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CAIE Further Paper 2 2020 November Q1
5 marks Standard +0.8
1 Find the Maclaurin's series for \(\tan \left( x + \frac { 1 } { 4 } \pi \right)\) up to and including the term in \(x ^ { 2 }\).
CAIE Further Paper 2 2020 November Q2
6 marks Challenging +1.2
2 A curve has equation \(\mathrm { y } = \cosh \mathrm { x }\), for \(0 \leqslant x \leqslant \frac { 1 } { 2 }\).
Find, in terms of \(\pi\) and e, the area of the surface generated when the curve is rotated through \(2 \pi\) radians about the \(x\)-axis.
CAIE Further Paper 2 2020 November Q3
4 marks Standard +0.3
3 Find all the roots of the equation \(( w + 1 ) ^ { 6 } = 1\), giving your answers in the form \(\mathrm { x } + \mathrm { iy }\) where \(x\) and \(y\) are real and exact.
CAIE Further Paper 2 2020 November Q4
8 marks Standard +0.3
4 Find the solution of the differential equation $$x \frac { d y } { d x } + 2 y = e ^ { x }$$ for which \(y = 3\) when \(x = 1\). Give your answer in the form \(y = f ( x )\).
CAIE Further Paper 2 2020 November Q5
8 marks Standard +0.8
5 The curve \(C\) has equation $$y ^ { 2 } + ( x y + 1 ) ^ { 2 } = 5$$
  1. Show that, at the point \(( 1,1 )\) on \(C , \frac { \mathrm { dy } } { \mathrm { dx } } = - \frac { 2 } { 3 }\).
  2. Find the value of \(\frac { d ^ { 2 } y } { d x ^ { 2 } }\) at the point \(( 1,1 )\).
CAIE Further Paper 2 2020 November Q6
11 marks Standard +0.8
6 Find the particular solution of the differential equation $$\frac { d ^ { 2 } x } { d t ^ { 2 } } + 8 \frac { d x } { d t } + 15 x = 102 \cos 3 t$$ given that, when \(t = 0 , x = 1\) and \(\frac { \mathrm { dx } } { \mathrm { dt } } = 0\).
CAIE Further Paper 2 2020 November Q7
7 marks Challenging +1.2
7
  1. Show that \(\sum _ { r = 1 } ^ { n } z ^ { 2 r } = \frac { z ^ { 2 n + 1 } - z } { z - z ^ { - 1 } }\), for \(z \neq 0,1 , - 1\).
  2. By letting \(z = \cos \theta + i \sin \theta\), show that, if \(\sin \theta \neq 0\), $$1 + 2 \sum _ { r = 1 } ^ { n } \cos ( 2 r \theta ) = \frac { \sin ( 2 n + 1 ) \theta } { \sin \theta }$$
CAIE Further Paper 2 2020 November Q8
10 marks Challenging +1.8
8 \includegraphics[max width=\textwidth, alt={}, center]{5b43cb39-7560-4484-ba6f-17303e986f47-10_369_1531_260_306} The diagram shows the curve \(\mathrm { y } = \frac { 1 } { \sqrt { \mathrm { x } ^ { 2 } + \mathrm { x } + 1 } }\) for \(x \geqslant 0\), together with a set of \(n\) rectangles of unit width. By considering the sum of the areas of these rectangles, show that $$\sum _ { r = 1 } ^ { n } \frac { 1 } { \sqrt { r ^ { 2 } + r + 1 } } < \ln \left( \frac { 1 } { 3 } + \frac { 2 } { 3 } n + \frac { 2 } { 3 } \sqrt { n ^ { 2 } + n + 1 } \right)$$
CAIE Further Paper 2 2020 November Q9
16 marks Standard +0.8
9 It is given that \(a\) is a positive constant.
  1. Show that the system of equations $$\begin{aligned} a x + ( 2 a + 5 ) y + ( a + 1 ) z & = 1 \\ - 4 y & = 2 \\ 3 y - z & = 3 \end{aligned}$$ has a unique solution and interpret this situation geometrically.
