| Exam Board | Edexcel |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2022 |
| Session | October |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Polynomial with rational/modulus curves |
| Difficulty | Moderate -0.3 This is a standard P1 curve sketching question requiring routine techniques: finding intercepts, identifying key features (repeated root at x=-4), and sketching a reciprocal curve. Part (b) involves counting intersections visually from the sketches. While multi-part, it requires only straightforward application of basic sketching principles with no novel problem-solving or deep conceptual insight, making it slightly easier than average. |
| Spec | 1.05b Sine and cosine rules: including ambiguous case1.05c Area of triangle: using 1/2 ab sin(C)1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Positive cubic shape with local max and local min | B1 | Do not be concerned about location of minimum; condone aspects appearing linear |
| Intersects x-axis at \(x = \frac{k}{2}\) to right of y-axis; local max/min on \(x = -4\) to left of y-axis; labelled \((-4, 0)\) and \(\left(\frac{k}{2}, 0\right)\) | B1 | If contradiction between graph labels and separately stated values, graph labels take precedence |
| y-intercept is \(-16k\) (below x-axis); labelled \((0, -16k)\) | B1 | Graph label takes precedence if contradiction exists |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Correct shaped curve in first or second quadrant; must not cross either axis | B1 | Mark intention to draw graph without clear turning point |
| Correct curves in first and second quadrants; no curves in quadrants 3 and 4; graph tends towards axes | B1 | Withhold if non-coordinate-axis asymptotes are labelled |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| One root because the two graphs intersect each other once | B1 | Only awarded if both graphs correct shape/position in (a); condone alternative wording implying "meet" once; do not allow references to intersecting axes |
## Question 6(a)(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Positive cubic shape with local max and local min | B1 | Do not be concerned about location of minimum; condone aspects appearing linear |
| Intersects x-axis at $x = \frac{k}{2}$ to right of y-axis; local max/min on $x = -4$ to left of y-axis; labelled $(-4, 0)$ and $\left(\frac{k}{2}, 0\right)$ | B1 | If contradiction between graph labels and separately stated values, graph labels take precedence |
| y-intercept is $-16k$ (below x-axis); labelled $(0, -16k)$ | B1 | Graph label takes precedence if contradiction exists |
## Question 6(a)(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Correct shaped curve in first **or** second quadrant; must not cross either axis | B1 | Mark intention to draw graph without clear turning point |
| Correct curves in first **and** second quadrants; no curves in quadrants 3 and 4; graph tends towards axes | B1 | Withhold if non-coordinate-axis asymptotes are labelled |
## Question 6(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| One root because the two graphs intersect each other once | B1 | Only awarded if both graphs correct shape/position in (a); condone alternative wording implying "meet" once; do not allow references to intersecting axes |\begin{enumerate}
\item (a) Given that $k$ is a positive constant such that $0 < k < 4$ sketch, on separate axes, the graphs of\\
(i) $y = ( 2 x - k ) ( x + 4 ) ^ { 2 }$\\
(ii) $y = \frac { k } { x ^ { 2 } }$\\
showing the coordinates of any points where the graphs cross or meet the coordinate axes, leaving coordinates in terms of $k$, where appropriate.\\
(b) State, with a reason, the number of roots of the equation
\end{enumerate}
$$( 2 x - k ) ( x + 4 ) ^ { 2 } = \frac { k } { x ^ { 2 } }$$
\hfill \mbox{\textit{Edexcel P1 2022 Q6 [6]}}