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CAIE Further Paper 1 2020 June Q1
1 Let \(a\) be a positive constant.
  1. Sketch the curve with equation \(\mathrm { y } = \frac { \mathrm { ax } } { \mathrm { x } + 7 }\).
  2. Sketch the curve with equation \(y = \left| \frac { a x } { x + 7 } \right|\) and find the set of values of \(x\) for which \(\left| \frac { a x } { x + 7 } \right| > \frac { a } { 2 }\).
CAIE Further Paper 1 2020 June Q2
2 The cubic equation \(6 \mathrm { x } ^ { 3 } + \mathrm { px } ^ { 2 } - 3 \mathrm { x } - 5 = 0\), where \(p\) is a constant, has roots \(\alpha , \beta , \gamma\).
  1. Find a cubic equation whose roots are \(\alpha ^ { 2 } , \beta ^ { 2 } , \gamma ^ { 2 }\).
  2. It is given that \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 2 ( \alpha + \beta + \gamma )\).
    1. Find the value of \(p\).
    2. Find the value of \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 }\).
CAIE Further Paper 1 2020 June Q3
3 The curve \(C\) has equation \(\mathrm { y } = \frac { \mathrm { x } ^ { 2 } } { 2 \mathrm { x } + 1 }\).
  1. Find the equations of the asymptotes of \(C\).
  2. Find the coordinates of the stationary points on \(C\).
  3. Sketch \(C\).
CAIE Further Paper 1 2020 June Q4
4
  1. By first expressing \(\frac { 1 } { r ^ { 2 } - 1 }\) in partial fractions, show that $$\sum _ { r = 2 } ^ { n } \frac { 1 } { r ^ { 2 } - 1 } = \frac { 3 } { 4 } - \frac { a n + b } { 2 n ( n + 1 ) }$$ where \(a\) and \(b\) are integers to be found.
  2. Deduce the value of \(\sum _ { r = 2 } ^ { \infty } \frac { 1 } { r ^ { 2 } - 1 }\).
  3. Find the limit, as \(n \rightarrow \infty\), of \(\sum _ { r = n + 1 } ^ { 2 n } \frac { n } { r ^ { 2 } - 1 }\).
CAIE Further Paper 1 2020 June Q5
5 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations \(\mathbf { r } = 3 \mathbf { i } + 3 \mathbf { k } + \lambda ( \mathbf { i } + 4 \mathbf { j } + 4 \mathbf { k } )\) and \(\mathbf { r } = 3 \mathbf { i } - 5 \mathbf { j } - 6 \mathbf { k } + \mu ( 5 \mathbf { j } + 6 \mathbf { k } )\) respectively.
  1. Find the shortest distance between \(l _ { 1 }\) and \(l _ { 2 }\).
    The plane \(\Pi\) contains \(l _ { 1 }\) and is parallel to the vector \(\mathbf { i } + \mathbf { k }\).
  2. Find the equation of \(\Pi\), giving your answer in the form \(a x + b y + c z = d\).
  3. Find the acute angle between \(l _ { 2 }\) and \(\Pi\).
CAIE Further Paper 1 2020 June Q6
6 Let \(\mathbf { A } = \left( \begin{array} { l l } 2 & 0
1 & 1 \end{array} \right)\).
  1. The transformation in the \(x - y\) plane represented by \(\mathbf { A } ^ { - 1 }\) transforms a triangle of area \(30 \mathrm {~cm} ^ { 2 }\) into a triangle of area \(d \mathrm {~cm} ^ { 2 }\). Find the value of \(d\).
  2. Prove by mathematical induction that, for all positive integers \(n\), $$\mathbf { A } ^ { n } = \left( \begin{array} { c c } 2 ^ { n } & 0
    2 ^ { n } - 1 & 1 \end{array} \right)$$
  3. The line \(y = 2 x\) is invariant under the transformation in the \(x - y\) plane represented by \(\mathbf { A } ^ { n } \mathbf { B }\), where \(\mathbf { B } = \left( \begin{array} { r l } 1 & 0
    33 & 0 \end{array} \right)\). Find the value of \(n\).
