Questions — CAIE FP1 (594 questions)

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CAIE FP1 2012 November Q10
12 marks Challenging +1.2
10 The curve \(C\) has equation \(x ^ { 3 } + y ^ { 3 } = 3 x y\), for \(x > 0\) and \(y > 0\). Find a relationship between \(x\) and \(y\) when \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\). Find the exact coordinates of the turning point of \(C\), and determine the nature of this turning point.
CAIE FP1 2012 November Q11
13 marks Challenging +1.2
11 Show that \(\int x \left( 1 - x ^ { 2 } \right) ^ { \frac { 1 } { 2 } } \mathrm {~d} x = - \frac { 1 } { 3 } \left( 1 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } + c\), where \(c\) is a constant. Given that \(I _ { n } = \int _ { 0 } ^ { 1 } x ^ { n } \left( 1 - x ^ { 2 } \right) ^ { \frac { 1 } { 2 } } \mathrm {~d} x\), prove that, for \(n \geqslant 2\), $$( n + 2 ) I _ { n } = ( n - 1 ) I _ { n - 2 }$$ Use the substitution \(x = \sin u\) to show that $$\int _ { 0 } ^ { 1 } \left( 1 - x ^ { 2 } \right) ^ { \frac { 1 } { 2 } } \mathrm {~d} x = \frac { 1 } { 4 } \pi$$ Find \(I _ { 4 }\).
CAIE FP1 2012 November Q12 EITHER
Standard +0.8
The vector \(\mathbf { e }\) is an eigenvector of each of the \(n \times n\) matrices \(\mathbf { A }\) and \(\mathbf { B }\), with corresponding eigenvalues \(\lambda\) and \(\mu\) respectively. Prove that \(\mathbf { e }\) is an eigenvector of the matrix \(\mathbf { A B }\) with eigenvalue \(\lambda \mu\). It is given that the matrix \(\mathbf { A }\), where $$\mathbf { A } = \left( \begin{array} { r r r } 3 & 2 & 2 \\ - 2 & - 2 & - 2 \\ 1 & 2 & 2 \end{array} \right) ,$$ has eigenvectors \(\left( \begin{array} { r } 0 \\ 1 \\ - 1 \end{array} \right)\) and \(\left( \begin{array} { r } 1 \\ 0 \\ - 1 \end{array} \right)\). Find the corresponding eigenvalues. Given that 2 is also an eigenvalue of \(\mathbf { A }\), find a corresponding eigenvector. The matrix \(\mathbf { B }\), where $$\mathbf { B } = \left( \begin{array} { r r r } - 1 & 2 & 2 \\ 2 & 2 & 2 \\ - 3 & - 6 & - 6 \end{array} \right) ,$$ has the same eigenvectors as \(\mathbf { A }\). Given that \(\mathbf { A B } = \mathbf { C }\), find a non-singular matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that $$\mathbf { P } ^ { - 1 } \mathbf { C } ^ { 2 } \mathbf { P } = \mathbf { D }$$
CAIE FP1 2012 November Q12 OR
Challenging +1.2
Obtain the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 6 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 13 x = 75 \cos 2 t$$ Given that \(x = 5\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 0\) when \(t = 0\), find \(x\) in terms of \(t\). Show that, for large positive values of \(t\) and for any initial conditions, $$x \approx 5 \cos ( 2 t - \phi ) ,$$ where the constant \(\phi\) is such that \(\tan \phi = \frac { 4 } { 3 }\).
CAIE FP1 2013 November Q1
5 marks Standard +0.3
1 The curve \(C\) has polar equation \(r = 2 \mathrm { e } ^ { \theta }\), for \(\frac { 1 } { 6 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\). Find
  1. the area of the region bounded by the half-lines \(\theta = \frac { 1 } { 6 } \pi , \theta = \frac { 1 } { 2 } \pi\) and \(C\),
  2. the length of \(C\).
CAIE FP1 2013 November Q2
6 marks Standard +0.8
2 The cubic equation \(x ^ { 3 } - p x - q = 0\), where \(p\) and \(q\) are constants, has roots \(\alpha , \beta , \gamma\). Show that
  1. \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 2 p\),
  2. \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } = 3 q\),
  3. \(6 \left( \alpha ^ { 5 } + \beta ^ { 5 } + \gamma ^ { 5 } \right) = 5 \left( \alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } \right) \left( \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } \right)\).
