| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2018 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Discriminant and conditions for roots |
| Type | Find range for no real roots |
| Difficulty | Moderate -0.8 Part (i) is a direct application of the discriminant condition (b² - 4ac < 0) for no real roots, requiring minimal algebraic manipulation. Part (ii) involves setting the curve equal to the line, forming a quadratic, and using discriminant = 0 for tangency—a standard textbook exercise with straightforward steps. Both parts test routine understanding of discriminants with no novel problem-solving required. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.02g Inequalities: linear and quadratic in single variable1.07m Tangents and normals: gradient and equations |
| Answer | Marks | Guidance |
|---|---|---|
| 2 | Use correct tan (A ± B) formula and obtain an equation in tan θ | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| Obtain a correct equation in any form | A1 | With values substituted |
| Reduce to 3 tan2θ = 1, or equivalent | A1 | |
| Obtain answer x = 30° | A1 | One correct solution |
| Obtain answer x = 150° | A1 | Second correct solution and no others in range |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 2:
2 | Use correct tan (A ± B) formula and obtain an equation in tan θ | M1 | 1 +1−tanθtan45 =2 Allow M1 withtan45°
tanθ tanθ+tan45
= 1 +1−tanθ
tanθ tanθ+1
Obtain a correct equation in any form | A1 | With values substituted
Reduce to 3 tan2θ = 1, or equivalent | A1
Obtain answer x = 30° | A1 | One correct solution
Obtain answer x = 150° | A1 | Second correct solution and no others in range
OR: use correct sin ( A±B ) and cos ( A±B ) to form
equation in sinθ and cosθ M1A1
Reduce to tan2θ=1 ,sin2θ= 1 ,cos2θ= 3 orcot2θ=3 A1
3 4 4
etc.
5
Question | Answer | Marks | GuidanceThe equation of a curve is $y = x^2 - 6x + k$, where $k$ is a constant.
\begin{enumerate}[label=(\roman*)]
\item Find the set of values of $k$ for which the whole of the curve lies above the $x$-axis. [2]
\item Find the value of $k$ for which the line $y + 2x = 7$ is a tangent to the curve. [3]
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2018 Q2 [5]}}