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CAIE FP1 2018 June Q2
6 marks Standard +0.8
2 It is given that \(\mathrm { f } ( n ) = 2 ^ { 3 n } + 8 ^ { n - 1 }\). By simplifying \(\mathrm { f } ( k ) + \mathrm { f } ( k + 1 )\), or otherwise, prove by mathematical induction that \(\mathrm { f } ( n )\) is divisible by 9 for every positive integer \(n\).
CAIE FP1 2018 June Q3
8 marks Standard +0.8
3 The curve \(C\) has polar equation \(r = \cos 2 \theta\), for \(- \frac { 1 } { 4 } \pi \leqslant \theta \leqslant \frac { 1 } { 4 } \pi\).
  1. Sketch \(C\).
  2. Find the area of the region enclosed by \(C\), showing full working.
  3. Find a cartesian equation of \(C\).
CAIE FP1 2018 June Q4
8 marks Standard +0.3
4 It is given that the equation $$x ^ { 3 } - 21 x ^ { 2 } + k x - 216 = 0$$ where \(k\) is a constant, has real roots \(a , a r\) and \(a r ^ { - 1 }\).
  1. Find the numerical values of the roots.
  2. Deduce the value of \(k\).
CAIE FP1 2018 June Q5
8 marks Standard +0.8
5 Let \(S _ { n } = \sum _ { r = 1 } ^ { n } ( - 1 ) ^ { r - 1 } r ^ { 2 }\).
  1. Use the standard result for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) given in the List of Formulae (MF10) to show that $$S _ { 2 n } = - n ( 2 n + 1 )$$
  2. State the value of \(\lim _ { n \rightarrow \infty } \frac { S _ { 2 n } } { n ^ { 2 } }\) and find \(\lim _ { n \rightarrow \infty } \frac { S _ { 2 n + 1 } } { n ^ { 2 } }\).
CAIE FP1 2018 June Q6
9 marks Standard +0.3
6 The curve \(C\) has equation $$y = \frac { x ^ { 2 } + b } { x + b }$$ where \(b\) is a positive constant.
  1. Find the equations of the asymptotes of \(C\).
  2. Show that \(C\) does not intersect the \(x\)-axis.
  3. Justifying your answer, find the number of stationary points on \(C\).
  4. Sketch C. Your sketch should indicate the coordinates of any points of intersection with the \(y\)-axis. You do not need to find the coordinates of any stationary points.
CAIE FP1 2018 June Q7
10 marks Standard +0.3
7 Find the particular solution of the differential equation $$49 \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 14 \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = 49 x + 735$$ given that when \(x = 0 , y = 0\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\).
CAIE FP1 2018 June Q8
10 marks Challenging +1.2
8 The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 3 }\) is represented by the matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { r r c r } 1 & 2 & \alpha & - 1 \\ 2 & 6 & - 3 & - 3 \\ 3 & 10 & - 6 & - 5 \end{array} \right)$$ and \(\alpha\) is a constant. When \(\alpha \neq 0\) the null space of T is denoted by \(K _ { 1 }\).
  1. Find a basis for \(K _ { 1 }\).
    When \(\alpha = 0\) the null space of T is denoted by \(K _ { 2 }\).
  2. Find a basis for \(K _ { 2 }\).
  3. Determine, justifying your answer, whether \(K _ { 1 }\) is a subspace of \(K _ { 2 }\).
CAIE FP1 2018 June Q9
10 marks Challenging +1.8
9
  1. Using the substitution \(u = \tan x\), or otherwise, find \(\int \sec ^ { 2 } x \tan ^ { 2 } x \mathrm {~d} x\).
    It is given that, for \(n \geqslant 0\), $$I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sec ^ { n } x \tan ^ { 2 } x \mathrm {~d} x$$
  2. Using the result that \(\frac { \mathrm { d } } { \mathrm { d } x } ( \sec x ) = \tan x \sec x\), show that, for \(n \geqslant 2\), $$( n + 1 ) I _ { n } = ( \sqrt { } 2 ) ^ { n - 2 } + ( n - 2 ) I _ { n - 2 }$$
  3. Hence find the mean value of \(\sec ^ { 4 } x \tan ^ { 2 } x\) with respect to \(x\) over the interval \(0 \leqslant x \leqslant \frac { 1 } { 4 } \pi\), giving your answer in exact form.
