October 2021 Paper 3 Q3
3
(a) Determine, in terms of \(k\), the coordinates of the point where the lines with the following equations intersect.
\(x + y = k\)
\(2x - y = 1\) [3]
\(x + y = k\)
\(2x - y = 1\) [3]
(b) Determine, in terms of \(k\), the coordinates of the points where the line \(x + y = k\) crosses the curve \(y = x^2 + k\). [4]
| Scheme | Marks | AO |
|---|---|---|
| \(3x = k + 1\) or \(2k - 3y = 1\) | M1 | 1.1a |
| \(x = \dfrac{k+1}{3}\) oe isw | A1 | 1.1 |
| \(y = \dfrac{2k-1}{3}\) oe isw | A1 | 2.2a |
| [3] |
Notes
M1: Elimination of one of \(x\), \(y\)
A1: May be as part of a pair of coordinates
A1: May be as part of a pair of coordinates
Allow correct unsimplified answers
| Scheme | Marks | AO |
|---|---|---|
| \(k - x = x^2 + k\) | M1 | 3.1a |
| \(x^2 + x = 0 \Rightarrow x = 0,\ x = -1\) | A1 | 1.1 |
| \((0, k)\) | A1 | 1.1 |
| \((-1, k + 1)\) | A1 | 2.1 |
| [4] |
Notes
\((0, k)\) unsupported earns B1 SC
Need not be as coordinates but must be clear which \(y\) goes with which \(x\)
Substituting for \(x\) gets M0 unless it leads to correct values of \(y\)