CAIE Further Paper 1 2020 June Q7
7 marks
7 The curve \(C _ { 1 }\) has polar equation \(r = \theta \cos \theta\), for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
  1. The point on \(C _ { 1 }\) furthest from the line \(\theta = \frac { 1 } { 2 } \pi\) is denoted by \(P\). Show that, at \(P\), $$2 \theta \tan \theta - 1 = 0$$ and verify that this equation has a root between 0.6 and 0.7 .
    The curve \(C _ { 2 }\) has polar equation \(r = \theta \sin \theta\), for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\). The curves \(C _ { 1 }\) and \(C _ { 2 }\) intersect at the pole, denoted by \(O\), and at another point \(Q\).
  2. Find the polar coordinates of \(Q\), giving your answers in exact form.
  3. Sketch \(C _ { 1 }\) and \(C _ { 2 }\) on the same diagram.
  4. Find, in terms of \(\pi\), the area of the region bounded by the arc \(O Q\) of \(C _ { 1 }\) and the arc \(O Q\) of \(C _ { 2 }\). [7]
    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 June Q1
1 The cubic equation \(7 x ^ { 3 } + 3 x ^ { 2 } + 5 x + 1 = 0\) has roots \(\alpha , \beta , \gamma\).
  1. Find a cubic equation whose roots are \(\alpha ^ { - 1 } , \beta ^ { - 1 } , \gamma ^ { - 1 }\).
  2. Find the value of \(\alpha ^ { - 2 } + \beta ^ { - 2 } + \gamma ^ { - 2 }\).
  3. Find the value of \(\alpha ^ { - 3 } + \beta ^ { - 3 } + \gamma ^ { - 3 }\).
CAIE Further Paper 1 2020 June Q2
2 The sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is such that \(u _ { 1 } = 1\) and \(\mathrm { u } _ { \mathrm { n } + 1 } = 2 \mathrm { u } _ { \mathrm { n } } + 1\) for \(n \geqslant 1\).
  1. Prove by induction that \(u _ { n } = 2 ^ { n } - 1\) for all positive integers \(n\).
  2. Deduce that \(\mathrm { u } _ { 2 \mathrm { n } }\) is divisible by \(\mathrm { u } _ { \mathrm { n } }\) for \(n \geqslant 1\).
CAIE Further Paper 1 2020 June Q3
3 Let \(S _ { n } = 2 ^ { 2 } + 6 ^ { 2 } + 10 ^ { 2 } + \ldots + ( 4 n - 2 ) ^ { 2 }\).
  1. Use standard results from the List of Formulae (MF19) to show that \(S _ { n } = \frac { 4 } { 3 } n \left( 4 n ^ { 2 } - 1 \right)\).
  2. Express \(\frac { \mathrm { n } } { \mathrm { S } _ { \mathrm { n } } }\) in partial fractions and find \(\sum _ { \mathrm { n } = 1 } ^ { \mathrm { N } } \frac { \mathrm { n } } { \mathrm { S } _ { \mathrm { n } } }\) in terms of \(N\).
  3. Deduce the value of \(\sum _ { n = 1 } ^ { \infty } \frac { n } { S _ { n } }\).
CAIE Further Paper 1 2020 June Q4
4 The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r r } k & 0 & 2
0 & - 1 & - 1
1 & 1 & - k \end{array} \right)$$ where \(k\) is a real constant.
  1. Show that \(\mathbf { A }\) is non-singular.
    The matrices \(\mathbf { B }\) and \(\mathbf { C }\) are given by $$\mathbf { B } = \left( \begin{array} { r r } 0 & - 3
    - 1 & 3
    0 & 0 \end{array} \right) \text { and } \mathbf { C } = \left( \begin{array} { r r r } - 3 & - 1 & 1
    1 & 1 & 2 \end{array} \right)$$ It is given that \(\mathbf { C A B } = \left( \begin{array} { l l } - 2 & - \frac { 3 } { 2 }
    - 1 & - \frac { 3 } { 2 } \end{array} \right)\).
