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CAIE FP1 2016 November Q11 OR
A curve \(C\) has parametric equations $$x = 1 - 3 t ^ { 2 } , \quad y = t \left( 1 - 3 t ^ { 2 } \right) , \quad \text { for } 0 \leqslant t \leqslant \frac { 1 } { \sqrt { 3 } }$$ Show that \(\left( \frac { \mathrm { d } x } { \mathrm {~d} t } \right) ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} t } \right) ^ { 2 } = \left( 1 + 9 t ^ { 2 } \right) ^ { 2 }\). Hence find
  1. the arc length of \(C\),
  2. the surface area generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis. Use the fact that \(t = \frac { y } { x }\) to find a cartesian equation of \(C\). Hence show that the polar equation of \(C\) is \(r = \sec \theta \left( 1 - 3 \tan ^ { 2 } \theta \right)\), and state the domain of \(\theta\). Find the area of the region enclosed between \(C\) and the initial line. \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at \href{http://www.cie.org.uk}{www.cie.org.uk} after the live examination series. Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }
CAIE FP1 2016 November Q1
1 Use the method of differences to find \(\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 2 r ) ^ { 2 } - 1 }\). Deduce the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { ( 2 r ) ^ { 2 } - 1 }\).
CAIE FP1 2016 November Q2
2 Find the cubic equation with roots \(\alpha , \beta\) and \(\gamma\) such that $$\begin{aligned} \alpha + \beta + \gamma & = 3 \\ \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } & = 1 \\ \alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } & = - 30 \end{aligned}$$ giving your answer in the form \(x ^ { 3 } + p x ^ { 2 } + q x + r = 0\), where \(p , q\) and \(r\) are integers to be found.
CAIE FP1 2016 November Q3
3 Find a matrix \(\mathbf { A }\) whose eigenvalues are \(- 1,1,2\) and for which corresponding eigenvectors are $$\left( \begin{array} { l } 1 \\ 0 \\ 0 \end{array} \right) , \quad \left( \begin{array} { l } 1 \\ 1 \\ 0 \end{array} \right) , \quad \left( \begin{array} { l } 0 \\ 1 \\ 1 \end{array} \right) ,$$ respectively.
CAIE FP1 2016 November Q4
4 Using factorials, show that \(\binom { n } { r - 1 } + \binom { n } { r } = \binom { n + 1 } { r }\). Hence prove by mathematical induction that $$( a + x ) ^ { n } = \binom { n } { 0 } a ^ { n } + \binom { n } { 1 } a ^ { n - 1 } x + \ldots + \binom { n } { r } a ^ { n - r } x ^ { r } + \ldots + \binom { n } { n } x ^ { n }$$ for every positive integer \(n\).
CAIE FP1 2016 November Q5
5 The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix \(\mathbf { A }\), where $$\mathbf { A } = \left( \begin{array} { r r r r } 1 & 3 & 5 & 7 \\ 2 & 8 & 7 & 9 \\ 3 & 13 & 9 & 11 \\ 6 & 24 & 21 & 27 \end{array} \right)$$ Find
  1. the rank of \(\mathbf { A }\),
  2. a basis for the range space of T ,
  3. a basis for the null space of T .
CAIE FP1 2016 November Q6
6 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 7 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 10 x = 116 \sin 2 t$$ State an approximate solution for large positive values of \(t\).
CAIE FP1 2016 November Q7
7 The curve \(C\) has equation \(y = \mathrm { e } ^ { - 2 x }\). Find, giving your answers correct to 3 significant figures,
  1. the mean value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) over the interval \(0 \leqslant x \leqslant 2\),
  2. the coordinates of the centroid of the region bounded by \(C\), \(x = 0\), \(x = 2\) and \(y = 0\).
CAIE FP1 2016 November Q8
8 A curve \(C\) has equation \(x ^ { 2 } + 4 x y - y ^ { 2 } + 20 = 0\). Show that, at stationary points on \(C , x = - 2 y\). Find the coordinates of the stationary points on \(C\), and determine their nature by considering the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at the stationary points.
CAIE FP1 2016 November Q9
9 Evaluate \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } x \sin x \mathrm {~d} x\). Given that \(I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } x ^ { n } \sin x \mathrm {~d} x\), prove that, for \(n > 1\), $$I _ { n } = n \left( \frac { 1 } { 2 } \pi \right) ^ { n - 1 } - n ( n - 1 ) I _ { n - 2 }$$ By first using the substitution \(x = \cos ^ { - 1 } u\), find the value of $$\int _ { 0 } ^ { 1 } \left( \cos ^ { - 1 } u \right) ^ { 3 } \mathrm {~d} u$$ giving your answer in an exact form.
