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CAIE Further Paper 2 2020 June Q1
1 Find the solution of the differential equation $$\frac { d y } { d x } + 5 y = e ^ { - 7 x }$$ for which \(y = 0\) when \(x = 0\). Give your answer in the form \(y = f ( x )\).
CAIE Further Paper 2 2020 June Q2
2 It is given that \(y = 2 ^ { x }\).
  1. By differentiating \(\ln y\) with respect to \(x\), show that \(\frac { \mathrm { dy } } { \mathrm { dx } } = 2 ^ { \mathrm { x } } \ln 2\).
  2. Write down \(\frac { d ^ { 2 } y } { d x ^ { 2 } }\).
  3. Hence find the first three terms in the Maclaurin's series for \(2 ^ { X }\).
CAIE Further Paper 2 2020 June Q3
3
  1. Find the roots of the equation \(z ^ { 3 } = - 1 - \mathrm { i }\), giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(0 \leqslant \theta < 2 \pi\).
    Let \(\mathbf { w } = \mathbf { z } _ { 1 } ^ { 3 \mathrm { k } } + \mathbf { z } _ { 2 } ^ { 3 \mathrm { k } } + \mathbf { z } _ { 3 } ^ { 3 \mathrm { k } }\), where \(k\) is a positive integer and \(\mathrm { z } _ { 1 } , \mathrm { z } _ { 2 } , \mathrm { z } _ { 3 }\) are the roots of \(\mathrm { z } ^ { 3 } = - 1 - \mathrm { i }\).
  2. Express \(w\) in the form \(R \mathrm { e } ^ { \mathrm { i } \alpha }\), where \(R > 0\), giving \(R\) and \(\alpha\) in terms of \(k\).
    \includegraphics[max width=\textwidth, alt={}, center]{20e14db3-0eb0-4954-91cf-027e16f8bf14-06_889_824_267_616} The diagram shows the curve with equation \(\mathrm { y } = \mathrm { x } ^ { 2 }\) for \(0 \leqslant x \leqslant 1\), together with a set of \(n\) rectangles of width \(\frac { 1 } { n }\).
CAIE Further Paper 2 2020 June Q5
5 The curves \(C _ { 1 } : y = \cosh x\) and \(C _ { 2 } : y = \sinh 2 x\) intersect at the point where \(x = a\).
  1. Find the exact value of \(a\), giving your answer in logarithmic form.
  2. Sketch \(C _ { 1 }\) and \(C _ { 2 }\) on the same diagram.
  3. Find the exact value of the length of the arc of \(C _ { 1 }\) from \(x = 0\) to \(\mathrm { x } = \mathrm { a }\).
CAIE Further Paper 2 2020 June Q6
6 The integral \(\mathrm { I } _ { \mathrm { n } }\), where \(n\) is an integer, is defined by \(\mathrm { I } _ { \mathrm { n } } = \int _ { 0 } ^ { \frac { 1 } { 2 } } \left( 1 - \mathrm { x } ^ { 2 } \right) ^ { - \frac { 1 } { 2 } \mathrm { n } } \mathrm { dx }\).
  1. Find the exact value of \(I _ { 1 }\).
  2. By considering \(\frac { \mathrm { d } } { \mathrm { dx } } \left( \mathrm { x } \left( 1 - \mathrm { x } ^ { 2 } \right) ^ { - \frac { 1 } { 2 } \mathrm { n } } \right)\), or otherwise, show that $$\mathrm { nl } _ { \mathrm { n } + 2 } = 2 ^ { \mathrm { n } - 1 } 3 ^ { - \frac { 1 } { 2 } \mathrm { n } } + ( \mathrm { n } - 1 ) \mathrm { I } _ { \mathrm { n } } .$$
  3. Find the exact value of \(I _ { 5 }\) giving the answer in the form \(k \sqrt { 3 }\), where \(k\) is a rational number to be determined.