  2. Find the value of \(k\).
  3. Find the equations of the invariant lines, through the origin, of the transformation in the \(x - y\) plane represented by \(\mathbf { C A B }\).
CAIE Further Paper 1 2020 June Q5
5 The curve \(C\) has polar equation \(r = \operatorname { atan } \theta\), where \(a\) is a positive constant and \(0 \leqslant \theta \leqslant \frac { 1 } { 4 } \pi\).
  1. Sketch \(C\) and state the greatest distance of a point on \(C\) from the pole.
  2. Find the exact value of the area of the region bounded by \(C\) and the half-line \(\theta = \frac { 1 } { 4 } \pi\).
  3. Show that \(C\) has Cartesian equation \(\mathrm { y } = \frac { \mathrm { x } ^ { 2 } } { \sqrt { \mathrm { a } ^ { 2 } - \mathrm { x } ^ { 2 } } }\).
  4. Using your answer to part (b), deduce the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 2 } a \sqrt { 2 } } \frac { x ^ { 2 } } { \sqrt { a ^ { 2 } - x ^ { 2 } } } d x\).
CAIE Further Paper 1 2020 June Q6
6 The curve \(C\) has equation \(\mathrm { y } = \frac { 10 + \mathrm { x } - 2 \mathrm { x } ^ { 2 } } { 2 \mathrm { x } - 3 }\).
  1. Find the equations of the asymptotes of \(C\).
  2. Show that \(C\) has no turning points.
  3. Sketch \(C\), stating the coordinates of the intersections with the axes.
  4. Sketch the curve with equation \(y = \left| \frac { 10 + x - 2 x ^ { 2 } } { 2 x - 3 } \right|\) and find in exact form the set of values of \(x\) for which \(\left| \frac { 10 + x - 2 x ^ { 2 } } { 2 x - 3 } \right| < 4\).
CAIE Further Paper 1 2020 June Q7
7 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations \(\mathbf { r } = - 5 \mathbf { j } + \lambda ( 5 \mathbf { i } + 2 \mathbf { k } )\) and \(\mathbf { r } = 4 \mathbf { i } + 2 \mathbf { j } - 2 \mathbf { k } + \mu ( \mathbf { j } + \mathbf { k } )\) respectively. The plane \(\Pi\) contains \(l _ { 1 }\) and is parallel to \(l _ { 2 }\).
  1. Find the equation of \(\Pi\), giving your answer in the form \(a x + b y + c z = d\).
  2. Find the distance between \(l _ { 2 }\) and \(\Pi\).
    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 \(l _ { 2 }\).
  3. Show that \(P\) has position vector \(\frac { 55 } { 27 } \mathbf { i } - 5 \mathbf { j } + \frac { 22 } { 27 } \mathbf { k }\) and state a vector equation for \(P Q\).
    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 June Q1
1 Prove by mathematical induction that \(2 ^ { 4 n } + 31 ^ { n } - 2\) is divisible by 15 for all positive integers \(n\).
CAIE Further Paper 1 2021 June Q2
2
  1. Use standard results from the List of formulae (MF19) to find \(\sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } \left( 1 - \mathrm { r } - \mathrm { r } ^ { 2 } \right)\) in terms of \(n\),
    simplifying your answer. simplifying your answer.
  2. Show that $$\frac { 1 - r - r ^ { 2 } } { \left( r ^ { 2 } + 2 r + 2 \right) \left( r ^ { 2 } + 1 \right) } = \frac { r + 1 } { ( r + 1 ) ^ { 2 } + 1 } - \frac { r } { r ^ { 2 } + 1 }$$ and hence use the method of differences to find \(\sum _ { r = 1 } ^ { n } \frac { 1 - r - r ^ { 2 } } { \left( r ^ { 2 } + 2 r + 2 \right) \left( r ^ { 2 } + 1 \right) }\).
  3. Deduce the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 1 - r - r ^ { 2 } } { \left( r ^ { 2 } + 2 r + 2 \right) \left( r ^ { 2 } + 1 \right) }\).
