| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2022 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Inequalities |
| Type | Quadratic equation real roots |
| Difficulty | Moderate -0.8 Part (a) is a standard discriminant problem requiring b²-4ac < 0, yielding a simple inequality. Part (b) is a routine quadratic-in-cosθ substitution with straightforward solutions. Both parts involve direct application of well-practiced techniques with no problem-solving insight required, making this easier than average. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.02f Solve quadratic equations: including in a function of unknown1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(k^2 - 4 \times 8 \times 2\ [< 0]\) | M1 | Use of \(b^2 - 4ac\) but not just in the quadratic formula. |
| \(-8 < k < 8\) or \(-8 < k,\ k < 8\) or \( | k | < 8\) or \((-8, 8)\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(2(4\cos\theta - 1)(\cos\theta - 1)\) or \((4\cos\theta - 1)(\cos\theta - 1)\) | M1 | OE. Or use of formula or completing the square. Allow use of replacement variable. |
| \(\cos\theta = \frac{2}{8},\ \cos\theta = 1\) | A1 | OE. For both answers. SC: If M0, SC B1 available for sight of \(\cos\theta = \frac{2}{8}\) and \(1\) |
| \([\theta =]\ 0°,\ 75.5°\) | A1 | AWRT ISW rejection of \(0°\). For both answers and no others in range \(0° \leqslant \theta \leqslant 180°\), must be in degrees. SC: If M0 B1 scored, SC B1 available for correct answers. SC: If M1 A0 scored, SC B1 available for \(\cos\theta = \frac{2}{8}\) and \(\theta = 75.5°\) only, WWW. |
## Question 3(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $k^2 - 4 \times 8 \times 2\ [< 0]$ | M1 | Use of $b^2 - 4ac$ but not just in the quadratic formula. |
| $-8 < k < 8$ or $-8 < k,\ k < 8$ or $|k| < 8$ or $(-8, 8)$ | A1 | Condone '$-8 < k$ or $k < 8$', '$-8 < k$ and $k < 8$' but not $\sqrt{64}$. |
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## Question 3(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $2(4\cos\theta - 1)(\cos\theta - 1)$ or $(4\cos\theta - 1)(\cos\theta - 1)$ | M1 | OE. Or use of formula or completing the square. Allow use of replacement variable. |
| $\cos\theta = \frac{2}{8},\ \cos\theta = 1$ | A1 | OE. For both answers. **SC:** If M0, **SC B1** available for sight of $\cos\theta = \frac{2}{8}$ and $1$ |
| $[\theta =]\ 0°,\ 75.5°$ | A1 | AWRT ISW rejection of $0°$. For both answers and no others in range $0° \leqslant \theta \leqslant 180°$, must be in degrees. **SC:** If M0 B1 scored, **SC B1** available for correct answers. **SC:** If M1 A0 scored, **SC B1** available for $\cos\theta = \frac{2}{8}$ and $\theta = 75.5°$ only, WWW. |
---3
\begin{enumerate}[label=(\alph*)]
\item Find the set of values of $k$ for which the equation $8 x ^ { 2 } + k x + 2 = 0$ has no real roots.
\item Solve the equation $8 \cos ^ { 2 } \theta - 10 \cos \theta + 2 = 0$ for $0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2022 Q3 [5]}}