Higher June 2025 Paper 1R Q24
24 The first 3 terms of an arithmetic series are
\((2x + 5)\) \((3y - 4)\) \((4x - 2)\)
where \(x\) and \(y\) are constants.
The sum of the first 9 terms of the series is 216
Find the value of \(x\) and the value of \(y\)
Show clear algebraic working.
(6)
| Scheme | Marks |
|---|---|
| eg \((d =)\;(3y - 4) - (2x + 5)\;(= 3y - 2x - 9)\) or \((2x + 5) + d = 3y - 4\) oe or \((d =)\;(4x - 2) - (3y - 4)\;(= 4x - 3y + 2)\) or \((3y - 4) + d = (4x - 2)\) or \((2d =)\;(4x - 2) - (2x + 5)\;(= 2x - 7)\) or \((2x + 5) + 2d = (4x - 2)\) oe | M1 |
eg \(216 = \dfrac{9}{2}\left[2(2x + 5) + (9 - 1)d\right]\) or \(216 = \dfrac{9}{2}\left[2(2x + 5) + (9 - 1)\text{``}{3y - 2x - 9}\text{''}\right]\) or \(216 = \dfrac{9}{2}\left[2(2x + 5) + (9 - 1)\text{``}{4x - 3y + 2}\text{''}\right]\) or \(216 = \dfrac{9}{2}\left[2(2x + 5) + (9 - 1)\text{``}{x - 3.5}\text{''}\right]\) or \(216 = \dfrac{9}{2}\left[2(2\text{``}{d + 3.5}\text{''} + 5) + 8d\right]\) | M1 |
| eg \(6x - 6y = -11\) oe and \(12y - 6x = 55\) oe or \(6x - 6y = -11\) oe and \(18x - 12y = 11\) oe or \(2x - 2d = 7\) oe and \(8d + 4x = 38\) oe or eg \(d = x - 3.5\) oe and \(48 = 12x - 18\) oe or \(x = d + 3.5\) oe and \(48 = 12d + 24\) oe or \(d = 4.75 - 0.5x\) oe and \(3x - 2 = 14.5\) oe one equation must be from the differences and one equation must be from the sum of the terms | M2 |
Working required Answer: \(x = \dfrac{11}{2}\) \(y = \dfrac{22}{3}\) | A2 |
| (6) | |
| (6 marks) |
Notes
M1: for a correct expression or equation using the common difference, may be in terms of \(d\) for this mark we will allow an expression for \(-d\) or \(-2d\)
M1: for a correct equation for the sum of 9 terms in \(x\) and \(d\) or in terms of \(x\) and \(y\) or in terms of \(x\) or in terms of \(d\)
for “\(3y - 2x - 9\)” we will allow \((3y - 4) - (2x + 5)\) or for using their incorrect simplification from \((3y - 4) - (2x + 5)\) shown
for “\(4x - 3y + 2\)” we will allow \((4x - 2) - (3y - 4)\) or for using their incorrect simplification from \((4x - 2) - (3y - 4)\) shown
similarly for their “\(x - 3.5\)” and their “\(d + 3.5\)”
M2: left hand column 2 correct equations in terms of \(x\) and \(y\) in the form \(px + qy = r\) oe or 2 correct equations in terms of \(x\) and \(d\) in the form \(px + qd = r\) oe
right hand column 2 correct equations in terms of \(x\) and \(y\) where one is substituted into the other to get a correct equation in the form \(px + q = r\) or \(py + q = r\) or 2 correct equations in terms of \(x\) and \(d\) where one is substituted into the other to get a correct equation in the form \(px + q = r\) or \(pd + q = r\)
If not M2 then M1 for one correct equation in any of the required forms from the left hand or right hand column