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CAIE Further Paper 1 2020 November Q6
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
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 2022 November Q1
1 The cubic equation \(x ^ { 3 } + b x ^ { 2 } + d = 0\) has roots \(\alpha , \beta , \gamma\), where \(\alpha = \beta\) and \(d \neq 0\).
  1. Show that \(4 b ^ { 3 } + 27 d = 0\).
  2. Given that \(2 \alpha ^ { 2 } + \gamma ^ { 2 } = 3 b\), find the values of \(b\) and \(d\).
CAIE Further Paper 1 2022 November Q2
6 marks
2 Prove by mathematical induction that, for all positive integers \(n , 7 ^ { 2 n } + 97 ^ { n } - 50\) is divisible by 48. [6]
CAIE Further Paper 1 2022 November Q3
3
  1. By considering \(( 2 r + 1 ) ^ { 3 } - ( 2 r - 1 ) ^ { 3 }\), use the method of differences to prove that $$\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )$$ Let \(S _ { n } = 1 ^ { 2 } + 3 \times 2 ^ { 2 } + 3 ^ { 2 } + 3 \times 4 ^ { 2 } + 5 ^ { 2 } + 3 \times 6 ^ { 2 } + \ldots + \left( 2 + ( - 1 ) ^ { n } \right) n ^ { 2 }\).
  2. Show that \(\mathrm { S } _ { 2 \mathrm { n } } = \frac { 1 } { 3 } \mathrm { n } ( 2 \mathrm { n } + 1 ) ( \mathrm { an } + \mathrm { b } )\), where \(a\) and \(b\) are integers to be determined.
  3. State the value of \(\lim _ { n \rightarrow \infty } \frac { S _ { 2 n } } { n ^ { 3 } }\).
CAIE Further Paper 1 2022 November Q4
4 The plane \(\Pi\) contains the lines \(\mathbf { r } = 3 \mathbf { i } - 2 \mathbf { j } + \mathbf { k } + \lambda ( - \mathbf { i } + 2 \mathbf { j } + \mathbf { k } )\) and \(\mathbf { r } = 4 \mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k } + \mu ( 3 \mathbf { i } + 2 \mathbf { j } - \mathbf { k } )\).
  1. Find a Cartesian equation of \(\Pi\), giving your answer in the form \(a x + b y + c z = d\).
    The line \(l\) passes through the point \(P\) with position vector \(2 \mathbf { i } + 3 \mathbf { j } + \mathbf { k }\) and is parallel to the vector \(\mathbf { j } + \mathbf { k }\).
  2. Find the acute angle between \(I\) and \(\Pi\).
  3. Find the position vector of the foot of the perpendicular from \(P\) to \(\Pi\).
CAIE Further Paper 1 2022 November Q5
5 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { r r } \frac { 1 } { 2 } \sqrt { 2 } & - \frac { 1 } { 2 } \sqrt { 2 } \\ \frac { 1 } { 2 } \sqrt { 2 } & \frac { 1 } { 2 } \sqrt { 2 } \end{array} \right) \left( \begin{array} { c c } 1 & k \\ 0 & 1 \end{array} \right)\), where \(k\) is a constant.
  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. The triangle \(A B C\) in the \(x - y\) plane is transformed by \(\mathbf { M }\) onto triangle \(D E F\). Find, in terms of \(k\), the single matrix which transforms triangle \(D E F\) onto triangle \(A B C\).
  3. Find the set of values of \(k\) for which the transformation represented by \(\mathbf { M }\) has no invariant lines through the origin.
CAIE Further Paper 1 2022 November Q6
6
  1. Show that the curve with Cartesian equation $$\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 36 \left( x ^ { 2 } - y ^ { 2 } \right)$$ has polar equation \(r ^ { 2 } = 36 \cos 2 \theta\).
    The curve \(C\) has polar equation \(r ^ { 2 } = 36 \cos 2 \theta\), for \(- \frac { 1 } { 4 } \pi \leqslant \theta \leqslant \frac { 1 } { 4 } \pi\).
