |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| \(x = 2\) only \(\Rightarrow \frac{d^2V}{dx^2} = -84\) (or < 0) | B1 | Correct \(\frac{d^2V}{dx^2}\) value OE full test. |
| Maximum value | E1✓ | ft if their test implies minimum |
| Total: 2 |
| | **Total: 7**
## Question 6(a)
Sides $24 - 2x, 9 - 2x$ | B1 | Either correct
$V = x(24-2x)(9-2x)$ | M1 | 3 sides involving x multiplied together
$= 4x^3 - 66x^2 + 216x$ | A1 | AG (be convinced)
## Question 6(b)(i)
$\frac{dV}{dx} = 12x^2 - 132x + 216$ | M1 | Power decreased by 1
| A1 | One term correct
| A1 | All correct (no +C etc)
## Question 6(b)(ii)
Putting their $\frac{dV}{dx} = 0$ (must see this first) | M1 | Or their $12x^2 - 132x + 216 = 0$ or $12(x^2 - 11x + 18) = 0$ or statement
$\Rightarrow x^2 - 11x + 18 = 0$ | A1 | AG (be convinced)
## Question 6(b)(iii)
$(x - 2)(x - 9) = 0$ | M1 | Factors, comp sq or formulae used (1 slip)
$\Rightarrow x = 2, x = 9$ | A1 |
| | **Total: 2**
## Question 6(b)(iv)
Reject $x = 9$, since $9 - 2x < 0$ | E1 | $x = 2$ is only possible value
| | **Total: 1**
## Question 6(c)(i)
$\frac{d^2V}{dx^2} = 24x - 132$ | M1 | Differentiating their $\frac{dV}{dx}$ (eg 2x-11)
| A1 | Correct
| | **Total: 2**
## Question 6(c)(ii)
$x = 2$ only $\Rightarrow \frac{d^2V}{dx^2} = -84$ (or < 0) | B1 | Correct $\frac{d^2V}{dx^2}$ value OE full test.
Maximum value | E1✓ | ft if their test implies minimum
| | **Total: 2**5
\begin{enumerate}[label=(\alph*)]
\item Simplify $( \sqrt { 12 } + 2 ) ( \sqrt { 12 } - 2 )$.
\item Express $\sqrt { 12 }$ in the form $m \sqrt { 3 }$, where $m$ is an integer.
\item Express $\frac { \sqrt { 12 } + 2 } { \sqrt { 12 } - 2 }$ in the form $a + b \sqrt { 3 }$, where $a$ and $b$ are integers.
\end{enumerate}
\hfill \mbox{\textit{AQA C1 2005 Q5 [7]}}