| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2016 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Inequalities |
| Type | Solve quadratic inequality |
| Difficulty | Moderate -0.8 Part (i) is a routine quadratic inequality requiring factorisation or the quadratic formula, while part (ii) involves standard discriminant work to find when a line is tangent to a parabola. Both are textbook exercises with straightforward methods and minimal problem-solving required, making this easier than average. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.02g Inequalities: linear and quadratic in single variable1.07a Derivative as gradient: of tangent to curve |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(2x^2 - 6x + 5 > 13 \rightarrow 2x^2 - 6x - 8 (>0)\) | M1 | Sets to \(0+\) attempts to solve |
| \((x =) -1\) and \(4\) | A1 | Both values required |
| \(x > 4\), \(x < -1\) | A1 | Allow all recognisable notation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(2x^2 - 6x + 5 = 2x + k \rightarrow 2x^2 - 8x + 5 - k (=0)\) | M1* | Equates and sets to 0 |
| Use of \(b^2 - 4ac \rightarrow -3\) | DM1 | Use of discriminant |
| A1 | ||
| OR | ||
| \(\frac{dy}{dx} = 4x - 6\), \(4x - 6 = 2\), \(x = 2\) | M1* | Sets their \(\frac{dy}{dx} = 2\) |
| \(x = 2 \rightarrow y = 1\) | ||
| Using their \((2,1)\) in \(y = 2x + k\) or \(y = 2x^2 - 6x + 5 \rightarrow k = -3\) | DM1, A1 | Uses their \(x = 2\) and their \(y = 1\) |
## Question 3(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $2x^2 - 6x + 5 > 13 \rightarrow 2x^2 - 6x - 8 (>0)$ | M1 | Sets to $0+$ attempts to solve |
| $(x =) -1$ and $4$ | A1 | Both values required |
| $x > 4$, $x < -1$ | A1 | Allow all recognisable notation |
## Question 3(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $2x^2 - 6x + 5 = 2x + k \rightarrow 2x^2 - 8x + 5 - k (=0)$ | M1* | Equates and sets to 0 |
| Use of $b^2 - 4ac \rightarrow -3$ | DM1 | Use of discriminant |
| | A1 | |
| **OR** | | |
| $\frac{dy}{dx} = 4x - 6$, $4x - 6 = 2$, $x = 2$ | M1* | Sets their $\frac{dy}{dx} = 2$ |
| $x = 2 \rightarrow y = 1$ | | |
| Using their $(2,1)$ in $y = 2x + k$ or $y = 2x^2 - 6x + 5 \rightarrow k = -3$ | DM1, A1 | Uses their $x = 2$ and their $y = 1$ |
---3 A curve has equation $y = 2 x ^ { 2 } - 6 x + 5$.\\
(i) Find the set of values of $x$ for which $y > 13$.\\
(ii) Find the value of the constant $k$ for which the line $y = 2 x + k$ is a tangent to the curve.
\hfill \mbox{\textit{CAIE P1 2016 Q3 [6]}}