June 2022 Paper 1 Q11
11 Expand and simplify fully \(\qquad (3 + 2x)^5\) [4 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 | ||
| Evidence of 1 5 10 10 5 1 used for all six coefficients (terms could be written incorrectly) | M1 | the 1s can be ignored but 5 10 10 5 must be seen and used (don’t accept it just being written in Pascal’s triangle) |
| \((3)^5 + 5(3)^4(2x) + 10(3)^3(2x)^2 + 10(3)^2(2x)^3 + 5(3)(2x)^4 + (2x)^5\) | M1dep | oe eg \((3)^5(2x)^0\) written for first term at least 4 terms correct (could already be simplified and missing brackets recovered) |
| \((3)^5 + 5(3)^4(2x) + 10(3)^3(2x)^2 + 10(3)^2(2x)^3 + 5(3)(2x)^4 + (2x)^5\) | M1dep | oe eg \((3)^5(2x)^0\) written for first term all correct |
| \(243 + 810x + 1080x^2 + 720x^3 + 240x^4 + 32x^5\) | A1 | |
| Alternative method 2 | ||
| \((3 + 2x)^2 = 9 + 12x + 4x^2\) | M1 | |
| \((3 + 2x)^3 = 27 + 54x + 36x^2 + 8x^3\) | M1dep | oe the terms may not have been collected could do \((3 + 2x)^2 \times (3 + 2x)^2\). If they use this method (doesn’t refer to \((3 + 2x)^3\)) then award this mark for answer expanded correctly but with one numerical error. Terms must be collected |
| \((3 + 2x)^4 = 81 + 216x + 216x^2 + 96x^3 + 16x^4\) | M1dep | terms must be collected could do \((3 + 2x)^2 \times (3 + 2x)^3\). If they use this method (doesn’t refer to \((3 + 2x)^4\)) then award this mark for answer expanded correctly but with one numerical error. Terms must be collected would imply first 2 M marks if done correctly |
| \(243 + 810x + 1080x^2 + 720x^3 + 240x^4 + 32x^5\) | A1 | |
| Alternative method 3 | ||
| Evidence of 1 5 10 10 5 1 used for all six coefficients (could be written incorrectly) | M1 | the 1s can be ignored but 5 10 10 5 must be seen and used |
| \(a^5 + 5a^4b + 10a^3b^2 + 10a^2b^3 + 5ab^4 + b^5\) | M1dep | from using a general expansion of \((a + b)^5\) |
| \((3)^5 + 5(3)^4(2x) + 10(3)^3(2x)^2 + 10(3)^2(2x)^3 + 5(3)(2x)^4 + (2x)^5\) | M1dep | oe all correct |
| \(243 + 810x + 1080x^2 + 720x^3 + 240x^4 + 32x^5\) | A1 | |
Additional guidance
| Working could be seen as a list or a grid. This can be awarded full marks if done correctly Candidates could use a combination of methods. Use whichever alt method works best (probably alt 2) Missing brackets must be recovered | M3A1 |