CAIE FP1 2013 November Q3
7 marks Standard +0.3
3 It is given that $$S _ { n } = \sum _ { r = 1 } ^ { n } u _ { r } = 2 n ^ { 2 } + n$$ Write down the values of \(S _ { 1 } , S _ { 2 } , S _ { 3 } , S _ { 4 }\). Express \(u _ { r }\) in terms of \(r\), justifying your answer. Find $$\sum _ { r = n + 1 } ^ { 2 n } u _ { r } .$$
CAIE FP1 2013 November Q4
7 marks Challenging +1.2
4 It is given that $$I _ { n } = \int _ { 0 } ^ { 1 } \frac { x ^ { n } } { \sqrt { } ( 1 + 2 x ) } \mathrm { d } x$$ Show that, for \(n \geqslant 1\), $$( 2 n + 1 ) I _ { n } = \sqrt { } 3 - n I _ { n - 1 }$$ Show that $$I _ { 3 } = \frac { 2 } { 35 } ( \sqrt { } 3 + 1 )$$
CAIE FP1 2013 November Q5
8 marks Challenging +1.2
5 It is given that \(y = ( 1 + x ) ^ { 2 } \ln ( 1 + x )\). Find \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } }\). Prove by mathematical induction that, for every integer \(n \geqslant 3\), $$\frac { \mathrm { d } ^ { n } y } { \mathrm {~d} x ^ { n } } = ( - 1 ) ^ { n - 1 } \frac { 2 ( n - 3 ) ! } { ( 1 + x ) ^ { n - 2 } }$$
CAIE FP1 2013 November Q6
9 marks Challenging +1.2
6 The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { r r r r } 1 & - 3 & - 1 & 2 \\ 4 & - 10 & 0 & 2 \\ 1 & - 1 & 3 & - 4 \\ 5 & - 12 & 1 & 1 \end{array} \right)$$ Find, in either order, the rank of \(\mathbf { M }\) and a basis for the null space \(K\) of T . Evaluate $$\mathbf { M } \left( \begin{array} { r } 1 \\ - 2 \\ - 3 \\ - 4 \end{array} \right)$$ and hence show that every solution of $$\mathbf { M x } = \left( \begin{array} { r } 2 \\ 16 \\ 10 \\ 22 \end{array} \right)$$ has the form $$\mathbf { x } = \left( \begin{array} { r } 1 \\ - 2 \\ - 3 \\ - 4 \end{array} \right) + \lambda \mathbf { e } _ { 1 } + \mu \mathbf { e } _ { 2 } ,$$ where \(\lambda\) and \(\mu\) are real numbers and \(\left\{ \mathbf { e } _ { 1 } , \mathbf { e } _ { 2 } \right\}\) is a basis for \(K\).
CAIE FP1 2013 November Q7
10 marks Standard +0.3
7 The square matrix \(\mathbf { A }\) has \(\lambda\) as an eigenvalue with \(\mathbf { e }\) as a corresponding eigenvector. Show that \(\mathbf { e }\) is an eigenvector of \(\mathbf { A } ^ { 2 }\) and state the corresponding eigenvalue. Find the eigenvalues of the matrix \(\mathbf { B }\), where $$\mathbf { B } = \left( \begin{array} { l l l } 1 & 3 & 0 \\ 2 & 0 & 2 \\ 1 & 1 & 2 \end{array} \right)$$ Find the eigenvalues of \(\mathbf { B } ^ { 4 } + 2 \mathbf { B } ^ { 2 } + 3 \mathbf { I }\), where \(\mathbf { I }\) is the \(3 \times 3\) identity matrix.
CAIE FP1 2013 November Q8
10 marks Standard +0.3
8 The plane \(\Pi _ { 1 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } 2 \\ 3 \\ - 1 \end{array} \right) + s \left( \begin{array} { l } 1 \\ 0 \\ 1 \end{array} \right) + t \left( \begin{array} { r } 1 \\ - 1 \\ - 2 \end{array} \right)\). Find a cartesian equation of \(\Pi _ { 1 }\). The plane \(\Pi _ { 2 }\) has equation \(2 x - y + z = 10\). Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\). Find an equation of the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }\).
