Questions — CAIE Further Paper 2 (195 questions)

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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 \(m\).
CAIE Further Paper 2 2022 November Q8
14 marks Challenging +1.2
8
  1. Use the substitution \(u = 1 - ( \theta - 1 ) ^ { 2 }\) to find $$\int \frac { \theta - 1 } { \sqrt { 1 - ( \theta - 1 ) ^ { 2 } } } \mathrm {~d} \theta$$
  2. Find the solution of the differential equation $$\theta \frac { d y } { d \theta } - y = \theta ^ { 2 } \sin ^ { - 1 } ( \theta - 1 ) ,$$ where \(0 < \theta < 2\), given that \(y = 1\) when \(\theta = 1\). Give your answer in the form \(y = \mathrm { f } ( \theta )\).
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE Further Paper 2 2022 November Q1
5 marks Standard +0.3
1
  1. Find the set of values of \(k\) for which the system of equations $$\begin{aligned} x + 2 y + 3 z & = 1 \\ k x + 4 y + 6 z & = 0 \\ 7 x + 8 y + 9 z & = 3 \end{aligned}$$ has a unique solution.
  2. Interpret the situation geometrically in the case where the system of equations does not have a unique solution.
CAIE Further Paper 2 2022 November Q2
7 marks Standard +0.3
2 A curve has equation $$( x + 1 ) y + y ^ { 2 } = 2$$
  1. Show that \(\frac { \mathrm { dy } } { \mathrm { dx } } = - \frac { 2 } { 3 }\) at the point \(( 0 , - 2 )\).
  2. Find the value of \(\frac { d ^ { 2 } y } { d x ^ { 2 } }\) at the point \(( 0 , - 2 )\).
CAIE Further Paper 2 2022 November Q3
8 marks Challenging +1.2
3
  1. A curve has equation \(\mathrm { y } = \mathrm { e } ^ { \mathrm { x } } + \frac { 1 } { 4 } \mathrm { e } ^ { - \mathrm { x } }\), for \(0 \leqslant x \leqslant 1\). 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.
  2. Using standard results from the list of formulae (MF19), or otherwise, find the Maclaurin's series for \(\mathrm { e } ^ { x } + \frac { 1 } { 4 } \mathrm { e } ^ { - x }\) up to and including the term in \(x ^ { 2 }\).
CAIE Further Paper 2 2022 November Q4
9 marks Standard +0.8
4 Find the solution of the differential equation $$\left( 4 t ^ { 2 } - 1 \right) \frac { d x } { d t } + 4 x = 4 t ^ { 2 } - 1$$ for which \(x = 3\) when \(t = 1\). Give your answer in the form \(\mathrm { x } = \mathrm { f } ( \mathrm { t } )\).
CAIE Further Paper 2 2022 November Q5
10 marks Challenging +1.2
5
  1. Write down the fourth roots of unity.
  2. Use de Moivre's theorem to show that $$\cos 4 \theta = 8 \cos ^ { 4 } \theta - 8 \cos ^ { 2 } \theta + 1$$
  3. Hence obtain the real roots of the equation $$16 \left( 8 x ^ { 4 } - 8 x ^ { 2 } + 1 \right) ^ { 4 } - 9 = 0$$ in the form \(\cos ( q \pi )\), where \(q\) is a rational number.
CAIE Further Paper 2 2022 November Q6
10 marks Challenging +1.2
6 \includegraphics[max width=\textwidth, alt={}, center]{323ac7a5-4690-441d-87fc-325a393098fa-10_585_1349_258_358} The diagram shows the curve \(\mathrm { y } = \frac { 1 } { \sqrt { \mathrm { x } ^ { 2 } + 2 \mathrm { x } } }\) for \(x > 0\), together with a set of \(( n - 1 )\) rectangles of unit
width. By considering the sum of the areas of these rectangles, show that $$\sum _ { r = 1 } ^ { n } \frac { 1 } { \sqrt { r ^ { 2 } + 2 r } } < \ln \left( n + 1 + \sqrt { n ^ { 2 } + 2 n } \right) + \frac { 1 } { 3 } \sqrt { 3 } - \ln ( 2 + \sqrt { 3 } )$$
CAIE Further Paper 2 2022 November Q7
12 marks Challenging +1.2
7
  1. It is given that \(\lambda\) is an eigenvalue of the non-singular square matrix \(\mathbf { A }\), with corresponding eigenvector \(\mathbf { e }\). Show that \(\lambda ^ { - 1 }\) is an eigenvalue of \(\mathbf { A } ^ { - 1 }\) for which \(\mathbf { e }\) is a corresponding eigenvector.
