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CAIE Further Paper 1 2020 November Q2
8 marks Standard +0.3
2
  1. Use standard results from the List of Formulae (MF19) to show that $$\sum _ { r = 1 } ^ { n } ( 7 r + 1 ) ( 7 r + 8 ) = a n ^ { 3 } + b n ^ { 2 } + c n$$ where \(a , b\) and \(c\) are constants to be determined.
  2. Use the method of differences to find \(\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 7 r + 1 ) ( 7 r + 8 ) }\) in terms of \(n\).
  3. Deduce the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { ( 7 r + 1 ) ( 7 r + 8 ) }\).
CAIE Further Paper 1 2020 November Q3
11 marks Challenging +1.8
3 The cubic equation \(\mathrm { x } ^ { 3 } + \mathrm { cx } + 1 = 0\), where \(c\) is a constant, has roots \(\alpha , \beta , \gamma\).
  1. Find a cubic equation whose roots are \(\alpha ^ { 3 } , \beta ^ { 3 } , \gamma ^ { 3 }\).
  2. Show that \(\alpha ^ { 6 } + \beta ^ { 6 } + \gamma ^ { 6 } = 3 - 2 c ^ { 3 }\).
  3. Find the real value of \(c\) for which the matrix \(\left( \begin{array} { c c c } 1 & \alpha ^ { 3 } & \beta ^ { 3 } \\ \alpha ^ { 3 } & 1 & \gamma ^ { 3 } \\ \beta ^ { 3 } & \gamma ^ { 3 } & 1 \end{array} \right)\) is singular.
CAIE Further Paper 1 2020 November Q4
11 marks Standard +0.3
4 The points \(A , B , C\) have position vectors $$- \mathbf { i } + \mathbf { j } + 2 \mathbf { k } , \quad - 2 \mathbf { i } - \mathbf { j } , \quad 2 \mathbf { i } + 2 \mathbf { k } ,$$ respectively, relative to the origin \(O\).
  1. Find the equation of the plane \(A B C\), giving your answer in the form \(a x + b y + c z = d\).
  2. Find the perpendicular distance from \(O\) to the plane \(A B C\).
  3. Find the acute angle between the planes \(O A B\) and \(A B C\).
CAIE Further Paper 1 2020 November Q5
7 marks Challenging +1.2
5 Prove by mathematical induction that, for every positive integer \(n\), $$\frac { d ^ { 2 n - 1 } } { d x ^ { 2 n - 1 } } ( x \sin x ) = ( - 1 ) ^ { n - 1 } ( x \cos x + ( 2 n - 1 ) \sin x )$$
CAIE Further Paper 1 2020 November Q6
12 marks Standard +0.8
6 The curve \(C\) has equation \(\mathrm { y } = \frac { \mathrm { x } ^ { 2 } + \mathrm { x } - 1 } { \mathrm { x } - 1 }\).
  1. Find the equations of the asymptotes of \(C\).
  2. Show that there is no point on \(C\) for which \(1 < y < 5\).
  3. Find the coordinates of the intersections of \(C\) with the axes, and sketch \(C\).
  4. Sketch the curve with equation \(\mathrm { y } = \left| \frac { \mathrm { x } ^ { 2 } + \mathrm { x } - 1 } { \mathrm { x } - 1 } \right|\).
CAIE Further Paper 1 2020 November Q7
17 marks Challenging +1.2
7
  1. Show that the curve with Cartesian equation $$\left( x ^ { 2 } + y ^ { 2 } \right) ^ { \frac { 5 } { 2 } } = 4 x y \left( x ^ { 2 } - y ^ { 2 } \right)$$ has polar equation \(r = \sin 4 \theta\).
    The curve \(C\) has polar equation \(r = \sin 4 \theta\), for \(0 \leqslant \theta \leqslant \frac { 1 } { 4 } \pi\).
  2. Sketch \(C\) and state the equation of the line of symmetry.
  3. Find the exact value of the area of the region enclosed by \(C\).
  4. Using the identity \(\sin 4 \theta \equiv 4 \sin \theta \cos ^ { 3 } \theta - 4 \sin ^ { 3 } \theta \cos \theta\), find the maximum distance of \(C\) from the line \(\theta = \frac { 1 } { 2 } \pi\). Give your answer correct to 2 decimal places.
    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 Further Paper 1 2020 November Q1
8 marks Standard +0.8
1 The cubic equation \(\mathrm { x } ^ { 3 } + \mathrm { bx } ^ { 2 } + \mathrm { cx } + \mathrm { d } = 0\), where \(b , c\) and \(d\) are constants, has roots \(\alpha , \beta , \gamma\). It is given that \(\alpha \beta \gamma = - 1\).