CAIE FP1 2013 November Q10
12 marks Standard +0.3
10
22 \end{array} \right)$$ has the form $$\mathbf { x } = \left( \begin{array} { r } 1
- 2
- 3
- 4 \end{array} \right) + \lambda \mathbf { e } _ { 1 } + \mu \mathbf { e } _ { 2 } ,$$ where \(\lambda\) and \(\mu\) are real numbers and \(\left\{ \mathbf { e } _ { 1 } , \mathbf { e } _ { 2 } \right\}\) is a basis for \(K\). 7 The square matrix \(\mathbf { A }\) has \(\lambda\) as an eigenvalue with \(\mathbf { e }\) as a corresponding eigenvector. Show that \(\mathbf { e }\) is an eigenvector of \(\mathbf { A } ^ { 2 }\) and state the corresponding eigenvalue. Find the eigenvalues of the matrix \(\mathbf { B }\), where $$\mathbf { B } = \left( \begin{array} { l l l } 1 & 3 & 0
2 & 0 & 2
1 & 1 & 2 \end{array} \right)$$ Find the eigenvalues of \(\mathbf { B } ^ { 4 } + 2 \mathbf { B } ^ { 2 } + 3 \mathbf { I }\), where \(\mathbf { I }\) is the \(3 \times 3\) identity matrix. 8 The plane \(\Pi _ { 1 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } 2 \\ 3 \\ - 1 \end{array} \right) + s \left( \begin{array} { l } 1 \\ 0 \\ 1 \end{array} \right) + t \left( \begin{array} { r } 1 \\ - 1 \\ - 2 \end{array} \right)\). Find a cartesian equation of \(\Pi _ { 1 }\). The plane \(\Pi _ { 2 }\) has equation \(2 x - y + z = 10\). Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\). Find an equation of the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }\). 9 The curve \(C\) has parametric equations $$x = t ^ { 2 } , \quad y = t - \frac { 1 } { 3 } t ^ { 3 } , \quad \text { for } 0 \leqslant t \leqslant 1 .$$ Find the surface area generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis. Find the coordinates of the centroid of the region bounded by \(C\), the \(x\)-axis and the line \(x = 1\). 10 The curve \(C\) has equation $$y = \frac { p x ^ { 2 } + 4 x + 1 } { x + 1 } ,$$ where \(p\) is a positive constant and \(p \neq 3\).
  1. Obtain the equations of the asymptotes of \(C\).
  2. Find the value of \(p\) for which the \(x\)-axis is a tangent to \(C\), and sketch \(C\) in this case.
  3. For the case \(p = 1\), show that \(C\) has no turning points, and sketch \(C\), giving the exact coordinates of the points of intersection of \(C\) with the \(x\)-axis.
CAIE FP1 2013 November Q16
Standard +0.3
16
10
22 \end{array} \right)$$ has the form $$\mathbf { x } = \left( \begin{array} { r } 1
- 2
- 3
- 4 \end{array} \right) + \lambda \mathbf { e } _ { 1 } + \mu \mathbf { e } _ { 2 } ,$$ where \(\lambda\) and \(\mu\) are real numbers and \(\left\{ \mathbf { e } _ { 1 } , \mathbf { e } _ { 2 } \right\}\) is a basis for \(K\). 7 The square matrix \(\mathbf { A }\) has \(\lambda\) as an eigenvalue with \(\mathbf { e }\) as a corresponding eigenvector. Show that \(\mathbf { e }\) is an eigenvector of \(\mathbf { A } ^ { 2 }\) and state the corresponding eigenvalue. Find the eigenvalues of the matrix \(\mathbf { B }\), where $$\mathbf { B } = \left( \begin{array} { l l l } 1 & 3 & 0
2 & 0 & 2
1 & 1 & 2 \end{array} \right)$$ Find the eigenvalues of \(\mathbf { B } ^ { 4 } + 2 \mathbf { B } ^ { 2 } + 3 \mathbf { I }\), where \(\mathbf { I }\) is the \(3 \times 3\) identity matrix. 8 The plane \(\Pi _ { 1 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } 2 \\ 3 \\ - 1 \end{array} \right) + s \left( \begin{array} { l } 1 \\ 0 \\ 1 \end{array} \right) + t \left( \begin{array} { r } 1 \\ - 1 \\ - 2 \end{array} \right)\). Find a cartesian equation of \(\Pi _ { 1 }\). The plane \(\Pi _ { 2 }\) has equation \(2 x - y + z = 10\). Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\). Find an equation of the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }\). 9 The curve \(C\) has parametric equations $$x = t ^ { 2 } , \quad y = t - \frac { 1 } { 3 } t ^ { 3 } , \quad \text { for } 0 \leqslant t \leqslant 1 .$$ Find the surface area generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis. Find the coordinates of the centroid of the region bounded by \(C\), the \(x\)-axis and the line \(x = 1\). 10 The curve \(C\) has equation $$y = \frac { p x ^ { 2 } + 4 x + 1 } { x + 1 } ,$$ where \(p\) is a positive constant and \(p \neq 3\).