CAIE FP1 2018 June Q10
12 marks Challenging +1.2
10 The line \(l _ { 1 }\) is parallel to the vector \(a \mathbf { i } - \mathbf { j } + \mathbf { k }\), where \(a\) is a constant, and passes through the point whose position vector is \(9 \mathbf { j } + 2 \mathbf { k }\). The line \(l _ { 2 }\) is parallel to the vector \(- a \mathbf { i } + 2 \mathbf { j } + 4 \mathbf { k }\) and passes through the point whose position vector is \(- 6 \mathbf { i } - 5 \mathbf { j } + 10 \mathbf { k }\).
  1. It is given that \(l _ { 1 }\) and \(l _ { 2 }\) intersect.
    (a) Show that \(a = - \frac { 6 } { 13 }\).
    (b) Find a cartesian equation of the plane containing \(l _ { 1 }\) and \(l _ { 2 }\).
  2. Given instead that the perpendicular distance between \(l _ { 1 }\) and \(l _ { 2 }\) is \(3 \sqrt { } ( 30 )\), find the value of \(a\).
CAIE FP1 2018 June Q11 EITHER
Challenging +1.2
  1. Show that if \(z = \mathrm { e } ^ { \mathrm { i } \theta }\) and \(z \neq - 1\) then $$\frac { z - 1 } { z + 1 } = \mathrm { i } \tan \frac { 1 } { 2 } \theta$$
  2. Hence, or otherwise, show that if \(z\) is a cube root of unity then $$\frac { z ^ { 3 } - 1 } { z ^ { 3 } + 1 } + \frac { z ^ { 2 } - 1 } { z ^ { 2 } + 1 } + \frac { z - 1 } { z + 1 } = 0$$
  3. Hence write down three roots of the equation $$\left( z ^ { 3 } - 1 \right) \left( z ^ { 2 } + 1 \right) ( z + 1 ) + \left( z ^ { 2 } - 1 \right) \left( z ^ { 3 } + 1 \right) ( z + 1 ) + ( z - 1 ) \left( z ^ { 3 } + 1 \right) \left( z ^ { 2 } + 1 \right) = 0$$ and find the other three roots. Give your answers in an exact form.
CAIE FP1 2018 June Q11 OR
Challenging +1.8
It is given that \(\mathbf { e }\) is an eigenvector of the matrix \(\mathbf { A }\), with corresponding eigenvalue \(\lambda\).
  1. Write down another eigenvector of \(\mathbf { A }\) corresponding to \(\lambda\).
  2. Write down an eigenvector and corresponding eigenvalue of \(\mathbf { A } ^ { n }\), where \(n\) is a positive integer.
    Let \(\mathbf { A } = \left( \begin{array} { l l l } 3 & 0 & 0 \\ 2 & 7 & 0 \\ 4 & 8 & 1 \end{array} \right)\).
  3. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { n } = \mathbf { P D P } \mathbf { P } ^ { - 1 }\).
  4. Determine the set of values of the real constant \(k\) such that $$\sum _ { n = 1 } ^ { \infty } k ^ { n } \left( \mathbf { A } ^ { n } - k \mathbf { A } ^ { n + 1 } \right) = k \mathbf { A } .$$ If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE FP1 2018 June Q1
5 marks Standard +0.8
1 The variables \(x\) and \(y\) are such that \(y = - 1\) when \(x = 0\) and $$\left( x + \frac { \mathrm { d } y } { \mathrm {~d} x } \right) ^ { 3 } = y ^ { 2 } + x$$
  1. Find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(x = 0\).
  2. Find also the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) when \(x = 0\).
CAIE FP1 2018 June Q2
6 marks Standard +0.8
2
  1. Verify that $$\frac { n ( \mathrm { e } - 1 ) + \mathrm { e } } { n ( n + 1 ) \mathrm { e } ^ { n + 1 } } = \frac { 1 } { n \mathrm { e } ^ { n } } - \frac { 1 } { ( n + 1 ) \mathrm { e } ^ { n + 1 } }$$ Let \(S _ { N } = \sum _ { n = 1 } ^ { N } \frac { n ( \mathrm { e } - 1 ) + \mathrm { e } } { n ( n + 1 ) \mathrm { e } ^ { n + 1 } }\).
  2. Express \(S _ { N }\) in terms of \(N\) and e.
    Let \(S = \lim _ { N \rightarrow \infty } S _ { N }\).
  3. Find the least value of \(N\) such that \(( N + 1 ) \left( S - S _ { N } \right) < 10 ^ { - 3 }\).
CAIE FP1 2018 June Q3
8 marks Challenging +1.2
3
  1. Use de Moivre's theorem to show that $$\cos 4 \theta = \cos ^ { 4 } \theta - 6 \cos ^ { 2 } \theta \sin ^ { 2 } \theta + \sin ^ { 4 } \theta$$
  2. Hence find all the roots of the equation $$x ^ { 4 } - 6 x ^ { 2 } + 1 = 0$$ in the form \(\tan q \pi\), where \(q\) is a positive rational number.
