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CAIE Further Paper 1 2022 June Q1
1 Let \(a\) be a positive constant.
  1. Use the method of differences to find \(\sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } \frac { 1 } { ( \mathrm { ar } + 1 ) ( \mathrm { ar } + \mathrm { a } + 1 ) }\) in terms of \(n\) and \(a\).
  2. Find the value of \(a\) for which \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { ( a r + 1 ) ( a r + a + 1 ) } = \frac { 1 } { 6 }\).
CAIE Further Paper 1 2022 June Q2
2 The points \(A , B , C\) have position vectors $$4 \mathbf { i } - 4 \mathbf { j } + \mathbf { k } , \quad - 4 \mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k } , \quad 4 \mathbf { i } - \mathbf { j } - 2 \mathbf { k } ,$$ 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. The point \(D\) has position vector \(2 \mathbf { i } + 3 \mathbf { j } - 3 \mathbf { k }\). Find the coordinates of the point of intersection of the line \(O D\) with the plane \(A B C\).
CAIE Further Paper 1 2022 June Q3
3 The sequence of positive numbers \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is such that \(u _ { 1 } > 4\) and, for \(n \geqslant 1\), $$u _ { n + 1 } = \frac { u _ { n } ^ { 2 } + u _ { n } + 12 } { 2 u _ { n } }$$
  1. By considering \(\mathrm { u } _ { \mathrm { n } + 1 } - 4\), or otherwise, prove by mathematical induction that \(\mathrm { u } _ { \mathrm { n } } > 4\) for all positive integers \(n\).
  2. Show that \(u _ { n + 1 } < u _ { n }\) for \(n \geqslant 1\).
CAIE Further Paper 1 2022 June Q4
4 The cubic equation \(2 x ^ { 3 } + 5 x ^ { 2 } - 6 = 0\) has roots \(\alpha , \beta , \gamma\).
  1. Find a cubic equation whose roots are \(\frac { 1 } { \alpha ^ { 3 } } , \frac { 1 } { \beta ^ { 3 } } , \frac { 1 } { \gamma ^ { 3 } }\).
  2. Find the value of \(\frac { 1 } { \alpha ^ { 6 } } + \frac { 1 } { \beta ^ { 6 } } + \frac { 1 } { \gamma ^ { 6 } }\).
  3. Find also the value of \(\frac { 1 } { \alpha ^ { 9 } } + \frac { 1 } { \beta ^ { 9 } } + \frac { 1 } { \gamma ^ { 9 } }\).
CAIE Further Paper 1 2022 June Q5
5 The curve \(C\) has equation \(y = \frac { 2 x ^ { 2 } - x - 1 } { x ^ { 2 } + x + 1 }\).
  1. Show that \(C\) has no vertical asymptotes and state the equation of the horizontal asymptote of \(C\).
  2. Find the coordinates of the stationary points on \(C\).
  3. Sketch \(C\), stating the coordinates of the intersections with the axes.
  4. Sketch the curve with equation \(y = \left| \frac { 2 x ^ { 2 } - x - 1 } { x ^ { 2 } + x + 1 } \right|\) and state the set of values of \(k\) for which \(\left| \frac { 2 x ^ { 2 } - x - 1 } { x ^ { 2 } + x + 1 } \right| = k\) has 4 distinct real solutions.
CAIE Further Paper 1 2022 June Q6
6 The curve \(C\) has polar equation \(r ^ { 2 } = \tan ^ { - 1 } \left( \frac { 1 } { 2 } \theta \right)\), where \(0 \leqslant \theta \leqslant 2\).
  1. Sketch \(C\) and state, in exact form, 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 \(\theta = 2\).
    Now consider the part of \(C\) where \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
  3. Show that, at the point furthest from the half-line \(\theta = \frac { 1 } { 2 } \pi\), $$\left( \theta ^ { 2 } + 4 \right) \tan ^ { - 1 } \left( \frac { 1 } { 2 } \theta \right) \sin \theta - \cos \theta = 0$$ and verify that this equation has a root between 0.6 and 0.7 . \(7 \quad\) The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l l } 1 & 2 & 3 \\ 4 & k & 6 \\ 7 & 8 & 9 \end{array} \right)\).
  4. Find the set of values of \(k\) for which \(\mathbf { A }\) is non-singular.
  5. Given that \(\mathbf { A }\) is non-singular, find, in terms of \(k\), the entries in the top row of \(\mathbf { A } ^ { - 1 }\).