  1. Obtain the equations of the asymptotes of \(C\).
  2. Find the value of \(p\) for which the \(x\)-axis is a tangent to \(C\), and sketch \(C\) in this case.
  3. For the case \(p = 1\), show that \(C\) has no turning points, and sketch \(C\), giving the exact coordinates of the points of intersection of \(C\) with the \(x\)-axis. 11 Answer only one of the following two alternatives. \section*{EITHER} State the fifth roots of unity in the form \(\cos \theta + \mathrm { i } \sin \theta\), where \(- \pi < \theta \leqslant \pi\). Simplify $$\left( x - \left[ \cos \frac { 2 } { 5 } \pi + i \sin \frac { 2 } { 5 } \pi \right] \right) \left( x - \left[ \cos \frac { 2 } { 5 } \pi - i \sin \frac { 2 } { 5 } \pi \right] \right)$$ Hence find the real factors of $$x ^ { 5 } - 1$$ Express the six roots of the equation $$x ^ { 6 } - x ^ { 3 } + 1 = 0$$ as three conjugate pairs, in the form \(\cos \theta \pm \mathrm { i } \sin \theta\). Hence find the real factors of $$x ^ { 6 } - x ^ { 3 } + 1$$ OR Given that $$y ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 6 y ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } + 3 y ^ { 3 } = 25 \mathrm { e } ^ { - 2 x }$$ and that \(v = y ^ { 3 }\), show that $$\frac { \mathrm { d } ^ { 2 } v } { \mathrm {~d} x ^ { 2 } } - 6 \frac { \mathrm {~d} v } { \mathrm {~d} x } + 9 v = 75 \mathrm { e } ^ { - 2 x }$$ Find the particular solution for \(y\) in terms of \(x\), given that when \(x = 0 , y = 2\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 1\).
CAIE FP1 2015 November Q5
8 marks Standard +0.8
5 The cubic equation \(x ^ { 3 } + p x ^ { 2 } + q x + r = 0\), where \(p , q\) and \(r\) are integers, has roots \(\alpha , \beta\) and \(\gamma\), such that $$\begin{aligned} \alpha + \beta + \gamma & = 15 \\ \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } & = 83 \end{aligned}$$ Write down the value of \(p\) and find the value of \(q\). Given that \(\alpha , \beta\) and \(\gamma\) are all real and that \(\alpha \beta + \alpha \gamma = 36\), find \(\alpha\) and hence find the value of \(r\).
CAIE FP1 2015 November Q6
10 marks Standard +0.8
6 The matrix \(\mathbf { A }\), where $$\mathbf { A } = \left( \begin{array} { r r r } 1 & 0 & 0 \\ 10 & - 7 & 10 \\ 7 & - 5 & 8 \end{array} \right)$$ has eigenvalues 1 and 3. Find corresponding eigenvectors. It is given that \(\left( \begin{array} { l } 0 \\ 2 \\ 1 \end{array} \right)\) is an eigenvector of \(\mathbf { A }\). Find the corresponding eigenvalue. Find a diagonal matrix \(\mathbf { D }\) and matrices \(\mathbf { P }\) and \(\mathbf { P } ^ { - 1 }\) such that \(\mathbf { P } ^ { - 1 } \mathbf { A P } = \mathbf { D }\).
CAIE FP1 2017 November Q8
11 marks Challenging +1.8
  1. Find the value of \(I _ { 2 }\).
  2. Show that, for \(n > 2\), $$( n - 1 ) I _ { n } = 2 ^ { \frac { 1 } { 2 } n - 1 } + ( n - 2 ) I _ { n - 2 }$$
  3. The curve \(C\) has equation \(y = \sec ^ { 3 } x\) for \(0 \leqslant x \leqslant \frac { 1 } { 4 } \pi\). The region \(R\) is bounded by \(C\), the \(x\)-axis, the \(y\)-axis and the line \(x = \frac { 1 } { 4 } \pi\). Find the volume of revolution generated when \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
CAIE FP1 2019 November Q5
9 marks Challenging +1.8
  1. Use standard results from the List of Formulae (MF10) to show that $$S _ { N } = \frac { 1 } { 3 } N \left( 25 N ^ { 2 } + 90 N + 83 \right)$$
