Moderate -0.3 This is a standard discriminant problem requiring students to set the equations equal, rearrange to standard form, and apply b²-4ac > 0. While it involves multiple algebraic steps and careful manipulation, it's a routine AS-level technique with no novel insight required—slightly easier than average due to its predictable structure.
A curve has equation
$$y = 2x^2 + px + 1$$
A line has equation
$$y = 5x - 2$$
Find the set of values of \(p\) for which the line intersects the curve at two distinct points.
Give your answer in exact form.
[5 marks]
Question 6:
6 | Equates the equation of the
curve to the equation of the line | 1.1a | M1 | 2x2 + px + 1 = 5x – 2
2x2 + (p – 5)x + 3 = 0
Discriminant is (p – 5)2 – 24
= p2 – 10p + 1
p2 – 10p + 1 > 0
p > 5 + 2 6 or p < 5 – 2 6
Obtains the correct quadratic in
form f(x) = 0
ACF | 1.1b | A1
Obtains (p – 5)2 – 24
ACF | 1.1b | A1
Sets their discriminant to be > 0
Condone non-strict inequality
here, but discriminant cannot
contain terms in x
Or
Solves their discriminant = 0 to
obtain exact values of p | 1.1a | M1
Obtains correct inequalities
ACF but must be exact | 1.1b | A1
Question 6 Total | 5
Q | Marking instructions | AO | Marks | Typical solution
A curve has equation
$$y = 2x^2 + px + 1$$
A line has equation
$$y = 5x - 2$$
Find the set of values of $p$ for which the line intersects the curve at two distinct points.
Give your answer in exact form.
[5 marks]
\hfill \mbox{\textit{AQA AS Paper 2 2023 Q6 [5]}}