  6. Given that \(\mathbf { B } = \left( \begin{array} { l l l } 1 & 0 & 0 \\ 0 & 1 & 0 \end{array} \right)\), give an example of a matrix \(\mathbf { C }\) such that \(\mathbf { B A C } = \left( \begin{array} { l l } 2 & 1 \\ k & 4 \end{array} \right)\).
  7. Find the set of values of \(k\) for which the transformation in the \(x - y\) plane represented by \(\left( \begin{array} { l l } 2 & 1 \\ k & 4 \end{array} \right)\) has two distinct invariant lines through the origin.
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE Further Paper 1 2023 June Q1
1 Let \(\mathbf { A } = \left( \begin{array} { l l } 3 & 0 \\ 1 & 1 \end{array} \right)\).
  1. Prove by mathematical induction that, for all positive integers \(n\), $$2 \mathbf { A } ^ { n } = \left( \begin{array} { l l } 2 \times 3 ^ { n } & 0 \\ 3 ^ { n } - 1 & 2 \end{array} \right)$$
  2. Find, in terms of \(n\), the inverse of \(\mathbf { A } ^ { n }\).
CAIE Further Paper 1 2023 June Q2
2 The cubic equation \(x ^ { 3 } + 4 x ^ { 2 } + 6 x + 1 = 0\) has roots \(\alpha , \beta , \gamma\).
  1. Find the value of \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\).
  2. Use standard results from the list of formulae (MF19) to show that $$\sum _ { r = 1 } ^ { n } \left( ( \alpha + r ) ^ { 2 } + ( \beta + r ) ^ { 2 } + ( \gamma + r ) ^ { 2 } \right) = n \left( n ^ { 2 } + a n + b \right)$$ where \(a\) and \(b\) are constants to be determined.
CAIE Further Paper 1 2023 June Q3
3
  1. Use the method of differences to find \(\sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } \frac { 1 } { ( \mathrm { kr } + 1 ) ( \mathrm { kr } - \mathrm { k } + 1 ) }\) in terms of \(n\) and \(k\), where \(k\) is a positive constant.
  2. Deduce the value of \(\sum _ { \mathrm { r } = 1 } ^ { \infty } \frac { 1 } { ( \mathrm { kr } + 1 ) ( \mathrm { kr } - \mathrm { k } + 1 ) }\).
  3. Find also \(\sum _ { \mathrm { r } = \mathrm { n } } ^ { \mathrm { n } ^ { 2 } } \frac { 1 } { ( \mathrm { kr } + 1 ) ( \mathrm { kr } - \mathrm { k } + 1 ) }\) in terms of \(n\) and \(k\).
CAIE Further Paper 1 2023 June Q4
4 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { l l } \mathrm { a } & \mathrm { b } ^ { 2 } \\ \mathrm { c } ^ { 2 } & \mathrm { a } \end{array} \right)\), where \(a , b , c\) are real constants and \(b \neq 0\).
  1. Show that \(\mathbf { M }\) does not represent a rotation about the origin.
  2. Find the equations of the invariant lines, through the origin, of the transformation in the \(x - y\) plane represented by \(\mathbf { M }\).
    It is given that \(\mathbf { M }\) represents the sequence of two transformations in the \(x - y\) plane given by an enlargement, centre the origin, scale factor 5 followed by a shear, \(x\)-axis fixed, with \(( 0,1 )\) mapped to \(( 5,1 )\).
  3. Find \(\mathbf { M }\).
  4. The triangle \(D E F\) in the \(x - y\) plane is transformed by \(\mathbf { M }\) onto triangle \(P Q R\). Given that the area of triangle \(D E F\) is \(12 \mathrm {~cm} ^ { 2 }\), find the area of triangle \(P Q R\).
CAIE Further Paper 1 2023 June Q5
5 The curve \(C\) has polar equation \(r ^ { 2 } = \frac { 1 } { \theta ^ { 2 } + 1 }\), for \(0 \leqslant \theta \leqslant \pi\).
  1. Sketch \(C\) and state the polar coordinates of the point of \(C\) furthest from the pole.
  2. Find the area of the region enclosed by \(C\), the initial line, and the half-line \(\theta = \pi\).
  3. Show that, at the point of \(C\) furthest from the initial line, $$\left( \theta + \frac { 1 } { \theta } \right) \cot \theta - 1 = 0$$ and verify that this equation has a root between 1.1 and 1.2.