    The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { c c c } a & 2 a + 5 & a + 1 \\ 0 & - 4 & 0 \\ 0 & 3 & - 1 \end{array} \right)$$
  2. Show that the eigenvalues of \(\mathbf { A }\) are \(a , - 1\) and - 4 .
  3. Find a matrix \(\mathbf { P }\) such that $$\mathbf { A } = \mathbf { P } \left( \begin{array} { r r r } a & 0 & 0 \\ 0 & - 1 & 0 \\ 0 & 0 & - 4 \end{array} \right) \mathbf { P } ^ { - 1 } .$$
  4. Use the characteristic equation of \(\mathbf { A }\) to find \(\mathbf { A } ^ { - 1 }\).
    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 2 2020 November Q3
9 marks Standard +0.8
3
  1. Show that the system of equations $$\begin{array} { r } x - 2 y - 4 z = 1 \\ x - 2 y + k z = 1 \\ - x + 2 y + 2 z = 1 \end{array}$$ where \(k\) is a constant, does not have a unique solution.
  2. Given that \(k = - 4\), show that the system of equations in part (a) is consistent. Interpret this situation geometrically.
  3. Given instead that \(k = - 2\), show that the system of equations in part (a) is inconsistent. Interpret this situation geometrically.
  4. For the case where \(k \neq - 2\) and \(k \neq - 4\), show that the system of equations in part (a) is inconsistent. Interpret this situation geometrically. \includegraphics[max width=\textwidth, alt={}, center]{23c7189f-850d-4745-a8ce-46a140ed0176-06_894_841_260_612} The diagram shows the curve with equation \(\mathrm { y } = 1 - \mathrm { x } ^ { 3 }\) for \(0 \leqslant x \leqslant 1\), together with a set of \(n\) rectangles of width \(\frac { 1 } { n }\).
CAIE Further Paper 2 2021 November Q1
5 marks Challenging +1.2
1 Find the Maclaurin's series for \(e ^ { x } \tan x\) from first principles up to and including the term in \(x ^ { 2 }\).
CAIE Further Paper 2 2021 November Q2
6 marks Standard +0.8
2 The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r r } - 1 & 2 & 12 \\ 0 & 1 & 0 \\ 0 & 0 & 3 \end{array} \right) .$$ Use the characteristic equation of \(\mathbf { A }\) to show that $$\mathbf { A } ^ { 4 } = p \mathbf { A } ^ { 2 } + q \mathbf { l }$$ where \(p\) and \(q\) are integers to be determined.
CAIE Further Paper 2 2021 November Q3
8 marks Standard +0.8
3 The curve \(C\) has equation $$x y ^ { 3 } - 4 x ^ { 3 } y = 3$$
  1. Show that, at the point \(( - 1,1 )\) on \(C , \frac { \mathrm { dy } } { \mathrm { dx } } = 11\).
  2. Find the value of \(\frac { d ^ { 2 } y } { d x ^ { 2 } }\) at the point \(( - 1,1 )\). \includegraphics[max width=\textwidth, alt={}, center]{37db1c60-0f94-413f-b29b-5872975eee9e-06_535_1584_276_276} The diagram shows the curve with equation \(\mathrm { y } = \frac { \ln \mathrm { x } } { \mathrm { x } ^ { 2 } }\) for \(x \geqslant 2\), together with a set of \(( N - 2 )\) rectangles
    of unit width.
CAIE Further Paper 2 2021 November Q5
11 marks Standard +0.8
5 Find the particular solution of the differential equation $$\frac { d ^ { 2 } y } { d x ^ { 2 } } - 2 \frac { d y } { d x } + y = 4 \cos x$$ given that, when \(x = 0 , y = - 4\) and \(\frac { d y } { d x } = 3\).
