Questions Further Paper 2 (305 questions)

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CAIE Further Paper 2 2020 Specimen Q3
8 marks Standard +0.8
3 Find the solution of the differential equation $$x \frac { \mathrm {~d} y } { \mathrm {~d} x } + 3 y = \frac { \sin x } { x }$$ for which \(y = 0\) when \(x = \frac { 1 } { 2 } \pi\). Give your answer in the form \(y = \mathrm { f } ( x )\).
CAIE Further Paper 2 2020 Specimen Q4
8 marks Challenging +1.2
4 \includegraphics[max width=\textwidth, alt={}, center]{b5503355-3952-47dc-91f4-80a674349b4a-06_538_949_269_557} The diagram shows the curve with equation \(y = \frac { 1 } { x ^ { 2 } }\) for \(x > 0\), together with a set of \(( n - 1 )\) rectangles of unit width.
  1. By considering the sum of the areas of these rectangles, show that $$\sum _ { r = 1 } ^ { n } \frac { 1 } { r ^ { 2 } } < \frac { 2 n - 1 } { n } .$$
  2. Use a similar method to find, in terms of \(n\), a lower bound for \(\sum _ { r = 1 } ^ { n } \frac { 1 } { r ^ { 2 } }\).
CAIE Further Paper 2 2020 Specimen Q5
10 marks Challenging +1.2
5 The curve \(C\) has parametric equations $$x = \mathrm { e } ^ { t } - 4 t + 3 , \quad y = 8 \mathrm { e } ^ { \frac { 1 } { 2 } t } , \quad \text { for } 0 \leqslant t \leqslant 2 .$$
  1. Find, in terms of e , the length of \(C\).
  2. Find, in terms of \(\pi\) and e , the area of the surface generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
CAIE Further Paper 2 2020 Specimen Q6
10 marks Challenging +1.8
6
  1. Using de Moivre's theorem, show that $$\tan 5 \theta = \frac { 5 \tan \theta - 10 \tan ^ { 3 } \theta + \tan ^ { 5 } \theta } { 1 - 10 \tan ^ { 2 } \theta + 5 \tan ^ { 4 } \theta } .$$
  2. Hence show that the equation \(x ^ { 2 } - 10 x + 5 = 0\) has roots \(\tan ^ { 2 } \left( \frac { 1 } { 5 } \pi \right)\) and \(\tan ^ { 2 } \left( \frac { 2 } { 5 } \pi \right)\).
CAIE Further Paper 2 2020 Specimen Q7
12 marks Challenging +1.8
7
  1. Starting from the definition of tanh in terms of exponentials, prove that \(\tanh ^ { - 1 } x = \frac { 1 } { 2 } \ln \left( \frac { 1 + x } { 1 - x } \right)\). [3]
  2. Given that \(y = \tanh ^ { - 1 } \left( \frac { 1 - x } { 2 + x } \right)\), show that \(( 2 x + 1 ) \frac { \mathrm { d } y } { \mathrm {~d} x } + 1 = 0\).
  3. Hence find the first three terms in the Maclaurin's series for \(\tanh ^ { - 1 } \left( \frac { 1 - x } { 2 + x } \right)\) in the form $$a \ln 3 + b x + c x ^ { 2 } ,$$ where \(a , b\) and \(c\) are constants to be determined.
CAIE Further Paper 2 2020 Specimen Q8
15 marks Standard +0.3
8
    1. Find the set of values of \(a\) for which the system of equations $$\begin{array} { r } x - 2 y - 2 z + 7 = 0
CAIE Further Paper 2 2024 November Q7
10 marks Challenging +1.8
7
  1. Show that an appropriate integrating factor for $$\sqrt { x ^ { 2 } + 16 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = x \sqrt { x ^ { 2 } + 16 }$$ is \(\frac { 1 } { 4 } x + \frac { 1 } { 4 } \sqrt { x ^ { 2 } + 16 }\) . \includegraphics[max width=\textwidth, alt={}, center]{374b91df-926d-4f7f-a1d3-a54c70e8ff0e-15_2723_33_99_22}
  2. Hence find the solution of the differential equation $$\sqrt { x ^ { 2 } + 16 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = x \sqrt { x ^ { 2 } + 16 }$$ for which \(y = 6\) when \(x = 3\).
CAIE Further Paper 2 2024 November Q8
14 marks Hard +2.3
8
  1. By considering the binomial expansion of \(\left( z + \frac { 1 } { z } \right) ^ { 7 }\) ,where \(z = \cos \theta + \mathrm { i } \sin \theta\) ,use de Moivre's theorem to show that $$\cos ^ { 7 } \theta = a \cos 7 \theta + b \cos 5 \theta + c \cos 3 \theta + d \cos \theta$$ where \(a , b , c\) and \(d\) are constants to be determined.
