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CAIE Further Paper 1 2023 June Q5
12 marks Challenging +1.3
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
15 marks Challenging +1.2
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
14 marks Challenging +1.2
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 2023 June Q1
6 marks Standard +0.3
1 Prove by mathematical induction that, for all positive integers \(n , 5 ^ { 3 n } + 32 ^ { n } - 33\) is divisible by 31 .
CAIE Further Paper 1 2023 June Q2
8 marks Standard +0.8
2
  1. Use standard results from the list of formulae (MF19) to show that $$\sum _ { r = 1 } ^ { n } \left( 6 r ^ { 2 } + 6 r - 5 \right) = a n ^ { 3 } + b n ^ { 2 } + c n$$ where \(a\), \(b\) and \(c\) are integers to be determined.
  2. Use the method of differences to find \(\sum _ { r = 1 } ^ { n } \frac { 6 r ^ { 2 } + 6 r - 5 } { r ^ { 2 } + r }\) in terms of \(n\).
  3. Find also \(\sum _ { r = n + 1 } ^ { 2 n } \frac { 6 r ^ { 2 } + 6 r - 5 } { r ^ { 2 } + r }\) in terms of \(n\).
CAIE Further Paper 1 2023 June Q3
9 marks Challenging +1.2
3 The equation \(x ^ { 4 } - x ^ { 2 } + 2 x + 5 = 0\) has roots \(\alpha , \beta , \gamma , \delta\).
  1. Find a quartic equation whose roots are \(\alpha ^ { 2 } , \beta ^ { 2 } , \gamma ^ { 2 } , \delta ^ { 2 }\) and state the value of \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 }\).
  2. Find the value of \(\frac { 1 } { \alpha ^ { 2 } } + \frac { 1 } { \beta ^ { 2 } } + \frac { 1 } { \gamma ^ { 2 } } + \frac { 1 } { \delta ^ { 2 } }\).
  3. Find the value of \(\alpha ^ { 4 } + \beta ^ { 4 } + \gamma ^ { 4 } + \delta ^ { 4 }\).
CAIE Further Paper 1 2023 June Q4
14 marks Challenging +1.2
4 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { r r } \cos 2 \theta & - \sin 2 \theta \\ \sin 2 \theta & \cos 2 \theta \end{array} \right) \left( \begin{array} { l l } 1 & k \\ 0 & 1 \end{array} \right)\), where \(0 < \theta < \pi\) and \(k\) is a non-zero constant. The matrix \(\mathbf { M }\) represents a sequence of two geometrical transformations, one of which is a shear.
  1. Describe fully the other transformation and state the order in which the transformations are applied.
  2. Write \(\mathbf { M } ^ { - 1 }\) as the product of two matrices, neither of which is \(\mathbf { I }\).
  3. Find, in terms of \(k\), the value of \(\tan \theta\) for which \(\mathbf { M - I }\) is singular.
  4. Given that \(k = 2 \sqrt { 3 }\) and \(\theta = \frac { 1 } { 3 } \pi\), show that the invariant points of the transformation represented by \(\mathbf { M }\) lie on the line \(3 y + \sqrt { 3 } x = 0\).
CAIE Further Paper 1 2023 June Q5
10 marks Standard +0.3
5
  1. Show that the curve with Cartesian equation $$x ^ { 2 } - y ^ { 2 } = a$$ where \(a\) is a positive constant, has polar equation \(r ^ { 2 } = a \sec 2 \theta\).
    The curve \(C\) has polar equation \(r ^ { 2 } = \operatorname { asec } 2 \theta\), where \(a\) is a positive constant, for \(0 \leqslant \theta < \frac { 1 } { 4 } \pi\).
  2. Sketch \(C\) and state the minimum distance of \(C\) from the pole.
  3. Find, in terms of \(a\), the exact value of the area of the region enclosed by \(C\), the initial line, and the half-line \(\theta = \frac { 1 } { 12 } \pi\). [You may use any result from the list of formulae (MF19) without proof.] [4]
CAIE Further Paper 1 2023 June Q6
15 marks Standard +0.3
6 The points \(A , B , C\) have position vectors $$\mathbf { i } + \mathbf { j } , \quad - \mathbf { i } + 2 \mathbf { j } + 4 \mathbf { k } , \quad - 2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k } ,$$ 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 a vector equation of the common perpendicular to the lines \(O C\) and \(A B\).