CAIE FP1 2016 November Q10
10 Let \(z = \cos \theta + \mathrm { i } \sin \theta\). Show that $$z ^ { n } + \frac { 1 } { z ^ { n } } = 2 \cos n \theta \quad \text { and } \quad z ^ { n } - \frac { 1 } { z ^ { n } } = 2 \mathrm { i } \sin n \theta$$ By considering \(\left( z - \frac { 1 } { z } \right) ^ { 4 } \left( z + \frac { 1 } { z } \right) ^ { 2 }\), show that $$\sin ^ { 4 } \theta \cos ^ { 2 } \theta = \frac { 1 } { 32 } ( \cos 6 \theta - 2 \cos 4 \theta - \cos 2 \theta + 2 ) .$$ Hence find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sin ^ { 4 } \theta \cos ^ { 2 } \theta d \theta\).
[0pt] [Question 11 is printed on the next page.]
CAIE FP1 2016 November Q11 EITHER
The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations $$\mathbf { r } = 6 \mathbf { i } - 3 \mathbf { j } + s ( 3 \mathbf { i } - 4 \mathbf { j } - 2 \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = 2 \mathbf { i } - \mathbf { j } - 4 \mathbf { k } + t ( \mathbf { i } - 3 \mathbf { j } - \mathbf { k } )$$ respectively. The point \(P\) on \(l _ { 1 }\) and the point \(Q\) on \(l _ { 2 }\) are such that \(P Q\) is perpendicular to both \(l _ { 1 }\) and \(l _ { 2 }\). Show that the position vector of \(P\) is \(3 \mathbf { i } + \mathbf { j } + 2 \mathbf { k }\) and find the position vector of \(Q\). Find, in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b } + \mu \mathbf { c }\), an equation of the plane \(\Pi\) which passes through \(P\) and is perpendicular to \(l _ { 1 }\). The plane \(\Pi\) meets the plane \(\mathbf { r } = p \mathbf { i } + q \mathbf { j }\) in the line \(l _ { 3 }\). Find a vector equation of \(l _ { 3 }\).
CAIE FP1 2016 November Q11 OR
A curve \(C\) has parametric equations $$x = 1 - 3 t ^ { 2 } , \quad y = t \left( 1 - 3 t ^ { 2 } \right) , \quad \text { for } 0 \leqslant t \leqslant \frac { 1 } { \sqrt { 3 } }$$ Show that \(\left( \frac { \mathrm { d } x } { \mathrm {~d} t } \right) ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} t } \right) ^ { 2 } = \left( 1 + 9 t ^ { 2 } \right) ^ { 2 }\). Hence find
  1. the arc length of \(C\),
  2. the surface area generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis. Use the fact that \(t = \frac { y } { x }\) to find a cartesian equation of \(C\). Hence show that the polar equation of \(C\) is \(r = \sec \theta \left( 1 - 3 \tan ^ { 2 } \theta \right)\), and state the domain of \(\theta\). Find the area of the region enclosed between \(C\) and the initial line. \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge International Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at \href{http://www.cie.org.uk}{www.cie.org.uk} after the live examination series. Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }
CAIE FP1 2018 November Q1
1 The vectors \(\mathbf { a } , \mathbf { b } , \mathbf { c }\) and \(\mathbf { d }\) in \(\mathbb { R } ^ { 3 }\) are given by $$\mathbf { a } = \left( \begin{array} { l } 1 \\ 2 \\ 1 \end{array} \right) , \quad \mathbf { b } = \left( \begin{array} { l } 2 \\ 9 \\ 0 \end{array} \right) , \quad \mathbf { c } = \left( \begin{array} { l } 3 \\ 3 \\ 4 \end{array} \right) \quad \text { and } \quad \mathbf { d } = \left( \begin{array} { r } 0 \\ - 8 \\ 3 \end{array} \right) .$$
  1. Show that \(\{ \mathbf { a } , \mathbf { b } , \mathbf { c } \}\) is a basis for \(\mathbb { R } ^ { 3 }\).
  2. Express \(\mathbf { d }\) in terms of \(\mathbf { a } , \mathbf { b }\) and \(\mathbf { c }\).
CAIE FP1 2018 November Q2
2 The roots of the equation $$x ^ { 3 } + p x ^ { 2 } + q x + r = 0$$ are \(\alpha , 2 \alpha , 4 \alpha\), where \(p , q , r\) and \(\alpha\) are non-zero real constants.
