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CAIE Further Paper 1 2023 November Q2
6 marks Challenging +1.2
2 Prove by mathematical induction that, for all positive integers \(n\), $$\frac { d ^ { n } } { d x ^ { n } } \left( x ^ { 2 } e ^ { x } \right) = \left( x ^ { 2 } + 2 n x + n ( n - 1 ) \right) e ^ { x }$$
CAIE Further Paper 1 2023 November Q3
8 marks Standard +0.3
3 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { l l } k & 0 \\ 0 & 1 \end{array} \right) \left( \begin{array} { l l } 1 & 0 \\ 1 & 1 \end{array} \right)\), where \(k\) is a constant and \(k \neq 0\) and \(k \neq 1\).
  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 \(k\), the area of parallelogram \(O P Q R\) and the matrix which transforms \(O P Q R\) onto the unit square.
  3. Show that the line through the origin with gradient \(\frac { 1 } { k - 1 }\) is invariant under the transformation in the \(x - y\) plane represented by \(\mathbf { M }\).
CAIE Further Paper 1 2023 November Q4
10 marks Challenging +1.2
4 The cubic equation \(27 x ^ { 3 } + 18 x ^ { 2 } + 6 x - 1 = 0\) has roots \(\alpha , \beta , \gamma\).
  1. Show that a cubic equation with roots \(3 \alpha + 1,3 \beta + 1,3 \gamma + 1\) is $$y ^ { 3 } - y ^ { 2 } + y - 2 = 0$$ The sum \(( 3 \alpha + 1 ) ^ { n } + ( 3 \beta + 1 ) ^ { n } + ( 3 \gamma + 1 ) ^ { n }\) is denoted by \(\mathrm { S } _ { \mathrm { n } }\).
  2. Find the values of \(S _ { 2 }\) and \(S _ { 3 }\).
  3. Find the values of \(S _ { - 1 }\) and \(S _ { - 2 }\).
CAIE Further Paper 1 2023 November Q5
13 marks Standard +0.8
5 The plane \(\Pi _ { 1 }\) has equation \(\mathbf { r } = \mathbf { i } - \mathbf { j } - 2 \mathbf { k } + \lambda ( \mathbf { i } - 2 \mathbf { j } - 3 \mathbf { k } ) + \mu ( 3 \mathbf { i } - \mathbf { k } )\).
  1. Find an equation for \(\Pi _ { 1 }\) in the form \(\mathrm { ax } + \mathrm { by } + \mathrm { cz } = \mathrm { d }\).
    The line \(l\), which does not lie in \(\Pi _ { 1 }\), has equation \(\mathbf { r } = - 3 \mathbf { i } + \mathbf { k } + t ( \mathbf { i } + \mathbf { j } + \mathbf { k } )\).
  2. Show that \(l\) is parallel to \(\Pi _ { 1 }\).
  3. Find the distance between \(l\) and \(\Pi _ { 1 }\).
  4. The plane \(\Pi _ { 2 }\) has equation \(3 x + 3 y + 2 z = 1\). Find a vector equation of the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).
CAIE Further Paper 1 2023 November Q6
13 marks Challenging +1.8
6 The curve \(C\) has polar equation \(r = \mathrm { e } ^ { - \theta } - \mathrm { e } ^ { - \frac { 1 } { 2 } \pi }\), where \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
  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 initial line.
  3. Show that, at the point on \(C\) furthest from the initial line, $$1 - e ^ { \theta - \frac { 1 } { 2 } \pi } - \tan \theta = 0$$ and verify that this equation has a root between 0.56 and 0.57 .
CAIE Further Paper 1 2023 November Q7
16 marks Challenging +1.2
7 The curve \(C\) has equation \(y = f ( x )\), where \(f ( x ) = \frac { x ^ { 2 } } { x + 1 }\).
  1. Find the equations of the asymptotes of \(C\).
  2. Find the coordinates of any stationary points on \(C\).
  3. Sketch \(C\).
  4. Find the coordinates of any stationary points on the curve with equation \(\mathrm { y } = \frac { 1 } { \mathrm { f } ( \mathrm { x } ) }\).