    \includegraphics[max width=\textwidth, alt={}, center]{20e14db3-0eb0-4954-91cf-027e16f8bf14-11_78_1576_336_321}
CAIE Further Paper 2 2020 June Q7
7 It is given that \(x = t ^ { 3 } y\) and $$t ^ { 3 } \frac { d ^ { 2 } y } { d t ^ { 2 } } + \left( 4 t ^ { 3 } + 6 t ^ { 2 } \right) \frac { d y } { d t } + \left( 13 t ^ { 3 } + 12 t ^ { 2 } + 6 t \right) y = 61 e ^ { \frac { 1 } { 2 } t }$$
  1. Show that $$\frac { d ^ { 2 } x } { d t ^ { 2 } } + 4 \frac { d x } { d t } + 13 x = 61 e ^ { \frac { 1 } { 2 } t }$$
  2. Find the general solution for \(y\) in terms of \(t\).
CAIE Further Paper 2 2020 June Q8
8
  1. Find the values of \(a\) for which the system of equations $$\begin{aligned} 3 x + y + z & = 0
    a x + 6 y - z & = 0
    a y - 2 z & = 0 \end{aligned}$$ does not have a unique solution.
    The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r r } 3 & 1 & 1
    0 & 6 & - 1
    0 & 0 & - 2 \end{array} \right) .$$
  2. Use the characteristic equation of \(\mathbf { A }\) to find the inverse of \(\mathbf { A } ^ { 2 }\).
  3. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { 5 } = \mathbf { P D P } ^ { - 1 }\).
    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 June Q3
3
  1. Find the roots of the equation \(z ^ { 3 } = - 1 - \mathrm { i }\), giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(0 \leqslant \theta < 2 \pi\).
    Let \(\mathbf { w } = \mathbf { z } _ { 1 } ^ { 3 \mathrm { k } } + \mathbf { z } _ { 2 } ^ { 3 \mathrm { k } } + \mathbf { z } _ { 3 } ^ { 3 \mathrm { k } }\), where \(k\) is a positive integer and \(\mathrm { z } _ { 1 } , \mathrm { z } _ { 2 } , \mathrm { z } _ { 3 }\) are the roots of \(\mathrm { z } ^ { 3 } = - 1 - \mathrm { i }\).
  2. Express \(w\) in the form \(R \mathrm { e } ^ { \mathrm { i } \alpha }\), where \(R > 0\), giving \(R\) and \(\alpha\) in terms of \(k\).
    \includegraphics[max width=\textwidth, alt={}, center]{1de67949-6262-4ade-b986-02b6563ae404-06_889_824_267_616} The diagram shows the curve with equation \(\mathrm { y } = \mathrm { x } ^ { 2 }\) for \(0 \leqslant x \leqslant 1\), together with a set of \(n\) rectangles of width \(\frac { 1 } { n }\).
CAIE Further Paper 2 2020 June Q6
6 The integral \(\mathrm { I } _ { \mathrm { n } }\), where \(n\) is an integer, is defined by \(\mathrm { I } _ { \mathrm { n } } = \int _ { 0 } ^ { \frac { 1 } { 2 } } \left( 1 - \mathrm { x } ^ { 2 } \right) ^ { - \frac { 1 } { 2 } \mathrm { n } } \mathrm { dx }\).
  1. Find the exact value of \(I _ { 1 }\).
  2. By considering \(\frac { \mathrm { d } } { \mathrm { dx } } \left( \mathrm { x } \left( 1 - \mathrm { x } ^ { 2 } \right) ^ { - \frac { 1 } { 2 } \mathrm { n } } \right)\), or otherwise, show that $$\mathrm { nl } _ { \mathrm { n } + 2 } = 2 ^ { \mathrm { n } - 1 } 3 ^ { - \frac { 1 } { 2 } \mathrm { n } } + ( \mathrm { n } - 1 ) \mathrm { I } _ { \mathrm { n } } .$$
  3. Find the exact value of \(I _ { 5 }\) giving the answer in the form \(k \sqrt { 3 }\), where \(k\) is a rational number to be determined.