CAIE Further Paper 2 2021 November Q6
10 marks Challenging +1.8
6
  1. Use de Moivre's theorem to show that $$\operatorname { cosec } 5 \theta = \frac { \operatorname { cosec } ^ { 5 } \theta } { 5 \operatorname { cosec } ^ { 4 } \theta - 20 \operatorname { cosec } ^ { 2 } \theta + 16 }$$
  2. Hence obtain the roots of the equation $$x ^ { 5 } - 10 x ^ { 4 } + 40 x ^ { 2 } - 32 = 0$$ in the form \(\operatorname { cosec } ( q \pi )\), where \(q\) is rational.
CAIE Further Paper 2 2021 November Q7
11 marks Challenging +1.8
7
  1. Show that an appropriate integrating factor for $$\sqrt { x ^ { 2 } - 1 } \frac { d y } { d x } + y = x ^ { 2 } - x \sqrt { x ^ { 2 } - 1 }$$ is \(x + \sqrt { x ^ { 2 } - 1 }\).
  2. Hence find the solution of the differential equation $$\sqrt { x ^ { 2 } - 1 } \frac { d y } { d x } + y = x ^ { 2 } - x \sqrt { x ^ { 2 } - 1 }$$ for which \(y = 1\) when \(x = \frac { 5 } { 4 }\). Give your answer in the form \(y = f ( x )\).
CAIE Further Paper 2 2021 November Q8
14 marks Challenging +1.8
8
  1. Starting from the definition of cosh in terms of exponentials, prove that $$2 \cosh ^ { 2 } A = \cosh 2 A + 1$$ The curve \(C\) has parametric equations $$\mathrm { x } = 2 \cosh 2 \mathrm { t } + 3 \mathrm { t } , \quad \mathrm { y } = \frac { 3 } { 2 } \cosh 2 \mathrm { t } - 4 \mathrm { t } , \quad \text { for } - \frac { 1 } { 2 } \leqslant t \leqslant \frac { 1 } { 2 }$$ The area of the surface generated when \(C\) is rotated through \(2 \pi\) radians about the \(y\)-axis is denoted by \(A\).
    1. Show that \(A = 10 \pi \int _ { - \frac { 1 } { 2 } } ^ { \frac { 1 } { 2 } } ( 2 \cosh 2 t + 3 t ) \cosh 2 t d t\).
    2. Hence find \(A\) in terms of \(\pi\) and e.
      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 2 2021 November Q3
8 marks Standard +0.8
3 The curve \(C\) has equation $$x y ^ { 3 } - 4 x ^ { 3 } y = 3$$
  1. Show that, at the point \(( - 1,1 )\) on \(C , \frac { \mathrm { dy } } { \mathrm { dx } } = 11\).
  2. Find the value of \(\frac { d ^ { 2 } y } { d x ^ { 2 } }\) at the point \(( - 1,1 )\). \includegraphics[max width=\textwidth, alt={}, center]{59982339-c496-4bd7-8dcd-9b257f3afc02-06_535_1584_276_276} The diagram shows the curve with equation \(\mathrm { y } = \frac { \ln \mathrm { x } } { \mathrm { x } ^ { 2 } }\) for \(x \geqslant 2\), together with a set of \(( N - 2 )\) rectangles
    of unit width.
CAIE Further Paper 2 2022 November Q1
5 marks Standard +0.3
1 Find the Maclaurin's series for \(\ln \left( 1 + \mathrm { e } ^ { x } \right)\) up to and including the term in \(x ^ { 2 }\).
CAIE Further Paper 2 2022 November Q2
7 marks Standard +0.8
2
  1. Show that the system of equations $$\begin{aligned} & x - y + 2 z = 4 \\ & x - y - 3 z = a \\ & x - y + 7 z = 13 \end{aligned}$$ where \(a\) is a constant, does not have a unique solution.
  2. Given that \(a = - 5\), show that the system of equations in part (a) is consistent. Interpret this situation geometrically.
