C1 June 2013 Q6
6. The straight line \(L_1\) passes through the points \((-1, 3)\) and \((11, 12)\).
where \(a\), \(b\) and \(c\) are integers. (4)
The line \(L_2\) has equation \(3y + 4x - 30 = 0\).
| Scheme | Marks |
|---|---|
| \((-1, 3)\ ,\ (11, 12)\) | |
| \(m = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{12 - 3}{11 - (-1)},\ = \dfrac{3}{4}\) | M1,A1 |
| \(y - 3 = \tfrac{3}{4}(x + 1)\) or \(y - 12 = \tfrac{3}{4}(x - 11)\) or \(y = \tfrac{3}{4}x + c\) with attempt at substitution to find \(c\) | M1 |
| \(4y - 3x - 15 = 0\) | A1 |
| This A1 should only be awarded in (a) | |
| (4) |
Notes
M1: Correct method for the gradient
A1: Any correct fraction or decimal
M1: Correct straight line method using either of the given points and a numerical gradient.
A1: Or equivalent with integer coefficients (= 0 is required)
(a) Way 2
| Scheme | Marks |
|---|---|
| \(\dfrac{y - y_1}{y_2 - y_1} = \dfrac{x - x_1}{x_2 - x_1} \Rightarrow \dfrac{y - 3}{12 - 3} = \dfrac{x + 1}{11 + 1}\) | M1A1 |
| \(12(y - 3) = 9(x + 1)\) | M1 |
| \(4y - 3x - 15 = 0\) | A1 |
| (4) |
M1: Use of a correct formula for the straight line
A1: Correct equation
M1: Eliminates fractions
A1: Or equivalent with integer coefficients (= 0 is required)
(b) Way 2
(This method finds the equation of \(L_1\) for part (a); it is printed as “(b) Way 2” in the mark scheme.)
| Scheme | Marks |
|---|---|
| \((-1, 3) \to -a + 3b + c = 0\) \((11, 12) \to 11a + 12b + c = 0\) | M1 |
| \(\therefore a = -\dfrac{3}{4}b,\ b = -\dfrac{4}{15}c\) | A1 |
| e.g. \(c = 1 \Rightarrow b = -\dfrac{4}{15},\ a = \dfrac{3}{15}\) | M1 |
| \(\dfrac{3}{15}x - \dfrac{4}{15}y + 1 = 0 \Rightarrow 4y - 3x - 15 = 0\) | A1 |
| (4) |
M1: Substitutes the coordinates to obtain two equations
A1: Obtains sufficient equations to establish values for \(a\), \(b\) and \(c\)
M1: Obtains values for \(a\), \(b\) and \(c\)
A1: Correct equation
| Scheme | Marks |
|---|---|
| Solves their equation from part (a) and \(L_2\) simultaneously to eliminate one variable | M1 |
| \(x = 3\) or \(y = 6\) | A1 |
| Both \(x = 3\) and \(y = 6\) | A1 |
| Fully correct answers with no working can score 3/3 in (b) | |
| (3) | |
| (7 marks) |
Notes
M1: Must reach as far as an equation in \(x\) only or in \(y\) only. (Allow slips in the algebra)
A1: One of \(x = 3\) or \(y = 6\)
A1: Values can be un-simplified fractions.