Questions — CAIE Further Paper 2 (186 questions)

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CAIE Further Paper 2 2020 November Q8
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
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
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
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
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
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
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
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
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
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 Q1
1 It is given that \(y = \sinh \left( x ^ { 2 } \right) + \cosh \left( x ^ { 2 } \right)\).
  1. Use standard results from the list of formulae (MF19) to find the Maclaurin's series for \(y\) in terms of \(x\) up to and including the term in \(x ^ { 4 }\).
  2. Deduce the value of \(\frac { \mathrm { d } ^ { 4 } \mathrm { y } } { \mathrm { dx } ^ { 4 } }\) when \(x = 0\).
  3. Use your answer to part (a) to find an approximation to \(\int _ { 0 } ^ { \frac { 1 } { 2 } } \mathrm { ydx }\), giving your answer as a rational
    fraction in its lowest terms. fraction in its lowest terms.
CAIE Further Paper 2 2021 November Q2
2 Find the solution of the differential equation $$\frac { d y } { d x } + \frac { 4 x ^ { 3 } y } { x ^ { 4 } + 5 } = 6 x$$ for which \(y = 1\) when \(x = 1\). Give your answer in the form \(y = f ( x )\).
\includegraphics[max width=\textwidth, alt={}, center]{a921c01f-4d8e-47cf-9f34-d7d7bf9c9fdd-04_867_812_278_621} The diagram shows the curve with equation \(\mathrm { y } = 1 - \mathrm { x } ^ { 2 }\) 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 the rectangles, show that $$\int _ { 0 } ^ { 1 } \left( 1 - x ^ { 2 } \right) d x < \frac { 4 n ^ { 2 } + 3 n - 1 } { 6 n ^ { 2 } }$$
  2. Use a similar method to find, in terms of \(n\), a lower bound for \(\int _ { 0 } ^ { 1 } \left( 1 - x ^ { 2 } \right) \mathrm { dx }\).
CAIE Further Paper 2 2021 November Q4
4
  1. Write down all the roots of the equation \(x ^ { 5 } - 1 = 0\).
  2. Use de Moivre's theorem to show that \(\cos 4 \theta = 8 \cos ^ { 4 } \theta - 8 \cos ^ { 2 } \theta + 1\).
  3. Use the results of parts (a) and (b) to express each real root of the equation $$8 x ^ { 9 } - 8 x ^ { 7 } + x ^ { 5 } - 8 x ^ { 4 } + 8 x ^ { 2 } - 1 = 0$$ in the form \(\cos k \pi\), where \(k\) is a rational number.
CAIE Further Paper 2 2021 November Q5
4 marks
5 The curve \(C\) has parametric equations $$x = 3 t + 2 t ^ { - 1 } + a t ^ { 3 } , \quad y = 4 t - \frac { 3 } { 2 } t ^ { - 1 } + b t ^ { 3 } , \quad \text { for } 1 \leqslant t \leqslant 2$$ where \(a\) and \(b\) are constants.
  1. It is given that \(a = \frac { 2 } { 3 }\) and \(b = - \frac { 1 } { 2 }\). Show that \(\left( \frac { d x } { d t } \right) ^ { 2 } + \left( \frac { d y } { d t } \right) ^ { 2 } = \frac { 25 } { 4 } \left( t ^ { 2 } + t ^ { - 2 } \right) ^ { 2 }\) and find the exact length of \(C\).
  2. It is given instead that \(\mathrm { a } = \mathrm { b } = 0\). Find the value of \(\frac { d ^ { 2 } y } { d x ^ { 2 } }\) when \(t = 1\).
    [0pt] [4]
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    \includegraphics[max width=\textwidth, alt={}, center]{a921c01f-4d8e-47cf-9f34-d7d7bf9c9fdd-09_62_1570_475_324}
    \includegraphics[max width=\textwidth, alt={}, center]{a921c01f-4d8e-47cf-9f34-d7d7bf9c9fdd-09_63_1570_566_324}
    \includegraphics[max width=\textwidth, alt={}, center]{a921c01f-4d8e-47cf-9f34-d7d7bf9c9fdd-09_61_1570_657_324}
    \includegraphics[max width=\textwidth, alt={}, center]{a921c01f-4d8e-47cf-9f34-d7d7bf9c9fdd-09_69_1570_740_324}
    ........................................................................................................................................ .
    \includegraphics[max width=\textwidth, alt={}, center]{a921c01f-4d8e-47cf-9f34-d7d7bf9c9fdd-09_70_1570_920_324}
    \includegraphics[max width=\textwidth, alt={}, center]{a921c01f-4d8e-47cf-9f34-d7d7bf9c9fdd-09_72_1572_1009_322}
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CAIE Further Paper 2 2021 November Q6
6 The matrix \(\mathbf { P }\) is given by $$\mathbf { P } = \left( \begin{array} { r r r } 1 & 6 & 6
0 & 2 & 6
0 & 0 & - 3 \end{array} \right) .$$
  1. Use the characteristic equation of \(\mathbf { P }\) to find \(\mathbf { P } ^ { - 1 }\).
  2. Find the matrix \(\mathbf { A }\) such that $$\mathbf { P } ^ { - 1 } \mathbf { A } \mathbf { P } = \left( \begin{array} { l l l } 4 & 0 & 0
    0 & 5 & 0
    0 & 0 & 6 \end{array} \right) .$$
  3. State the eigenvalues and corresponding eigenvectors of \(\mathbf { A } ^ { 3 }\).
CAIE Further Paper 2 2021 November Q7
7 It is given that \(y = x ^ { 2 } w\) and $$x ^ { 2 } \frac { d ^ { 2 } w } { d x ^ { 2 } } + 4 x ( x + 1 ) \frac { d w } { d x } + \left( 5 x ^ { 2 } + 8 x + 2 \right) w = 5 x ^ { 2 } + 4 x + 2$$
  1. Show that $$\frac { d ^ { 2 } y } { d x ^ { 2 } } + 4 \frac { d y } { d x } + 5 y = 5 x ^ { 2 } + 4 x + 2$$
  2. Find the general solution for \(w\) in terms of \(x\).
CAIE Further Paper 2 2021 November Q8
8
  1. Starting from the definitions of tanh and sech in terms of exponentials, prove that $$1 - \tanh ^ { 2 } x = \operatorname { sech } ^ { 2 } x$$
  2. Using the substitution \(\mathrm { u } = \tanh \mathrm { x }\), or otherwise, find \(\int \operatorname { sech } ^ { 2 } x \tanh ^ { 2 } x \mathrm {~d} x\).
    It is given that, for \(n \geqslant 0 , \mathrm { I } _ { \mathrm { n } } = \int _ { 0 } ^ { \ln 3 } \operatorname { sech } ^ { \mathrm { n } } x \tanh ^ { 2 } x \mathrm { dx }\).
  3. Show that, for \(n \geqslant 2\), $$( n + 1 ) I _ { n } = \left( \frac { 4 } { 5 } \right) ^ { 3 } \left( \frac { 3 } { 5 } \right) ^ { n - 2 } + ( n - 2 ) I _ { n - 2 }$$ [You may use the result that \(\frac { \mathrm { d } } { \mathrm { d } x } ( \operatorname { sech } x ) = - \tanh x \operatorname { sech } x\).]
  4. Find the value of \(I _ { 4 }\).
    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
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
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
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
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
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
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
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
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 \(m\).