CAIE Further Paper 1 2023 June Q6
6 The curve \(C\) has equation \(\mathrm { y } = \frac { \mathrm { x } ^ { 2 } + 2 \mathrm { x } - 15 } { \mathrm { x } - 2 }\).
  1. Find the equations of the asymptotes of \(C\).
  2. Show that \(C\) has no stationary points.
  3. Sketch \(C\), stating the coordinates of the intersections with the axes.
  4. Sketch the curve with equation \(\mathrm { y } = \left| \frac { \mathrm { x } ^ { 2 } - 2 \mathrm { x } - 15 } { \mathrm { x } - 2 } \right|\).
  5. Find the set of values of \(x\) for which \(\left| \frac { 2 x ^ { 2 } + 4 x - 30 } { x - 2 } \right| < 15\).
CAIE Further Paper 1 2023 June Q7
7 The plane \(\Pi _ { 1 }\) has equation \(r = - 4 \mathbf { j } - 3 \mathbf { k } + \lambda ( \mathbf { i } - \mathbf { j } + \mathbf { k } ) + \mu ( \mathbf { i } + \mathbf { j } - \mathbf { k } )\).
  1. Obtain an equation of \(\Pi _ { 1 }\) in the form \(\mathrm { px } + \mathrm { qy } + \mathrm { rz } = \mathrm { d }\).
  2. The plane \(\Pi _ { 2 }\) has equation \(\mathbf { r } . ( - 5 \mathbf { i } + 3 \mathbf { j } + 5 \mathbf { k } ) = 4\). Find a vector equation of the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).
    The line \(l\) passes through the point \(A\) with position vector \(a \mathbf { i } + a \mathbf { j } + ( a - 7 ) \mathbf { k }\) and is parallel to \(( 1 - b ) \mathbf { i } + b \mathbf { j } + b \mathbf { k }\), where \(a\) and \(b\) are positive constants.
  3. Given that the perpendicular distance from \(A\) to \(\Pi _ { 1 }\) is \(\sqrt { 2 }\), find the value of \(a\).
  4. Given that the obtuse angle between \(l\) and \(\Pi _ { 1 }\) is \(\frac { 3 } { 4 } \pi\), find the exact value of \(b\).
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE Further Paper 1 2024 June Q1
1 The cubic equation \(2 x ^ { 3 } + x ^ { 2 } - p x - 5 = 0\), where \(p\) is a positive constant, has roots \(\alpha , \beta , \gamma\).
  1. State, in terms of \(p\), the value of \(\alpha \beta + \beta \gamma + \gamma \alpha\).
  2. Find the value of \(\alpha ^ { 2 } \beta \gamma + \alpha \beta ^ { 2 } \gamma + \alpha \beta \gamma ^ { 2 }\).
  3. Deduce a cubic equation whose roots are \(\alpha \beta , \beta \gamma , \alpha \gamma\).
  4. Given that \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = \frac { 1 } { 3 }\), find the value of \(p\).
CAIE Further Paper 1 2024 June Q2
2 Prove by mathematical induction that \(6 ^ { 4 n } + 38 ^ { n } - 2\) is divisible by 74 for all positive integers \(n\).
CAIE Further Paper 1 2024 June Q3
3
  1. Use standard results from the list of formulae (MF19) to show that $$\sum _ { r = 1 } ^ { N } r ( r + 1 ) ( 3 r + 4 ) = \frac { 1 } { 12 } N ( N + 1 ) ( N + 2 ) ( 9 N + 19 )$$
  2. Express \(\frac { 3 r + 4 } { r ( r + 1 ) }\) in partial fractions and hence use the method of differences to find $$\sum _ { r = 1 } ^ { N } \frac { 3 r + 4 } { r ( r + 1 ) } \left( \frac { 1 } { 4 } \right) ^ { r + 1 }$$ in terms of \(N\).
  3. Deduce the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 3 r + 4 } { r ( r + 1 ) } \left( \frac { 1 } { 4 } \right) ^ { r + 1 }\).
CAIE Further Paper 1 2024 June Q4
4 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { c c } \frac { 1 } { 2 } & - \frac { 1 } { 2 } \sqrt { 3 } \\ \frac { 1 } { 2 } \sqrt { 3 } & \frac { 1 } { 2 } \end{array} \right) \left( \begin{array} { c c } 14 & 0 \\ 0 & 1 \end{array} \right)\).