    Let \(I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \cos ^ { n } \theta \mathrm {~d} \theta\).
  2. Show that $$n I _ { n } = 2 ^ { - \frac { 1 } { 2 } n } + ( n - 1 ) I _ { n - 2 }$$ \includegraphics[max width=\textwidth, alt={}, center]{374b91df-926d-4f7f-a1d3-a54c70e8ff0e-18_2718_42_107_2007}
  3. Using the results given in parts (a) and (b), find the exact value of \(I _ { 9 }\).
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE Further Paper 2 2024 November Q7
10 marks Challenging +1.8
7
  1. Show that an appropriate integrating factor for $$\sqrt { x ^ { 2 } + 16 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = x \sqrt { x ^ { 2 } + 16 }$$ is \(\frac { 1 } { 4 } x + \frac { 1 } { 4 } \sqrt { x ^ { 2 } + 16 }\) . \includegraphics[max width=\textwidth, alt={}, center]{4af32247-c1f9-4c1f-bdf8-bafe17aca1dc-15_2723_33_99_22}
  2. Hence find the solution of the differential equation $$\sqrt { x ^ { 2 } + 16 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = x \sqrt { x ^ { 2 } + 16 }$$ for which \(y = 6\) when \(x = 3\).
CAIE Further Paper 2 2020 June Q4
8 marks Challenging +1.2
  1. By considering the sum of the areas of these rectangles, show that $$\int _ { 0 } ^ { 1 } x ^ { 2 } d x < \frac { 2 n ^ { 2 } + 3 n + 1 } { 6 n ^ { 2 } }$$
  2. Use a similar method to find, in terms of \(n\), a lower bound for \(\int _ { 0 } ^ { 1 } x ^ { 2 } \mathrm {~d} x\).
CAIE Further Paper 2 2020 November Q4
8 marks Challenging +1.2
  1. By considering the sum of the areas of the rectangles, show that $$\int _ { 0 } ^ { 1 } \left( 1 - x ^ { 3 } \right) d x \leqslant \frac { 3 n ^ { 2 } + 2 n - 1 } { 4 n ^ { 2 } }$$
  2. Use a similar method to find, in terms of \(n\), a lower bound for \(\int _ { 0 } ^ { 1 } \left( 1 - x ^ { 3 } \right) \mathrm { dx }\).
CAIE Further Paper 2 2021 November Q4
10 marks Challenging +1.8
  1. By considering the sum of the areas of these rectangles, show that $$\sum _ { r = 1 } ^ { N } \frac { \ln r } { r ^ { 2 } } < \frac { 2 + 3 \ln 2 } { 4 } - \frac { 1 + \ln N } { N }$$
  2. Use a similar method to find, in terms of \(N\), a lower bound for \(\sum _ { \mathrm { r } = 1 } ^ { \mathrm { N } } \frac { \ln \mathrm { r } } { \mathrm { r } ^ { 2 } }\).
AQA Further Paper 2 2021 June Q1
1 marks Easy -1.8
1 Which of the following matrices is singular?
Circle your answer. \(\left[ \begin{array} { l l } 1 & 0 \\ 0 & 1 \end{array} \right]\) \(\left[ \begin{array} { l l } 1 & 1 \\ 2 & 2 \end{array} \right]\) \(\left[ \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right]\) \(\left[ \begin{array} { c c } 1 & - 2 \\ 1 & 2 \end{array} \right]\)
AQA Further Paper 2 2021 June Q2
1 marks Easy -1.2
2 Find arg ( \(- 4 - 7 \mathrm { i }\) ) to the nearest degree.
Circle your answer.
[0pt] [1 mark] \(- 120 ^ { \circ }\) \(- 60 ^ { \circ }\) \(30 ^ { \circ }\) \(60 ^ { \circ }\)
AQA Further Paper 2 2021 June Q3
1 marks Easy -1.2
3 The line \(L\) has equation \(\mathbf { r } = \left[ \begin{array} { l } 3 \\ 2 \\ 0 \end{array} \right] + \lambda \left[ \begin{array} { c } - 1 \\ - 2 \\ 5 \end{array} \right]\) Which of the following lines is perpendicular to the line \(L\) ?
