Questions FP1 (1385 questions)

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CAIE FP1 2018 June Q10
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
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
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
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
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
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
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
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
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
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
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
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 \(C\).
CAIE FP1 2019 June Q2
Challenging +1.2
2 Let \(u _ { n } = \frac { 4 \sin \left( n - \frac { 1 } { 2 } \right) \sin \frac { 1 } { 2 } } { \cos ( 2 n - 1 ) + \cos 1 }\).
  1. Using the formulae for \(\cos P \pm \cos Q\) given in the List of Formulae MF10, show that $$u _ { n } = \frac { 1 } { \cos n } - \frac { 1 } { \cos ( n - 1 ) }$$
  2. Use the method of differences to find \(\sum _ { n = 1 } ^ { N } u _ { n }\).
  3. Explain why the infinite series \(u _ { 1 } + u _ { 2 } + u _ { 3 } + \ldots\) does not converge.
CAIE FP1 2019 June Q3
Challenging +1.2
3 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations \(\mathbf { r } = 6 \mathbf { i } + 2 \mathbf { j } + 7 \mathbf { k } + \lambda ( \mathbf { i } + \mathbf { j } )\) and \(\mathbf { r } = 4 \mathbf { i } + 4 \mathbf { j } + \mu ( - 6 \mathbf { j } + \mathbf { k } )\) 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 \(l _ { 2 }\). Find the position vectors of \(P\) and \(Q\).
CAIE FP1 2019 June Q4
Challenging +1.2
4 It is given that, for \(n \geqslant 0\), $$I _ { n } = \int _ { 0 } ^ { 1 } x ^ { n } \mathrm { e } ^ { x ^ { 3 } } \mathrm {~d} x$$
  1. Show that \(I _ { 2 } = \frac { 1 } { 3 } ( \mathrm { e } - 1 )\).
  2. Show that, for \(n \geqslant 3\), $$3 I _ { n } = \mathrm { e } - ( n - 2 ) I _ { n - 3 }$$
  3. Hence find the exact value of \(I _ { 8 }\).
CAIE FP1 2019 June Q5
Challenging +1.2
5 A curve \(C\) is defined parametrically by $$x = \frac { 2 } { \mathrm { e } ^ { t } + \mathrm { e } ^ { - t } } \quad \text { and } \quad y = \frac { \mathrm { e } ^ { t } - \mathrm { e } ^ { - t } } { \mathrm { e } ^ { t } + \mathrm { e } ^ { - t } }$$ for \(0 \leqslant t \leqslant 1\). The area of the surface generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis is denoted by \(S\).
  1. Show that \(S = 4 \pi \int _ { 0 } ^ { 1 } \frac { \mathrm { e } ^ { t } - \mathrm { e } ^ { - t } } { \left( \mathrm { e } ^ { t } + \mathrm { e } ^ { - t } \right) ^ { 2 } } \mathrm {~d} t\).
  2. Using the substitution \(u = \mathrm { e } ^ { t } + \mathrm { e } ^ { - t }\), or otherwise, find \(S\) in terms of \(\pi\) and e .
CAIE FP1 2019 June Q6
Challenging +1.2
6 The equation $$x ^ { 3 } - x + 1 = 0$$ has roots \(\alpha , \beta , \gamma\).
  1. Use the relation \(x = y ^ { \frac { 1 } { 3 } }\) to show that the equation $$y ^ { 3 } + 3 y ^ { 2 } + 2 y + 1 = 0$$ has roots \(\alpha ^ { 3 } , \beta ^ { 3 } , \gamma ^ { 3 }\). Hence write down the value of \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 }\).
    Let \(S _ { n } = \alpha ^ { n } + \beta ^ { n } + \gamma ^ { n }\).
  2. Find the value of \(S _ { - 3 }\).
  3. Show that \(S _ { 6 } = 5\) and find the value of \(S _ { 9 }\).
CAIE FP1 2019 June Q7
Standard +0.8
7 Find the particular solution of the differential equation $$10 \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 3 \frac { \mathrm {~d} x } { \mathrm {~d} t } - x = t + 2$$ given that when \(t = 0 , x = 0\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 0\).
CAIE FP1 2019 June Q8
Challenging +1.2
8
  1. Prove by mathematical induction that, for \(z \neq 1\) and all positive integers \(n\), $$1 + z + z ^ { 2 } + \ldots + z ^ { n - 1 } = \frac { z ^ { n } - 1 } { z - 1 }$$
  2. By letting \(z = \frac { 1 } { 2 } ( \cos \theta + \mathrm { i } \sin \theta )\), use de Moivre's theorem to deduce that $$\sum _ { m = 1 } ^ { \infty } \left( \frac { 1 } { 2 } \right) ^ { m } \sin m \theta = \frac { 2 \sin \theta } { 5 - 4 \cos \theta }$$
CAIE FP1 2019 June Q9
Standard +0.8
9 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 \(\mathbf { A } ^ { 2 }\), with corresponding eigenvalue \(\lambda ^ { 2 }\).
    The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by $$\mathbf { A } = \left( \begin{array} { c c c } n & 1 & 3 \\ 0 & 2 n & 0 \\ 0 & 0 & 3 n \end{array} \right) \quad \text { and } \quad \mathbf { B } = ( \mathbf { A } + n \mathbf { I } ) ^ { 2 }$$ where \(\mathbf { I }\) is the \(3 \times 3\) identity matrix and \(n\) is a non-zero integer.
  2. Find, in terms of \(n\), a non-singular matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { B } = \mathbf { P D P } \mathbf { P } ^ { - 1 }\).