  2. Sketch \(C\) and state the maximum distance of a point on \(C\) from the pole.
  3. Find the area of the region enclosed by \(C\).
  4. Find the maximum distance of a point on \(C\) from the initial line, giving the answer in exact form.
CAIE Further Paper 1 2022 November Q7
7 The curve \(C\) has equation \(y = \frac { 5 x ^ { 2 } } { 5 x - 2 }\).
  1. Find the equations of the asymptotes of \(C\).
  2. Find the coordinates of the stationary points on \(C\).
  3. Sketch \(C\).
  4. Sketch the curve with equation \(y = \left| \frac { 5 x ^ { 2 } } { 5 x - 2 } \right|\) and find in exact form the set of values of \(x\) for which \(\left| \frac { 5 x ^ { 2 } } { 5 x - 2 } \right| < 2\).
    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 November Q1
1
  1. By considering \(( r + 1 ) ^ { 2 } - r ^ { 2 }\), use the method of differences to prove that $$\sum _ { r = 1 } ^ { n } r = \frac { 1 } { 2 } n ( n + 1 )$$
  2. Given that \(\sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } ( \mathrm { r } + \mathrm { a } ) = \mathrm { n }\), find \(a\) in terms of \(n\).
CAIE Further Paper 1 2023 November Q2
2 Prove by mathematical induction that, for all positive integers \(n\), $$1 + 2 x + 3 x ^ { 2 } + \ldots + n x ^ { n - 1 } = \frac { 1 - ( n + 1 ) x ^ { n } + n x ^ { n + 1 } } { ( 1 - x ) ^ { 2 } }$$
CAIE Further Paper 1 2023 November Q3
3 The quartic equation \(\mathrm { x } ^ { 4 } + \mathrm { bx } ^ { 3 } + \mathrm { cx } ^ { 2 } + \mathrm { dx } - 2 = 0\) has roots \(\alpha , \beta , \gamma , \delta\). It is given that $$\alpha + \beta + \gamma + \delta = 3 , \quad \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 } = 5 , \quad \alpha ^ { - 1 } + \beta ^ { - 1 } + \gamma ^ { - 1 } + \delta ^ { - 1 } = 6$$
  1. Find the values of \(b , c\) and \(d\).
  2. Given also that \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } + \delta ^ { 3 } = - 27\), find the value of \(\alpha ^ { 4 } + \beta ^ { 4 } + \gamma ^ { 4 } + \delta ^ { 4 }\).
CAIE Further Paper 1 2023 November Q4
4 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations $$\mathbf { r } = - 2 \mathbf { i } - 3 \mathbf { j } - 5 \mathbf { k } + \lambda ( - 4 \mathbf { i } + 3 \mathbf { j } + 5 \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = 2 \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k } + \mu ( 2 \mathbf { i } - 3 \mathbf { j } + \mathbf { k } )$$ respectively.
  1. Find the shortest distance between \(l _ { 1 }\) and \(l _ { 2 }\).
    The plane \(\Pi\) contains \(l _ { 1 }\) and the point with position vector \(- \mathbf { i } - 3 \mathbf { j } - 4 \mathbf { k }\).
  2. Find an equation of \(\Pi\), giving your answer in the form \(a x + b y + c z = d\).
CAIE Further Paper 1 2023 November Q5
5 Let \(k\) be a constant. The matrices \(\mathbf { A } , \mathbf { B }\) and \(\mathbf { C }\) are given by $$\mathbf { A } = \left( \begin{array} { l l l } 1 & k & 3 \\ 2 & 1 & 3 \\ 3 & 2 & 5 \end{array} \right) , \quad \mathbf { B } = \left( \begin{array} { r r } 0 & - 2 \\ - 1 & 3 \\ 0 & 0 \end{array} \right) \quad \text { and } \quad \mathbf { C } = \left( \begin{array} { r r r } - 2 & - 1 & 1 \\ 1 & 1 & 3 \end{array} \right)$$ It is given that \(\mathbf { A }\) is singular.
  1. Show that \(\mathbf { C A B } = \left( \begin{array} { r r } 3 & - 7 \\ - 9 & 3 \end{array} \right)\).
  2. Find the equations of the invariant lines, through the origin, of the transformation in the \(x - y\) plane represented by \(\mathbf { C A B }\).