Tick \(( \checkmark )\) one box. $$\begin{aligned} & \mathbf { r } = \left[ \begin{array} { c } 2 \\ - 3 \\ 4 \end{array} \right] + \mu \left[ \begin{array} { c } 1 \\ 2 \\ - 5 \end{array} \right] \\ & \mathbf { r } = \left[ \begin{array} { l } 1 \\ 0 \\ 1 \end{array} \right] + \mu \left[ \begin{array} { c } 2 \\ - 3 \\ 1 \end{array} \right] \\ & \mathbf { r } = \left[ \begin{array} { l } 1 \\ 2 \\ 1 \end{array} \right] + \mu \left[ \begin{array} { l } 1 \\ 1 \\ 2 \end{array} \right] \\ & \mathbf { r } = \left[ \begin{array} { l } 0 \\ 3 \\ 2 \end{array} \right] + \mu \left[ \begin{array} { l } 4 \\ 3 \\ 2 \end{array} \right] \end{aligned}$$ □
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AQA Further Paper 2 2021 June Q4
7 marks Moderate -0.8
4
  1. Show that $$( r + 1 ) ^ { 2 } - r ^ { 2 } = 2 r + 1$$ 4
  2. Use the method of differences to show that $$\sum _ { r = 1 } ^ { n } ( 2 r + 1 ) = n ^ { 2 } + 2 n$$ 4
  3. Verify that using the formula for \(\sum _ { r = 1 } ^ { n } r\) gives the same result as that given in part (b).
    [0pt] [3 marks]
AQA Further Paper 2 2021 June Q5
5 marks Standard +0.3
5 The equation $$z ^ { 3 } + 2 z ^ { 2 } - 5 z - 3 = 0$$ has roots \(\alpha , \beta\) and \(\gamma\) Find a cubic equation with roots $$\frac { 1 } { 2 } \alpha - 1 , \frac { 1 } { 2 } \beta - 1 \text { and } \frac { 1 } { 2 } \gamma - 1$$
AQA Further Paper 2 2021 June Q6
8 marks Challenging +1.2
6 The ellipse \(E _ { 1 }\) has equation $$x ^ { 2 } + \frac { y ^ { 2 } } { 4 } = 1$$ \(E _ { 1 }\) is translated by the vector \(\left[ \begin{array} { l } 3 \\ 0 \end{array} \right]\) to give the ellipse \(E _ { 2 }\) 6
  1. Write down the equation of \(E _ { 2 }\) 6
  2. The ellipse \(E _ { 3 }\) has equation $$\frac { x ^ { 2 } } { 4 } + ( y - 3 ) ^ { 2 } = 1$$ Describe the transformation that maps \(E _ { 2 }\) to \(E _ { 3 }\) 6
  3. Each of the lines \(L _ { A }\) and \(L _ { B }\) is a tangent to both \(E _ { 2 }\) and \(E _ { 3 }\) \(L _ { A }\) is closer to the origin than \(L _ { B }\) \(E _ { 2 }\) and \(E _ { 3 }\) both lie between \(L _ { A }\) and \(L _ { B }\) Sketch and label \(E _ { 2 } , E _ { 3 } , L _ { A }\) and \(L _ { B }\) on the axes below.
    You do not need to show the values of the axis intercepts for \(L _ { A }\) and \(L _ { B }\) \includegraphics[max width=\textwidth, alt={}, center]{13abb93f-2fef-465c-980c-3b412de06618-09_1095_1095_726_475} 6
  4. Explain, without doing any calculations, why \(L _ { A }\) has an equation of the form $$x + y = c$$ where \(c\) is a constant.
AQA Further Paper 2 2021 June Q7
7 marks Challenging +1.8
7 \includegraphics[max width=\textwidth, alt={}, center]{13abb93f-2fef-465c-980c-3b412de06618-10_854_1027_264_520} The diagram shows a curve known as an astroid.
The curve has parametric equations $$\begin{aligned} & x = 4 \cos ^ { 3 } t \\ & y = 4 \sin ^ { 3 } t \\ & ( 0 \leq t < 2 \pi ) \end{aligned}$$ The section of the curve from \(t = 0\) to \(t = \frac { \pi } { 2 }\) is rotated through \(2 \pi\) radians about the \(x\)-axis. Show that the curved surface area of the shape formed is equal to \(\frac { b \pi } { c }\), where \(b\) and \(c\) are integers.
AQA Further Paper 2 2021 June Q8
6 marks Challenging +1.8
8 The complex number \(z\) satisfies the equations $$\left| z ^ { * } - 1 - 2 i \right| = | z - 3 |$$ and $$| z - a | = 3$$ where \(a\) is real.
Show that \(a\) must lie in the interval \([ 1 - s \sqrt { t } , 1 + s \sqrt { t } ]\), where \(s\) and \(t\) are prime numbers.
[0pt] [6 marks]
AQA Further Paper 2 2021 June Q9
14 marks Challenging +1.8
9
  1. The line \(L\) has polar equation $$r = \frac { 7 } { 4 } \sec \theta \quad \left( - \frac { \pi } { 2 } < \theta < \frac { \pi } { 2 } \right)$$ Show that \(L\) is perpendicular to the initial line.
