| Exam Board | Edexcel |
|---|---|
| Module | C12 (Core Mathematics 1 & 2) |
| Year | 2016 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indices and Surds |
| Type | Solve exponential equations |
| Difficulty | Moderate -0.8 This is a straightforward C1/C2 question testing basic manipulation of indices and surds. Part (i) requires converting to the same base (powers of 2) and equating exponents—a standard textbook exercise. Part (ii) involves simplifying surds by factoring out perfect squares, then squaring to find n. Both parts are routine applications of fundamental techniques with no problem-solving insight required, making this easier than average but not trivial since it requires careful algebraic manipulation. |
| Spec | 1.02a Indices: laws of indices for rational exponents1.02b Surds: manipulation and rationalising denominators |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(4^{2x+1} = 2^{2(2x+1)}\) and \(8^{4x} = 2^{3\times4x}\) or \(8^{4x} = 4^{\frac{3}{2}\times4x}\) | M1 | Writes both sides as powers of 2 (or 4, 8, 64); note \(2^{+(2x+1)} = 2^{3+4x}\) is M0; condone poor/missing brackets but not incorrect index work |
| \(2(2x+1) = 12x \Rightarrow x = \frac{1}{4}\) | dM1A1 | dM1 (dep): equates indices and proceeds to \(x=\ldots\); condone sign/bracketing errors but not processing errors; A1: \(x=\frac{1}{4}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(3\sqrt{18} - \sqrt{32} = 9\sqrt{2} - 4\sqrt{2} = 5\sqrt{2}\) | M1A1 | M1: writes \(\sqrt{18}=3\sqrt{2}\) or \(3\sqrt{18}=9\sqrt{2}\) or \(\sqrt{32}=4\sqrt{2}\); A1: \(5\sqrt{2}\) or states \(k=5\); answer without working is 0 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\sqrt{n} = 5\sqrt{2} \Rightarrow n = (5\sqrt{2})^2 = 25\times2 = 50\) | M1A1 | M1: moves from \(\sqrt{n}=k\sqrt{2}\) to \(n=2k^2\); also accept \(\sqrt{n}=\sqrt{50}\); A1: \(n=50\) |
## Question 3:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $4^{2x+1} = 2^{2(2x+1)}$ and $8^{4x} = 2^{3\times4x}$ or $8^{4x} = 4^{\frac{3}{2}\times4x}$ | M1 | Writes both sides as powers of 2 (or 4, 8, 64); note $2^{+(2x+1)} = 2^{3+4x}$ is M0; condone poor/missing brackets but not incorrect index work |
| $2(2x+1) = 12x \Rightarrow x = \frac{1}{4}$ | dM1A1 | dM1 (dep): equates indices and proceeds to $x=\ldots$; condone sign/bracketing errors but not processing errors; A1: $x=\frac{1}{4}$ |
### Part (ii)(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $3\sqrt{18} - \sqrt{32} = 9\sqrt{2} - 4\sqrt{2} = 5\sqrt{2}$ | M1A1 | M1: writes $\sqrt{18}=3\sqrt{2}$ or $3\sqrt{18}=9\sqrt{2}$ or $\sqrt{32}=4\sqrt{2}$; A1: $5\sqrt{2}$ or states $k=5$; answer without working is 0 marks |
### Part (ii)(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\sqrt{n} = 5\sqrt{2} \Rightarrow n = (5\sqrt{2})^2 = 25\times2 = 50$ | M1A1 | M1: moves from $\sqrt{n}=k\sqrt{2}$ to $n=2k^2$; also accept $\sqrt{n}=\sqrt{50}$; A1: $n=50$ |
---3. Answer this question without a calculator, showing all your working and giving your answers in their simplest form.
\begin{enumerate}[label=(\roman*)]
\item Solve the equation
$$4 ^ { 2 x + 1 } = 8 ^ { 4 x }$$
\item (a) Express
$$3 \sqrt { 18 } - \sqrt { 32 }$$
in the form $k \sqrt { 2 }$, where $k$ is an integer.\\
(b) Hence, or otherwise, solve
$$3 \sqrt { 18 } - \sqrt { 32 } = \sqrt { n }$$
\end{enumerate}
\hfill \mbox{\textit{Edexcel C12 2016 Q3 [7]}}