| Exam Board | OCR |
|---|---|
| Module | H240/01 (Pure Mathematics) |
| Year | 2020 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Topic | Indices and Surds |
| Type | Simplify algebraic expressions with indices |
| Difficulty | Easy -1.3 This is a routine indices and surds question testing standard manipulation techniques. Part (a) requires basic surd multiplication, part (b) applies fractional and negative index laws, and part (c) involves recognizing powers of 3 to factorize. All are textbook exercises requiring recall and direct application of rules with minimal problem-solving, making this easier than average. |
| Spec | 1.02a Indices: laws of indices for rational exponents1.02b Surds: manipulation and rationalising denominators |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(6a^3\) | B1 | Obtain 6; B1 only for \(\pm6a^3\) |
| B1 | Obtain \(a^3\) | |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \((64b^3)^{\frac{1}{3}}=4b\) or \((4b^4)^{-\frac{1}{2}}=\frac{1}{2b^2}\) | B1 | Correct simplification of either term; allow \((2b^2)^{-1}\) for the second term |
| \(2b^{-1}\) or \(\frac{2}{b}\) | B1 | Correct final answer |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(9^{3c}=3^{6c}\) | B1 | Either \(9^c\) or \(27^{2c}\) correct as a power of 3 (or 729); ignore coefficient; index must be simplified |
| \(27^{2c}=3^{6c}\) | M1 | Attempt to write the other one of \(9^c\) and \(27^{2c}\) with the same base; ignore coefficient; allow unsimplified index |
| B2 | for \(27^{2c}=9^{3c}\) | |
| \(7\times3^{6c}-4\times3^{6c}=3\times3^{6c}\) | A1 | Combine to obtain correct single term; allow equiv eg \(3\times729^c\) or \(3\times27^{2c}\) or \(3\times9^{3c}\) |
| \(=3^{6c+1}\) | A1 | Obtain correct final answer; must be single power of 3 |
| [4] | ||
| OR: B1 \(9^{2c}(7\times9^c-4\times3^{2c})\); M1 \(9^{2c}(7\times3^{2c}-4\times3^{2c})\); \(9^{2c}\times3\times3^{2c}\); A1 \(3\times27^{2c}\); A1 \(3^{6c+1}\) |
# Question 2:
## Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $6a^3$ | B1 | Obtain 6; B1 only for $\pm6a^3$ |
| | B1 | Obtain $a^3$ |
| **[2]** | | |
## Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $(64b^3)^{\frac{1}{3}}=4b$ or $(4b^4)^{-\frac{1}{2}}=\frac{1}{2b^2}$ | B1 | Correct simplification of either term; allow $(2b^2)^{-1}$ for the second term |
| $2b^{-1}$ or $\frac{2}{b}$ | B1 | Correct final answer |
| **[2]** | | |
## Part (c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $9^{3c}=3^{6c}$ | B1 | Either $9^c$ or $27^{2c}$ correct as a power of 3 (or 729); ignore coefficient; index must be simplified |
| $27^{2c}=3^{6c}$ | M1 | Attempt to write the other one of $9^c$ and $27^{2c}$ with the same base; ignore coefficient; allow unsimplified index |
| | B2 | for $27^{2c}=9^{3c}$ |
| $7\times3^{6c}-4\times3^{6c}=3\times3^{6c}$ | A1 | Combine to obtain correct single term; allow equiv eg $3\times729^c$ or $3\times27^{2c}$ or $3\times9^{3c}$ |
| $=3^{6c+1}$ | A1 | Obtain correct final answer; must be single power of 3 |
| **[4]** | | |
| **OR:** B1 $9^{2c}(7\times9^c-4\times3^{2c})$; M1 $9^{2c}(7\times3^{2c}-4\times3^{2c})$; $9^{2c}\times3\times3^{2c}$; A1 $3\times27^{2c}$; A1 $3^{6c+1}$ | | |
---2 Simplify fully.
\begin{enumerate}[label=(\alph*)]
\item $\sqrt { 12 a } \times \sqrt { 3 a ^ { 5 } }$
\item $\left( 64 b ^ { 3 } \right) ^ { \frac { 1 } { 3 } } \times \left( 4 b ^ { 4 } \right) ^ { - \frac { 1 } { 2 } }$
\item $7 \times 9 ^ { 3 c } - 4 \times 27 ^ { 2 c }$
\end{enumerate}
\hfill \mbox{\textit{OCR H240/01 2020 Q2 [8]}}