Standard +0.3 This is a straightforward discriminant problem requiring substitution of the linear equation into the quadratic, rearranging to standard form, and showing the discriminant is non-negative for all k. While it involves multiple algebraic steps and the concept of discriminant conditions, it's a standard technique taught explicitly in P1 with no novel insight required—slightly easier than average.
Question 4:
4 | Substitute for y (or x) in first equation and simplify | *M1 | All terms to one side and brackets expanded.
Obtain 10x2 +3kx−40 [= 0] (or 10y2+11ky+k2−360 =0 ) | A1
Attempt b2 −4ac for 3-term quadratic involving k | DM1 | Not in quadratic formula unless b2 −4acis
isolated.
Obtain 9k2 +1600 (or 81k2 +14400) | A1
9k2 +1600 0 | A1 FT | FT for ak2 + b 0 with a, b 0.
5
Question | Answer | Marks | Guidance
Show that the curve with equation $x^2 - 3xy - 40 = 0$ and the line with equation $3x + y + k = 0$ meet for all values of the constant $k$. [5]
\hfill \mbox{\textit{CAIE P1 2024 Q4 [5]}}