| Exam Board | OCR |
|---|---|
| Module | PURE |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Topic | Polynomial Division & Manipulation |
| Type | Sketching Polynomial Curves |
| Difficulty | Moderate -0.3 This is a straightforward curve sketching question requiring knowledge of standard function shapes (ln x and rectangular hyperbola) and interpreting intersections graphically. The connection between intersections and roots is direct with no algebraic manipulation needed. Slightly easier than average due to being standard bookwork with well-known curves. |
| Spec | 1.02m Graphs of functions: difference between plotting and sketching1.02p Interpret algebraic solutions: graphically1.06d Natural logarithm: ln(x) function and properties |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Correct curve for \(y = kx^{-1}\) in both quadrants | B2 (1.1, 1.1) | Correct shape, rotational symmetry about \(O\), not touching axes; asymptote clearly the axes; not finite (extending to ends of axes); allow slight movement away from asymptote at one end but not more. N.B. Ignore 'feathering'. B1 only – correct shape in 1st and 3rd quadrants only. Graph must not touch axes more than once. Finite 'plotting' condoned. |
| Correct curve for \(y = \ln x\) in 1st and 4th quadrants – asymptote clearly the \(y\)-axis | B1 | For full marks curves should only intersect once in the 1st quadrant |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(x\ln x - k = 0\) is a rearrangement of \(\ln x = \frac{k}{x}\) | B1 | Either explained or shown algebraically |
| The curves intersect at a single point and so therefore the equation \(x\ln x - k = 0\) has only one real root | B1 FT | Follow through from incorrect curves provided they only intersect once. Intersection in the 1st quadrant |
# Question 2:
## Part (a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Correct curve for $y = kx^{-1}$ in both quadrants | B2 (1.1, 1.1) | Correct shape, rotational symmetry about $O$, not touching axes; asymptote clearly the axes; not finite (extending to ends of axes); allow slight movement away from asymptote at one end but not more. **N.B.** Ignore 'feathering'. **B1** only – correct shape in 1st and 3rd quadrants only. Graph must not touch axes more than once. Finite 'plotting' condoned. |
| Correct curve for $y = \ln x$ in 1st and 4th quadrants – asymptote clearly the $y$-axis | B1 | For full marks curves should only intersect once in the 1st quadrant |
**[3 marks]**
## Part (b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $x\ln x - k = 0$ is a rearrangement of $\ln x = \frac{k}{x}$ | B1 | Either explained or shown algebraically |
| The curves intersect at a single point and so therefore the equation $x\ln x - k = 0$ has only one real root | B1 FT | Follow through from incorrect curves provided they only intersect once. Intersection in the 1st quadrant |
**[2 marks]**
---2 Two curves have equations $y = \ln x$ and $y = \frac { k } { x }$, where $k$ is a positive constant.
\begin{enumerate}[label=(\alph*)]
\item Sketch the curves on a single diagram.
\item Explain how your diagram shows that the equation $x \ln x - k = 0$ has exactly one real root.
\end{enumerate}
\hfill \mbox{\textit{OCR PURE Q2 [5]}}