Higher June 2017 Paper 5 Q15
15
(a) Simplify fully.
(i) \(\sqrt{50} + \sqrt{2}\) [2]
(ii) \(\dfrac{10}{\sqrt{6}}\) [2]
(b) There are two errors in Sam’s method for finding the value of \(64^{-\frac{2}{3}}\) shown below.
Find the cube root of 64 and then multiply by 2.
The cube root of 64 is 4 and then 4 x 2 = 8.
The negative power makes the answer negative so answer equals –8.
The cube root of 64 is 4 and then 4 x 2 = 8.
The negative power makes the answer negative so answer equals –8.
Describe these errors and then give the correct value of \(64^{-\frac{2}{3}}\). [3]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| (i) \(6\sqrt{2}\) final answer | 2 | M1 for \(\sqrt{25 \times 2}\) or better seen | |
| (ii) \(\dfrac{5\sqrt{6}}{3}\) final answer | 2 | M1 for \(\frac{10}{\sqrt{6}} \times \frac{\sqrt{6}}{\sqrt{6}}\) oe | For 2 marks accept \(1\frac{2}{3}\sqrt{6}\) |
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| Identifies both errors and explains the correct steps e.g. Square not multiply by 2 oe negative power does not make answer negative it should be the reciprocal oe | 2 | B1 for each with no incorrect statement for either | Accept implication of error by a description of correct step e.g. should be squared should be reciprocal, should be 1/\(n\), should be 1/64 Descriptions must be in words do not accept numeric examples alone SEE APPENDIX |
| \(\dfrac{1}{16}\) | 1 | isw attempt to convert to decimal | |
Appendix: Exemplar responses for Q15b
Accept reciprocal or one over for description of negative power, must use squared. Descriptions must be in words
| Response | Mark | ||
|---|---|---|---|
| 1 | Make the fraction reciprocal | (Not clear as the fraction could be the index) | 0 |
| 2 | Negative power means find the inverse | 0 | |
| 3 | Make it a reciprocal | 1 | |
| 4 | The negative power makes it positive | 0 | |
| 5 | Make it one over | 1 | |
| 6 | He multiplied the power by 2 | 0 | |
| 7 | Need to turn into fraction ; The negative makes it a fraction | 0 | |
| 8 | The –2 is the power so it must be squared | 1 | |
| 9 | Square not times 2 | 1 | |
| 10 | Cube root should be squared and the negative power is 1 under | ‘1 under’ is not quite right | 1, 0 |
| 11 | Top fraction is to square not double, and the negative makes it 1 over | 1, 1 | |
| 12 | Negative powers don’t make negative numbers. | Need to describe corrective step | 0 |
| 13 | It is not × 2, it is to the power of 2 | No – must say ‘square’ | 0 |