  1. State the value of \(d\).
  2. Find a cubic equation, with coefficients in terms of \(b\) and \(c\), whose roots are \(\alpha + 1 , \beta + 1 , \gamma + 1\).
  3. Given also that \(\gamma + 1 = - \alpha - 1\), deduce that \(( \mathrm { c } - 2 \mathrm {~b} + 3 ) ( \mathrm { b } - 3 ) = \mathrm { b } - \mathrm { c }\).
CAIE Further Paper 1 2020 November Q2
5 marks Standard +0.3
2 Prove by mathematical induction that \(7 ^ { 2 n } - 1\) is divisible by 12 for every positive integer \(n\).
CAIE Further Paper 1 2020 November Q3
7 marks Challenging +1.2
3
  1. By simplifying \(\left( x ^ { n } - \sqrt { x ^ { 2 n } + 1 } \right) \left( x ^ { n } + \sqrt { x ^ { 2 n } + 1 } \right)\), show that \(\frac { 1 } { x ^ { n } - \sqrt { x ^ { 2 n } + 1 } } = - x ^ { n } - \sqrt { x ^ { 2 n } + 1 }\). [1]
    Let \(u _ { n } = x ^ { n + 1 } + \sqrt { x ^ { 2 n + 2 } + 1 } + \frac { 1 } { x ^ { n } - \sqrt { x ^ { 2 n } + 1 } }\).
  2. Use the method of differences to find \(\sum _ { \mathrm { n } = 1 } ^ { \mathrm { N } } \mathrm { u } _ { \mathrm { n } }\) in terms of \(N\) and \(x\).
  3. Deduce the set of values of \(x\) for which the infinite series $$u _ { 1 } + u _ { 2 } + u _ { 3 } + \ldots$$ is convergent and give the sum to infinity when this exists.
CAIE Further Paper 1 2020 November Q4
13 marks Standard +0.3
4 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by $$\mathbf { A } = \left( \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right) \text { and } \mathbf { B } = \left( \begin{array} { c c } \frac { 1 } { 2 } & - \frac { 1 } { 2 } \sqrt { 3 } \\ \frac { 1 } { 2 } \sqrt { 3 } & \frac { 1 } { 2 } \end{array} \right)$$
  1. Give full details of the geometrical transformation in the \(x - y\) plane represented by \(\mathbf { A }\).
  2. Give full details of the geometrical transformation in the \(x - y\) plane represented by \(\mathbf { B }\).
    The triangle \(D E F\) in the \(x - y\) plane is transformed by \(\mathbf { A B }\) onto triangle \(P Q R\).
  3. Show that the triangles \(D E F\) and \(P Q R\) have the same area.
  4. Find the matrix which transforms triangle \(P Q R\) onto triangle \(D E F\).
  5. Find the equations of the invariant lines, through the origin, of the transformation in the \(x - y\) plane represented by \(\mathbf { A B }\).
CAIE Further Paper 1 2020 November Q5
14 marks Challenging +1.8
5 The curve \(C\) has polar equation \(r = \ln ( 1 + \pi - \theta )\), for \(0 \leqslant \theta \leqslant \pi\).
  1. Sketch \(C\) and state the polar coordinates of the point of \(C\) furthest from the pole.
  2. Using the substitution \(u = 1 + \pi - \theta\), or otherwise, show that the area of the region enclosed by \(C\) and the initial line is $$\frac { 1 } { 2 } ( 1 + \pi ) \ln ( 1 + \pi ) ( \ln ( 1 + \pi ) - 2 ) + \pi$$
  3. Show that, at the point of \(C\) furthest from the initial line, $$( 1 + \pi - \theta ) \ln ( 1 + \pi - \theta ) - \tan \theta = 0$$ and verify that this equation has a root between 1.2 and 1.3.
CAIE Further Paper 1 2020 November Q6
13 marks Standard +0.8
6 Let \(a\) be a positive constant.
  1. The curve \(C _ { 1 }\) has equation \(\mathrm { y } = \frac { \mathrm { x } - \mathrm { a } } { \mathrm { x } - 2 \mathrm { a } }\). Sketch \(C _ { 1 }\). The curve \(C _ { 2 }\) has equation \(\mathrm { y } = \left( \frac { \mathrm { x } - \mathrm { a } } { \mathrm { x } - 2 \mathrm { a } } \right) ^ { 2 }\). The curve \(C _ { 3 }\) has equation \(\mathrm { y } = \left| \frac { \mathrm { x } - \mathrm { a } } { \mathrm { x } - 2 \mathrm { a } } \right|\).
    1. Find the coordinates of any stationary points of \(C _ { 2 }\).
    2. Find also the coordinates of any points of intersection of \(C _ { 2 }\) and \(C _ { 3 }\).
  3. Sketch \(C _ { 2 }\) and \(C _ { 3 }\) on a single diagram, clearly identifying each curve. Hence find the set of values of \(x\) for which \(\left( \frac { x - a } { x - 2 a } \right) ^ { 2 } \leqslant \left| \frac { x - a } { x - 2 a } \right|\).
