C1 June 2005 Q8
8. The line \(l_1\) passes through the point \((9, -4)\) and has gradient \(\frac{1}{3}\).
(a) Find an equation for \(l_1\) in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers. (3)
The line \(l_2\) passes through the origin \(O\) and has gradient \(-2\). The lines \(l_1\) and \(l_2\) intersect at the point \(P\).
(b) Calculate the coordinates of \(P\). (4)
Given that \(l_1\) crosses the \(y\)-axis at the point \(C\),
(c) calculate the exact area of \(\triangle OCP\). (3)
| Scheme | Marks |
|---|---|
| \(y - (-4) = \dfrac{1}{3}(x - 9)\) | M1 A1 |
| \(3y - x + 21 = 0\) (o.e.) (condone 3 terms with integer coefficients e.g. \(3y + 21 = x\)) | A1 |
| (3) |
Notes
(a) M1 for full method to find equation of \(l_1\)
1stA1 any unsimplified form
| Scheme | Marks |
|---|---|
| Equation of \(l_2\) is: \(y = -2x\) (o.e.) | B1 |
| Solving \(l_1\) and \(l_2\): \(-6x - x + 21 = 0\) | M1 |
| \(p\) is point where \(x_p = 3,\ \ y_p = -6\) \(x_p\) or \(y_p\) | A1 |
| \(y_p\) or \(x_p\) | A1f.t. |
| (4) |
Notes
(b) M1 Attempt to solve two linear equations leading to linear equation in one variable
2nd A1 f.t. only f.t. their \(x_p\) or \(y_p\) in \(y = -2x\)
| Scheme | Marks |
|---|---|
| \(\left(l_1 \text{ is } y = \dfrac{1}{3}x - 7\right)\) \(C\) is \((0, -7)\) or \(OC = 7\) | B1f.t. |
| Area of \(\Delta OCP = \dfrac{1}{2}OC\times x_p,\ = \dfrac{1}{2}\times 7\times 3 = 10.5\) or \(\dfrac{21}{2}\) | M1 A1c.a.o. |
| (3) | |
| (10 marks) |
Notes
(c) B1f.t. Either a correct \(OC\) or f.t. from their \(l_1\)
M1 for correct attempt in letters or symbols for \(\Delta OCP\)
A1 c.a.o.
\(-\dfrac{1}{2}\times 7\times 3\) scores M1 A0