Higher June 2023 Paper 2 Q17
17
(a) Expand and simplify \(\quad (x + 6)(3x - 2)(x + 6)\) (3)
(b) Make \(e\) the subject of \(\quad w = \sqrt{\dfrac{e + g}{ef - d}}\) (4)
| Scheme | Marks |
|---|---|
| eg \(3x^2 - 2x + 18x - 12\) (\(= 3x^2 + 16x - 12\)) or \(x^2 + 6x + 6x + 36\) (\(= x^2 + 12x + 36\)) (allow in a table with no sign indicating +) | M1 |
| eg \(3x^3 + 36x^2 + 108x - 2x^2 - 24x - 72\) or \(3x^3 + 18x^2 + 18x^2 + 108x - 2x^2 - 12x - 12x - 72\) or \(3x^3 + 16x^2 - 12x + 18x^2 + 96x - 72\) or \(3x^3 - 2x^2 + 18x^2 - 12x + 18x^2 - 12x + 108x - 72\) | M1ft |
| Correct answer scores full marks (unless from obvious incorrect working) Answer: \(3x^3 + 34x^2 + 84x - 72\) | A1 |
| (3) |
Notes
M1: for a correct method to expand two brackets with at least 3 terms correct out of 4 terms seen (or 2 terms correct out of 3 terms seen)
M1ft: ft dep on M1 and a quadratic for a correct method to multiply by the 3rd bracket allow one further error
A1: If no working shown then award B2 for 3 out of a maximum of 4 terms correct
17(a) ALTERNATIVE
| Scheme | Marks |
|---|---|
| \(3x^3 - 2x^2 + 18x^2 - 12x + 18x^2 - 12x + 108x - 72\) | M2 |
| \(3x^3 + 34x^2 + 84x - 72\) | A1 |
| (3) |
Notes
M2: For a complete expansion with 8 terms present of which 4 are correct
(M1 for 4 correct terms from any number of terms)
| Scheme | Marks |
|---|---|
| \(w^2 = \dfrac{e + g}{ef - d}\) | M1 |
| \(w^2ef - w^2d = e + g\) oe | M1 |
| \(w^2ef - e = g + w^2d\) oe | M1ft |
Correct answer scores full marks (unless from obvious incorrect working) Answer: \(e = \dfrac{g + w^2d}{w^2f - 1}\) | A1 |
| (4) | |
| (7 marks) |
Notes
M1: for removing square root
M1: for multiplying by denominator and expanding in a correct equation
M1ft: ft their equation dep on 2 terms in \(e\) and two other terms
for gathering terms in \(e\) on one side and other terms the other side
A1: oe eg \(e = \dfrac{-g - w^2d}{1 - w^2f}\), \(e = -\dfrac{g + w^2d}{1 - w^2f}\) oe
must see \(e\) = on answer line or in working.