    9
  2. The curve \(C\) has polar equation $$r = 3 + \cos \theta \quad ( - \pi < \theta \leq \pi )$$ Find the polar coordinates of the points of intersection of \(L\) and \(C\) Fully justify your answer.
    9
  3. The region \(R\) is the set of points such that
    and $$r > \frac { 7 } { 4 } \sec \theta$$ Find the exact area of \(R\) $$r < 3 + \cos \theta$$ Find the exact area of \(R\) [0pt] [7 marks]
AQA Further Paper 2 2021 June Q10
13 marks Standard +0.3
10 In a colony of seabirds, there are \(y\) birds at time \(t\) years. 10
  1. The rate of reduction in the number of birds due to birds dying or leaving the colony is proportional to the number of birds. In one year the reduction in the number of birds due to birds dying or leaving the colony is equal to \(16 \%\) of the number of birds at the start of the year. If no birds are born or join the colony, find the constant \(k\) such that $$\frac { \mathrm { d } y } { \mathrm {~d} t } = - k y$$ Give your answer to three significant figures.
    10
  2. A wildlife protection group takes measures to support the colony.
    The rate of reduction in the number of birds due to birds dying or leaving the colony is the same as in part (a), but in addition:
    Write down a first-order differential equation for \(y\) and \(t\) 10
  3. The initial number of birds is 340 Solve your differential equation from part (b) to find \(y\) in terms of \(t\) 10
  4. Describe two limitations of the model you have used. Limitation 1 \(\_\_\_\_\) Limitation 2 \(\_\_\_\_\)
AQA Further Paper 2 2021 June Q11
9 marks Hard +2.3
11 The Cartesian equation of the line \(L _ { 1 }\) is $$\frac { x + 1 } { 3 } = \frac { - y + 5 } { 2 } = \frac { 2 z + 5 } { 3 }$$ The Cartesian equation of the line \(L _ { 2 }\) is $$\frac { 2 x - 1 } { 2 } = \frac { y - 14 } { m } = \frac { z + 12 } { p }$$ The non-singular matrix \(\mathbf { N } = \left[ \begin{array} { c c c } - 0.5 & 1 & 2 \\ 1 & b & 4 \\ - 3 & - 2 & c \end{array} \right]\) maps the line \(L _ { 1 }\) onto the line \(L _ { 2 }\) Calculate the values of the constants \(b , c , m\) and \(p\) Fully justify your answers.
AQA Further Paper 2 2021 June Q12
12 marks Challenging +1.2
12 The integral \(S _ { n }\) is defined by $$S _ { n } = \int _ { 0 } ^ { a } x ^ { n } \sinh x \mathrm {~d} x \quad ( n \geq 0 )$$ 12
  1. Show that for \(n \geq 2\) $$S _ { n } = n ( n - 1 ) S _ { n - 2 } + a ^ { n } \cosh a - n a ^ { n - 1 } \sinh a$$
    12
  		2. Hence show that \(\int _ { 0 } ^ { 1 } x ^ { 4 } \sinh x d x = \frac { 9 } { 2 } e + \frac { 65 } { 2 } e ^ { - 1 } - 24\)
AQA Further Paper 2 2021 June Q13
16 marks Challenging +1.8
13
  1. Two of the solutions to the equation \(\cos 6 \theta = 0\) are \(\theta = \frac { \pi } { 4 }\) and \(\theta = \frac { 3 \pi } { 4 }\) Find the other solutions to the equation \(\cos 6 \theta = 0\) for \(0 \leq \theta \leq \pi\) 13
  2. Use de Moivre's theorem to show that $$\cos 6 \theta = 32 \cos ^ { 6 } \theta - 48 \cos ^ { 4 } \theta + 18 \cos ^ { 2 } \theta - 1$$ 13
  3. Use the fact that \(\theta = \frac { \pi } { 4 }\) and \(\theta = \frac { 3 \pi } { 4 }\) are solutions to the equation \(\cos 6 \theta = 0\) to find a factor of \(32 \cos ^ { 6 } \theta - 48 \cos ^ { 4 } \theta + 18 \cos ^ { 2 } \theta - 1\) in the form ( \(a \cos ^ { 2 } \theta + b\) ), where \(a\) and \(b\) are integers.
    [0pt] [4 marks]
  4. Hence show that $$\cos \left( \frac { 11 \pi } { 12 } \right) = - \sqrt { \frac { 2 + \sqrt { 3 } } { 4 } }$$ \includegraphics[max width=\textwidth, alt={}, center]{13abb93f-2fef-465c-980c-3b412de06618-25_2492_1721_217_150}