  5. Sketch the curve with equation \(y = \frac { 1 } { f ( x ) }\) and find, in exact form, 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 Further Paper 1 2024 November Q1
10 marks Standard +0.3
1 The matrix \(\mathbf { M }\) represents the sequence of two transformations in the \(x - y\) plane given by a stretch parallel to the \(x\)-axis, scale factor \(k ( k \neq 0 )\), followed by a shear, \(x\)-axis fixed, with \(( 0,1 )\) mapped to \(( k , 1 )\).
  1. Show that \(\mathbf { M } = \left( \begin{array} { c c } k & k \\ 0 & 1 \end{array} \right)\).
  2. The transformation represented by \(\mathbf { M }\) has a line of invariant points. Find, in terms of \(k\), the equation of this line. \includegraphics[max width=\textwidth, alt={}, center]{99ac7fe2-184b-4a72-89f6-03fe4b2af138-02_2722_43_107_2005} The unit square \(S\) in the \(x - y\) plane is transformed by \(\mathbf { M }\) onto the parallelogram \(P\).
  3. Find, in terms of \(k\), a matrix which transforms \(P\) onto \(S\).
  4. Given that the area of \(P\) is \(3 k ^ { 2 }\) units \({ } ^ { 2 }\), find the possible values of \(k\).
CAIE Further Paper 1 2024 November Q2
6 marks Challenging +1.8
2 Prove by mathematical induction that, for all positive integers \(n\), $$\frac { \mathrm { d } ^ { n } } { \mathrm {~d} x ^ { n } } \left( \tan ^ { - 1 } x \right) = P _ { n } ( x ) \left( 1 + x ^ { 2 } \right) ^ { - n } ,$$ where \(P _ { n } ( x )\) is a polynomial of degree \(n - 1\). \includegraphics[max width=\textwidth, alt={}, center]{99ac7fe2-184b-4a72-89f6-03fe4b2af138-05_2726_35_97_20}
CAIE Further Paper 1 2024 November Q3
10 marks Challenging +1.8
3 The quartic equation \(x ^ { 4 } + 2 x ^ { 3 } - 1 = 0\) has roots \(\alpha , \beta , \gamma , \delta\).
  1. Find a quartic equation whose roots are \(\alpha ^ { 4 } , \beta ^ { 4 } , \gamma ^ { 4 } , \delta ^ { 4 }\) and state the value of \(\alpha ^ { 4 } + \beta ^ { 4 } + \gamma ^ { 4 } + \delta ^ { 4 }\).
  2. Find the value of \(\alpha ^ { 5 } + \beta ^ { 5 } + \gamma ^ { 5 } + \delta ^ { 5 }\).
  3. Find the value of \(\alpha ^ { 8 } + \beta ^ { 8 } + \gamma ^ { 8 } + \delta ^ { 8 }\).
CAIE Further Paper 1 2024 November Q4
8 marks Challenging +1.2
4
  1. Use the method of differences to find \(\sum _ { r = 1 } ^ { n } \frac { 5 k } { ( 5 r + k ) ( 5 r + 5 + k ) }\) in terms of \(n\) and \(k\), where \(k\) is a positive constant. \includegraphics[max width=\textwidth, alt={}, center]{99ac7fe2-184b-4a72-89f6-03fe4b2af138-08_2720_35_109_2010} \includegraphics[max width=\textwidth, alt={}, center]{99ac7fe2-184b-4a72-89f6-03fe4b2af138-09_2723_35_101_20} It is given that \(\sum _ { r = 1 } ^ { \infty } \frac { 5 k } { ( 5 r + k ) ( 5 r + 5 + k ) } = \frac { 1 } { 3 }\).
  2. Find the value of \(k\).
  3. Hence find \(\sum _ { r = n } ^ { n ^ { 2 } } \frac { 5 k } { ( 5 r + k ) ( 5 r + 5 + k ) }\) in terms of \(n\). $$\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 6 x y$$ has polar equation \(r ^ { 2 } = 3 \sin 2 \theta\).