    \includegraphics[max width=\textwidth, alt={}, center]{1de67949-6262-4ade-b986-02b6563ae404-11_78_1576_336_321}
CAIE Further Paper 2 2020 June Q1
1 Find the general solution of the differential equation $$\frac { d ^ { 2 } x } { d t ^ { 2 } } - 8 \frac { d x } { d t } - 9 x = 9 e ^ { 8 t }$$
CAIE Further Paper 2 2020 June Q2
2 Let \(\mathrm { I } _ { \mathrm { n } } = \int _ { 0 } ^ { 1 } ( 1 + 3 \mathrm { x } ) ^ { \mathrm { n } } \mathrm { e } ^ { - 3 \mathrm { x } } \mathrm { dx }\), where \(n\) is an integer.
  1. Show that \(3 \mathrm { I } _ { \mathrm { n } } = 1 - 4 ^ { \mathrm { n } } \mathrm { e } ^ { - 3 } + 3 \mathrm { nl } _ { \mathrm { n } - 1 }\).
  2. Find the exact value of \(I _ { 2 }\).
CAIE Further Paper 2 2020 June Q3
3 The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r r } 5 & - 1 & 7
0 & 6 & 0
7 & 7 & 5 \end{array} \right) .$$
  1. Find the eigenvalues of \(\mathbf { A }\).
  2. Use the characteristic equation of \(\mathbf { A }\) to find \(\mathbf { A } ^ { - 1 }\).
    \includegraphics[max width=\textwidth, alt={}, center]{671d8d26-8c9b-40d5-bc59-97c3ccdcadf4-06_568_1614_294_262} The diagram shows the curve with equation \(\mathrm { y } = \ln \mathrm { x }\) for \(x \geqslant 1\), together with a set of ( \(N - 1\) ) rectangles of unit width.
  3. By considering the sum of the areas of these rectangles, show that $$\ln N ! > N \ln N - N + 1 .$$
  4. Use a similar method to find, in terms of \(N\), an upper bound for \(\operatorname { In } N\) !.
CAIE Further Paper 2 2020 June Q5
5 The curve \(C\) has parametric equations $$\mathrm { x } = \frac { 1 } { 2 } \mathrm { t } ^ { 2 } - \ln \mathrm { t } , \quad \mathrm { y } = 2 \mathrm { t } + 1 , \quad \text { for } \frac { 1 } { 2 } \leqslant t \leqslant 2$$
  1. Find the exact length of \(C\).
  2. Find \(\frac { \mathrm { d } ^ { 2 } \mathrm { y } } { \mathrm { dx } ^ { 2 } }\) in terms of \(t\), simplifying your answer.
CAIE Further Paper 2 2020 June Q6
6
  1. Starting from the definitions of tanh and sech in terms of exponentials, prove that $$1 - \tanh ^ { 2 } \theta = \operatorname { sech } ^ { 2 } \theta$$ \includegraphics[max width=\textwidth, alt={}, center]{671d8d26-8c9b-40d5-bc59-97c3ccdcadf4-10_72_1552_374_347}
    \includegraphics[max width=\textwidth, alt={}]{671d8d26-8c9b-40d5-bc59-97c3ccdcadf4-10_67_1569_466_328} ......................................................................................................................................... ........................................................................................................................................
    \includegraphics[max width=\textwidth, alt={}, center]{671d8d26-8c9b-40d5-bc59-97c3ccdcadf4-10_72_1573_735_324}
    \includegraphics[max width=\textwidth, alt={}]{671d8d26-8c9b-40d5-bc59-97c3ccdcadf4-10_72_1573_826_324} .......................................................................................................................................... . ........................................................................................................................................ . The variables \(x\) and \(y\) are such that \(\tanh y = \cos \left( x + \frac { 1 } { 4 } \pi \right)\), for \(- \frac { 1 } { 4 } \pi < x < \frac { 3 } { 4 } \pi\).
  2. By differentiating the equation \(\tanh y = \cos \left( x + \frac { 1 } { 4 } \pi \right)\) with respect to \(x\), show that $$\frac { \mathrm { dy } } { \mathrm { dx } } = - \operatorname { cosec } \left( \mathrm { x } + \frac { 1 } { 4 } \pi \right)$$
  3. Hence find the first three terms in the Maclaurin's series for \(\tanh ^ { - 1 } \left( \cos \left( x + \frac { 1 } { 4 } \pi \right) \right)\) in the form \(\frac { 1 } { 2 } \ln a + b x + c x ^ { 2 }\), giving the exact values of the constants \(a , b\) and \(c\).
