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Edexcel CP AS 2022 June Q7
6 marks Standard +0.3
  1. Prove by mathematical induction that, for \(n \in \mathbb { N }\)
$$\left( \begin{array} { l l } - 5 & 9 \\ - 4 & 7 \end{array} \right) ^ { n } = \left( \begin{array} { c c } 1 - 6 n & 9 n \\ - 4 n & 1 + 6 n \end{array} \right)$$
Edexcel CP AS 2022 June Q8
15 marks Standard +0.3
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{545661a6-8d78-488c-b73b-ab2ced60debf-28_663_531_210_258} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{545661a6-8d78-488c-b73b-ab2ced60debf-28_394_903_431_900} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 1 shows a sketch of a 16 cm tall vase which has a flat circular base with diameter 8 cm and a circular opening of diameter 8 cm at the top. A student measures the circular cross-section halfway up the vase to be 8 cm in diameter.
The student models the shape of the vase by rotating a curve, shown in Figure 2, through \(360 ^ { \circ }\) about the \(x\)-axis.
  1. State the value of \(a\) that should be used when setting up the model. Two possible equations are suggested for the curve in the model. $$\begin{array} { l l } \text { Model A } & y = a - 2 \sin \left( \frac { 45 } { 2 } x \right) ^ { \circ } \\ \text { Model B } & y = a + \frac { x ( x - 8 ) ( x + 8 ) } { 100 } \end{array}$$ For each model,
    1. find the distance from the base at which the widest part of the vase occurs,
    2. find the diameter of the vase at this widest point. The widest part of the vase has diameter 12 cm and is just over 3 cm from the base.
  3. Using this information and making your reasoning clear, suggest which model is more appropriate.
  4. Using algebraic integration, find the volume for the vase predicted by Model B. You must make your method clear. The student pours water from a full one litre jug into the vase and finds that there is 100 ml left in the jug when the vase is full.
  5. Comment on the suitability of Model B in light of this information.
Edexcel CP AS 2023 June Q1
4 marks Moderate -0.8
1. $$\left( \begin{array} { l l } x & 9 \\ y & z \end{array} \right) - 3 \left( \begin{array} { l l } z & y \\ z & y \end{array} \right) = k \mathbf { I }$$ where \(x , y , z\) and \(k\) are constants.
Determine the value of \(x\), the value of \(y\) and the value of \(z\).
Edexcel CP AS 2023 June Q2
7 marks Standard +0.3
  1. \(\mathrm { f } ( \mathrm { z } ) = \mathrm { z } ^ { 3 } + a \mathrm { z } ^ { 2 } + b \mathrm { z } + 175 \quad\) where \(a\) and \(b\) are real constants
Given that \(- 3 + 4 \mathrm { i }\) is a root of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)
  1. determine the value of \(a\) and the value of \(b\).
  2. Show all the roots of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\) on a single Argand diagram.
  3. Write down the roots of the equation \(\mathrm { f } ( \mathrm { z } + 2 ) = 0\)
Edexcel CP AS 2023 June Q3
4 marks Moderate -0.3
3. $$\mathbf { A } = \left( \begin{array} { c c c } 1 & 0 & 0 \\ 0 & \frac { \sqrt { 3 } } { 2 } & - \frac { 1 } { 2 } \\ 0 & \frac { 1 } { 2 } & \frac { \sqrt { 3 } } { 2 } \end{array} \right)$$
  1. Describe fully the single geometric transformation \(A\) represented by the matrix \(\mathbf { A }\). $$\mathbf { B } = \left( \begin{array} { c c c } 1 & 3 & 0 \\ \sqrt { 3 } & 0 & 5 \sqrt { 3 } \\ 1 & 2 & 0 \end{array} \right)$$ The transformation \(B\) is represented by the matrix \(\mathbf { B }\).
