Edexcel IALGeneralGeneral Applications10 Marks

The curve `C` has equation `y = 2x^2 + kx + 8`, where `k` is a constant. The line `L` has equation `y = x + 4`. (a) Show that the x-coordinates of any points of intersection of `L` and `C` satisfy the equation `2x^2 + (k-1)x + 4 = 0`. (b) Given that `L` is a tangent to `C`, (i) find the two possible values of `k`. (ii) for each of these values of `k`, find the coordinates of the point of intersection.

(a)Show that the x-coordinates of any points of intersection of `L` and `C` satisfy the equation `2x^2 + (k-1)x + 4 = 0`.[1]
(b)(i)find the two possible values of `k`.[4]
(b)(ii)for each of these values of `k`, find the coordinates of the point of intersection.[5]

Mark scheme

Marking points from the official mark scheme. The mark or mark type is shown in brackets.

B1 for `2x^2 + kx + 8 = x + 4 \Rightarrow 2x^2 + (k-1)x + 4 = 0` (cso, clearly shown).
M1 for using `b^2 - 4ac = 0`.
A1 for `(k-1)^2 - 4(2)(4) = 0`.
A1 for `(k-1)^2 = 32`.
A1 for `k-1 = ± √32 ⇒ k = 1 ± 4√2`.
M1 for substituting one of their `k` values into `2x^2 + (k-1)x + 4 = 0` (and solving for `x`) OR `y = x+4` and `dy/dx` of curve with `dy/dx` of line.
A1 for `x = - ((k-1))/4`. For `k = 1 + 4√2`, `x = -4√2/4 = -√2`.
A1 for `x = √2` (For `k = 1 - 4√2`).
M1 for finding `y` using `y = x+4` (or `y = 2x^2 + kx + 8`).
A1 for `(-√2, 4-√2)` and `(√2, 4+√2)`.

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