CAIE FP1 2018 June Q4
8 marks Standard +0.3
4 The curve \(C\) has equation $$y = \frac { x ^ { 2 } + 7 x + 6 } { x - 2 }$$
  1. Find the coordinates of the points of intersection of \(C\) with the axes.
  2. Find the equation of each of the asymptotes of \(C\).
  3. Sketch C.
CAIE FP1 2018 June Q5
8 marks Standard +0.3
5 It is given that \(\mathbf { e }\) is an eigenvector of the matrix \(\mathbf { A }\) with corresponding eigenvalue \(\lambda\).
  1. Show that \(\mathbf { e }\) is an eigenvector of \(\mathbf { A } ^ { 3 }\) and state the corresponding eigenvalue.
    It is given that $$\mathbf { A } = \left( \begin{array} { r r } 2 & 0 \\ - 1 & 3 \end{array} \right) .$$
  2. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that $$\mathbf { A } ^ { 3 } + \mathbf { I } = \mathbf { P } \mathbf { D } \mathbf { P } ^ { - 1 }$$ where \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.
CAIE FP1 2018 June Q6
8 marks Standard +0.8
6 The equation $$9 x ^ { 3 } - 9 x ^ { 2 } + x - 2 = 0$$ has roots \(\alpha , \beta , \gamma\).
  1. Use the substitution \(y = 3 x - 1\) to show that \(3 \alpha - 1,3 \beta - 1,3 \gamma - 1\) are the roots of the equation $$y ^ { 3 } - 2 y - 7 = 0$$ The sum \(( 3 \alpha - 1 ) ^ { n } + ( 3 \beta - 1 ) ^ { n } + ( 3 \gamma - 1 ) ^ { n }\) is denoted by \(S _ { n }\).
  2. Find the value of \(S _ { 3 }\).
  3. Find the value of \(S _ { - 2 }\).
CAIE FP1 2018 June Q7
11 marks Standard +0.8
7 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have vector equations $$\mathbf { r } = a \mathbf { i } + 9 \mathbf { j } + 13 \mathbf { k } + \lambda ( \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = - 3 \mathbf { i } + 7 \mathbf { j } - 2 \mathbf { k } + \mu ( - \mathbf { i } + 2 \mathbf { j } - 3 \mathbf { k } )$$ respectively. It is given that \(l _ { 1 }\) and \(l _ { 2 }\) intersect.
  1. Find the value of the constant \(a\).
    The point \(P\) has position vector \(3 \mathbf { i } + \mathbf { j } + 6 \mathbf { k }\).
  2. Find the perpendicular distance from \(P\) to the plane containing \(l _ { 1 }\) and \(l _ { 2 }\).
  3. Find the perpendicular distance from \(P\) to \(l _ { 2 }\).
CAIE FP1 2018 June Q8
10 marks Standard +0.8
8 The curves \(C _ { 1 }\) and \(C _ { 2 }\) have polar equations, for \(0 \leqslant \theta \leqslant \pi\), as follows: $$\begin{aligned} & C _ { 1 } : r = a \\ & C _ { 2 } : r = 2 a | \cos \theta | \end{aligned}$$ where \(a\) is a positive constant. The curves intersect at the points \(P _ { 1 }\) and \(P _ { 2 }\).
  1. Find the polar coordinates of \(P _ { 1 }\) and \(P _ { 2 }\).
  2. In a single diagram, sketch \(C _ { 1 } , C _ { 2 }\) and their line of symmetry.
  3. The region \(R\) enclosed by \(C _ { 1 }\) and \(C _ { 2 }\) is bounded by the \(\operatorname { arcs } O P _ { 1 } , P _ { 1 } P _ { 2 }\) and \(P _ { 2 } O\), where \(O\) is the pole. Find the area of \(R\), giving your answer in exact form.
CAIE FP1 2018 June Q9
10 marks Standard +0.8
9 For the sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\), it is given that \(u _ { 1 } = 8\) and $$u _ { r + 1 } = \frac { 5 u _ { r } - 3 } { 4 }$$ for all \(r\).
  1. Prove by mathematical induction that $$u _ { n } = 4 \left( \frac { 5 } { 4 } \right) ^ { n } + 3$$ for all positive integers \(n\).
  2. Deduce the set of values of \(x\) for which the infinite series $$\left( u _ { 1 } - 3 \right) x + \left( u _ { 2 } - 3 \right) x ^ { 2 } + \ldots + \left( u _ { r } - 3 \right) x ^ { r } + \ldots$$ is convergent.