CAIE Further Paper 1 2021 June Q3
4 marks
3 The equation \(x ^ { 4 } - 2 x ^ { 3 } - 1 = 0\) has roots \(\alpha , \beta , \gamma , \delta\).
  1. Find a quartic equation whose roots are \(\alpha ^ { 3 } , \beta ^ { 3 } , \gamma ^ { 3 } , \delta ^ { 3 }\) and state the value of \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } + \delta ^ { 3 }\). [4]
  2. Find the value of \(\frac { 1 } { \alpha ^ { 3 } } + \frac { 1 } { \beta ^ { 3 } } + \frac { 1 } { \gamma ^ { 3 } } + \frac { 1 } { \delta ^ { 3 } }\).
  3. Find the value of \(\alpha ^ { 4 } + \beta ^ { 4 } + \gamma ^ { 4 } + \delta ^ { 4 }\).
CAIE Further Paper 1 2021 June Q4
4 The matrix \(\mathbf { M }\) represents the sequence of two transformations in the \(x - y\) plane given by a rotation of \(60 ^ { \circ }\) anticlockwise about the origin followed by a one-way stretch in the \(x\)-direction, scale factor \(d ( d \neq 0 )\).
  1. Find \(\mathbf { M }\) in terms of \(d\).
  2. The unit square in the \(x - y\) plane is transformed by \(\mathbf { M }\) onto a parallelogram of area \(\frac { 1 } { 2 } d ^ { 2 }\) units \({ } ^ { 2 }\). Show that \(d = 2\).
    The matrix \(\mathbf { N }\) is such that \(\mathbf { M N } = \left( \begin{array} { l l } 1 & 1
    \frac { 1 } { 2 } & \frac { 1 } { 2 } \end{array} \right)\).
  3. Find \(\mathbf { N }\).
  4. Find the equations of the invariant lines, through the origin, of the transformation in the \(x - y\) plane represented by \(\mathbf { M N }\).
CAIE Further Paper 1 2021 June Q5
5 The curve \(C\) has polar equation \(r = \operatorname { acot } \left( \frac { 1 } { 3 } \pi - \theta \right)\), where \(a\) is a positive constant and \(0 \leqslant \theta \leqslant \frac { 1 } { 6 } \pi\). It is given that the greatest distance of a point on \(C\) from the pole is \(2 \sqrt { 3 }\).
  1. Sketch \(C\) and show that \(a = 2\).
  2. Find the exact value of the area of the region bounded by \(C\), the initial line and the half-line \(\theta = \frac { 1 } { 6 } \pi\).
  3. Show that \(C\) has Cartesian equation \(2 ( x + y \sqrt { 3 } ) = ( x \sqrt { 3 } - y ) \sqrt { x ^ { 2 } + y ^ { 2 } }\).
CAIE Further Paper 1 2021 June Q6
6 Let \(t\) be a positive constant.
The line \(l _ { 1 }\) passes through the point with position vector \(t \mathbf { i } + \mathbf { j }\) and is parallel to the vector \(- 2 \mathbf { i } - \mathbf { j }\). The line \(l _ { 2 }\) passes through the point with position vector \(\mathbf { j } + t \mathbf { k }\) and is parallel to the vector \(- 2 \mathbf { j } + \mathbf { k }\). It is given that the shortest distance between the lines \(l _ { 1 }\) and \(l _ { 2 }\) is \(\sqrt { \mathbf { 2 1 } }\).
  1. Find the value of \(t\).
    The plane \(\Pi _ { 1 }\) contains \(l _ { 1 }\) and is parallel to \(l _ { 2 }\).
  2. Write down an equation of \(\Pi _ { 1 }\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b } + \mu \mathbf { c }\).
    The plane \(\Pi _ { 2 }\) has Cartesian equation \(5 x - 6 y + 7 z = 0\).
  3. Find the acute angle between \(l _ { 2 }\) and \(\Pi _ { 2 }\).
  4. Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).
