10 November 2025
Instructions
- Answer all the questions.
- Use black or blue ink. Pencil may be used for graphs and diagrams only.
- There are blank pages at the end of the paper for additional working. You must clearly indicate when you have moved onto additional pages on the question itself. Make sure to include the question number.
- You are permitted to use a scientific or graphical calculator in this paper.
- Where appropriate, your answer should be supported with working. Marks might be given for using a correct method, even if your answer is wrong.
- Give non-exact numerical answers correct to 3 significant figures unless a different degree of accuracy is specified in the question.
- The acceleration due to gravity is denoted by \(g \mathrm {~ms} ^ { - 2 }\). When a numerical value is needed use \(g = 9.8\) unless a different value is specified in the question.
Information
- The total mark for this paper is \(\mathbf { 7 5 }\) marks.
- The marks for each question are shown in brackets.
- You are reminded of the need for clear presentation in your answers.
- You have \(\mathbf { 7 5 }\) minutes for this paper.
\section*{Arithmetic series}
\(S _ { n } = \frac { 1 } { 2 } n ( a + l ) = \frac { 1 } { 2 } n \{ 2 a + ( n - 1 ) d \}\)
\section*{Geometric series}
\(S _ { n } = \frac { a \left( 1 - r ^ { n } \right) } { 1 - r }\)
\(S _ { \infty } = \frac { a } { 1 - r }\) for \(| r | < 1\)
\section*{Binomial series}
\(( a + b ) ^ { n } = a ^ { n } + { } ^ { n } \mathrm { C } _ { 1 } a ^ { n - 1 } b + { } ^ { n } \mathrm { C } _ { 2 } a ^ { n - 2 } b ^ { 2 } + \ldots + { } ^ { n } \mathrm { C } _ { r } a ^ { n - r } b ^ { r } + \ldots + b ^ { n } \quad ( n \in \mathbb { N } )\),
where \({ } ^ { n } \mathrm { C } _ { r } = { } _ { n } \mathrm { C } _ { r } = \binom { n } { r } = \frac { n ! } { r ! ( n - r ) ! }\)
\(( 1 + x ) ^ { n } = 1 + n x + \frac { n ( n - 1 ) } { 2 ! } x ^ { 2 } + \ldots + \frac { n ( n - 1 ) \ldots ( n - r + 1 ) } { r ! } x ^ { r } + \ldots \quad ( | x | < 1 , n \in \mathbb { R } )\)
\section*{Series}
\(\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 ) , \sum _ { r = 1 } ^ { n } r ^ { 3 } = \frac { 1 } { 4 } n ^ { 2 } ( n + 1 ) ^ { 2 }\)
\section*{Maclaurin series}
\(\mathrm { f } ( x ) = \mathrm { f } ( 0 ) + \mathrm { f } ^ { \prime } ( 0 ) x + \frac { \mathrm { f } ^ { \prime \prime } ( 0 ) } { 2 ! } x ^ { 2 } + \ldots + \frac { \mathrm { f } ^ { ( r ) } ( 0 ) } { r ! } x ^ { r } + \ldots\)
\(\mathrm { e } ^ { x } = \exp ( x ) = 1 + x + \frac { x ^ { 2 } } { 2 ! } + \ldots + \frac { x ^ { r } } { r ! } + \ldots\) for all \(x\)
\(\ln ( 1 + x ) = x - \frac { x ^ { 2 } } { 2 } + \frac { x ^ { 3 } } { 3 } - \ldots + ( - 1 ) ^ { r + 1 } \frac { x ^ { r } } { r } + \ldots ( - 1 < x \leq 1 )\)
\(\sin x = x - \frac { x ^ { 3 } } { 3 ! } + \frac { x ^ { 5 } } { 5 ! } - \ldots + ( - 1 ) ^ { r } \frac { x ^ { 2 r + 1 } } { ( 2 r + 1 ) ! } + \ldots\) for all \(x\)
\(\cos x = 1 - \frac { x ^ { 2 } } { 2 ! } + \frac { x ^ { 4 } } { 4 ! } - \ldots + ( - 1 ) ^ { r } \frac { x ^ { 2 r } } { ( 2 r ) ! } + \ldots\) for all \(x\)
\(( 1 + x ) ^ { n } = 1 + n x + \frac { n ( n - 1 ) } { 2 ! } x ^ { 2 } + \ldots + \frac { n ( n - 1 ) \ldots ( n - r + 1 ) } { r ! } x ^ { r } + \ldots \quad ( | x | < 1 , n \in \mathbb { R } )\)
\section*{Differentiation}
| f(x) | \(\mathrm { f } ^ { \prime } ( x )\) |
| \(\tan k x\) | \(k \sec ^ { 2 } k x\) |
| \(\sec x\) | \(\sec x \tan x\) |
| \(\cot x\) | \(- \operatorname { cosec } ^ { 2 } x\) |
