| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2009 |
| Session | November |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Inequalities |
| Type | Line-curve intersection conditions |
| Difficulty | Standard +0.3 This is a straightforward multi-part question covering standard P1 techniques: solving simultaneous equations (quadratic), finding tangent equations using differentiation, calculating angles between lines using gradients, and applying discriminant conditions for non-intersection. All parts follow routine procedures with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.08b Integrate x^n: where n != -1 and sums |
| Answer | Marks | Guidance |
|---|---|---|
| Sim equations \(\rightarrow 2x^2 - 9x + 9 = 0 \rightarrow x = 3\) or \(x = 1\frac{1}{2}\). | M1, DM1 A1 [3] | Complete elimination of x or y. Correct method for quadratic. co. |
| (b) \(\frac{dy}{dx} = 2x - 4 \rightarrow y' - 4 = 2(x - 3)\) | B1, M1 A1 [3] | co. Correct form of eqn with m numeric. co |
| Answer | Marks | Guidance |
|---|---|---|
| \(m = \frac{1}{2} \rightarrow\) angle of \(26.6° \rightarrow\) angle between \(= 37°\) | M1, M1A1 [3] | Finds angle with x-axis once. Subtracts two angles. co. |
| \((i+2j).(2i+j) \rightarrow 4 = \sqrt{5}\sqrt{5}\cos\theta\) M1M1A1 or use of \(\tan(A-B)\) M2A1 or Cosine rule with 3 sides found. |
| Answer | Marks | Guidance |
|---|---|---|
| Key value is \(k = 3.875\) or \(31/8\). \(k < 3.875\) | M1 A1, M1, A1 [4] | Eliminates y or x completely. Co (= 0). Uses \(b^2 - 4ac = 0\), or \(< 0\) or \(> 0\). Co condone \(\leq\). |
**(i)** (a) $2y = x + 5$, $y = x^2 - 4x + 7$
Sim equations $\rightarrow 2x^2 - 9x + 9 = 0 \rightarrow x = 3$ or $x = 1\frac{1}{2}$. | M1, DM1 A1 [3] | Complete elimination of x or y. Correct method for quadratic. co.
(b) $\frac{dy}{dx} = 2x - 4 \rightarrow y' - 4 = 2(x - 3)$ | B1, M1 A1 [3] | co. Correct form of eqn with m numeric. co
nb use of $y + 4$ or x, y interchanged M1 A0
(c) $m = 2 \rightarrow$ angle of $63.4°$
$m = \frac{1}{2} \rightarrow$ angle of $26.6° \rightarrow$ angle between $= 37°$ | M1, M1A1 [3] | Finds angle with x-axis once. Subtracts two angles. co.
$(i+2j).(2i+j) \rightarrow 4 = \sqrt{5}\sqrt{5}\cos\theta$ M1M1A1 or use of $\tan(A-B)$ M2A1 or Cosine rule with 3 sides found. |
**(ii)** $y = x^2 - 4x + 7$ $2y = x + k$
Sim eqns $\rightarrow 2x^2 - 9x + 14 - k = 0$
Uses $b^2 - 4ac$, $81 - 8(14 - k)$
Key value is $k = 3.875$ or $31/8$. $k < 3.875$ | M1 A1, M1, A1 [4] | Eliminates y or x completely. Co (= 0). Uses $b^2 - 4ac = 0$, or $< 0$ or $> 0$. Co condone $\leq$.
**Total: [16]**10\\
\begin{tikzpicture}[>=latex, thick]
% Draw x and y axes
\draw[->] (-1.5, 0) -- (6.5, 0) node[right] {$x$};
\draw[->] (0, -1) -- (0, 9.5) node[above] {$y$};
% Origin label
\node[below left] at (0, 0) {$O$};
% Plot the parabola: y = x^2 - 4x + 7
\draw[domain=-0.4:4.4, smooth, samples=100]
plot ({\x}, {\x*\x - 4*\x + 7})
node[right] {$y = x^2 - 4x + 7$};
% Plot the straight line: 2y = x + 5
\draw[domain=-1.5:5.5]
plot ({\x}, {0.5*\x + 2.5})
node[right] {$2y = x + 5$};
% Intersection Points A and B
\node[below=2pt] at (1.5, 3.25) {$A$};
\node[below right=1pt] at (3, 4) {$B$};
\end{tikzpicture}\\
(i) The diagram shows the line $2 y = x + 5$ and the curve $y = x ^ { 2 } - 4 x + 7$, which intersect at the points $A$ and $B$. Find
\begin{enumerate}[label=(\alph*)]
\item the $x$-coordinates of $A$ and $B$,
\item the equation of the tangent to the curve at $B$,
\item the acute angle, in degrees correct to 1 decimal place, between this tangent and the line $2 y = x + 5$.\\
(ii) Determine the set of values of $k$ for which the line $2 y = x + k$ does not intersect the curve $y = x ^ { 2 } - 4 x + 7$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2009 Q10 [13]}}