CAIE Further Paper 1 2021 June Q7
7 The curve \(C\) has equation \(\mathrm { y } = \frac { \mathrm { x } ^ { 2 } + \mathrm { x } + 9 } { \mathrm { x } + 1 }\).
  1. Find the equations of the asymptotes of \(C\).
  2. Find the coordinates of the stationary points on \(C\).
  3. Sketch \(C\), stating the coordinates of any intersections with the axes.
  4. Sketch the curve with equation \(\mathrm { y } = \left| \frac { \mathrm { x } ^ { 2 } + \mathrm { x } + 9 } { \mathrm { x } + 1 } \right|\) and find the set of values of \(x\) for which \(2 \left| x ^ { 2 } + x + 9 \right| > 13 | x + 1 |\). If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE Further Paper 1 2021 June Q1
1
  1. Show that $$\tan ( r + 1 ) - \tan r = \frac { \sin 1 } { \cos ( r + 1 ) \cos r }$$ Let \(\mathrm { u } _ { \mathrm { r } } = \frac { 1 } { \cos ( \mathrm { r } + 1 ) \cos \mathrm { r } }\).
  2. Use the method of differences to find \(\sum _ { r = 1 } ^ { n } u _ { r }\).
  3. Explain why the infinite series \(u _ { 1 } + u _ { 2 } + u _ { 3 } + \ldots\) does not converge.
CAIE Further Paper 1 2021 June Q2
2 The cubic equation \(2 x ^ { 3 } - 4 x ^ { 2 } + 3 = 0\) has roots \(\alpha , \beta , \gamma\). Let \(\mathrm { S } _ { \mathrm { n } } = \alpha ^ { \mathrm { n } } + \beta ^ { \mathrm { n } } + \gamma ^ { \mathrm { n } }\).
  1. State the value of \(S _ { 1 }\) and find the value of \(S _ { 2 }\).
    1. Express \(\mathrm { S } _ { \mathrm { n } + 3 }\) in terms of \(\mathrm { S } _ { \mathrm { n } + 2 }\) and \(\mathrm { S } _ { \mathrm { n } }\).
    2. Hence, or otherwise, find the value of \(S _ { 4 }\).
  3. Use the substitution \(\mathrm { y } = \mathrm { S } _ { 1 } - \mathrm { x }\), where \(S _ { 1 }\) is the numerical value found in part (a), to find and simplify an equation whose roots are \(\alpha + \beta , \beta + \gamma , \gamma + \alpha\).
  4. Find the value of \(\frac { 1 } { \alpha + \beta } + \frac { 1 } { \beta + \gamma } + \frac { 1 } { \gamma + \alpha }\).
CAIE Further Paper 1 2021 June Q3
3
  1. Prove by mathematical induction that, for all positive integers \(n\), $$\sum _ { r = 1 } ^ { n } \left( 5 r ^ { 4 } + r ^ { 2 } \right) = \frac { 1 } { 2 } n ^ { 2 } ( n + 1 ) ^ { 2 } ( 2 n + 1 )$$
  2. Use the result given in part (a) together with the List of formulae (MF19) to find \(\sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } \mathrm { r } ^ { 4 }\) in terms of \(n\), fully factorising your answer.
CAIE Further Paper 1 2021 June Q4
4 The matrices \(\mathbf { A } , \mathbf { B }\) and \(\mathbf { C }\) are given by $$\mathbf { A } = \left( \begin{array} { c c c } 2 & k & k
5 & - 1 & 3
1 & 0 & 1 \end{array} \right) , \quad \mathbf { B } = \left( \begin{array} { c c } 1 & 0
0 & 1
1 & 0 \end{array} \right) \text { and } \quad \mathbf { C } = \left( \begin{array} { r c c } 0 & 1 & 1
- 1 & 2 & 0 \end{array} \right)$$ where \(k\) is a real constant.
  1. Find \(\mathbf { C A B }\).
  2. Given that \(\mathbf { A }\) is singular, find the value of \(k\).
  3. Using the value of \(k\) from part (b), find the equations of the invariant lines, through the origin, of the transformation in the \(x - y\) plane represented by \(\mathbf { C A B }\).