    The transformation \(A\) followed by the transformation \(B\) is the transformation \(C\), which is represented by the matrix \(\mathbf { C }\). To determine matrix \(\mathbf { C }\), a student attempts the following matrix multiplication. $$\left( \begin{array} { c c c } 1 & 0 & 0 \\ 0 & \frac { \sqrt { 3 } } { 2 } & - \frac { 1 } { 2 } \\ 0 & \frac { 1 } { 2 } & \frac { \sqrt { 3 } } { 2 } \end{array} \right) \left( \begin{array} { c c c } 1 & 3 & 0 \\ \sqrt { 3 } & 0 & 5 \sqrt { 3 } \\ 1 & 2 & 0 \end{array} \right)$$
  2. State the error made by the student.
  3. Determine the correct matrix \(\mathbf { C }\).
Edexcel CP AS 2023 June Q4
8 marks Standard +0.3
  1. (i) (a) Show that
$$\frac { 2 + 3 \mathrm { i } } { 5 + \mathrm { i } } = k ( 1 + \mathrm { i } )$$ where \(k\) is a constant to be determined.
(Solutions relying on calculator technology are not acceptable.) Given that
  • \(n\) is a positive integer
  • \(\left( \frac { 2 + 3 \mathrm { i } } { 5 + \mathrm { i } } \right) ^ { n }\) is a real number
    (b) use the answer to part (a) to write down the smallest possible value of \(n\).
    (ii) The complex number \(z = a + b \mathrm { i }\) where \(a\) and \(b\) are real constants.
Given that
  • \(\left| z ^ { 10 } \right| = 59049\)
  • \(\arg \left( z ^ { 10 } \right) = - \frac { 5 \pi } { 3 }\) determine the value of \(a\) and the value of \(b\).
Edexcel CP AS 2023 June Q5
8 marks Standard +0.3
  1. In this question you must show all stages of your working.
Solutions relying on calculator technology are not acceptable. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ab572f1e-2828-4ab3-b148-605f35ccd1db-14_385_526_447_420} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ab572f1e-2828-4ab3-b148-605f35ccd1db-14_485_433_388_1187} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A large pile of concrete waste is created on a building site.
Figure 1 shows a central vertical cross-section of the concrete waste.
The curve \(C\), shown in Figure 2, has equation $$y + x ^ { 2 } = 2 \quad 0 \leqslant x \leqslant \sqrt { 2 }$$ The region \(R\), shown shaded in Figure 2, is bounded by the \(y\)-axis, the \(x\)-axis and the curve \(C\). The volume of concrete waste is modelled by the volume of revolution formed when \(R\) is rotated through \(360 ^ { \circ }\) about the \(y\)-axis. The units are metres. The density of the concrete waste is \(900 \mathrm { kgm } ^ { - 3 }\)
  1. Use the model to estimate the mass of the concrete waste. Give your answer to 2 significant figures.
  2. Give a limitation of the model. The mass of the concrete waste is approximately 5500 kg .
  3. Use this information and your answer to part (a) to evaluate the model, giving a reason for your answer.
Edexcel CP AS 2023 June Q6
11 marks Standard +0.3
  1. The line \(l _ { 1 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } - 2 \\ 2 \\ 0 \end{array} \right) + \lambda \left( \begin{array} { l } 3 \\ 0 \\ 1 \end{array} \right)\) where \(\lambda\) is a scalar parameter.
The line \(l _ { 2 }\) is parallel to \(\left( \begin{array} { r } 1 \\ 2 \\ - 3 \end{array} \right)\)
  1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) are perpendicular. The plane \(\Pi\) contains the line \(l _ { 1 }\) and is perpendicular to \(\left( \begin{array} { r } 1 \\ 2 \\ - 3 \end{array} \right)\)
  2. Determine a Cartesian equation of \(\Pi\)
  3. Verify that the point \(A ( 3,1,1 )\) lies on \(\Pi\) Given that
    • the point of intersection of \(\Pi\) and \(l _ { 2 }\) has coordinates \(( 2,3,2 )\)
    • the point \(B ( p , q , r )\) lies on \(l _ { 2 }\)
    • the distance \(A B\) is \(2 \sqrt { 5 }\)
    • \(p , q\) and \(r\) are positive integers
    • determine the coordinates of \(B\).