  3. Use the result given in part (i) to find surds \(a\) and \(b\) such that $$\sum _ { n = 1 } ^ { N } \ln \left( u _ { n } - 3 \right) = N ^ { 2 } \ln a + N \ln b .$$
CAIE FP1 2018 June Q10
12 marks Challenging +1.2
10 It is given that \(t \neq 0\) and $$t \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 9 t x = 3 t ^ { 2 } + 1$$
  1. Show that if \(y = t x\) then $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } + 9 y = 3 t ^ { 2 } + 1$$
  2. Find \(x\) in terms of \(t\), given that \(x = \frac { 1 } { 9 } \pi\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { 2 } { 3 }\) when \(t = \frac { 1 } { 3 } \pi\).
CAIE FP1 2018 June Q11 EITHER
Challenging +1.8
  1. Show that $$\int _ { - \frac { 1 } { 2 } \pi } ^ { \frac { 1 } { 2 } \pi } \mathrm { e } ^ { x } \cos x \mathrm {~d} x = \frac { 1 } { 2 } \left( \mathrm { e } ^ { \frac { 1 } { 2 } \pi } + \mathrm { e } ^ { - \frac { 1 } { 2 } \pi } \right)$$
  2. It is given that, for \(n \geqslant 0\), $$I _ { n } = \int _ { - \frac { 1 } { 2 } \pi } ^ { \frac { 1 } { 2 } \pi } \mathrm { e } ^ { 2 x } \cos ^ { n } x \mathrm {~d} x$$ Show that, for \(n \geqslant 2\), $$4 I _ { n } = n ( n - 1 ) \int _ { - \frac { 1 } { 2 } \pi } ^ { \frac { 1 } { 2 } \pi } \mathrm { e } ^ { 2 x } \sin ^ { 2 } x \cos ^ { n - 2 } x \mathrm {~d} x - n I _ { n }$$ and deduce the reduction formula $$\left( n ^ { 2 } + 4 \right) I _ { n } = n ( n - 1 ) I _ { n - 2 }$$
  3. Using the result in part (i) and the reduction formula in part (ii), find the \(y\)-coordinate of the centroid of the region bounded by the \(x\)-axis and the arc of the curve \(y = \mathrm { e } ^ { x } \cos x\) from \(x = - \frac { 1 } { 2 } \pi\) to \(x = \frac { 1 } { 2 } \pi\). Give your answer correct to 3 significant figures.
CAIE FP1 2018 June Q11 OR
Hard +2.3
Let \(V\) be the subspace of \(\mathbb { R } ^ { 4 }\) spanned by $$\mathbf { v } _ { 1 } = \left( \begin{array} { l } 1
2
0
2 \end{array} \right) , \quad \mathbf { v } _ { 2 } = \left( \begin{array} { r } - 2
- 5
5
6 \end{array} \right) , \quad \mathbf { v } _ { 3 } = \left( \begin{array} { r } 0
- 3
CAIE FP1 2018 June Q18
Challenging +1.2
18 \end{array} \right) \quad \text { and } \quad \mathbf { v } _ { 4 } = \left( \begin{array} { r } 0
- 2
10
8 \end{array} \right) .$$
  1. Show that the dimension of \(V\) is 3 .
  2. Express \(\mathbf { v } _ { 4 }\) as a linear combination of \(\mathbf { v } _ { 1 } , \mathbf { v } _ { 2 }\) and \(\mathbf { v } _ { 3 }\).
  3. Write down a basis for \(V\).
    Let \(\mathbf { M } = \left( \begin{array} { r r r r } 1 & - 2 & 0 & 0 \\ 2 & - 5 & - 3 & - 2 \\ 0 & 5 & 15 & 10 \\ 2 & 6 & 18 & 8 \end{array} \right)\).
  4. Find the general solution of \(\mathbf { M x } = \mathbf { v } _ { 1 } + \mathbf { v } _ { 2 }\).
    The set of elements of \(\mathbb { R } ^ { 4 }\) which are not solutions of \(\mathbf { M x } = \mathbf { v } _ { 1 } + \mathbf { v } _ { 2 }\) is denoted by \(W\).
  5. State, with a reason, whether \(W\) is a vector space.
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE FP1 2019 June Q1
6 marks Standard +0.3
1 A curve \(C\) has equation \(\cos y = x\), for \(- \pi < x < \pi\).
  1. Use implicit differentiation to show that $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = - \cot y \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 }$$
  2. Hence find the exact value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at the point \(\left( \frac { 1 } { 2 } , \frac { 1 } { 3 } \pi \right)\) on \(C\).