    The curve \(C\) has polar equation \(r ^ { 2 } = 3 \sin 2 \theta\), for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
  4. Sketch \(C\) and state the maximum distance of a point on \(C\) from the pole. \includegraphics[max width=\textwidth, alt={}, center]{99ac7fe2-184b-4a72-89f6-03fe4b2af138-10_2716_35_108_2012}
  5. Find the area of the region enclosed by \(C\).
  6. Find the maximum distance of a point on \(C\) from the initial line.
CAIE Further Paper 1 2024 November Q6
13 marks Challenging +1.2
6 The curve \(C\) has equation \(y = \frac { 4 x ^ { 2 } + x + 1 } { 2 x ^ { 2 } - 7 x + 3 }\).
  1. Find the equations of the asymptotes of \(C\).
  2. Find the coordinates of any stationary points on \(C\). \includegraphics[max width=\textwidth, alt={}, center]{99ac7fe2-184b-4a72-89f6-03fe4b2af138-13_2720_40_106_18}
  3. Sketch \(C\), stating the coordinates of any intersections with the axes.
  4. Sketch the curve with equation \(y = \left| \frac { 4 x ^ { 2 } + x + 1 } { 2 x ^ { 2 } - 7 x + 3 } \right|\) and state the set of values of \(k\) for which \(\left| \frac { 4 x ^ { 2 } + x + 1 } { 2 x ^ { 2 } - 7 x + 3 } \right| = k\) has 4 distinct real solutions.
CAIE Further Paper 1 2024 November Q7
15 marks Challenging +1.2
7 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations \(\mathbf { r } = \mathbf { i } + 3 \mathbf { j } - 2 \mathbf { k } + \lambda ( 2 \mathbf { i } + \mathbf { j } + \mathbf { k } )\) and \(\mathbf { r } = \mathbf { i } - 2 \mathbf { j } + 9 \mathbf { k } + \mu ( \mathbf { i } - 4 \mathbf { j } + 2 \mathbf { k } )\) respectively. The plane \(\Pi _ { 1 }\) contains \(l _ { 1 }\) and is parallel to \(l _ { 2 }\).
  1. Find the equation of \(\Pi _ { 1 }\), giving your answer in the form \(a x + b y + c z = d\).
    The plane \(\Pi _ { 2 }\) contains \(l _ { 2 }\) and the point with coordinates \(( 2 , - 1,7 )\).
  2. Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\). \includegraphics[max width=\textwidth, alt={}, center]{99ac7fe2-184b-4a72-89f6-03fe4b2af138-15_2723_35_101_20} 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. Find a vector equation for \(P Q\).
    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 November Q1
5 marks Moderate -0.3
1 The sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is such that \(u _ { 1 } = 4\) and \(u _ { n + 1 } = 3 u _ { n } - 2\) for \(n \geqslant 1\).
Prove by induction that \(u _ { n } = 3 ^ { n } + 1\) for all positive integers \(n\).
CAIE Further Paper 1 2024 November Q2
7 marks Standard +0.3
2 The line \(l _ { 1 }\) has equation \(\mathbf { r } = \mathbf { i } + 3 \mathbf { j } - \mathbf { k } + \lambda ( \mathbf { i } - \mathbf { j } - 4 \mathbf { k } )\).
The plane \(\Pi\) contains \(l _ { 1 }\) and is parallel to the vector \(2 \mathbf { i } + 5 \mathbf { j } - 4 \mathbf { k }\).
  1. Find the equation of \(\Pi\), giving your answer in the form \(a x + b y + c z = d\). \includegraphics[max width=\textwidth, alt={}, center]{beb9c1f1-1676-4432-a42a-c418ff9f45d8-05_2723_33_99_22} The line \(l _ { 2 }\) is parallel to the vector \(5 \mathbf { i } - 5 \mathbf { j } - 2 \mathbf { k }\).
  2. Find the acute angle between \(l _ { 2 }\) and \(\Pi\).
CAIE Further Paper 1 2024 November Q3
10 marks Standard +0.8
3 It is given that $$\begin{aligned} & \alpha + \beta + \gamma + \delta = 2 \\ & \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 } = 3 \\ & \alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } + \delta ^ { 3 } = 4 \end{aligned}$$