CAIE Further Paper 1 2020 November Q7
15 marks Challenging +1.2
7 The points \(A , B , C\) have position vectors $$- 2 \mathbf { i } + 2 \mathbf { j } - \mathbf { k } , \quad - 2 \mathbf { i } + \mathbf { j } + 2 \mathbf { k } , \quad - 2 \mathbf { j } + \mathbf { k } ,$$ respectively, relative to the origin \(O\).
  1. Find the equation of the plane \(A B C\), giving your answer in the form \(a x + b y + c z = d\).
  2. Find the acute angle between the planes \(O B C\) and \(A B C\).
    The point \(D\) has position vector \(t \mathbf { i } - \mathbf { j }\).
  3. Given that the shortest distance between the lines \(A B\) and \(C D\) is \(\sqrt { \mathbf { 1 0 } }\), find the value of \(t\).
    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 Further Paper 1 2021 November Q1
6 marks Challenging +1.2
1 It is given that $$\alpha + \beta + \gamma = 3 , \quad \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 5 , \quad \alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } = 6 .$$ The cubic equation \(\mathrm { x } ^ { 3 } + \mathrm { bx } ^ { 2 } + \mathrm { cx } + \mathrm { d } = 0\) has roots \(\alpha , \beta , \gamma\).
Find the values of \(b , c\) and \(d\).
CAIE Further Paper 1 2021 November Q2
9 marks Standard +0.8
2
  1. Use standard results from the list of formulae (MF19) to find \(\sum _ { r = 1 } ^ { n } r ( r + 1 ) ( r + 2 )\) in terms of \(n\),
    fully factorising your answer. fully factorising your answer.
  2. Express \(\frac { 1 } { r ( r + 1 ) ( r + 2 ) }\) in partial fractions and hence use the method of differences to find $$\sum _ { r = 1 } ^ { n } \frac { 1 } { r ( r + 1 ) ( r + 2 ) }$$
  3. Deduce the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { r ( r + 1 ) ( r + 2 ) }\).
CAIE Further Paper 1 2021 November Q3
8 marks Challenging +1.8
3 The sequence of real numbers \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is such that \(a _ { 1 } = 1\) and $$a _ { n + 1 } = \left( a _ { n } + \frac { 1 } { a _ { n } } \right) ^ { 3 }$$
  1. Prove by mathematical induction that \(\ln a _ { n } \geqslant 3 ^ { n - 1 } \ln 2\) for all integers \(n \geqslant 2\).
    [0pt] [You may use the fact that \(\ln \left( x + \frac { 1 } { x } \right) > \ln x\) for \(x > 0\).]
  2. Show that \(\ln \mathrm { a } _ { \mathrm { n } + 1 } - \ln \mathrm { a } _ { \mathrm { n } } > 3 ^ { \mathrm { n } - 1 } \ln 4\) for \(n \geqslant 2\).
CAIE Further Paper 1 2021 November Q4
11 marks Challenging +1.2
4 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { c c } \cos \theta & - \sin \theta \\ \sin \theta & \cos \theta \end{array} \right) \left( \begin{array} { l l } 3 & 0 \\ 0 & 1 \end{array} \right)\).
  1. The matrix \(\mathbf { M }\) represents a sequence of two geometrical transformations. State the type of each transformation, and make clear the order in which they are applied.
  2. Find the values of \(\theta\), for \(0 \leqslant \theta \leqslant \pi\), for which the transformation represented by \(\mathbf { M }\) has exactly one invariant line through the origin, giving your answers in terms of \(\pi\).
CAIE Further Paper 1 2021 November Q5
13 marks Standard +0.3
5 The plane \(\Pi\) has equation \(\mathbf { r } = - 2 \mathbf { i } + 3 \mathbf { j } + 3 \mathbf { k } + \lambda ( \mathbf { i } + \mathbf { k } ) + \mu ( 2 \mathbf { i } + 3 \mathbf { j } )\).
  1. Find a Cartesian equation of \(\Pi\), giving your answer in the form \(a x + b y + c = d\).
    The line \(l\) passes through the point \(P\) with position vector \(2 \mathbf { i } - 3 \mathbf { j } + 5 \mathbf { k }\) and is parallel to the vector \(\mathbf { k }\).
