AQA Paper 3 2022 June — Question 10 13 marks

Exam BoardAQA
ModulePaper 3 (Paper 3)
Year2022
SessionJune
Marks13
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicComposite & Inverse Functions
TypeState domain or range
DifficultyStandard +0.3 This is a multi-part question covering standard A-level topics (domain, many-to-one functions, intersection points, stationary points, range). Parts (a)-(c) are routine algebraic manipulation. Part (d) requires showing intersection points are stationary points, which needs the quotient rule and some algebraic verification but follows a clear method. Part (e) requires interpreting the graph. While lengthy (13 marks total), each component uses familiar techniques without requiring novel insight or particularly sophisticated reasoning.
Spec1.02c Simultaneous equations: two variables by elimination and substitution1.02f Solve quadratic equations: including in a function of unknown1.02p Interpret algebraic solutions: graphically1.02u Functions: definition and vocabulary (domain, range, mapping)1.07n Stationary points: find maxima, minima using derivatives1.08h Integration by substitution
The function f is defined by $$f(x) = \frac{x^2 + 10}{2x + 5}$$ where f has its maximum possible domain. The curve \(y = f(x)\) intersects the line \(y = x\) at the points P and Q as shown below. \includegraphics{figure_10}
  1. State the value of \(x\) which is not in the domain of f. [1 mark]
  2. Explain how you know that the function f is many-to-one. [2 marks]
    1. Show that the \(x\)-coordinates of P and Q satisfy the equation $$x^2 + 5x - 10 = 0$$ [2 marks]
    2. Hence, find the exact \(x\)-coordinate of P and the exact \(x\)-coordinate of Q. [1 mark]
  4. Show that P and Q are stationary points of the curve. Fully justify your answer. [5 marks]
  5. Using set notation, state the range of f. [2 marks]
Question 10:
AnswerMarks Guidance
10(a)States –2.5
OE2.2a B1
Subtotal1
QMarking instructions AO
AnswerMarks
10(b)Explains that many-to-one
function is when distinct values
AnswerMarks Guidance
of x give the same value for y2.4 E1
or more x values give the same y
value.
This graph is many-to-one because
you can draw a horizontal line and
it will cross the graph twice.
Uses the shape of the graph to
justify their answer
or
gives an example of two x
values eg f( 0 )=f( 4 )
or
states turning or minimum or
maximum points indicate many-
AnswerMarks Guidance
to-one2.4 E1
Subtotal2
QMarking instructions AO
AnswerMarks
10(c)(i)x2 +10
Equates x and
AnswerMarks Guidance
2x+53.1a M1
x=
2x+5
x ( 2x+5 )= x2 +10
2x2 +5x= x2 +10
x2 +5x−10=0
Rearranges with at least one
intermediate step to obtain
quadratic equation AG
AnswerMarks Guidance
Condone 0 = x2 + 5x –102.1 R1
Subtotal2
QMarking instructions AO
AnswerMarks
10(c)(ii)−5± 65
Obtains
2
Ignore any labels
AnswerMarks Guidance
ISW1.1b B1
x =
2
AnswerMarks Guidance
Subtotal1
QMarking instructions AO
AnswerMarks
10(d)Uses quotient rule to obtain an
expression in the form of
Ax ( 2x+5 )+B ( x2 +10 )
( 2x+5 )2
or
uses product rule to obtain an
expression in the form of
Cx( 2x+5 )−1+D ( x2 +10 )( 2x+5 )−2
or
uses implicit differentiation to
obtain an equation of the form
𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑
𝐴𝐴𝐴𝐴 +𝐵𝐵𝑑𝑑+𝐶𝐶 = 𝐷𝐷𝐴𝐴
A, B𝑑𝑑,𝐴𝐴 C and D c𝑑𝑑a𝐴𝐴n be any values
but not 0
AnswerMarks Guidance
Condone missing brackets3.1a M1
f′( )=
x
( 2x+5 )2
2x2 +10x−20
=
( 2x+5 )2
f′( x )=0⇔2x2 +10x−20=0
x2 +5x−10=0
This is the same equation solved in
part c(i) so P and Q must be
stationary points.
Obtains fully correct f′(x)
or
obtains
𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑
2𝐴𝐴𝑑𝑑𝑑𝑑+2𝑑𝑑+5𝑑𝑑𝑑𝑑 = 2𝐴𝐴
ACF
AnswerMarks Guidance
May be unsimplified1.1b A1
Equates their ′ or their
numerator of ′ to 0
𝑓𝑓(𝐴𝐴)
or sets = 0
𝑓𝑓(𝐴𝐴)
AnswerMarks Guidance
d𝑑𝑑1.1a M1
d𝑑𝑑
Rearranges to obtain
x2 + 5x –10 = 0 or
2x2 + 10x – 20 = 0 and links it to the
equation in part c(i) or their answer
to c(ii)
or
solves their quadratic f′(x)=0
correctly
or
deduces y = x and substitutes to
x2 +10
get x= then rearranges to
2x+5
AnswerMarks Guidance
get x2 + 5x –10 = 01.1a M1
Completes a reasoned argument
−5± 65
x =
by using to
2
conclude that P and Q are
stationary points
CSO
Must have brackets correct
AnswerMarks Guidance
throughout2.1 R1
Subtotal5
QMarking instructions AO
AnswerMarks
10(e)Deduces critical regions from
their answer to c(ii)
condone strict inequalities or
AnswerMarks Guidance
poor notation or decimal values2.2a M1
x≤ and x≥
2 2
 −5− 65  −5+ 65
x:x≤ ∪x:x≥ 
 2   2 
Writes correct range in correct
set notation
eg
 −5− 65  −5+ 65 
−∞, ∪ ,∞
 
