Question 2 - A-Level Maths

Edexcel CP1 2023 June — Question 2 6 marks

Exam Board	Edexcel
Module	CP1 (Core Pure 1)
Year	2023
Session	June
Marks	6
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Integration using inverse trig and hyperbolic functions
Type	Completing square then standard inverse trig
Difficulty	Standard +0.3 This is a straightforward application of completing the square followed by recognition of a standard inverse hyperbolic integral from the formula book. Part (a) is routine algebra, part (b) requires pattern matching to arsinh form, and part (c) applies the mean value definition. While it involves Further Maths content (inverse hyperbolic functions), the question is highly structured with no novel problem-solving required—students simply execute learned procedures step-by-step.
Spec	1.02e Complete the square: quadratic polynomials and turning points 1.08h Integration by substitution 4.08e Mean value of function: using integral

(a) Write $x ^ { 2 } + 4 x - 5$ in the form $( x + p ) ^ { 2 } + q$ where $p$ and $q$ are integers.
(b) Hence use a standard integral from the formula book to find

$$\int \frac { 1 } { \sqrt { x ^ { 2 } + 4 x - 5 } } \mathrm {~d} x$$ (c) Determine the mean value of the function $$\mathrm { f } ( x ) = \frac { 1 } { \sqrt { x ^ { 2 } + 4 x - 5 } } \quad 3 \leqslant x \leqslant 13$$ giving your answer in the form $A \ln B$ where $A$ and $B$ are constants in simplest form.

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Question 2:

Part (a):

Answer	Marks	Guidance
$x^2 + 4x - 5 = (x+2)^2 - 9$	B1	Correct completed square form. Allow $3^2$ for 9.

Part (b):

Answer	Marks	Guidance
$\int \frac{1}{\sqrt{(x+p)^2 - q}} dx = \text{arcosh}\left(\frac{x+p}{\sqrt{q}}\right)(+c)$ or $\ln\left(x+p+\sqrt{(x+p)^2-q}\right)(+c)$	M1	Achieves correct form for integration using their $p$ and $q$ from part (a), where $p \neq 0$, $q \neq 1$. Allow substitution approach e.g. $x+p = \sqrt{q}\cosh u$. Allow $\cosh^{-1}$ for arcosh.
$= \text{arcosh}\left(\frac{x+2}{3}\right)$ or $\ln\left(x+2+\sqrt{(x+2)^2-9}\right)$	A1	Correct integration. The "$+c$" is not required. Note $\ln\left(\frac{x+2}{3}+\sqrt{\left(\frac{x+2}{3}\right)^2-1}\right)(+c)$ is also correct.

Part (c):

Answer	Marks	Guidance
$\text{Mean} = \frac{1}{13-3}\int_3^{13} \frac{1}{\sqrt{x^2+4x-5}}\, dx$	B1	Recalls definition of mean function accurately. $\frac{1}{13-3}\int_3^{13} \frac{1}{\sqrt{x^2+4x-5}}\,dx$ seen or implied. Note $\frac{1}{13-3}$ may appear at the end.
$\frac{1}{10}\int_3^{13} \frac{1}{\sqrt{x^2+4x-5}}\,dx = \frac{1}{10}\left(\text{arcosh}\left(\frac{15}{3}\right) - \text{arcosh}\left(\frac{5}{3}\right)\right)$ or $\frac{1}{10}\left(\ln(15+\sqrt{216}) - \ln(5+\sqrt{16})\right)$	M1	Applies correct limits the right way round to their answer from part (b). Can be awarded whether or not the $\frac{1}{10}$ is present.
$= \frac{1}{10}\ln\left(\frac{5+2\sqrt{6}}{3}\right)$ or $\frac{1}{20}\ln\left(\frac{49+20\sqrt{6}}{9}\right)$	A1	Correct answer in correct form. Allow equivalents e.g. $\frac{1}{10}\ln\left(\frac{5}{3}+\frac{2\sqrt{6}}{3}\right)$, $\frac{1}{20}\ln\left(\frac{49}{9}+\frac{20\sqrt{6}}{9}\right)$. Allow unsimplified surd e.g. $\frac{1}{10}\ln\left(\frac{5+\sqrt{24}}{3}\right)$. Brackets must be present in ln forms. If extra values offered score A0.

# Question 2:

## Part (a):
| $x^2 + 4x - 5 = (x+2)^2 - 9$ | B1 | Correct completed square form. Allow $3^2$ for 9. |

## Part (b):
| $\int \frac{1}{\sqrt{(x+p)^2 - q}} dx = \text{arcosh}\left(\frac{x+p}{\sqrt{q}}\right)(+c)$ or $\ln\left(x+p+\sqrt{(x+p)^2-q}\right)(+c)$ | M1 | Achieves correct form for integration using their $p$ and $q$ from part (a), where $p \neq 0$, $q \neq 1$. Allow substitution approach e.g. $x+p = \sqrt{q}\cosh u$. Allow $\cosh^{-1}$ for arcosh. |
| $= \text{arcosh}\left(\frac{x+2}{3}\right)$ or $\ln\left(x+2+\sqrt{(x+2)^2-9}\right)$ | A1 | Correct integration. The "$+c$" is not required. Note $\ln\left(\frac{x+2}{3}+\sqrt{\left(\frac{x+2}{3}\right)^2-1}\right)(+c)$ is also correct. |