  3. The matrices \(\mathbf { D } , \mathbf { E }\) and \(\mathbf { F }\) represent geometrical transformations in the \(x - y\) plane.
    • D represents an enlargement, centre the origin.
    • E represents a stretch parallel to the \(x\)-axis.
    • F represents a reflection in the line \(y = x\).
    Given that \(\mathbf { C A B } = \mathbf { D } - 9 \mathbf { E F }\), find \(\mathbf { D } , \mathbf { E }\) and \(\mathbf { F }\).
CAIE Further Paper 1 2023 November Q6
6
  1. Show that the curve with Cartesian equation $$\left( x - \frac { 1 } { 2 } \right) ^ { 2 } + y ^ { 2 } = \frac { 1 } { 4 }$$ has polar equation \(r = \cos \theta\).
    The curves \(C _ { 1 }\) and \(C _ { 2 }\) have polar equations $$r = \cos \theta \quad \text { and } \quad r = \sin 2 \theta$$ respectively, where \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\). The curves \(C _ { 1 }\) and \(C _ { 2 }\) intersect at the pole and at another point \(P\).
  2. Find the polar coordinates of \(P\).
  3. In a single diagram sketch \(C _ { 1 }\) and \(C _ { 2 }\), clearly identifying each curve, and mark the point \(P\).
  4. The region \(R\) is enclosed by \(C _ { 1 }\) and \(C _ { 2 }\) and includes the line \(O P\). Find, in exact form, the area of \(R\).
CAIE Further Paper 1 2023 November Q7
7 The curve \(C\) has equation \(y = f ( x )\), where \(f ( x ) = \frac { x ^ { 2 } + 2 } { x ^ { 2 } - x - 2 }\).
  1. Find the equations of the asymptotes of \(C\).
  2. Find the coordinates of any stationary points on \(C\), giving your answers correct to 1 decimal place.
  3. Sketch \(C\), stating the coordinates of any intersections with the axes.
  4. Sketch the curve with equation \(\mathrm { y } = \frac { 1 } { \mathrm { f } ( \mathrm { x } ) }\).
  5. Find the set of values for which \(\frac { 1 } { \mathrm { f } ( x ) } < \mathrm { f } ( x )\).
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE FP1 2010 June Q1
1 The variables \(x\) and \(y\) are such that \(y = - 1\) when \(x = 1\) and $$x ^ { 2 } + y ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) ^ { 3 } = 29$$ Find the values of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) when \(x = 1\).
CAIE FP1 2010 June Q2
2 The curve \(C\) has polar equation $$r = a \left( 1 - \mathrm { e } ^ { - \theta } \right)$$ where \(a\) is a positive constant and \(0 \leqslant \theta < 2 \pi\).
  1. Draw a sketch of \(C\).
  2. Show that the area of the region bounded by \(C\) and the lines \(\theta = \ln 2\) and \(\theta = \ln 4\) is $$\frac { 1 } { 2 } a ^ { 2 } \left( \ln 2 - \frac { 13 } { 32 } \right)$$
CAIE FP1 2010 June Q3
3 At any point \(( x , y )\) on the curve \(C\), $$\frac { \mathrm { d } x } { \mathrm {~d} t } = t \sqrt { } \left( t ^ { 2 } + 4 \right) \quad \text { and } \quad \frac { \mathrm { d } y } { \mathrm {~d} t } = - t \sqrt { } \left( 4 - t ^ { 2 } \right)$$ where the parameter \(t\) is such that \(0 \leqslant t \leqslant 2\). Show that the length of \(C\) is \(4 \sqrt { } 2\). Given that \(y = 0\) when \(t = 2\), determine the area of the surface generated when \(C\) is rotated through one complete revolution about the \(x\)-axis, leaving your answer in an exact form.
CAIE FP1 2010 June Q4
4 The sum \(S _ { N }\) is defined by \(S _ { N } = \sum _ { n = 1 } ^ { N } n ^ { 5 }\). Using the identity $$\left( n + \frac { 1 } { 2 } \right) ^ { 6 } - \left( n - \frac { 1 } { 2 } \right) ^ { 6 } \equiv 6 n ^ { 5 } + 5 n ^ { 3 } + \frac { 3 } { 8 } n$$ find \(S _ { N }\) in terms of \(N\). [You need not simplify your result.] Hence find \(\lim _ { N \rightarrow \infty } N ^ { - \lambda } S _ { N }\), for each of the two cases
  1. \(\lambda = 6\),
  2. \(\lambda > 6\).