CAIE Further Paper 1 2023 June Q7
13 marks Challenging +1.2
7 The curve \(C\) has equation \(\mathrm { y } = \frac { \mathrm { x } ^ { 2 } + 2 \mathrm { x } + 1 } { \mathrm { x } - 3 }\).
  1. Find the equations of the asymptotes of \(C\).
  2. Find the coordinates of the turning points on \(C\).
  3. Sketch \(C\).
  4. Sketch the curves with equations \(y = \left| \frac { x ^ { 2 } + 2 x + 1 } { x - 3 } \right|\) and \(y ^ { 2 } = \frac { x ^ { 2 } + 2 x + 1 } { x - 3 }\) on a single diagram, clearly identifying each curve. 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
6 marks Standard +0.8
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
6 marks Standard +0.8
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
8 marks Challenging +1.2
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
13 marks Standard +0.3
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
12 marks Standard +0.3
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
15 marks Challenging +1.2
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
15 marks Challenging +1.8
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 2024 June Q1
5 marks Standard +0.3
1 The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { c c c } k & 1 & 0 \\ 6 & 5 & 2 \\ - 1 & 3 & - k \end{array} \right)$$ where \(k\) is a real constant.
  1. Show that \(\mathbf { A }\) is non-singular.
  2. Given that \(\mathbf { A } ^ { - 1 } = \left( \begin{array} { c c c } 3 & 0 & - 1 \\ 1 & 0 & 0 \\ - \frac { 23 } { 2 } & \frac { 1 } { 2 } & 3 \end{array} \right)\), find the value of \(k\).
CAIE Further Paper 1 2024 June Q2
7 marks Challenging +1.2
2 The cubic equation \(x ^ { 3 } + 2 x ^ { 2 } + 3 x + 1 = 0\) has roots \(\alpha , \beta , \gamma\).
  1. Find a cubic equation whose roots are \(\alpha ^ { 2 } + 1 , \beta ^ { 2 } + 1 , \gamma ^ { 2 } + 1\). \includegraphics[max width=\textwidth, alt={}, center]{7eb2abb1-68f4-4cc8-8314-f436906d6c4e-04_2714_34_143_2012}
  2. Find the value of \(\left( \alpha ^ { 2 } + 1 \right) ^ { 2 } + \left( \beta ^ { 2 } + 1 \right) ^ { 2 } + \left( \gamma ^ { 2 } + 1 \right) ^ { 2 }\).
  3. Find the value of \(\left( \alpha ^ { 2 } + 1 \right) ^ { 3 } + \left( \beta ^ { 2 } + 1 \right) ^ { 3 } + \left( \gamma ^ { 2 } + 1 \right) ^ { 3 }\).
CAIE Further Paper 1 2024 June Q3
14 marks Standard +0.3
3 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { l l } 1 & 2 \\ 0 & 1 \end{array} \right) \left( \begin{array} { l l } 7 & 0 \\ 0 & 1 \end{array} \right)\).
  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. [4]
  2. Find the equations of the invariant lines, through the origin, of the transformation represented by \(\mathbf { M }\).
    The triangle \(D E F\) in the \(x - y\) plane is transformed by \(\mathbf { M }\) onto triangle \(P Q R\) .
  3. Given that the area of triangle \(P Q R\) is \(35 \mathrm {~cm} ^ { 2 }\) ,find the area of triangle \(D E F\) .
CAIE Further Paper 1 2024 June Q4
13 marks Standard +0.3
4
  1. Prove by mathematical induction that, for all positive integers \(n\), $$\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )$$ \includegraphics[max width=\textwidth, alt={}, center]{7eb2abb1-68f4-4cc8-8314-f436906d6c4e-08_2716_35_143_2012} The sum \(S _ { n }\) is defined by \(S _ { n } = \sum _ { r = 1 } ^ { n } r ^ { 4 }\).
  2. Using the identity $$( 2 r + 1 ) ^ { 5 } - ( 2 r - 1 ) ^ { 5 } \equiv 160 r ^ { 4 } + 80 r ^ { 2 } + 2$$ show that \(S _ { n } = \frac { 1 } { 30 } n ( n + 1 ) ( 2 n + 1 ) \left( 3 n ^ { 2 } + 3 n - 1 \right)\).