Edexcel CP AS 2023 June Q7
9 marks Standard +0.8
  1. (i) Shade, on an Argand diagram, the set of points for which
$$| z - 3 | \leqslant | z + 6 i |$$ (ii) Determine the exact complex number \(w\) which satisfies both $$\arg ( w - 2 ) = \frac { \pi } { 3 } \quad \text { and } \quad \arg ( w + 1 ) = \frac { \pi } { 6 }$$
Edexcel CP AS 2023 June Q8
8 marks Moderate -0.3
  1. (a) Use the standard results for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r\) to show that, for all positive integers \(n\),
$$\sum _ { r = 1 } ^ { n } ( 2 r - 1 ) ^ { 2 } = \frac { n } { 3 } \left( a n ^ { 2 } - 1 \right)$$ where \(a\) is a constant to be determined.
(b) Hence determine the sum of the squares of all positive odd three-digit integers.
Edexcel CP AS 2023 June Q9
9 marks Standard +0.3
  1. (i)
$$\mathbf { P } = \left( \begin{array} { r r r } k & - 2 & 7 \\ - 3 & - 5 & 2 \\ k & k & 4 \end{array} \right)$$ where \(k\) is a constant Show that \(\mathbf { P }\) is non-singular for all real values of \(k\).
(ii) $$\mathbf { Q } = \left( \begin{array} { r r } 2 & - 1 \\ - 3 & 0 \end{array} \right)$$ The matrix \(\mathbf { Q }\) represents a linear transformation \(T\) Under \(T\), the point \(A ( a , 2 )\) and the point \(B ( 4 , - a )\), where \(a\) is a constant, are transformed to the points \(A ^ { \prime }\) and \(B ^ { \prime }\) respectively. Given that the distance \(A ^ { \prime } B ^ { \prime }\) is \(\sqrt { 58 }\), determine the possible values of \(a\).
Edexcel CP AS 2023 June Q10
12 marks Challenging +1.2
  1. In this question you must show all stages of your working.
Solutions relying on calculator technology are not acceptable.
  1. The quartic equation $$z ^ { 4 } + 5 z ^ { 2 } - 30 = 0$$ has roots \(p , q , r\) and \(s\).
    Without solving the equation, determine the quartic equation whose roots are $$( 3 p - 1 ) , ( 3 q - 1 ) , ( 3 r - 1 ) \text { and } ( 3 s - 1 )$$ Give your answer in the form \(w ^ { 4 } + a w ^ { 3 } + b w ^ { 2 } + c w + d = 0\), where \(a , b , c\) and \(d\) are integers to be found.
  2. The roots of the cubic equation $$4 x ^ { 3 } + n x + 81 = 0 \quad \text { where } n \text { is a real constant }$$ are \(\alpha , 2 \alpha\) and \(\alpha - \beta\) Determine
    (a) the values of the roots of the equation,
    (b) the value of \(n\).
Edexcel CP AS 2024 June Q1
9 marks Standard +0.3
  1. The cubic equation
$$2 x ^ { 3 } - 3 x ^ { 2 } + 5 x + 7 = 0$$ has roots \(\alpha , \beta\) and \(\gamma\).
Without solving the equation, determine the exact value of
  1. \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\)
  2. \(\frac { 3 } { \alpha } + \frac { 3 } { \beta } + \frac { 3 } { \gamma }\)
  3. \(( 5 - \alpha ) ( 5 - \beta ) ( 5 - \gamma )\)
Edexcel CP AS 2024 June Q2
10 marks
  1. \(\left[ \begin{array} { l } \text { With respect to the right-hand rule, a rotation through } \theta ^ { \circ } \text { anticlockwise about } \\ \text { the } z \text {-axis is represented by the matrix } \\ \qquad \left( \begin{array} { c c c } \cos \theta & - \sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{array} \right) \end{array} \right]\)
Given that the matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { c c c } - \frac { \sqrt { 3 } } { 2 } & \frac { 1 } { 2 } & 0 \\ - \frac { 1 } { 2 } & - \frac { \sqrt { 3 } } { 2 } & 0 \\ 0 & 0 & 1 \end{array} \right)$$ represents a rotation through \(\alpha ^ { \circ }\) anticlockwise about the \(z\)-axis with respect to the right-hand rule,
  1. determine the value of \(\alpha\).