## Part (c):
| $\text{Mean} = \frac{1}{13-3}\int_3^{13} \frac{1}{\sqrt{x^2+4x-5}}\, dx$ | B1 | Recalls definition of mean function accurately. $\frac{1}{13-3}\int_3^{13} \frac{1}{\sqrt{x^2+4x-5}}\,dx$ seen or implied. Note $\frac{1}{13-3}$ may appear at the end. |
| $\frac{1}{10}\int_3^{13} \frac{1}{\sqrt{x^2+4x-5}}\,dx = \frac{1}{10}\left(\text{arcosh}\left(\frac{15}{3}\right) - \text{arcosh}\left(\frac{5}{3}\right)\right)$ or $\frac{1}{10}\left(\ln(15+\sqrt{216}) - \ln(5+\sqrt{16})\right)$ | M1 | Applies correct limits the right way round to their answer from part (b). Can be awarded whether or not the $\frac{1}{10}$ is present. |
| $= \frac{1}{10}\ln\left(\frac{5+2\sqrt{6}}{3}\right)$ or $\frac{1}{20}\ln\left(\frac{49+20\sqrt{6}}{9}\right)$ | A1 | Correct answer in correct form. Allow equivalents e.g. $\frac{1}{10}\ln\left(\frac{5}{3}+\frac{2\sqrt{6}}{3}\right)$, $\frac{1}{20}\ln\left(\frac{49}{9}+\frac{20\sqrt{6}}{9}\right)$. Allow unsimplified surd e.g. $\frac{1}{10}\ln\left(\frac{5+\sqrt{24}}{3}\right)$. Brackets must be present in ln forms. If extra values offered score A0. |

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Show LaTeX source

\begin{enumerate}
  \item (a) Write $x ^ { 2 } + 4 x - 5$ in the form $( x + p ) ^ { 2 } + q$ where $p$ and $q$ are integers.\\
(b) Hence use a standard integral from the formula book to find
\end{enumerate}

$$\int \frac { 1 } { \sqrt { x ^ { 2 } + 4 x - 5 } } \mathrm {~d} x$$

(c) Determine the mean value of the function

$$\mathrm { f } ( x ) = \frac { 1 } { \sqrt { x ^ { 2 } + 4 x - 5 } } \quad 3 \leqslant x \leqslant 13$$

giving your answer in the form $A \ln B$ where $A$ and $B$ are constants in simplest form.

\hfill \mbox{\textit{Edexcel CP1 2023 Q2 [6]}}

This paper (8 questions)

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Q1 5 Q2 6 Q3 10 Q4 5 Q5 12 Q6 12 Q7 12 Q10

Answer	Marks	Guidance
\(\int \frac{1}{\sqrt{(x+p)^2 - q}} dx = \text{arcosh}\left(\frac{x+p}{\sqrt{q}}\right)(+c)\) or \(\ln\left(x+p+\sqrt{(x+p)^2-q}\right)(+c)\)	M1	Achieves correct form for integration using their \(p\) and \(q\) from part (a), where \(p \neq 0\), \(q \neq 1\). Allow substitution approach e.g. \(x+p = \sqrt{q}\cosh u\). Allow \(\cosh^{-1}\) for arcosh.
\(= \text{arcosh}\left(\frac{x+2}{3}\right)\) or \(\ln\left(x+2+\sqrt{(x+2)^2-9}\right)\)	A1	Correct integration. The "\(+c\)" is not required. Note \(\ln\left(\frac{x+2}{3}+\sqrt{\left(\frac{x+2}{3}\right)^2-1}\right)(+c)\) is also correct.

Answer	Marks	Guidance
\(\text{Mean} = \frac{1}{13-3}\int_3^{13} \frac{1}{\sqrt{x^2+4x-5}}\, dx\)	B1	Recalls definition of mean function accurately. \(\frac{1}{13-3}\int_3^{13} \frac{1}{\sqrt{x^2+4x-5}}\,dx\) seen or implied. Note \(\frac{1}{13-3}\) may appear at the end.
\(\frac{1}{10}\int_3^{13} \frac{1}{\sqrt{x^2+4x-5}}\,dx = \frac{1}{10}\left(\text{arcosh}\left(\frac{15}{3}\right) - \text{arcosh}\left(\frac{5}{3}\right)\right)\) or \(\frac{1}{10}\left(\ln(15+\sqrt{216}) - \ln(5+\sqrt{16})\right)\)	M1	Applies correct limits the right way round to their answer from part (b). Can be awarded whether or not the \(\frac{1}{10}\) is present.
\(= \frac{1}{10}\ln\left(\frac{5+2\sqrt{6}}{3}\right)\) or \(\frac{1}{20}\ln\left(\frac{49+20\sqrt{6}}{9}\right)\)	A1	Correct answer in correct form. Allow equivalents e.g. \(\frac{1}{10}\ln\left(\frac{5}{3}+\frac{2\sqrt{6}}{3}\right)\), \(\frac{1}{20}\ln\left(\frac{49}{9}+\frac{20\sqrt{6}}{9}\right)\). Allow unsimplified surd e.g. \(\frac{1}{10}\ln\left(\frac{5+\sqrt{24}}{3}\right)\). Brackets must be present in ln forms. If extra values offered score A0.