  2. Use the method of differences to express \(T _ { N }\) in terms of \(N\).
  3. Find \(\lim _ { N \rightarrow \infty } \left( N ^ { - 3 } S _ { N } T _ { N } \right)\).
CAIE FP1 2015 November Q1
Standard +0.8
1 The curve \(C\) is defined parametrically by $$x = 2 \cos ^ { 3 } t \quad \text { and } \quad y = 2 \sin ^ { 3 } t , \quad \text { for } 0 < t < \frac { 1 } { 2 } \pi .$$ Show that, at the point with parameter \(t\), $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { 1 } { 6 } \sec ^ { 4 } t \operatorname { cosec } t$$
CAIE FP1 2015 November Q4
Challenging +1.2
4 The sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is such that, for all positive integers \(n\), $$a _ { n } = \frac { n + 5 } { \sqrt { } \left( n ^ { 2 } - n + 1 \right) } - \frac { n + 6 } { \sqrt { } \left( n ^ { 2 } + n + 1 \right) }$$ The sum \(\sum _ { n = 1 } ^ { N } a _ { n }\) is denoted by \(S _ { N }\). Find
  1. the value of \(S _ { 30 }\) correct to 3 decimal places,
  2. the least value of \(N\) for which \(S _ { N } > 4.9\).
CAIE FP1 2015 November Q5
Standard +0.8
5 The cubic equation \(x ^ { 3 } + p x ^ { 2 } + q x + r = 0\), where \(p , q\) and \(r\) are integers, has roots \(\alpha , \beta\) and \(\gamma\), such that $$\begin{aligned} \alpha + \beta + \gamma & = 15 , \\ \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } & = 83 . \end{aligned}$$ Write down the value of \(p\) and find the value of \(q\). Given that \(\alpha , \beta\) and \(\gamma\) are all real and that \(\alpha \beta + \alpha \gamma = 36\), find \(\alpha\) and hence find the value of \(r\).
CAIE FP1 2015 November Q6
Standard +0.3
6 The matrix A, where $$\mathbf { A } = \left( \begin{array} { r r r } 1 & 0 & 0 \\ 10 & - 7 & 10 \\ 7 & - 5 & 8 \end{array} \right)$$ has eigenvalues 1 and 3. Find corresponding eigenvectors. It is given that \(\left( \begin{array} { l } 0 \\ 2 \\ 1 \end{array} \right)\) is an eigenvector of \(\mathbf { A }\). Find the corresponding eigenvalue. Find a diagonal matrix \(\mathbf { D }\) and matrices \(\mathbf { P }\) and \(\mathbf { P } ^ { - 1 }\) such that \(\mathbf { P } ^ { - 1 } \mathbf { A P } = \mathbf { D }\).
CAIE FP1 2015 November Q7
Challenging +1.2
7 The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { r r r r } 1 & - 2 & - 3 & 1 \\ 3 & - 5 & - 7 & 7 \\ 5 & - 9 & - 13 & 9 \\ 7 & - 13 & - 19 & 11 \end{array} \right)$$ Find the rank of \(\mathbf { M }\) and a basis for the null space of T . The vector \(\left( \begin{array} { l } 1 \\ 2 \\ 3 \\ 4 \end{array} \right)\) is denoted by \(\mathbf { e }\). Show that there is a solution of the equation \(\mathbf { M x } = \mathbf { M e }\) of the form \(\mathbf { x } = \left( \begin{array} { c } a \\ b \\ - 1 \\ - 1 \end{array} \right)\), where the constants \(a\) and \(b\) are to be found.
CAIE FP1 2015 November Q8
Standard +0.8
8 The curve \(C\) has equation \(y = \frac { 2 x ^ { 2 } + k x } { x + 1 }\), where \(k\) is a constant. Find the set of values of \(k\) for which \(C\) has no stationary points. For the case \(k = 4\), find the equations of the asymptotes of \(C\) and sketch \(C\), indicating the coordinates of the points where \(C\) intersects the coordinate axes.
CAIE FP1 2015 November Q9
Challenging +1.3
9 It is given that \(I _ { n } = \int _ { 1 } ^ { \mathrm { e } } ( \ln x ) ^ { n } \mathrm {~d} x\) for \(n \geqslant 0\). Show that $$I _ { n } = ( n - 1 ) \left[ I _ { n - 2 } - I _ { n - 1 } \right] \text { for } n \geqslant 2$$ Hence find, in an exact form, the mean value of \(( \ln x ) ^ { 3 }\) with respect to \(x\) over the interval \(1 \leqslant x \leqslant \mathrm { e }\).