June 2018 Paper 2 Q14
14
(a) Factorise fully \(\qquad 12pq^3r - 18pq^2r^2 + 24pq^2r\) [2 marks]
(b) Factorise fully \(\qquad 6(y + 3)^5 + 4(y + 3)^4\)
Give your answer in its simplest form.
Do not attempt to expand \((y + 3)^5\) or \((y + 3)^4\) [3 marks]
(c) Factorise fully \(\qquad 48 - 75x^2\) [2 marks]
| Answer | Mark | Comments |
|---|---|---|
| \(6pq^2r\,(2q - 3r + 4)\) | B2 | B1 correct factorised expression with a common factor involving at least two variables eg \(pq(12q^2r - 18qr^2 + 24qr)\) or \(2q^2r\,(6pq - 9pr + 12p)\) or common factor \(6pq^2r\) with two out of the three terms in the bracket correct eg \(6pq^2r\,(2q - 3r + 4p)\) |
Additional guidance
| B2 answer followed by further work | B1 |
| \(6pq^2r\,(2q - 3r + 4)\) in working with \(6qp^2r\,(2q - 3r + 4)\) on answer line | B1 |
| B1 answer followed by further work | B1 |
| \(2q^2r\,(6pq - 9pr + 12p)\) in working with \(2p^2r\,(6pq - 9pr + 12p)\) on answer line | B1 |
| Use of multiplication signs scores a maximum of B1 | |
| \(qpq(12qr - 18r^2 + 24r)\) | B1 |
| \(6pqrq\,(2q - 3r + 4)\) | B1 |
| Answer | Mark | Comments |
|---|---|---|
| Correct factorised expression with a common factor | M1 | eg \((y + 3)\,[6(y + 3)^4 + 4(y + 3)^3]\) or \(2[3(y + 3)^5 + 2(y + 3)^4]\) or \(2(y + 3)^2\,[3(y + 3)^3 + 2(y + 3)^2]\) |
| \(2(y + 3)^4\,[3(y + 3) + 2]\) or \(2(y + 3)^4\,(3y + 9 + 2)\) or \((y + 3)^4\,[6(y + 3) + 4]\) or \((y + 3)^4\,(6y + 18 + 4)\) or \((y + 3)^4\,(6y + 22)\) | A1 | |
| \(2(y + 3)^4\,(3y + 11)\) | A1 |
Additional guidance
| Use of multiplication signs scores a maximum of M1A1A0 | |
| Any combination of bracket shape may be used | |
| Correct answer followed by further work | M1A1A0 |
| Incorrect notation eg \((y + 3)^4\,2(3y + 11)\) | M1A1A0 |
| \((2)(y + 3)^4\,(3y + 11)\) or \((2(y + 3)^4)(3y + 11)\) | M1A1A1 |
| Allow substitution eg \(n = (y + 3)\) for M1A1 but must revert to \((y + 3)\) for final mark | |
| Missing brackets must be recovered eg \((y + 3)^4\,6y + 22\) with M1 not seen | Zero |
| Answer | Mark | Comments |
|---|---|---|
| \(3(4 + 5x)(4 - 5x)\) or \(3(-4 - 5x)(5x - 4)\) or \(-3(4 + 5x)(5x - 4)\) or \(-3(-4 - 5x)(4 - 5x)\) | B2 | B1 Partial factorisation eg \(3(16 - 25x^2)\) or \(-3(25x^2 - 16)\) or \((12 + 15x)(4 - 5x)\) or \((12 - 15x)(4 + 5x)\) |
Additional guidance
| Brackets in either order for B2 or B1 | |
| \(-(75x^2 - 48)\) | B0 |
| \((-5x + 4)\) is equivalent to \((4 - 5x)\) etc | |
| Incorrect notation eg \((4 + 5x)3(4 - 5x)\) | B1 |
| Use of surds eg \((\sqrt{48} + \sqrt{75}\,x)(\sqrt{48} - \sqrt{75}\,x)\) or \((4\sqrt{3} + 5\sqrt{3}\,x)(4\sqrt{3} - 5\sqrt{3}\,x)\) | B1 |
| Use of multiplication signs scores a maximum of B1 eg \(3 \times (4 + 5x)(4 - 5x)\) | B1 |
| B2 answer followed by further work | B1 |
| B1 answer followed by further work | B1 |
| Missing brackets must be recovered eg \(3 \times 16 - 25x^2\) | B0 |