  1. Show that $$2 p \alpha + q = 0$$
  2. Show that $$p ^ { 3 } r - q ^ { 3 } = 0$$
CAIE FP1 2018 November Q3
3 The sequence of positive numbers \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is such that \(u _ { 1 } < 3\) and, for \(n \geqslant 1\), $$u _ { n + 1 } = \frac { 4 u _ { n } + 9 } { u _ { n } + 4 }$$
  1. By considering \(3 - u _ { n + 1 }\), or otherwise, prove by mathematical induction that \(u _ { n } < 3\) for all positive integers \(n\).
  2. Show that \(u _ { n + 1 } > u _ { n }\) for \(n \geqslant 1\).
CAIE FP1 2018 November Q4
4 A curve is defined parametrically by $$x = t - \frac { 1 } { 2 } \sin 2 t \quad \text { and } \quad y = \sin ^ { 2 } t$$ The arc of the curve joining the point where \(t = 0\) to the point where \(t = \pi\) is rotated through one complete revolution about the \(x\)-axis. The area of the surface generated is denoted by \(S\).
  1. Show that $$S = a \pi \int _ { 0 } ^ { \pi } \sin ^ { 3 } t \mathrm {~d} t$$ where the constant \(a\) is to be found.
  2. Using the result \(\sin 3 t = 3 \sin t - 4 \sin ^ { 3 } t\), find the exact value of \(S\).
CAIE FP1 2018 November Q5
5 It is given that \(\lambda\) is an eigenvalue of the matrix \(\mathbf { A }\) with \(\mathbf { e }\) as a corresponding eigenvector, and \(\mu\) is an eigenvalue of the matrix \(\mathbf { B }\) for which \(\mathbf { e }\) is also a corresponding eigenvector.
  1. Show that \(\lambda + \mu\) is an eigenvalue of the matrix \(\mathbf { A } + \mathbf { B }\) with \(\mathbf { e }\) as a corresponding eigenvector.
    The matrix \(\mathbf { A }\), given by $$\mathbf { A } = \left( \begin{array} { r r r } 2 & 0 & 1 \\ - 1 & 2 & 3 \\ 1 & 0 & 2 \end{array} \right)$$ has \(\left( \begin{array} { l } 1 \\ 2 \\ 1 \end{array} \right) , \left( \begin{array} { r } 1 \\ 4 \\ - 1 \end{array} \right)\) and \(\left( \begin{array} { l } 0 \\ 1 \\ 0 \end{array} \right)\) as eigenvectors.
  2. Find the corresponding eigenvalues.
    The matrix \(\mathbf { B }\) has eigenvalues 4, 5 and 1 with corresponding eigenvectors \(\left( \begin{array} { l } 1 \\ 2 \\ 1 \end{array} \right) , \left( \begin{array} { r } 1 \\ 4 \\ - 1 \end{array} \right)\) and \(\left( \begin{array} { l } 0 \\ 1 \\ 0 \end{array} \right)\) respectively.
  3. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(( \mathbf { A } + \mathbf { B } ) ^ { 3 } = \mathbf { P D P } ^ { - 1 }\).
CAIE FP1 2018 November Q6
6 The curve \(C\) has equation $$y = \frac { x ^ { 2 } + a x - 1 } { x + 1 }$$ where \(a\) is constant and \(a > 1\).
  1. Find the equations of the asymptotes of \(C\).
  2. Show that \(C\) intersects the \(x\)-axis twice.
  3. Justifying your answer, find the number of stationary points on \(C\).
  4. Sketch \(C\), stating the coordinates of its point of intersection with the \(y\)-axis.
CAIE FP1 2018 November Q7
7
  1. Use de Moivre's theorem to show that $$\sin 8 \theta = 8 \sin \theta \cos \theta \left( 1 - 10 \sin ^ { 2 } \theta + 24 \sin ^ { 4 } \theta - 16 \sin ^ { 6 } \theta \right) .$$
  2. Use the equation \(\frac { \sin 8 \theta } { \sin 2 \theta } = 0\) to find the roots of $$16 x ^ { 6 } - 24 x ^ { 4 } + 10 x ^ { 2 } - 1 = 0$$ in the form \(\sin k \pi\), where \(k\) is rational.