  2. Hence determine the smallest possible positive integer value of \(k\) for which \(\mathbf { M } ^ { k } = \mathbf { I }\) The \(3 \times 3\) matrix \(\mathbf { N }\) represents a reflection in the plane with equation \(y = 0\)
  3. Write down the matrix \(\mathbf { N }\). The point \(A\) has coordinates (-2, 4, 3)
    The point \(B\) is the image of the point \(A\) under the transformation represented by matrix \(\mathbf { M }\) followed by the transformation represented by matrix \(\mathbf { N }\).
  4. Show that the coordinates of \(B\) are \(( 2 + \sqrt { 3 } , 2 \sqrt { 3 } - 1,3 )\) Given that \(O\) is the origin,
  5. show that, to 3 significant figures, the size of angle \(A O B\) is \(66.9 ^ { \circ }\)
  6. Hence determine the area of triangle \(A O B\), giving your answer to 3 significant figures.
Edexcel CP AS 2024 June Q3
10 marks Standard +0.3
  1. (a) Use the standard results for summations to show that, for all positive integers \(n\),
$$\sum _ { r = 1 } ^ { n } r ^ { 2 } ( r + 1 ) = \frac { 1 } { 12 } n ( n + 1 ) ( n + 2 ) ( a n + b )$$ where \(a\) and \(b\) are integers to be determined.
(b) Hence show that, for all positive integers \(k\), $$\sum _ { r = k + 1 } ^ { 3 k } r ^ { 2 } ( r + 1 ) = \frac { 1 } { 3 } k ( 3 k + 1 ) \left( A k ^ { 2 } + B k + C \right)$$ where \(A , B\) and \(C\) are integers to be determined.
(c) Hence, using algebra and making your method clear, determine the value of \(k\) for which $$25 \sum _ { r = k + 1 } ^ { 3 k } r ^ { 2 } ( r + 1 ) = 192 k ^ { 3 } ( 3 k + 1 )$$
Edexcel CP AS 2024 June Q4
8 marks Standard +0.3
4. $$\mathbf { A } = \left( \begin{array} { r r r } - 1 & - 2 & - 7 \\ 3 & k & 2 \\ 1 & 1 & 4 \end{array} \right) \quad \mathbf { B } = \left( \begin{array} { c c c } 4 k - 2 & 1 & 7 k - 4 \\ - 10 & 3 & - 19 \\ 3 - k & - 1 & 6 - k \end{array} \right)$$ where \(k\) is a constant.
  1. Determine the value of the constant \(c\) for which $$\mathbf { A B } = ( 3 k + c ) \mathbf { I }$$
  2. Hence determine the value of \(k\) for which \(\mathbf { A } ^ { - 1 }\) does not exist. Given that \(\mathbf { A } ^ { - 1 }\) does exist,
  3. write down \(\mathbf { A } ^ { - 1 }\) in terms of \(k\).
  4. Use the answer to part (c) to solve the simultaneous equations $$\begin{aligned} - x - 2 y - 7 z & = 10 \\ 3 x + k y + 2 z & = 3 \\ x + y + 4 z & = 1 \end{aligned}$$ giving the values of \(x , y\) and \(z\) in simplest form in terms of \(k\).
Edexcel CP AS 2024 June Q5
10 marks Standard +0.8
  1. Given that on an Argand diagram the locus of points defined by \(| z + 5 - 12 i | = 10\) is a circle,
    1. write down,
      1. the coordinates of the centre of this circle,
      2. the radius of this circle.