  1. Find the value of \(\alpha \beta + \alpha \gamma + \alpha \delta + \beta \gamma + \beta \delta + \gamma \delta\).
  2. Find the value of \(\alpha ^ { 2 } \beta + \alpha ^ { 2 } \gamma + \alpha ^ { 2 } \delta + \beta ^ { 2 } \alpha + \beta ^ { 2 } \gamma + \beta ^ { 2 } \delta + \gamma ^ { 2 } \alpha + \gamma ^ { 2 } \beta + \gamma ^ { 2 } \delta + \delta ^ { 2 } \alpha + \delta ^ { 2 } \beta + \delta ^ { 2 } \gamma\). \includegraphics[max width=\textwidth, alt={}, center]{beb9c1f1-1676-4432-a42a-c418ff9f45d8-06_2717_33_109_2014} \includegraphics[max width=\textwidth, alt={}, center]{beb9c1f1-1676-4432-a42a-c418ff9f45d8-07_2723_33_99_22}
  3. It is given that \(\alpha , \beta , \gamma , \delta\) are the roots of the equation $$6 x ^ { 4 } - 12 x ^ { 3 } + 3 x ^ { 2 } + 2 x + 6 = 0 .$$
    1. Find the value of \(\alpha ^ { 4 } + \beta ^ { 4 } + \gamma ^ { 4 } + \delta ^ { 4 }\).
    2. Find the value of \(\alpha ^ { 5 } + \beta ^ { 5 } + \gamma ^ { 5 } + \delta ^ { 5 }\).
CAIE Further Paper 1 2024 November Q4
13 marks Standard +0.8
4 The matrices \(\mathbf { A } , \mathbf { B }\) and \(\mathbf { C }\) are given by $$\mathbf { A } = \left( \begin{array} { l l l } 1 & 2 & 3 \\ 2 & 1 & 3 \\ 3 & 2 & 5 \end{array} \right) , \mathbf { B } = \left( \begin{array} { r r } 0 & - 2 \\ - 1 & 3 \\ 0 & 0 \end{array} \right) \text { and } \mathbf { C } = \left( \begin{array} { r r r } - 2 & - 1 & 1 \\ 1 & 1 & 3 \end{array} \right)$$
  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 \(\mathbf { C A B }\). \includegraphics[max width=\textwidth, alt={}, center]{beb9c1f1-1676-4432-a42a-c418ff9f45d8-08_2715_31_106_2016} Let \(\mathbf { M } = \left( \begin{array} { l l } 3 & 0 \\ 0 & 1 \end{array} \right)\).
  3. Give full details of the transformation represented by \(\mathbf { M }\).
  4. Find the matrix \(\mathbf { N }\) such that \(\mathbf { N M } = \mathbf { C A B }\).
CAIE Further Paper 1 2024 November Q5
9 marks Challenging +1.2
5 It is given that \(S _ { n } = \sum _ { r = 1 } ^ { n } u _ { r }\), where \(u _ { r } = x ^ { \mathrm { f } ( r ) } - x ^ { \mathrm { f } ( r + 1 ) }\) and \(x > 0\).
  1. Find \(S _ { n }\) in terms of \(n , x\) and the function f .
  2. Given that \(\mathrm { f } ( r ) = \ln r\), find 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. \includegraphics[max width=\textwidth, alt={}, center]{beb9c1f1-1676-4432-a42a-c418ff9f45d8-10_2716_31_106_2016} \includegraphics[max width=\textwidth, alt={}, center]{beb9c1f1-1676-4432-a42a-c418ff9f45d8-11_2723_35_101_20}
  3. Given instead that \(\mathrm { f } ( r ) = 2 \log _ { x } r\) where \(x \neq 1\), use standard results from the List of formulae (MF19) to find \(\sum _ { n = 1 } ^ { N } S _ { n }\) in terms of \(N\). Fully factorise your answer.
CAIE Further Paper 1 2024 November Q6
15 marks Standard +0.8
6 The curve \(C\) has equation \(y = \frac { x ^ { 2 } + 3 } { x ^ { 2 } + 1 }\).
  1. Show that \(C\) has no vertical asymptotes and state the equation of the horizontal asymptote.
  2. Show that \(1 < y \leqslant 3\) for all real values of \(x\).
  3. Find the coordinates of any stationary points on \(C\). \includegraphics[max width=\textwidth, alt={}, center]{beb9c1f1-1676-4432-a42a-c418ff9f45d8-12_2718_42_107_2007} \includegraphics[max width=\textwidth, alt={}, center]{beb9c1f1-1676-4432-a42a-c418ff9f45d8-13_2720_40_106_18}
  4. Sketch \(C\), stating the coordinates of any intersections with the axes and labelling the asymptote.
  5. Sketch the curve with equation \(y = \frac { x ^ { 2 } + 1 } { x ^ { 2 } + 3 }\) and find the set of values of \(x\) for which \(\frac { x ^ { 2 } + 1 } { x ^ { 2 } + 3 } < \frac { 1 } { 2 }\).
