Higher November 2024 Paper 1 Q19
19
(a) Expand and simplify fully \(\quad (x + 2)(2x + 3)(3x + 4)\) [3 marks]
(b) Use your answer from part (a) to work out \(\quad 102 \times 203 \times 304\) [2 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1: multiplies \((x + 2)(2x + 3)\) first | ||
| \(2x^2 + 3x + 4x + 6\) or \(2x^2 + 7x + 6\) | M1 | oe four terms with at least three correct implied by \(2x^2 + 7x + k\) or \(px^2 + 7x + 6\) where \(k\) and \(p\) are a non-zero constants may be seen in a grid |
| \(6x^3 + 9x^2 + 12x^2 + 18x + 8x^2 + 12x + 16x + 24\) or \(6x^3 + 21x^2 + 18x + 8x^2 + 28x + 24\) | M1dep | oe full expansion with correct multiplication of their 3 or 4 terms by \(3x\) or 4 may be seen in a grid |
| \(6x^3 + 29x^2 + 46x + 24\) | A1 | |
| Alternative method 2: multiplies \((2x + 3)(3x + 4)\) first | ||
| \(6x^2 + 8x + 9x + 12\) or \(6x^2 + 17x + 12\) | M1 | oe four terms with at least three correct implied by \(6x^2 + 17x + k\) or \(px^2 + 17x + 12\) where \(k\) and \(p\) are non-zero constants may be seen in a grid |
| \(6x^3 + 8x^2 + 9x^2 + 12x + 12x^2 + 16x + 18x + 24\) or \(6x^3 + 17x^2 + 12x + 12x^2 + 34x + 24\) | M1dep | oe full expansion with correct multiplication of their 3 or 4 terms by \(x\) or 2 may be seen in a grid |
| \(6x^3 + 29x^2 + 46x + 24\) | A1 | |
| Alternative method 3: multiplies \((x + 2)(3x + 4)\) first | ||
| \(3x^2 + 4x + 6x + 8\) or \(3x^2 + 10x + 8\) | M1 | oe four terms with at least three correct implied by \(3x^2 + 10x + k\) or \(px^2 + 10x + 8\) where \(k\) and \(p\) are non-zero constants may be seen in a grid |
| \(6x^3 + 8x^2 + 12x^2 + 16x + 9x^2 + 12x + 18x + 24\) or \(6x^3 + 20x^2 + 16x + 9x^2 + 30x + 24\) | M1dep | oe full expansion with correct multiplication of their 3 or 4 terms by \(2x\) or 3 may be seen in a grid |
| \(6x^3 + 29x^2 + 46x + 24\) | A1 | |
Additional guidance
| M1 may be awarded for correct work with no answer or incorrect answer, even if this is seen amongst multiple attempts | |
| Terms may be in any order throughout | |
| For method marks, terms may be given in a table, with minus signs shown where necessary | |
| 2nd M1 A full expansion will be 8 terms if 4 terms are used in first expansion A full expansion will be 6 terms if 3 terms are used in first expansion | |
| 1st M1 scored with a 4-term expansion followed by incorrect simplification to 3 terms can still score the 2nd M1 using their 3 terms | |
| eg \(2x^2 + 3x + 4x + 6 = 2x^2 + 9x + 6\) (\(9x\) should be \(7x\)) | M1 |
| \((2x^2 + 9x + 6)(3x + 4) = 6x^3 + 27x^2 + 18x + 8x^2 + 36x + 24\) | M1 |
| Answer | Mark | Comments |
|---|---|---|
| Substitution of \(x = 100\) into their cubic expansion of at least three terms | M1 | eg with part (a) correct, \(6 \times 100^3 + 29 \times 100^2 + 46 \times 100 + 24\) or \(6\,000\,000 + 290\,000 + 4600 + 24\) |
| 6 294 624 | A1ft | ft their cubic expansion, which must contain at least 3 terms |
Additional guidance
| Ignore any rounding or truncation after correct answer seen | |
| 6 294 624 from long multiplication | M1A1 |