| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2016 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Function Transformations |
| Type | Completing square for transformations |
| Difficulty | Moderate -0.3 This is a standard multi-part Pure 1 question covering routine techniques: solving a quadratic inequality, tangent condition (discriminant = 0), completing the square, and finding an inverse function. All parts follow textbook methods with no novel insight required, making it slightly easier than average but still requiring competent execution across multiple topics. |
| Spec | 1.02e Complete the square: quadratic polynomials and turning points1.02g Inequalities: linear and quadratic in single variable1.02h Express solutions: using 'and', 'or', set and interval notation1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence1.07m Tangents and normals: gradient and equations |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(6x-x^2-5\leqslant 3\) | ||
| \(\rightarrow x^2-6x+8\geqslant 0\) | M1 | \(\pm(6x-x^2-8)=,\leqslant,\geqslant 0\) and attempts to solve |
| \(\rightarrow x=2,\ x=4\) | A1 | Needs both values whether \(=2\), \(<2\), \(>2\) |
| \(x\leqslant 2,\ x\geqslant 4\) | A1 [3] | Accept all recognisable notation; condone \(<\) and/or \(>\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Equate \(mx+c\) and \(6x-x^2-5\); use of \(b^2-4ac\) | M1, DM1 | Equates, sets to \(0\); use of discriminant with values of \(a,b,c\) independent of \(x\) |
| \(4c=m^2-12m+16\) AG | A1 [3] | \(=0\) must appear before last line |
| OR | ||
| \(\frac{dy}{dx}=6-2x=m \rightarrow x=\left(\frac{6-m}{2}\right)\) | M1 | Equates \(\frac{dy}{dx}\) to \(m\) and rearrange |
| \(m\!\left(\frac{6-m}{2}\right)+c=6\!\left(\frac{6-m}{2}\right)-\!\left(\frac{6-m}{2}\right)^2\!-5\) | M1 | Equates \(mx+c\) and \(6x-x^2-5\) and substitutes for \(x\) |
| \(4c=m^2-12m+16\) AG | A1 [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(6x-x^2-5=4-(x-3)^2\) | B1 B1 [2] | \(4\,\text{B1}\); \(-(x-3)^2\,\text{B1}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(k=3\) | B1\(^\checkmark\) [1] | \(\checkmark\) for "\(b\)" |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(g^{-1}(x)=\sqrt{4-x}+3\) | M1 A1 [2] | Correct order of operations; \(\pm\sqrt{4-x}+3\) M1A0; \(\sqrt{x-4}+3\) M1A0; \(\sqrt{4-y}+3\) M1A0 |
# Question 11(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $6x-x^2-5\leqslant 3$ | | |
| $\rightarrow x^2-6x+8\geqslant 0$ | **M1** | $\pm(6x-x^2-8)=,\leqslant,\geqslant 0$ and attempts to solve |
| $\rightarrow x=2,\ x=4$ | **A1** | Needs both values whether $=2$, $<2$, $>2$ |
| $x\leqslant 2,\ x\geqslant 4$ | **A1** [3] | Accept all recognisable notation; condone $<$ and/or $>$ |
---
# Question 11(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Equate $mx+c$ and $6x-x^2-5$; use of $b^2-4ac$ | **M1, DM1** | Equates, sets to $0$; use of discriminant with values of $a,b,c$ independent of $x$ |
| $4c=m^2-12m+16$ **AG** | **A1** [3] | $=0$ must appear before last line |
| **OR** | | |
| $\frac{dy}{dx}=6-2x=m \rightarrow x=\left(\frac{6-m}{2}\right)$ | **M1** | Equates $\frac{dy}{dx}$ to $m$ and rearrange |
| $m\!\left(\frac{6-m}{2}\right)+c=6\!\left(\frac{6-m}{2}\right)-\!\left(\frac{6-m}{2}\right)^2\!-5$ | **M1** | Equates $mx+c$ and $6x-x^2-5$ and substitutes for $x$ |
| $4c=m^2-12m+16$ **AG** | **A1** [3] | |
---
# Question 11(iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $6x-x^2-5=4-(x-3)^2$ | **B1 B1** [2] | $4\,\text{B1}$; $-(x-3)^2\,\text{B1}$ |
---
# Question 11(iv):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $k=3$ | **B1$^\checkmark$** [1] | $\checkmark$ for "$b$" |
---
# Question 11(v):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $g^{-1}(x)=\sqrt{4-x}+3$ | **M1 A1** [2] | Correct order of operations; $\pm\sqrt{4-x}+3$ M1A0; $\sqrt{x-4}+3$ M1A0; $\sqrt{4-y}+3$ M1A0 |11 The function f is defined by $\mathrm { f } : x \mapsto 6 x - x ^ { 2 } - 5$ for $x \in \mathbb { R }$.\\
(i) Find the set of values of $x$ for which $\mathrm { f } ( x ) \leqslant 3$.\\
(ii) Given that the line $y = m x + c$ is a tangent to the curve $y = \mathrm { f } ( x )$, show that $4 c = m ^ { 2 } - 12 m + 16$.
The function g is defined by $\mathrm { g } : x \mapsto 6 x - x ^ { 2 } - 5$ for $x \geqslant k$, where $k$ is a constant.\\
(iii) Express $6 x - x ^ { 2 } - 5$ in the form $a - ( x - b ) ^ { 2 }$, where $a$ and $b$ are constants.\\
(iv) State the smallest value of $k$ for which g has an inverse.\\
(v) For this value of $k$, find an expression for $\mathrm { g } ^ { - 1 } ( x )$.
{www.cie.org.uk} after the live examination series.
}
\hfill \mbox{\textit{CAIE P1 2016 Q11 [11]}}