Question 2 - A-Level Maths

CAIE P1 2023 June — Question 2 4 marks

Exam Board	CAIE
Module	P1 (Pure Mathematics 1)
Year	2023
Session	June
Marks	4
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Inequalities
Type	Quadratic always positive/negative
Difficulty	Standard +0.3 This is a straightforward application of the discriminant condition for a quadratic inequality. Students need to rearrange to x² - 6x + (c-2) > 0 for all x, then apply b² - 4ac < 0, requiring only routine algebraic manipulation. It's slightly easier than average as it's a standard textbook exercise with a clear method.
Spec	1.02f Solve quadratic equations: including in a function of unknown 1.02g Inequalities: linear and quadratic in single variable

2 The function f is defined for \(x \in \mathbb { R }\) by \(\mathrm { f } ( x ) = x ^ { 2 } - 6 x + c\), where \(c\) is a constant. It is given that \(\mathrm { f } ( x ) > 2\) for all values of \(x\). Find the set of possible values of \(c\).

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Question 2:

Answer	Marks	Guidance
Answer	Marks	Guidance
\(x^2 - 6x + c > 2\) leading to \((x-3)^2 - 9 + c > 2\)	M1 A1	M1 for completion of the square with an equation or in equality with the '2'
\(c > 11 - (x-3)^2\) and \((x-3)^2 \geq 0\)	M1	SOI
\(c > 11\)	A1

Alternative Method 1:

Answer	Marks	Guidance
Answer	Marks	Guidance
\(\frac{dy}{dx} = 2x - 6 = 0\)	M1	M1 for differentiating and setting \(\frac{dy}{dx} = 0\)
\(x = 3\)	A1
When \(x = 3\), \(y = 9 - 18 + c\)	M1
\([-9 + c > 2]\) \(c > 11\)	A1

Alternative Method 2:

Answer	Marks	Guidance
Answer	Marks	Guidance
\(x^2 - 6x + c > 2\) leading to \(x^2 - 6x + c - 2 [> 0]\), then use of \(b^2 - 4ac\)	M1
\(36 - 4(1)(c-2) < 0\)	M1 A1	OE Must be correct inequality for M1
\(c > 11\)	A1

## Question 2:

| Answer | Marks | Guidance |
|--------|-------|----------|
| $x^2 - 6x + c > 2$ leading to $(x-3)^2 - 9 + c > 2$ | M1 A1 | M1 for completion of the square with an equation or in equality with the '2' |
| $c > 11 - (x-3)^2$ and $(x-3)^2 \geq 0$ | M1 | SOI |
| $c > 11$ | A1 | |

**Alternative Method 1:**

| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{dy}{dx} = 2x - 6 = 0$ | M1 | M1 for differentiating and setting $\frac{dy}{dx} = 0$ |
| $x = 3$ | A1 | |
| When $x = 3$, $y = 9 - 18 + c$ | M1 | |
| $[-9 + c > 2]$ $c > 11$ | A1 | |

**Alternative Method 2:**

| Answer | Marks | Guidance |
|--------|-------|----------|
| $x^2 - 6x + c > 2$ leading to $x^2 - 6x + c - 2 [> 0]$, then use of $b^2 - 4ac$ | M1 | |
| $36 - 4(1)(c-2) < 0$ | M1 A1 | OE Must be correct inequality for M1 |
| $c > 11$ | A1 | |

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2 The function f is defined for $x \in \mathbb { R }$ by $\mathrm { f } ( x ) = x ^ { 2 } - 6 x + c$, where $c$ is a constant. It is given that $\mathrm { f } ( x ) > 2$ for all values of $x$.

Find the set of possible values of $c$.\\

\hfill \mbox{\textit{CAIE P1 2023 Q2 [4]}}

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