    2. Show, by shading on an Argand diagram, the set of points defined by
    $$| z + 5 - 12 i | \leqslant 10$$
  2. For the set of points defined in part (b), determine the maximum value of \(| z |\) The set of points \(A\) is defined by $$A = \{ z : 0 \leqslant \arg ( z + 5 - 20 i ) \leqslant \pi \} \cap \{ z : | z + 5 - 12 i | \leqslant 10 \}$$
  3. Determine the area of the region defined by \(A\), giving your answer to 3 significant figures.
Edexcel CP AS 2024 June Q6
12 marks Moderate -0.3
  1. The drainage system for a sports field consists of underground pipes.
This situation is modelled with respect to a fixed origin \(O\).
According to the model,
  • the surface of the sports field is a plane with equation \(z = 0\)
  • the pipes are straight lines
  • one of the pipes, \(P _ { 1 }\), passes through the points \(A ( 3,4 , - 2 )\) and \(B ( - 2 , - 8 , - 3 )\)
  • a different pipe, \(P _ { 2 }\), has equation \(\frac { x - 1 } { 2 } = \frac { y - 3 } { 4 } = \frac { z + 1 } { - 2 }\)
  • the units are metres
    1. Determine a vector equation of the line representing the pipe \(P _ { 1 }\)
    2. Determine the coordinates of the point at which the pipe \(P _ { 1 }\) meets the surface of the playing field, according to the model.
Determine, according to the model,
  • the acute angle between pipes \(P _ { 1 }\) and \(P _ { 2 }\), giving your answer in degrees to 3 significant figures,
  • the shortest distance between pipes \(P _ { 1 }\) and \(P _ { 2 }\)
    •
    Edexcel CP AS 2024 June Q7
    10 marks Standard +0.3
    1. (i) Prove by induction that, for all positive integers \(n\),
    $$\sum _ { r = 1 } ^ { n } \frac { 1 } { r ( r + 1 ) } = \frac { n } { n + 1 }$$ (ii) Prove by induction that, for all positive integers \(n\), $$f ( n ) = 3 ^ { 2 n + 4 } - 2 ^ { 2 n }$$ is divisible by 5
    Edexcel CP AS 2024 June Q8
    11 marks Standard +0.3
    8. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{18386c8a-6d2d-4c63-972a-bb9f78786b36-30_634_264_319_374} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{18386c8a-6d2d-4c63-972a-bb9f78786b36-30_762_609_260_1080} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 1 shows the central vertical cross-section, \(O A B C D E O\), of the design for a solid glass ornament. Figure 2 shows the finite region, \(R\), which is bounded by the \(y\)-axis, the horizontal line \(C B\), the vertical line \(B A\), and the curve \(A O\). The ornament is formed by rotating the region \(R\) through \(360 ^ { \circ }\) about the \(y\)-axis.
    The curve \(A O\) is modelled by the equation $$x = k y ^ { 2 } + \sqrt { y } \quad 0 \leqslant y \leqslant 4$$ where \(k\) is a constant.
    The point \(A\) has coordinates ( \(0.4,4\) ) and the point \(B\) has coordinates ( \(0.4,4.5\) )
    The units are centimetres.
    1. Determine the value of \(k\) according to this model.
    2. Use algebraic integration to determine the exact volume of glass that would be required to make the ornament, according to the model.
    3. State a limitation of the model. When the ornament was manufactured, \(9 \mathrm {~cm} ^ { 3 }\) of glass was required.
    4. Use this information and your answer to part (b) to evaluate the model, explaining your reasoning.
    Edexcel CP AS Specimen Q1
    7 marks Standard +0.8
    1. $$f ( z ) = z ^ { 3 } + p z ^ { 2 } + q z - 15$$ where \(p\) and \(q\) are real constants.