| \(\operatorname { cosec } x\) | \(- \operatorname { cosec } x \cot x\) |
| \(\arcsin x\) or \(\sin ^ { - 1 } x\) | \(\frac { 1 } { \sqrt { 1 - x ^ { 2 } } }\) |
| \(\arccos x\) or \(\cos ^ { - 1 } x\) | \(- \frac { 1 } { \sqrt { 1 - x ^ { 2 } } }\) |
| \(\arctan x\) or \(\tan ^ { - 1 } x\) | \(\frac { 1 } { 1 + x ^ { 2 } }\) |
Quotient rule \(y = \frac { u } { v } , \frac { \mathrm {~d} y } { \mathrm {~d} x } = \frac { v \frac { \mathrm {~d} u } { \mathrm {~d} x } - u \frac { \mathrm {~d} v } { \mathrm {~d} x } } { v ^ { 2 } }\)
\section*{Differentiation from first principles}
\(\mathrm { f } ^ { \prime } ( x ) = \lim _ { h \rightarrow 0 } \frac { \mathrm { f } ( x + h ) - \mathrm { f } ( x ) } { h }\)
\section*{Integration}
\(\int \frac { \mathrm { f } ^ { \prime } ( x ) } { \mathrm { f } ( x ) } \mathrm { d } x = \ln | \mathrm { f } ( x ) | + c\)
\(\int \mathrm { f } ^ { \prime } ( x ) ( \mathrm { f } ( x ) ) ^ { n } \mathrm {~d} x = \frac { 1 } { n + 1 } ( \mathrm { f } ( x ) ) ^ { n + 1 } + c\)
Integration by parts \(\int u \frac { \mathrm {~d} v } { \mathrm {~d} x } \mathrm {~d} x = u v - \int v \frac { \mathrm {~d} u } { \mathrm {~d} x } \mathrm {~d} x\)
Area of sector enclosed by polar curve is \(\frac { 1 } { 2 } \int r ^ { 2 } \mathrm {~d} \theta\)
| \(\mathrm { f } ( x )\) | \(\int \mathrm { f } ( \mathrm { x } ) \mathrm { d } x\) |
| \(\frac { 1 } { \sqrt { a ^ { 2 } - x ^ { 2 } } }\) | \(\sin ^ { - 1 } \left( \frac { x } { a } \right) \quad ( | x | < a )\) |
| \(\frac { 1 } { a ^ { 2 } + x ^ { 2 } }\) | \(\frac { 1 } { a } \tan ^ { - 1 } \left( \frac { x } { a } \right)\) |
| \(\frac { 1 } { \sqrt { a ^ { 2 } + x ^ { 2 } } }\) | \(\sinh ^ { - 1 } \left( \frac { x } { a } \right)\) or \(\ln \left( x + \sqrt { x ^ { 2 } + a ^ { 2 } } \right)\) |
| \(\frac { 1 } { \sqrt { x ^ { 2 } - a ^ { 2 } } }\) | \(\cosh ^ { - 1 } \left( \frac { x } { a } \right)\) or \(\ln \left( x + \sqrt { x ^ { 2 } - a ^ { 2 } } \right) \quad ( x > a )\) |
\section*{Numerical methods}
Trapezium rule: \(\int _ { a } ^ { b } y \mathrm {~d} x \approx \frac { 1 } { 2 } h \left\{ \left( y _ { 0 } + y _ { n } \right) + 2 \left( y _ { 1 } + y _ { 2 } + \ldots + y _ { n - 1 } \right) \right\}\), where \(h = \frac { b - a } { n }\)
The Newton-Raphson iteration for solving \(\mathrm { f } ( x ) = 0 : x _ { n + 1 } = x _ { n } - \frac { \mathrm { f } \left( x _ { n } \right) } { \mathrm { f } ^ { \prime } \left( x _ { n } \right) }\)
\section*{Complex numbers}
Circles: \(| z - a | = k\)
Half lines: \(\arg ( z - a ) = \alpha\)
Lines: \(| z - a | = | z - b |\)
\section*{Small angle approximations}
\(\sin \theta \approx \theta , \cos \theta \approx 1 - \frac { 1 } { 2 } \theta ^ { 2 } , \tan \theta \approx \theta\) where \(\theta\) is small and measured in radians
\section*{Trigonometric identities}
\(\sin ( A \pm B ) = \sin A \cos B \pm \cos A \sin B\)
\(\cos ( A \pm B ) = \cos A \cos B \mp \sin A \sin B\)
\(\tan ( A \pm B ) = \frac { \tan A \pm \tan B } { 1 \mp \tan A \tan B } \quad \left( A \pm B \neq \left( k + \frac { 1 } { 2 } \right) \pi \right)\)
\section*{Hyperbolic functions}
$$\begin{aligned}
& \cosh ^ { 2 } x - \sinh ^ { 2 } x = 1
& \sinh ^ { - 1 } x = \ln \left[ x + \sqrt { \left( x ^ { 2 } + 1 \right) } \right]
& \cosh ^ { - 1 } x = \ln \left[ x + \sqrt { \left( x ^ { 2 } - 1 \right) } \right] , x \geq 1
& \tanh ^ { - 1 } x = \frac { 1 } { 2 } \ln \left( \frac { 1 + x } { 1 - x } \right) , - 1 < x < 1
\end{aligned}$$
- The complex number \(z\) satisfies the equation \(z + 2 \mathrm { i } z ^ { * } + 1 - 4 \mathrm { i } = 0\).
