AQA Further Paper 1 (Further Paper 1) 2022 June

1 The displacement of a particle from its equilibrium position is \(x\) metres at time \(t\) seconds. The motion of the particle obeys the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - 9 x$$ Calculate the period of its motion in seconds.
Circle your answer.
[0pt] [1 mark]
\(\frac { \pi } { 9 }\)
\(\frac { 2 \pi } { 9 }\)
\(\frac { \pi } { 3 }\)
\(\frac { 2 \pi } { 3 }\)

2 Simplify $$\frac { \cos \left( \frac { 6 \pi } { 13 } \right) + i \sin \left( \frac { 6 \pi } { 13 } \right) } { \cos \left( \frac { 2 \pi } { 13 } \right) - i \sin \left( \frac { 2 \pi } { 13 } \right) }$$ Tick ( \(\checkmark\) ) one box. $$\begin{array} { l l } \cos \left( \frac { 8 \pi } { 13 } \right) + i \sin \left( \frac { 8 \pi } { 13 } \right) & \square
\cos \left( \frac { 8 \pi } { 13 } \right) - i \sin \left( \frac { 8 \pi } { 13 } \right) & \square
\cos \left( \frac { 4 \pi } { 13 } \right) + i \sin \left( \frac { 4 \pi } { 13 } \right) & \square
\cos \left( \frac { 4 \pi } { 13 } \right) - i \sin \left( \frac { 4 \pi } { 13 } \right) & \square \end{array}$$

4 The vector \(\mathbf { v }\) is an eigenvector of the matrix \(\mathbf { N }\) with corresponding eigenvalue 4
The vector \(\mathbf { v }\) is also an eigenvector of the matrix \(\mathbf { M }\) with corresponding eigenvalue 3
Given that $$\mathbf { N M } ^ { 2 } \mathbf { v } = \lambda \mathbf { v }$$ find the value of \(\lambda\)
Circle your answer.
[0pt] [1 mark]
102436144

5 It is given that \(z = - \frac { 3 } { 2 } + \mathrm { i } \frac { \sqrt { 11 } } { 2 }\) is a root of the equation $$z ^ { 4 } - 3 z ^ { 3 } - 5 z ^ { 2 } + k z + 40 = 0$$ where \(k\) is a real number.
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1. Find the other three roots.
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2. Given that \(x \in \mathbb { R }\), solve $$x ^ { 4 } - 3 x ^ { 3 } - 5 x ^ { 2 } + k x + 40 < 0$$

6

1. Given that \(| x | < 1\), prove that $$\tanh ^ { - 1 } x = \frac { 1 } { 2 } \ln \left( \frac { 1 + x } { 1 - x } \right)$$ 6
2. Solve the equation $$20 \operatorname { sech } ^ { 2 } x - 11 \tanh x = 16$$ Give your answer in logarithmic form.
   \(7 \quad\) The matrix \(\mathbf { M }\) is defined as $$\mathbf { M } = \left[ \begin{array} { c c c } 1 & 7 & - 3
   3 & 6 & k + 1
   1 & 3 & 2 \end{array} \right]$$ where \(k\) is a constant.

7

1. 1. Given that \(\mathbf { M }\) is a non-singular matrix, find \(\mathbf { M } ^ { - 1 }\) in terms of \(k\)
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2. (ii) State any restrictions on the value of \(k\) 7
3. Using your answer to part (a)(i), solve $$\begin{array} { r } x + 7 y - 3 z = 6
   3 x + 6 y + 6 z = 3
   x + 3 y + 2 z = 1 \end{array}$$

8

1. The complex number \(w\) is such that $$\arg ( w + 2 \mathrm { i } ) = \tan ^ { - 1 } \frac { 1 } { 2 }$$ It is given that \(w = x + \mathrm { i } y\), where \(x\) and \(y\) are real and \(x > 0\)
   Find an equation for \(y\) in terms of \(x\)
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2. The complex number \(z\) satisfies both $$- \frac { \pi } { 2 } \leq \arg ( z + 2 \mathrm { i } ) \leq \tan ^ { - 1 } \frac { 1 } { 2 } \quad \text { and } \quad | z - 2 + 3 \mathrm { i } | \leq 2$$ The region \(R\) is the locus of \(z\)
   Sketch the region \(R\) on the Argand diagram below.
   \includegraphics[max width=\textwidth, alt={}, center]{a889963c-266c-497e-b7fc-99a249ba9e58-10_1015_1020_1683_511} 8
3. \(\quad z _ { 1 }\) is the point in \(R\) at which \(| z |\) is minimum. 8
4. 1. Calculate the exact value of \(\left| z _ { 1 } \right|\)
      8
5. (ii) Express \(z _ { 1 }\) in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are real.

9 Roberto is solving this mathematics problem: The curve \(C _ { 1 }\) has polar equation $$r ^ { 2 } = 9 \sin 2 \theta$$ for all possible values of \(\theta\)
Find the area enclosed by \(C _ { 1 }\) Roberto's solution is as follows: $$\begin{aligned} A & = \frac { 1 } { 2 } \int _ { - \pi } ^ { \pi } 9 \sin 2 \theta \mathrm {~d} \theta
& = \left[ - \frac { 9 } { 4 } \cos 2 \theta \right] _ { - \pi } ^ { \pi }
& = 0 \end{aligned}$$ 9

1. \(\quad\) Sketch the curve \(C _ { 1 }\) 9
2. Explain what Roberto has done wrong.
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3. \(\quad\) Find the area enclosed by \(C _ { 1 }\)
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4. \(\quad P\) and \(Q\) are distinct points on \(C _ { 1 }\) for which \(r\) is a maximum. \(P\) is above the initial line. Find the polar coordinates of \(P\) and \(Q\)
   9
5. The matrix \(\mathbf { M } = \left[ \begin{array} { l l } 1 & 2
   0 & 1 \end{array} \right]\) represents the transformation T T maps \(C _ { 1 }\) onto a curve \(C _ { 2 }\)
   9
6. 1. T maps \(P\) onto the point \(P ^ { \prime }\)
      Find the polar coordinates of \(P ^ { \prime }\)
      [0pt] [4 marks]
      9
7. (ii) Find the area enclosed by \(C _ { 2 }\) Fully justify your answer.