CAIE FP1 2018 November Q8
5 marks
8 The plane \(\Pi _ { 1 }\) has equation $$\mathbf { r } = \left( \begin{array} { l } 5 \\ 1 \\ 0 \end{array} \right) + s \left( \begin{array} { r } - 4 \\ 1 \\ 3 \end{array} \right) + t \left( \begin{array} { l } 0 \\ 1 \\ 2 \end{array} \right)$$
  1. Find a cartesian equation of \(\Pi _ { 1 }\).
    The plane \(\Pi _ { 2 }\) has equation \(3 x + y - z = 3\).
  2. Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), giving your answer in degrees.
  3. Find an equation of the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }\). [5]
CAIE FP1 2018 November Q9
9 The curve \(C\) has polar equation $$r = 5 \sqrt { } ( \cot \theta ) ,$$ where \(0.01 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
  1. Find the area of the finite region bounded by \(C\) and the line \(\theta = 0.01\), showing full working. Give your answer correct to 1 decimal place.
    Let \(P\) be the point on \(C\) where \(\theta = 0.01\).
  2. Find the distance of \(P\) from the initial line, giving your answer correct to 1 decimal place.
  3. Find the maximum distance of \(C\) from the initial line.
  4. Sketch \(C\).
CAIE FP1 2018 November Q10
10
  1. Find the particular solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 10 x = 37 \sin 3 t$$ given that \(x = 3\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 0\) when \(t = 0\).
  2. Show that, for large positive values of \(t\) and for any initial conditions, $$x \approx \sqrt { } ( 37 ) \sin ( 3 t - \phi ) ,$$ where the constant \(\phi\) is such that \(\tan \phi = 6\).
CAIE FP1 2018 November Q11 EITHER
  1. By considering \(( 2 r + 1 ) ^ { 2 } - ( 2 r - 1 ) ^ { 2 }\), use the method of differences to prove that $$\sum _ { r = 1 } ^ { n } r = \frac { 1 } { 2 } n ( n + 1 )$$
  2. By considering \(( 2 r + 1 ) ^ { 4 } - ( 2 r - 1 ) ^ { 4 }\), use the method of differences and the result given in part (i) to prove that $$\sum _ { r = 1 } ^ { n } r ^ { 3 } = \frac { 1 } { 4 } n ^ { 2 } ( n + 1 ) ^ { 2 }$$ The sums \(S\) and \(T\) are defined as follows: $$\begin{aligned} & S = 1 ^ { 3 } + 2 ^ { 3 } + 3 ^ { 3 } + 4 ^ { 3 } + \ldots + ( 2 N ) ^ { 3 } + ( 2 N + 1 ) ^ { 3 } , \\ & T = 1 ^ { 3 } + 3 ^ { 3 } + 5 ^ { 3 } + 7 ^ { 3 } + \ldots + ( 2 N - 1 ) ^ { 3 } + ( 2 N + 1 ) ^ { 3 } . \end{aligned}$$
  3. Use the result given in part (ii) to show that \(S = ( 2 N + 1 ) ^ { 2 } ( N + 1 ) ^ { 2 }\).
  4. Hence, or otherwise, find an expression in terms of \(N\) for \(T\), factorising your answer as far as possible.
  5. Deduce the value of \(\frac { S } { T }\) as \(N \rightarrow \infty\).
CAIE FP1 2018 November Q11 OR
The curve \(C\) has equation $$x ^ { 2 } + 2 x y = y ^ { 3 } - 2$$
  1. Show that \(A ( - 1,1 )\) is the only point on \(C\) with \(x\)-coordinate equal to - 1 .
    For \(n \geqslant 1\), let \(A _ { n }\) denote the value of \(\frac { \mathrm { d } ^ { n } y } { \mathrm {~d} x ^ { n } }\) at the point \(A ( - 1,1 )\).
  2. Show that \(A _ { 1 } = 0\).
  3. Show that \(A _ { 2 } = \frac { 2 } { 5 }\).
    Let \(I _ { n } = \int _ { - 1 } ^ { 0 } x ^ { n } \frac { \mathrm {~d} ^ { n } y } { \mathrm {~d} x ^ { n } } \mathrm {~d} x\).
  4. Show that for \(n \geqslant 2\), $$I _ { n } = ( - 1 ) ^ { n + 1 } A _ { n - 1 } - n I _ { n - 1 } .$$
  5. Deduce the value of \(I _ { 3 }\) in terms of \(I _ { 1 }\).
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.