CAIE Further Paper 2 2020 June Q7
7
  1. Show that an appropriate integrating factor for $$\left( x ^ { 2 } + 1 \right) \frac { d y } { d x } + y \sqrt { x ^ { 2 } + 1 } = x ^ { 2 } - x \sqrt { x ^ { 2 } + 1 }$$ is \(x + \sqrt { x ^ { 2 } + 1 }\).
  2. Hence find the solution of the differential equation $$\left( x ^ { 2 } + 1 \right) \frac { d y } { d x } + y \sqrt { x ^ { 2 } + 1 } = x ^ { 2 } - x \sqrt { x ^ { 2 } + 1 }$$ for which \(y = \ln 2\) when \(x = 0\). Give your answer in the form \(y = f ( x )\).
CAIE Further Paper 2 2020 June Q8
8
  1. Use de Moivre's theorem to show that \(\sin ^ { 6 } \theta = - \frac { 1 } { 32 } ( \cos 6 \theta - 6 \cos 4 \theta + 15 \cos 2 \theta - 10 )\).
    It is given that \(\cos ^ { 6 } \theta = \frac { 1 } { 32 } ( \cos 6 \theta + 6 \cos 4 \theta + 15 \cos 2 \theta + 10 )\).
  2. Find the exact value of \(\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \left( \cos ^ { 6 } \left( \frac { 1 } { 4 } x \right) + \sin ^ { 6 } \left( \frac { 1 } { 4 } x \right) \right) \mathrm { d } x\).
  3. Express each root of the equation \(16 c ^ { 6 } + 16 \left( 1 - c ^ { 2 } \right) ^ { 3 } - 13 = 0\) in the form \(\cos k \pi\), where \(k\) is a rational number.
    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 June Q1
1
  1. Given that \(a\) is an integer, show that the system of equations $$\begin{aligned} a x + 3 y + z & = 14
    2 x + y + 3 z & = 0
    - x + 2 y - 5 z & = 17 \end{aligned}$$ has a unique solution and interpret this situation geometrically.
  2. Find the value of \(a\) for which \(x = 1 , y = 4 , z = - 2\) is the solution to the system of equations in part (a).
CAIE Further Paper 2 2021 June Q2
2 The variables \(x\) and \(y\) are related by the differential equation $$\frac { d ^ { 2 } y } { d x ^ { 2 } } + 3 \frac { d y } { d x } + 2 y = 2 x + 1$$
  1. Find the general solution for \(y\) in terms of \(x\).
  2. State an approximate solution for large positive values of \(x\).
CAIE Further Paper 2 2021 June Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{fa2213b3-480c-44cb-8ba0-ebd2b94d3d90-04_851_805_251_616} The diagram shows the curve with equation \(\mathrm { y } = \mathrm { x } ^ { 3 }\) for \(0 \leqslant x \leqslant 1\), together with a set of \(n\) rectangles of width \(\frac { 1 } { n }\).
  1. By considering the sum of the areas of these rectangles, show that \(\int _ { 0 } ^ { 1 } x ^ { 3 } d x < U _ { n }\), where $$\mathrm { U } _ { \mathrm { n } } = \left( \frac { \mathrm { n } + 1 } { 2 \mathrm { n } } \right) ^ { 2 }$$
  2. Use a similar method to find, in terms of \(n\), a lower bound \(L _ { n }\) for \(\int _ { 0 } ^ { 1 } x ^ { 3 } d x\).
  3. Find the least value of \(n\) such that \(\mathrm { U } _ { \mathrm { n } } - \mathrm { L } _ { \mathrm { n } } < 10 ^ { - 3 }\).
CAIE Further Paper 2 2021 June Q4
4 Find the solution of the differential equation $$\sin \theta \frac { d y } { d \theta } + y = \tan \frac { 1 } { 2 } \theta$$ where \(0 < \theta < \pi\), given that \(y = 1\) when \(\theta = \frac { 1 } { 2 } \pi\). Give your answer in the form \(y = \mathrm { f } ( \theta )\). [You may use without proof the result that \(\int \operatorname { cosec } \theta d \theta = \ln \tan \frac { 1 } { 2 } \theta\).]