  2. Find the position vector of the point where \(l\) meets \(\Pi\).
  3. Find the acute angle between \(l\) and \(\Pi\).
  4. Find the perpendicular distance from \(P\) to \(\Pi\).
CAIE Further Paper 1 2021 November Q6
13 marks Challenging +1.2
6 The curve \(C\) has polar equation \(r = 2 \cos \theta ( 1 + \sin \theta )\), for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
  1. Find the polar coordinates of the point on \(C\) that is furthest from the pole.
  2. Sketch C.
  3. Find the area of the region bounded by \(C\) and the initial line, giving your answer in exact form.
CAIE Further Paper 1 2021 November Q7
15 marks Challenging +1.2
7 The curve \(C\) has equation \(y = \frac { 4 x + 5 } { 4 - 4 x ^ { 2 } }\).
  1. Find the equations of the asymptotes of \(C\).
  2. Find the coordinates of any stationary points on \(C\).
  3. Sketch \(C\), stating the coordinates of the intersections with the axes.
  4. Sketch the curve with equation \(y = \left| \frac { 4 x + 5 } { 4 - 4 x ^ { 2 } } \right|\) and find in exact form the set of values of \(x\) for which \(4 | 4 x + 5 | > 5 \left| 4 - 4 x ^ { 2 } \right|\).
    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 Further Paper 1 2021 November Q1
9 marks Moderate -0.3
1
  1. Give full details of the geometrical transformation in the \(x - y\) plane represented by the matrix \(\left( \begin{array} { l l } 6 & 0 \\ 0 & 6 \end{array} \right)\). Let \(\mathbf { A } = \left( \begin{array} { l l } 3 & 4 \\ 2 & 2 \end{array} \right)\).
  2. The triangle \(D E F\) in the \(x - y\) plane is transformed by \(\mathbf { A }\) onto triangle \(P Q R\). Given that the area of triangle \(D E F\) is \(13 \mathrm {~cm} ^ { 2 }\), find the area of triangle \(P Q R\).
  3. Find the matrix \(\mathbf { B }\) such that \(\mathbf { A B } = \left( \begin{array} { l l } 6 & 0 \\ 0 & 6 \end{array} \right)\).
  4. Show that the origin is the only invariant point of the transformation in the \(x - y\) plane represented by \(\mathbf { A }\).
CAIE Further Paper 1 2021 November Q2
6 marks Challenging +1.2
2 It is given that \(\mathrm { y } = \mathrm { xe } ^ { \mathrm { ax } }\), where \(a\) is a constant.
Prove by mathematical induction that, for all positive integers \(n\), $$\frac { d ^ { n } y } { d x ^ { n } } = \left( a ^ { n } x + n a ^ { n - 1 } \right) e ^ { a x }$$
CAIE Further Paper 1 2021 November Q3
7 marks Challenging +1.2
3 Let \(S _ { n } = \sum _ { r = 1 } ^ { n } \ln \frac { r ( r + 2 ) } { ( r + 1 ) ^ { 2 } }\).
  1. Using the method of differences, or otherwise, show that \(S _ { n } = \ln \frac { n + 2 } { 2 ( n + 1 ) }\).
    Let \(S = \sum _ { r = 1 } ^ { \infty } \ln \frac { r ( r + 2 ) } { ( r + 1 ) ^ { 2 } }\).
  2. Find the least value of \(n\) such that \(\mathrm { S } _ { \mathrm { n } } - \mathrm { S } < 0.01\).
CAIE Further Paper 1 2021 November Q4
10 marks Challenging +1.2
4 The cubic equation \(x ^ { 3 } + 2 x ^ { 2 } + 3 x + 3 = 0\) has roots \(\alpha , \beta , \gamma\).
  1. Find the value of \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\).
  2. Show that \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } = 1\).
  3. Use standard results from the list of formulae (MF19) to show that $$\sum _ { r = 1 } ^ { n } \left( ( \alpha + r ) ^ { 3 } + ( \beta + r ) ^ { 3 } + ( \gamma + r ) ^ { 3 } \right) = n + \frac { 1 } { 4 } n ( n + 1 ) \left( a n ^ { 2 } + b n + c \right)$$ where \(a\), \(b\) and \(c\) are constants to be determined.
CAIE Further Paper 1 2021 November Q5
12 marks Challenging +1.2
5 The curve \(C\) has polar equation \(r = 3 + 2 \sin \theta\), for \(- \pi < \theta \leqslant \pi\).
  1. The diagram shows part of \(C\). Sketch the rest of \(C\) on the diagram. \includegraphics[max width=\textwidth, alt={}, center]{3dbf7021-79c0-4ebf-b96a-5ddeeed45011-08_865_702_408_1023} The straight line \(l\) has polar equation \(r \sin \theta = 2\).
  2. Add \(l\) to the diagram in part (a) and find the polar coordinates of the points of intersection of \(C\) and \(l\).
  3. The region \(R\) is enclosed by \(C\) and \(l\), and contains the pole. Find the area of \(R\), giving your answer in exact form.