  3. Given instead that \(a \neq - 5\), show that the system of equations in part (a) is inconsistent. Interpret this situation geometrically.
CAIE Further Paper 2 2022 November Q3
6 marks Challenging +1.2
3 The curve \(C\) has parametric equations $$\mathrm { x } = \mathrm { e } ^ { \mathrm { t } } - \frac { 1 } { 3 } \mathrm { t } ^ { 3 } , \quad \mathrm { y } = 4 \mathrm { e } ^ { \frac { 1 } { 2 } \mathrm { t } } ( \mathrm { t } - 2 ) , \quad \text { for } 0 \leqslant t \leqslant 2$$ Find, in terms of e , the length of \(C\).
CAIE Further Paper 2 2022 November Q4
12 marks Challenging +1.2
4
  1. Starting from the definitions of cosh and sinh in terms of exponentials, prove that $$\cosh ^ { 2 } x - \sinh ^ { 2 } x = 1 .$$
  2. Show that \(\frac { \mathrm { d } } { \mathrm { dx } } \left( \tan ^ { - 1 } ( \sinh x ) \right) = \operatorname { sech } x\).
  3. Sketch the graph of \(y = \operatorname { sechx }\), stating the equation of the asymptote.
  4. By considering a suitable set of \(n\) rectangles of unit width, use your sketch to show that $$\sum _ { r = 1 } ^ { n } \operatorname { sechr } < \tan ^ { - 1 } ( \operatorname { sinhn } )$$
  5. Hence state an upper bound, in terms of \(\pi\), for \(\sum _ { r = 1 } ^ { \infty }\) sech \(r\).
CAIE Further Paper 2 2022 November Q5
10 marks Standard +0.8
5 Find the particular solution of the differential equation $$2 \frac { d ^ { 2 } y } { d x ^ { 2 } } + 2 \frac { d y } { d x } + y = 4 x ^ { 2 } + 3 x + 3$$ given that, when \(x = 0 , y = \frac { d y } { d x } = 0\).
CAIE Further Paper 2 2022 November Q6
11 marks Challenging +1.2
6 The matrix \(\mathbf { A }\) is given by $$A = \left( \begin{array} { r r r } 2 & - 3 & - 7 \\ 0 & 5 & 7 \\ 0 & 0 & - 2 \end{array} \right) .$$
  1. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { 5 } = \mathbf { P D P } ^ { - 1 }\).
  2. Use the characteristic equation of \(\mathbf { A }\) to show that $$\mathbf { A } ^ { 4 } = a \mathbf { A } ^ { 2 } + b \mathbf { I } ,$$ where \(a\) and \(b\) are integers to be determined.
CAIE Further Paper 2 2022 November Q7
10 marks Challenging +1.8
7
  1. State the sum of the series \(1 + \mathrm { w } + \mathrm { w } ^ { 2 } + \mathrm { w } ^ { 3 } + \ldots + \mathrm { w } ^ { \mathrm { n } - 1 }\), for \(w \neq 1\).
  2. Show that \(( 1 + i \tan \theta ) ^ { k } = \sec ^ { k } \theta ( \cos k \theta + i \sin k \theta )\), where \(\theta\) is not an integer multiple of \(\frac { 1 } { 2 } \pi\).
  3. By considering \(\sum _ { \mathrm { k } = 0 } ^ { \mathrm { n } - 1 } ( 1 + \mathrm { i } \tan \theta ) ^ { \mathrm { k } }\), show that $$\sum _ { k = 0 } ^ { n - 1 } \sec ^ { k } \theta \sin k \theta = \cot \theta \left( 1 - \sec ^ { n } \theta \cos n \theta \right)$$ provided \(\theta\) is not an integer multiple of \(\frac { 1 } { 2 } \pi\).
  4. Hence find \(\sum _ { k = 0 } ^ { 6 m - 1 } 2 ^ { k } \sin \left( \frac { 1 } { 3 } k \pi \right)\) in terms of \(m\).