  1. The matrix \(\mathbf { M }\) represents a sequence of two geometrical transformations in the \(x - y\) plane. Give full details of each transformation, and make clear the order in which they are applied.
  2. Write \(\mathbf { M } ^ { - 1 }\) as the product of two matrices, neither of which is \(\mathbf { I }\).
  3. Find the equations of the invariant lines, through the origin, of the transformation represented by \(\mathbf { M }\).
  4. The triangle \(A B C\) in the \(x - y\) plane is transformed by \(\mathbf { M }\) onto triangle \(D E F\). Given that the area of triangle \(D E F\) is \(28 \mathrm {~cm} ^ { 2 }\), find the area of triangle \(A B C\).
CAIE Further Paper 1 2024 June Q5
5 The points \(A , B , C\) have position vectors $$2 \mathbf { i } + 2 \mathbf { j } + 4 \mathbf { k } , \quad 2 \mathbf { i } + 4 \mathbf { j } - \mathbf { k } , \quad - 3 \mathbf { i } - 3 \mathbf { j } + 4 \mathbf { k }$$ respectively, relative to the origin \(O\).
  1. Find the equation of the plane \(A B C\), giving your answer in the form \(a x + b y + c z = d\).
    The point \(D\) has position vector \(2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k }\).
  2. Find the perpendicular distance from \(D\) to the plane \(A B C\).
  3. Find the shortest distance between the lines \(A B\) and \(C D\).
CAIE Further Paper 1 2024 June Q6
6 The curve \(C\) has equation \(\mathrm { y } = \frac { \mathrm { x } ^ { 2 } + \mathrm { ax } + 1 } { \mathrm { x } + 2 }\), where \(a > \frac { 5 } { 2 }\).
  1. Find the equations of the asymptotes of \(C\).
  2. Show that \(C\) has no stationary points.
  3. Sketch \(C\), stating the coordinates of the point of intersection with the \(y\)-axis and labelling the asymptotes.
    1. Sketch the curve with equation \(\mathrm { y } = \left| \frac { \mathrm { x } ^ { 2 } + \mathrm { ax } + 1 } { \mathrm { x } + 2 } \right|\).
    2. On your sketch in part (i), draw the line \(\mathrm { y } = \mathrm { a }\).
    3. It is given that \(\left| \frac { \mathrm { x } ^ { 2 } + \mathrm { ax } + 1 } { \mathrm { x } + 2 } \right| < \mathrm { a }\) for \(- 5 - \sqrt { 14 } < x < - 3\) and \(- 5 + \sqrt { 14 } < x < 3\). Find the value of \(a\).
CAIE Further Paper 1 2024 June Q7
7 The curve \(C\) has polar equation \(r ^ { 2 } = ( \pi - \theta ) \tan ^ { - 1 } ( \pi - \theta )\), 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 \(u = \pi - \theta\), or otherwise, find the area of the region enclosed by \(C\) and the initial line.
  3. Show that, at the point of \(C\) furthest from the initial line, $$2 ( \pi - \theta ) \tan ^ { - 1 } ( \pi - \theta ) \cot \theta - \frac { \pi - \theta } { 1 + ( \pi - \theta ) ^ { 2 } } - \tan ^ { - 1 } ( \pi - \theta ) = 0$$ and verify that this equation has a root for \(\theta\) between 1.2 and 1.3.
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE Further Paper 1 2020 November Q1
1 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { l l } 1 & b \\ 0 & 1 \end{array} \right) \left( \begin{array} { l l } a & 0 \\ 0 & 1 \end{array} \right)\), where \(a\) and \(b\) are positive constants.
  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.
    The unit square in the \(x - y\) plane is transformed by \(\mathbf { M }\) onto parallelogram \(O P Q R\).
  2. Find, in terms of \(a\) and \(b\), the matrix which transforms parallelogram \(O P Q R\) onto the unit square.
    It is given that the area of \(O P Q R\) is \(2 \mathrm {~cm} ^ { 2 }\) and that the line \(\mathrm { x } + 3 \mathrm { y } = 0\) is invariant under the transformation represented by \(\mathbf { M }\).
  3. Find the values of \(a\) and \(b\).
CAIE Further Paper 1 2020 November Q2
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
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
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
5 Prove by mathematical induction that, for every positive integer \(n\), $$\frac { d ^ { 2 n - 1 } } { d x ^ { 2 n - 1 } } ( x \sin x ) = ( - 1 ) ^ { n - 1 } ( x \cos x + ( 2 n - 1 ) \sin x )$$