  3. Find the value of \(\lim _ { n \rightarrow \infty } \left( n ^ { - 5 } S _ { n } \right)\).
CAIE Further Paper 1 2024 June Q5
10 marks
5 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations \(\mathbf { r } = \mathbf { i } + 4 \mathbf { j } - \mathbf { k } + \lambda ( \mathbf { j } - 2 \mathbf { k } )\) and \(\mathbf { r } = - 3 \mathbf { i } + 4 \mathbf { j } + \mu ( \mathbf { i } + 2 \mathbf { j } + \mathbf { k } )\) respectively.
  1. Find the shortest distance between \(l _ { 1 }\) and \(l _ { 2 }\).
    The plane \(\Pi _ { 1 }\) contains \(l _ { 1 }\) and is parallel to \(l _ { 2 }\).
  2. Obtain an equation of \(\Pi _ { 1 }\) in the form \(p x + q y + r z = s\). \includegraphics[max width=\textwidth, alt={}, center]{7eb2abb1-68f4-4cc8-8314-f436906d6c4e-10_2715_40_144_2007}
  3. The point \(( 1,1,1 )\) lies on the plane \(\Pi _ { 2 }\). It is given that the line of intersection of the planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) passes through the point ( \(0,0,2\) ) and is parallel to the vector \(\mathbf { i } + 4 \mathbf { j } - 3 \mathbf { k }\). Obtain an equation of \(\Pi _ { 2 }\) in the form \(a x + b y + c z = d\).
CAIE Further Paper 1 2024 June Q6
13 marks Moderate -0.3
6 The curve \(C\) has equation \(y = \frac { x + 1 } { x ^ { 2 } + 3 }\).
  1. Show that \(C\) has no vertical asymptotes and state the equation of the horizontal asymptote. [2]
  2. Find the coordinates of any stationary points on \(C\). \includegraphics[max width=\textwidth, alt={}, center]{7eb2abb1-68f4-4cc8-8314-f436906d6c4e-12_2715_35_144_2012} \includegraphics[max width=\textwidth, alt={}, center]{7eb2abb1-68f4-4cc8-8314-f436906d6c4e-13_2718_33_141_23}
  3. Sketch \(C\), stating the coordinates of the intersections with the axes.
  4. Sketch \(y ^ { 2 } = \frac { x + 1 } { x ^ { 2 } + 3 }\), stating the coordinates of the stationary points and the intersections with the axes.
CAIE Further Paper 1 2024 June Q7
13 marks Challenging +1.8
7 The curve \(C\) has polar equation \(r ^ { 2 } = \sin 2 \theta \cos \theta\), for \(0 \leqslant \theta \leqslant \pi\).
  1. Sketch \(C\) and state the equation of the line of symmetry.
  2. Find a Cartesian equation for \(C\).
  3. Find the total area enclosed by \(C\).
  4. Find the greatest distance of a point on \(C\) from the pole. \includegraphics[max width=\textwidth, alt={}, center]{7eb2abb1-68f4-4cc8-8314-f436906d6c4e-16_2718_36_141_2011} If you use the following page to complete the answer to any question, the question number must be clearly shown.
    \includegraphics[max width=\textwidth, alt={}]{7eb2abb1-68f4-4cc8-8314-f436906d6c4e-18_436_29_143_2014}\includegraphics[max width=\textwidth, alt={}]{7eb2abb1-68f4-4cc8-8314-f436906d6c4e-18_436_29_714_2014}\includegraphics[max width=\textwidth, alt={}]{7eb2abb1-68f4-4cc8-8314-f436906d6c4e-18_438_29_1283_2014}\includegraphics[max width=\textwidth, alt={}]{7eb2abb1-68f4-4cc8-8314-f436906d6c4e-18_436_29_1852_2014}\includegraphics[max width=\textwidth, alt={}]{7eb2abb1-68f4-4cc8-8314-f436906d6c4e-18_436_29_2423_2014}
CAIE Further Paper 1 2020 November Q1
9 marks Standard +0.3
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 \(b\).