    The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r r } 2 & 0 & 3 \\ 15 & - 4 & 3 \\ 3 & 0 & 2 \end{array} \right)$$
  2. Given that - 1 is an eigenvalue of \(\mathbf { A }\), find a corresponding eigenvector.
  3. It is also given that \(\left( \begin{array} { l } 0 \\ 1 \\ 0 \end{array} \right)\) and \(\left( \begin{array} { l } 1 \\ 2 \\ 1 \end{array} \right)\) are eigenvectors of \(\mathbf { A }\). Find the corresponding eigenvalues.
  4. Hence find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { - 1 } = \mathbf { P D P } ^ { - 1 }\).
  5. Use the characteristic equation of \(\mathbf { A }\) to show that \(\mathbf { A } ^ { - 1 } = p \mathbf { A } ^ { 2 } + q l\), where \(p\) and \(q\) are rational numbers to be determined.
CAIE Further Paper 2 2022 November Q8
14 marks Challenging +1.8
8 It is given that \(\mathrm { y } = \operatorname { coshu }\), where \(u > 0\), and $$\sqrt { \cosh ^ { 2 } u - 1 } \left( \frac { d ^ { 2 } u } { d x ^ { 2 } } + \frac { d u } { d x } \right) + \cosh u \left( \frac { d u } { d x } \right) ^ { 2 } - 2 \cosh u = 4 e ^ { - x }$$
  1. Show that $$\frac { d ^ { 2 } y } { d x ^ { 2 } } + \frac { d y } { d x } - 2 y = 4 e ^ { - x }$$
  2. Find \(u\) in terms of \(x\), given that, when \(x = 0 , u = \ln 3\) and \(\frac { d u } { d x } = 3\).
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE Further Paper 2 2023 November Q1
4 marks Standard +0.8
1 Show that the system of equations $$\begin{aligned} 14 x - 4 y + 6 z & = 5 \\ x + y + k z & = 3 \\ - 21 x + 6 y - 9 z & = 14 \end{aligned}$$ where \(k\) is a constant, does not have a unique solution and interpret this situation geometrically.
CAIE Further Paper 2 2023 November Q2
5 marks Standard +0.8
2 Find the roots of the equation \(( z + 5 i ) ^ { 3 } = 4 + 4 \sqrt { 3 } i\), giving your answers in the form \(r \cos \theta + i ( r \sin \theta - 5 )\), where \(r > 0\) and \(0 < \theta < 2 \pi\).
CAIE Further Paper 2 2023 November Q3
6 marks Challenging +1.2
3 Find the first three terms in the Maclaurin's series for \(\tanh ^ { - 1 } \left( \frac { 1 } { 2 } e ^ { x } \right)\) 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 2023 November Q4
10 marks Standard +0.8
4 Find the particular solution of the differential equation $$\frac { d ^ { 2 } y } { d x ^ { 2 } } + 2 \frac { d y } { d x } + 3 y = 27 x ^ { 2 }$$ given that, when \(x = 0 , y = 2\) and \(\frac { \mathrm { dy } } { \mathrm { dx } } = - 8\).
CAIE Further Paper 2 2023 November Q5
10 marks Standard +0.8
5 The curve \(C\) has parametric equations $$\mathrm { x } = \frac { 2 } { 3 } \mathrm { t } ^ { \frac { 3 } { 2 } } - 2 \mathrm { t } ^ { \frac { 1 } { 2 } } , \quad \mathrm { y } = 2 \mathrm { t } + 5 , \quad \text { for } 0 < t \leqslant 3$$
  1. Find the exact length of \(C\).
  2. Find the set of values of \(t\) for which \(\frac { d ^ { 2 } y } { d x ^ { 2 } } > 0\).