    Given that the equation \(\mathrm { f } ( \mathrm { z } ) = 0\) has roots $$\alpha , \frac { 5 } { \alpha } \text { and } \left( \alpha + \frac { 5 } { \alpha } - 1 \right)$$
    1. solve completely the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)
    2. Hence find the value of \(p\).
    Edexcel CP AS Specimen Q2
    10 marks Standard +0.2
    1. The plane \(\Pi\) passes through the point \(A\) and is perpendicular to the vector \(\mathbf { n }\)
    Given that $$\overrightarrow { O A } = \left( \begin{array} { r } 5 \\ - 3 \\ - 4 \end{array} \right) \quad \text { and } \quad \mathbf { n } = \left( \begin{array} { r } 3 \\ - 1 \\ 2 \end{array} \right)$$ where \(O\) is the origin,
    1. find a Cartesian equation of \(\Pi\). With respect to the fixed origin \(O\), the line \(l\) is given by the equation $$\mathbf { r } = \left( \begin{array} { r } 7 \\ 3 \\ - 2 \end{array} \right) + \lambda \left( \begin{array} { r } - 1 \\ - 5 \\ 3 \end{array} \right)$$ The line \(l\) intersects the plane \(\Pi\) at the point \(X\).
    2. Show that the acute angle between the plane \(\Pi\) and the line \(l\) is \(21.2 ^ { \circ }\) correct to one decimal place.
    3. Find the coordinates of the point \(X\).
    Edexcel CP AS Specimen Q3
    7 marks Standard +0.3
    1. Tyler invested a total of \(\pounds 5000\) across three different accounts; a savings account, a property bond account and a share dealing account.
    Tyler invested \(\pounds 400\) more in the property bond account than in the savings account.
    After one year
    • the savings account had increased in value by \(1.5 \%\)
    • the property bond account had increased in value by \(3.5 \%\)
    • the share dealing account had decreased in value by \(2.5 \%\)
    • the total value across Tyler's three accounts had increased by \(\pounds 79\)
    Form and solve a matrix equation to find out how much money was invested by Tyler in each account.
    Edexcel CP AS Specimen Q5
    7 marks Moderate -0.5
    5. $$\mathbf { M } = \left( \begin{array} { c c } 1 & - \sqrt { 3 } \\ \sqrt { 3 } & 1 \end{array} \right)$$
    1. Show that \(\mathbf { M }\) is non-singular. The hexagon \(R\) is transformed to the hexagon \(S\) by the transformation represented by the matrix \(\mathbf { M }\). Given that the area of hexagon \(R\) is 5 square units,
    2. find the area of hexagon \(S\). The matrix \(\mathbf { M }\) represents an enlargement, with centre \(( 0,0 )\) and scale factor \(k\), where \(k > 0\), followed by a rotation anti-clockwise through an angle \(\theta\) about \(( 0,0 )\).
    3. Find the value of \(k\).
    4. Find the value of \(\theta\).
    Edexcel CP AS Specimen Q7
    12 marks Challenging +1.2
    7. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{75a62878-dd50-4d52-915a-fe329935d97a-14_577_716_360_296} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{75a62878-dd50-4d52-915a-fe329935d97a-14_630_705_296_1153} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 1 shows the central cross-section \(A O B C D\) of a circular bird bath, which is made of concrete. Measurements of the height and diameter of the bird bath, and the depth of the bowl of the bird bath have been taken in order to estimate the amount of concrete that was required to make this bird bath. Using these measurements, the cross-sectional curve CD, shown in Figure 2, is modelled as a curve with equation $$y = 1 + k x ^ { 2 } \quad - 0.2 \leqslant x \leqslant 0.2$$ where \(k\) is a constant and where \(O\) is the fixed origin.
    The height of the bird bath measured 1.16 m and the diameter, \(A B\), of the base of the bird bath measured 0.40 m , as shown in Figure 1.
    1. Suggest the maximum depth of the bird bath.
    2. Find the value of \(k\).
    3. Hence find the volume of concrete that was required to make the bird bath according to this model. Give your answer, in \(\mathrm { m } ^ { 3 }\), correct to 3 significant figures.
    4. State a limitation of the model. It was later discovered that the volume of concrete used to make the bird bath was \(0.127 \mathrm {~m} ^ { 3 }\) correct to 3 significant figures.
    5. Using this information and the answer to part (c), evaluate the model, explaining your reasoning.