CAIE Further Paper 1 2024 November Q7
16 marks Challenging +1.2
7 The curve \(C _ { 1 }\) has polar equation \(r = a ( \cos \theta + \sin \theta )\) for \(- \frac { 1 } { 4 } \pi \leqslant \theta \leqslant \frac { 3 } { 4 } \pi\), where \(a\) is a positive constant.
  1. Find a Cartesian equation for \(C _ { 1 }\) and show that it represents a circle, stating its radius and the Cartesian coordinates of its centre.
  2. Sketch \(C _ { 1 }\) and state the greatest distance of a point on \(C _ { 1 }\) from the pole. \includegraphics[max width=\textwidth, alt={}, center]{beb9c1f1-1676-4432-a42a-c418ff9f45d8-14_2721_40_107_2010} The curve \(C _ { 2 }\) with polar equation \(r = a \theta\) intersects \(C _ { 1 }\) at the pole and the point with polar coordinates \(( a \phi , \phi )\).
  3. Verify that \(1.25 < \phi < 1.26\).
  4. Show that the area of the smaller region enclosed by \(C _ { 1 }\) and \(C _ { 2 }\) is equal to $$\frac { 1 } { 2 } a ^ { 2 } \left( \frac { 3 } { 4 } \pi + \frac { 1 } { 3 } \phi ^ { 3 } - \phi + \frac { 1 } { 2 } \cos 2 \phi \right)$$ and deduce, in terms of \(a\) and \(\phi\), the area of the larger region enclosed by \(C _ { 1 }\) and \(C _ { 2 }\).
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE Further Paper 1 2024 November Q1
10 marks Standard +0.3
1 The matrix \(\mathbf { M }\) represents the sequence of two transformations in the \(x - y\) plane given by a stretch parallel to the \(x\)-axis, scale factor \(k ( k \neq 0 )\), followed by a shear, \(x\)-axis fixed, with \(( 0,1 )\) mapped to \(( k , 1 )\).
  1. Show that \(\mathbf { M } = \left( \begin{array} { c c } k & k \\ 0 & 1 \end{array} \right)\).
  2. The transformation represented by \(\mathbf { M }\) has a line of invariant points. Find, in terms of \(k\), the equation of this line.
    The unit square \(S\) in the \(x - y\) plane is transformed by \(\mathbf { M }\) onto the parallelogram \(P\).
  3. Find, in terms of \(k\), a matrix which transforms \(P\) onto \(S\).
  4. Given that the area of \(P\) is \(3 k ^ { 2 }\) units \({ } ^ { 2 }\), find the possible values of \(k\).
CAIE Further Paper 1 2024 November Q2
6 marks Challenging +1.2
2 Prove by mathematical induction that, for all positive integers \(n\), $$\frac { \mathrm { d } ^ { n } } { \mathrm {~d} x ^ { n } } \left( \tan ^ { - 1 } x \right) = P _ { n } ( x ) \left( 1 + x ^ { 2 } \right) ^ { - n } ,$$ where \(P _ { n } ( x )\) is a polynomial of degree \(n - 1\). \includegraphics[max width=\textwidth, alt={}, center]{beb69d0f-3f83-49bf-b9a2-329ddc7243fa-04_2718_42_107_2007} \includegraphics[max width=\textwidth, alt={}, center]{beb69d0f-3f83-49bf-b9a2-329ddc7243fa-05_2726_35_97_20}
CAIE Further Paper 1 2024 November Q4
8 marks Challenging +1.2
4
  1. Use the method of differences to find \(\sum _ { r = 1 } ^ { n } \frac { 5 k } { ( 5 r + k ) ( 5 r + 5 + k ) }\) in terms of \(n\) and \(k\), where \(k\) is a positive constant. \includegraphics[max width=\textwidth, alt={}, center]{beb69d0f-3f83-49bf-b9a2-329ddc7243fa-08_2715_35_110_2012} \includegraphics[max width=\textwidth, alt={}, center]{beb69d0f-3f83-49bf-b9a2-329ddc7243fa-09_2723_35_101_20} It is given that \(\sum _ { r = 1 } ^ { \infty } \frac { 5 k } { ( 5 r + k ) ( 5 r + 5 + k ) } = \frac { 1 } { 3 }\).