CAIE Further Paper 2 2021 June Q5
5
  1. State the sum of the series \(z + z ^ { 2 } + z ^ { 3 } + \ldots + z ^ { n }\), for \(z \neq 1\).
  2. Given that \(z\) is an \(n\)th root of unity and \(z \neq 1\), deduce that \(1 + z + z ^ { 2 } + \ldots + z ^ { n - 1 } = 0\).
  3. Given instead that \(z = \frac { 1 } { 3 } ( \cos \theta + \mathrm { i } \sin \theta )\), use de Moivre's theorem to show that $$\sum _ { m = 1 } ^ { \infty } 3 ^ { - m } \cos m \theta = \frac { 3 \cos \theta - 1 } { 10 - 6 \cos \theta }$$
CAIE Further Paper 2 2021 June Q6
6 The matrix \(\mathbf { A }\) is given by $$A = \left( \begin{array} { r r r } 5 & - \frac { 22 } { 3 } & 8
0 & - 6 & 0
0 & 0 & 1 \end{array} \right)$$
  1. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { 2 } = \mathbf { P D P } ^ { - 1 }\).
  2. Use the characteristic equation of \(\mathbf { A }\) to find \(\mathbf { A } ^ { 3 }\).
CAIE Further Paper 2 2021 June Q7
7
  1. It is given that \(\mathrm { y } = \operatorname { sech } ^ { - 1 } \left( \mathrm { x } + \frac { 1 } { 2 } \right)\).
    Express cosh \(y\) in terms of \(x\) and hence show that \(\sinh y \frac { d y } { d x } = - \frac { 1 } { \left( x + \frac { 1 } { 2 } \right) ^ { 2 } }\).
  2. Find the first three terms in the Maclaurin's series for \(\operatorname { sech } ^ { - 1 } \left( x + \frac { 1 } { 2 } \right)\) in the form $$\ln a + b x + c x ^ { 2 }$$ where \(a\), \(b\) and \(c\) are constants to be determined.
CAIE Further Paper 2 2021 June Q8
8 The curve \(C\) has parametric equations $$\mathbf { x } = 2 \cosh t , \quad \mathbf { y } = \frac { 3 } { 2 } \mathbf { t } - \frac { 1 } { 4 } \sinh 2 \mathbf { t } , \text { for } 0 \leqslant t \leqslant 1$$
  1. Find \(\frac { \mathrm { dx } } { \mathrm { dt } }\) and show that \(\frac { \mathrm { dy } } { \mathrm { dt } } = 1 - \sinh ^ { 2 } \mathrm { t }\).
    The area of the surface generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis is denoted by \(A\).
    1. Show that \(\mathrm { A } = \pi \int _ { 0 } ^ { 1 } \left( \frac { 3 } { 2 } \mathrm { t } - \frac { 1 } { 4 } \sinh 2 \mathrm { t } \right) ( 1 + \cosh 2 \mathrm { t } ) \mathrm { dt }\).
    2. Hence find \(A\) in terms of \(\pi , \sinh 2\) and \(\cosh 2\).
      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 June Q3
3
\includegraphics[max width=\textwidth, alt={}, center]{fd247a71-4680-45d8-89d2-fef17ed3a5e9-04_851_805_251_616} The diagram shows the curve with equation \(\mathrm { y } = \mathrm { x } ^ { 3 }\) for \(0 \leqslant x \leqslant 1\), together with a set of \(n\) rectangles of width \(\frac { 1 } { n }\).
  1. By considering the sum of the areas of these rectangles, show that \(\int _ { 0 } ^ { 1 } x ^ { 3 } d x < U _ { n }\), where $$\mathrm { U } _ { \mathrm { n } } = \left( \frac { \mathrm { n } + 1 } { 2 \mathrm { n } } \right) ^ { 2 }$$
  2. Use a similar method to find, in terms of \(n\), a lower bound \(L _ { n }\) for \(\int _ { 0 } ^ { 1 } x ^ { 3 } d x\).
  3. Find the least value of \(n\) such that \(\mathrm { U } _ { \mathrm { n } } - \mathrm { L } _ { \mathrm { n } } < 10 ^ { - 3 }\).