CAIE FP1 2010 June Q5
5 Let $$I _ { n } = \int _ { 1 } ^ { \mathrm { e } } x ( \ln x ) ^ { n } \mathrm {~d} x$$ where \(n \geqslant 1\). Show that $$I _ { n + 1 } = \frac { 1 } { 2 } \mathrm { e } ^ { 2 } - \frac { 1 } { 2 } ( n + 1 ) I _ { n }$$ Hence prove by induction that, for all positive integers \(n , I _ { n }\) is of the form \(A _ { n } \mathrm { e } ^ { 2 } + B _ { n }\), where \(A _ { n }\) and \(B _ { n }\) are rational numbers.
CAIE FP1 2010 June Q6
6 The equation $$x ^ { 3 } + x - 1 = 0$$ has roots \(\alpha , \beta , \gamma\). Use the relation \(x = \sqrt { } y\) to show that the equation $$y ^ { 3 } + 2 y ^ { 2 } + y - 1 = 0$$ has roots \(\alpha ^ { 2 } , \beta ^ { 2 } , \gamma ^ { 2 }\). Let \(S _ { n } = \alpha ^ { n } + \beta ^ { n } + \gamma ^ { n }\).
  1. Write down the value of \(S _ { 2 }\) and show that \(S _ { 4 } = 2\).
  2. Find the values of \(S _ { 6 }\) and \(S _ { 8 }\).
CAIE FP1 2010 June Q7
7 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have vector equations $$\mathbf { r } = 4 \mathbf { i } - 2 \mathbf { j } + \lambda ( 2 \mathbf { i } + \mathbf { j } - 4 \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = 4 \mathbf { i } - 5 \mathbf { j } + 2 \mathbf { k } + \mu ( \mathbf { i } - \mathbf { j } - \mathbf { k } )$$ respectively.
  1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) intersect.
  2. Find the perpendicular distance from the point \(P\) whose position vector is \(3 \mathbf { i } - 5 \mathbf { j } + 6 \mathbf { k }\) to the plane containing \(l _ { 1 }\) and \(l _ { 2 }\).
  3. Find the perpendicular distance from \(P\) to \(l _ { 1 }\).
CAIE FP1 2010 June Q8
8 The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r r } 4 & 1 & - 1 \\ - 4 & - 1 & 4 \\ 0 & - 1 & 5 \end{array} \right)$$ Given that one eigenvector of \(\mathbf { A }\) is \(\left( \begin{array} { r } 1 \\ - 2 \\ - 1 \end{array} \right)\), find the corresponding eigenvalue. Given also that another eigenvalue of \(\mathbf { A }\) is 4, find a corresponding eigenvector. Given further that \(\left( \begin{array} { r } 1 \\ - 4 \\ - 1 \end{array} \right)\) is an eigenvector of \(\mathbf { A }\), with corresponding eigenvalue 1 , find matrices \(\mathbf { P }\) and \(\mathbf { Q }\), together with a diagonal matrix \(\mathbf { D }\), such that \(\mathbf { A } ^ { 5 } = \mathbf { P D Q }\).
CAIE FP1 2010 June Q9
9
  1. Write down the five fifth roots of unity.
  2. Hence find all the roots of the equation $$z ^ { 5 } + 16 + ( 16 \sqrt { } 3 ) i = 0$$ giving answers in the form \(r \mathrm { e } ^ { \mathrm { i } q \pi }\), where \(r > 0\) and \(q\) is a rational number. Show these roots on an Argand diagram. Let \(w\) be a root of the equation in part (ii).
  3. Show that $$\sum _ { k = 0 } ^ { 4 } \left( \frac { w } { 2 } \right) ^ { k } = \frac { 3 + \mathrm { i } \sqrt { } 3 } { 2 - w }$$
  4. Identify the root for which \(| 2 - w |\) is least.