You are given that \(z = x + \mathrm { i } y\), where \(x\) and \(y\) are real numbers.
Determine the values of \(x\) and \(y\).
2. Prove by induction that, for all positive integers \(n\),
$$\sum _ { r = 1 } ^ { n } ( 2 r - 1 ) ^ { 2 } = \frac { 1 } { 3 } n \left( 4 n ^ { 2 } - 1 \right)$$
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3. The figure below shows the curve with cartesian equation \(\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = x y\).
\includegraphics[max width=\textwidth, alt={}, center]{f42517a5-d7ed-40f3-bb04-faea97d4b19b-08_830_997_228_262}
- Show that the polar equation of the curve is \(r ^ { 2 } = a \sin b \theta\), where \(a\) and \(b\) are positive constants to be determined.
- Determine the exact maximum value of \(r\).
- Determine the area enclosed by one of the loops.
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\section*{4. In this question you must show detailed reasoning.}
- The curves with equations
$$y = \frac { 3 } { 4 } \sinh x \text { and } y = \tanh x + \frac { 1 } { 5 }$$
intersect at just one point \(P\)
- Use algebra to show that the \(x\) coordinate of \(P\) satisfies the equation
$$15 \mathrm { e } ^ { 4 x } - 48 \mathrm { e } ^ { 3 x } + 32 \mathrm { e } ^ { x } - 15 = 0$$
- Show that \(\mathrm { e } ^ { x } = 3\) is a solution of this equation.
- Hence state the exact coordinates of \(P\).
(ii) Show that
$$\int _ { - 4 } ^ { 0 } \frac { \mathrm { e } ^ { \frac { 1 } { x } } } { x ^ { 2 } } \mathrm {~d} x = \mathrm { e } ^ { - \frac { 1 } { 4 } }$$
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5. Use the method of differences to prove that for \(n > 2\)
$$\sum _ { r = 2 } ^ { n } \frac { 4 } { r ^ { 2 } - 1 } = \frac { ( p n + q ) ( n - 1 ) } { n ( n + 1 ) }$$
where \(p\) and \(q\) are constants to be determined.
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6.
(i)
$$z _ { 1 } = a + b \mathrm { i } \text { and } z _ { 2 } = c + d \mathrm { i }$$
where \(a , b , c\) and \(d\) are real constants.
Given that
- \(b > d\)
- \(z _ { 1 } + z _ { 2 }\) is real
- \(\left| z _ { 1 } \right| = \sqrt { 13 }\)
- \(\left| z _ { 2 } \right| = 5\)
- \(\operatorname { Re } \left( z _ { 2 } - z _ { 1 } \right) = 2\)
show that \(a = 2\) and determine the value of each of \(b , c\) and \(d\)
(ii) (a) On the same Argand diagram - sketch the locus of points \(z\) which satisfy \(| z - 12 | = 7\)
- sketch the locus of points \(w\) which satisfy \(| w - 5 \mathrm { i } | = 4\) showing the coordinates of any points of intersection with the axes.
- Determine the range of possible values of \(| z - w |\)
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- In this question you must show detailed reasoning.
Evaluate \(\int _ { 0 } ^ { \frac { 1 } { 2 } } \frac { 2 } { x ^ { 2 } - x + 1 } \mathrm {~d} x\). Give your answer in exact form.
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\section*{8. In this question you must show detailed reasoning.}
The diagram shows the curve with equation \(y = \frac { x + 3 } { \sqrt { x ^ { 2 } + 9 } }\).
\includegraphics[max width=\textwidth, alt={}, center]{f42517a5-d7ed-40f3-bb04-faea97d4b19b-18_890_1010_367_242}
The region R , shown shaded in the diagram, is bounded by the curve, the \(x\)-axis, the \(y\)-axis, and the line \(x = 4\). - Determine the area of R . Give your answer in the form \(p + \ln q\) where \(p\) and \(q\) are integers to be determined.
The region R is rotated through \(2 \pi\) radians about the \(x\)-axis.
- Determine the volume of the solid of revolution formed. Give your answer in the form \(\pi \left( a + b \ln \left( \frac { c } { d } \right) \right)\) where \(a , b , c\) and \(d\) are integers to be determined.
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9. Given that
$$y = \cos x \sinh x \quad x \in \mathbb { R }$$ - show that
$$\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } } = k y$$
where \(k\) is a constant to be determined.
- Hence determine the first three non-zero terms of the Maclaurin series for \(y\), giving each coefficient in simplest form.
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10. The quartic equation
$$2 x ^ { 4 } + A x ^ { 3 } - A x ^ { 2 } - 5 x + 6 = 0$$
where \(A\) is a real constant, has roots \(\alpha , \beta , \gamma\) and \(\delta\) - Determine the value of
$$\frac { 3 } { \alpha } + \frac { 3 } { \beta } + \frac { 3 } { \gamma } + \frac { 3 } { \delta }$$
Given that \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 } = - \frac { 3 } { 4 }\)
- determine the possible values of \(A\)
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