  2. Find the value of \(k\).
  3. Hence find \(\sum _ { r = n } ^ { n ^ { 2 } } \frac { 5 k } { ( 5 r + k ) ( 5 r + 5 + k ) }\) in terms of \(n\). $$\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 6 x y$$ has polar equation \(r ^ { 2 } = 3 \sin 2 \theta\).
    The curve \(C\) has polar equation \(r ^ { 2 } = 3 \sin 2 \theta\), for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
  4. Sketch \(C\) and state the maximum distance of a point on \(C\) from the pole. \includegraphics[max width=\textwidth, alt={}, center]{beb69d0f-3f83-49bf-b9a2-329ddc7243fa-10_2716_35_108_2012}
  5. Find the area of the region enclosed by \(C\).
  6. Find the maximum distance of a point on \(C\) from the initial line.
CAIE Further Paper 1 2024 November Q6
13 marks Challenging +1.2
6 The curve \(C\) has equation \(y = \frac { 4 x ^ { 2 } + x + 1 } { 2 x ^ { 2 } - 7 x + 3 }\).
  1. Find the equations of the asymptotes of \(C\).
  2. Find the coordinates of any stationary points on \(C\). \includegraphics[max width=\textwidth, alt={}, center]{beb69d0f-3f83-49bf-b9a2-329ddc7243fa-13_2720_40_106_18}
  3. Sketch \(C\), stating the coordinates of any intersections with the axes.
  4. Sketch the curve with equation \(y = \left| \frac { 4 x ^ { 2 } + x + 1 } { 2 x ^ { 2 } - 7 x + 3 } \right|\) and state the set of values of \(k\) for which \(\left| \frac { 4 x ^ { 2 } + x + 1 } { 2 x ^ { 2 } - 7 x + 3 } \right| = k\) has 4 distinct real solutions.
CAIE Further Paper 1 2024 November Q7
15 marks Challenging +1.2
7 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations \(\mathbf { r } = \mathbf { i } + 3 \mathbf { j } - 2 \mathbf { k } + \lambda ( 2 \mathbf { i } + \mathbf { j } + \mathbf { k } )\) and \(\mathbf { r } = \mathbf { i } - 2 \mathbf { j } + 9 \mathbf { k } + \mu ( \mathbf { i } - 4 \mathbf { j } + 2 \mathbf { k } )\) respectively. The plane \(\Pi _ { 1 }\) contains \(l _ { 1 }\) and is parallel to \(l _ { 2 }\).
  1. Find the equation of \(\Pi _ { 1 }\), giving your answer in the form \(a x + b y + c z = d\).
    The plane \(\Pi _ { 2 }\) contains \(l _ { 2 }\) and the point with coordinates \(( 2 , - 1,7 )\).
  2. Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\). \includegraphics[max width=\textwidth, alt={}, center]{beb69d0f-3f83-49bf-b9a2-329ddc7243fa-15_2723_35_101_20} 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. Find a vector equation for \(P Q\).
    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 Specimen Q1
6 marks Standard +0.3
1
  1. Gie it h \(\mathrm { f } ( r ) = \frac { 1 } { ( r + 1 ) ( r + 2 ) } , \mathrm { s }\) th t $$\mathrm { f } \left( r - 1 \quad \mathrm { f } ( r ) = \frac { 2 } { r ( r + 1 ) ( r + 2 ) } \right.$$
  2. Hen e fid \(\sum _ { r = 1 } ^ { n } \frac { 1 } { r ( r + 1 ) ( r + 2 ) }\).
  3. Ded e th le \(6 \sum _ { r = 1 } ^ { \infty } \frac { 1 } { r ( r + 1 ) ( r + 2 ) }\).