 2   2 
Accept other letters for x or
using f(x) provided consistent
throughout set
Follow through their answer to
AnswerMarks Guidance
c(ii)2.5 A1F
Subtotal2
Question 10 Total13
QMarking instructions AO 
Question 10:
--- 10(a) ---
10(a) | States –2.5
OE | 2.2a | B1 | –2.5
Subtotal | 1
Q | Marking instructions | AO | Marks | Typical solution
--- 10(b) ---
10(b) | Explains that many-to-one
function is when distinct values
of x give the same value for y | 2.4 | E1 | Many-to-one function is when two
or more x values give the same y
value.
This graph is many-to-one because
you can draw a horizontal line and
it will cross the graph twice.
Uses the shape of the graph to
justify their answer
or
gives an example of two x
values eg f( 0 )=f( 4 )
or
states turning or minimum or
maximum points indicate many-
to-one | 2.4 | E1
Subtotal | 2
Q | Marking instructions | AO | Marks | Typical solution
--- 10(c)(i) ---
10(c)(i) | x2 +10
Equates x and
2x+5 | 3.1a | M1 | x2 +10
x=
2x+5
x ( 2x+5 )= x2 +10
2x2 +5x= x2 +10
x2 +5x−10=0
Rearranges with at least one
intermediate step to obtain
quadratic equation AG
Condone 0 = x2 + 5x –10 | 2.1 | R1
Subtotal | 2
Q | Marking instructions | AO | Marks | Typical solution
--- 10(c)(ii) ---
10(c)(ii) | −5± 65
Obtains
2
Ignore any labels
ISW | 1.1b | B1 | −5± 65
x =
2
Subtotal | 1
Q | Marking instructions | AO | Marks | Typical solution
--- 10(d) ---
10(d) | Uses quotient rule to obtain an
expression in the form of
Ax ( 2x+5 )+B ( x2 +10 )
( 2x+5 )2
or
uses product rule to obtain an
expression in the form of
Cx( 2x+5 )−1+D ( x2 +10 )( 2x+5 )−2
or
uses implicit differentiation to
obtain an equation of the form
𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑
𝐴𝐴𝐴𝐴 +𝐵𝐵𝑑𝑑+𝐶𝐶 = 𝐷𝐷𝐴𝐴
A, B𝑑𝑑,𝐴𝐴 C and D c𝑑𝑑a𝐴𝐴n be any values
but not 0
Condone missing brackets | 3.1a | M1 | 2x ( 2x+5 )−2 ( x2 +10 )
f′( )=
x
( 2x+5 )2
2x2 +10x−20
=
( 2x+5 )2
f′( x )=0⇔2x2 +10x−20=0
x2 +5x−10=0
This is the same equation solved in
part c(i) so P and Q must be
stationary points.
Obtains fully correct f′(x)
or
obtains
𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑
2𝐴𝐴𝑑𝑑𝑑𝑑+2𝑑𝑑+5𝑑𝑑𝑑𝑑 = 2𝐴𝐴
ACF
May be unsimplified | 1.1b | A1
Equates their ′ or their
numerator of ′ to 0
𝑓𝑓(𝐴𝐴)
or sets = 0
𝑓𝑓(𝐴𝐴)
d𝑑𝑑 | 1.1a | M1
d𝑑𝑑
Rearranges to obtain
x2 + 5x –10 = 0 or
2x2 + 10x – 20 = 0 and links it to the
equation in part c(i) or their answer
to c(ii)
or
solves their quadratic f′(x)=0
correctly
or
deduces y = x and substitutes to
x2 +10
get x= then rearranges to
2x+5
get x2 + 5x –10 = 0 | 1.1a | M1
Completes a reasoned argument
−5± 65
x =
by using to
2
conclude that P and Q are
stationary points
CSO
Must have brackets correct
throughout | 2.1 | R1
Subtotal | 5
Q | Marking instructions | AO | Marks | Typical solution
--- 10(e) ---
10(e) | Deduces critical regions from
their answer to c(ii)
condone strict inequalities or
poor notation or decimal values | 2.2a | M1 | −5− 65 −5+ 65
x≤ and x≥
2 2
 −5− 65  −5+ 65
x:x≤ ∪x:x≥ 
 2   2 
Writes correct range in correct
set notation
eg
 −5− 65  −5+ 65 
−∞, ∪ ,∞
 
 2   2 
Accept other letters for x or
using f(x) provided consistent
throughout set
Follow through their answer to
c(ii) | 2.5 | A1F
Subtotal | 2
Question 10 Total | 13
Q | Marking instructions | AO | Marks | Typical solution
The function f is defined by
$$f(x) = \frac{x^2 + 10}{2x + 5}$$
where f has its maximum possible domain.

The curve $y = f(x)$ intersects the line $y = x$ at the points P and Q as shown below.

\includegraphics{figure_10}

\begin{enumerate}[label=(\alph*)]
\item State the value of $x$ which is not in the domain of f.
[1 mark]

\item Explain how you know that the function f is many-to-one.
[2 marks]

\item \begin{enumerate}[label=(\roman*)]
\item Show that the $x$-coordinates of P and Q satisfy the equation
$$x^2 + 5x - 10 = 0$$
[2 marks]

\item Hence, find the exact $x$-coordinate of P and the exact $x$-coordinate of Q.
[1 mark]
\end{enumerate}

\item Show that P and Q are stationary points of the curve.

Fully justify your answer.
[5 marks]

\item Using set notation, state the range of f.
[2 marks]
\end{enumerate}

\hfill \mbox{\textit{AQA Paper 3 2022 Q10 [13]}}
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