10 In this question all measurements are in centimetres. A small, thin laser pen is set up with one end at \(A ( 7,2 , - 3 )\) and the other end at \(B ( 9 , - 3 , - 2 )\) A laser beam travels from \(A\) to \(B\) and continues in a straight line towards a large thin sheet of glass. The sheet of glass lies within a plane \(\Pi _ { 1 }\) which is modelled by the equation $$4 x + p y + 5 z = 9$$ where \(p\) is an integer.
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1. The laser beam hits \(\Pi _ { 1 }\) at an acute angle \(\alpha\), where \(\sin \alpha = \frac { \sqrt { 15 } } { 75 }\)
   Find the value of \(p\)
   10
2. A second large sheet of glass lies on the other side of \(\Pi _ { 1 }\) This second sheet lies within a plane \(\Pi _ { 2 }\) which is modelled by the equation $$4 x + p y + 5 z = - 5$$ Calculate the distance between the sheets of glass.
   10
3. The point \(A ( 7,2 , - 3 )\) is reflected in \(\Pi _ { 1 }\)
   Find the coordinates of the image of \(A\) after reflection in \(\Pi _ { 1 }\)

11 In this question use \(g\) as \(10 \mathrm {~m \mathrm {~s} ^ { - 2 }\)} A smooth plane is inclined at \(30 ^ { \circ }\) to the horizontal.
The fixed points \(A\) and \(B\) are 3.6 metres apart on the line of greatest slope of the plane, with \(A\) higher than \(B\) A particle \(P\) of mass 0.32 kg is attached to one end of each of two light elastic strings. The other ends of these strings are attached to the points \(A\) and \(B\) respectively. The particle \(P\) moves on a straight line that passes through \(A\) and \(B\)
\includegraphics[max width=\textwidth, alt={}, center]{a889963c-266c-497e-b7fc-99a249ba9e58-18_417_709_774_669} The natural length of the string \(A P\) is 1.4 metres.
When the extension of the string \(A P\) is \(e _ { A }\) metres, the tension in the string \(A P\) is \(7 e _ { A }\) newtons.
The natural length of the string \(B P\) is 1 metre.
When the extension of the string \(B P\) is \(e _ { B }\) metres, the tension in the string \(B P\) is \(9 e _ { B }\) newtons. The particle \(P\) is held at the point between \(A\) and \(B\) which is 0.2 metres from its equilibrium position and lower than its equilibrium position.
The particle \(P\) is then released from rest.
At time \(t\) seconds after \(P\) is released, its displacement towards \(B\) from its equilibrium position is \(x\) metres. 11

1. Show that during the subsequent motion the object satisfies the equation $$\ddot { x } + 50 x = 0$$ Fully justify your answer. 11
2. The experiment is repeated in a large tank of oil.
   During the motion the oil causes a resistive force of \(k v\) newtons to act on the particle, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the particle. The oil causes critical damping to occur.
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3. 1. Show that \(k = \frac { 16 \sqrt { 2 } } { 5 }\)
      11
4. (ii) Find \(x\) in terms of \(t\), giving your answer in exact form.
   11
5. (iii) Calculate the maximum speed of the particle.

12 The Argand diagram shows the solutions to the equation \(z ^ { 5 } = 1\)
\includegraphics[max width=\textwidth, alt={}, center]{a889963c-266c-497e-b7fc-99a249ba9e58-22_1079_995_354_520} 12

1. Solve the equation $$z ^ { 5 } = 1$$ giving your answers in the form \(z = \cos \theta + \mathrm { i } \sin \theta\), where \(0 \leq \theta < 2 \pi\)
   [0pt] [2 marks] 12
2. Explain why the points on an Argand diagram which represent the solutions found in part (a) are the vertices of a regular pentagon.
   [0pt] [2 marks]
   \includegraphics[max width=\textwidth, alt={}]{a889963c-266c-497e-b7fc-99a249ba9e58-23_2484_1726_219_141}
   12
3. The Argand diagram on page 22 is repeated below.
   \includegraphics[max width=\textwidth, alt={}, center]{a889963c-266c-497e-b7fc-99a249ba9e58-24_1079_1000_354_520} Explain, with reference to the Argand diagram, why the expression $$16 c ^ { 5 } - 20 c ^ { 3 } + 5 c - 1$$ has a repeated quadratic factor.
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4. \(O\) is the centre of a regular pentagon \(A B C D E\) such that \(O A = O B = O C = O D =\) \(O E = 1\) unit.
   The distance from \(O\) to \(A B\) is \(h\)
   By solving the equation \(16 c ^ { 5 } - 20 c ^ { 3 } + 5 c - 1 = 0\), show that $$h = \frac { \sqrt { 5 } + 1 } { 4 }$$ \includegraphics[max width=\textwidth, alt={}, center]{a889963c-266c-497e-b7fc-99a249ba9e58-26_2492_1721_217_150}