AQA Paper 1 (Paper 1) 2022 June

1 A curve is defined by the parametric equations $$x = \cos \theta \text { and } y = \sin \theta \quad \text { where } 0 \leq \theta \leq 2 \pi$$ Which of the options shown below is a Cartesian equation for this curve?
Circle your answer. $$\frac { y } { x } = \tan \theta \quad x ^ { 2 } + y ^ { 2 } = 1 \quad x ^ { 2 } - y ^ { 2 } = 1 \quad x ^ { 2 } y ^ { 2 } = 1$$

2 A periodic sequence is defined by $$U _ { n } = ( - 1 ) ^ { n }$$ State the period of the sequence.
Circle your answer.

3 The curve $$y = \log _ { 4 } x$$ is transformed by a stretch, scale factor 2 , parallel to the \(y\)-axis.
State the equation of the curve after it has been transformed.
Circle your answer.
[0pt] [1 mark] $$y = \frac { 1 } { 2 } \log _ { 4 } x \quad y = 2 \log _ { 4 } x \quad y = \log _ { 4 } 2 x \quad y = \log _ { 8 } x$$

4 The graph of $$y = \mathrm { f } ( x )$$ where $$f ( x ) = a x ^ { 2 } + b x + c$$ is shown in Figure 1. \begin{figure}[h] \end{figure} Which of the following shows the graph of \(y = \mathrm { f } ^ { \prime } ( x )\) ? Tick \(( \checkmark )\) one box.
\includegraphics[max width=\textwidth, alt={}, center]{22ff390e-1360-43bd-8c7f-3d2b58627e91-05_2272_437_429_557}
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\includegraphics[max width=\textwidth, alt={}, center]{22ff390e-1360-43bd-8c7f-3d2b58627e91-05_117_117_1151_1133}
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5 Find an equation of the tangent to the curve $$y = ( x - 2 ) ^ { 4 }$$ at the point where \(x = 0\)

6

1. Find the first two terms, in ascending powers of \(x\), of the binomial expansion of $$\left( 1 - \frac { x } { 2 } \right) ^ { \frac { 1 } { 2 } }$$ 6
2. Hence, for small values of \(x\), show that $$\sin 4 x + \sqrt { \cos x } \approx A + B x + C x ^ { 2 }$$ where \(A , B\) and \(C\) are constants to be found.

7 Sketch the graph of $$y = \cot \left( x - \frac { \pi } { 2 } \right)$$ for \(0 \leq x \leq 2 \pi\)
[0pt] [3 marks]
\includegraphics[max width=\textwidth, alt={}, center]{22ff390e-1360-43bd-8c7f-3d2b58627e91-08_1650_1226_587_408}
\includegraphics[max width=\textwidth, alt={}, center]{22ff390e-1360-43bd-8c7f-3d2b58627e91-09_2488_1716_219_153}

8 The lines \(L _ { 1 }\) and \(L _ { 2 }\) are parallel.
\(L _ { 1 }\) has equation $$5 x + 3 y = 15$$ and \(L _ { 2 }\) has equation $$5 x + 3 y = 83$$ \(L _ { 1 }\) intersects the \(y\)-axis at the point \(P\).
The point \(Q\) is the point on \(L _ { 2 }\) closest to \(P\), as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{22ff390e-1360-43bd-8c7f-3d2b58627e91-10_849_917_945_561} 8

1. 1. Find the coordinates of \(Q\).
      8
2. (ii) Hence show that \(P Q = k \sqrt { 34 }\), where \(k\) is an integer to be found. 8
3. A circle, \(C\), has centre ( \(a , - 17\) ).
   \(L _ { 1 }\) and \(L _ { 2 }\) are both tangents to \(C\).
   8
4. 1. Find \(a\).
      8
5. (ii) Find the equation of \(C\).
   \includegraphics[max width=\textwidth, alt={}, center]{22ff390e-1360-43bd-8c7f-3d2b58627e91-13_2493_1732_214_139}

9 The first three terms of an arithmetic sequence are given by $$2 x + 5 \quad 5 x + 1 \quad 6 x + 7$$ 9

1. Show that \(x = 5\) is the only value which gives an arithmetic sequence.
   9
2. 1. Write down the value of the first term of the sequence.
      9
3. (ii) Find the value of the common difference of the sequence.
   9
4. The sum of the first \(N\) terms of the arithmetic sequence is \(S _ { N }\) where $$\begin{array} { r } S _ { N } < 100000
   S _ { N + 1 } > 100000 \end{array}$$ Find the value of \(N\).
   [0pt] [4 marks]

10 The diagram shows a sector of a circle \(O A B\).
\includegraphics[max width=\textwidth, alt={}, center]{22ff390e-1360-43bd-8c7f-3d2b58627e91-16_758_796_360_623} The point \(C\) lies on \(O B\) such that \(A C\) is perpendicular to \(O B\).
Angle \(A O B\) is \(\theta\) radians.
10

1. Given the area of the triangle \(O A C\) is half the area of the sector \(O A B\), show that $$\theta = \sin 2 \theta$$ 10
2. Use a suitable change of sign to show that a solution to the equation $$\theta = \sin 2 \theta$$ lies in the interval given by \(\theta \in \left[ \frac { \pi } { 5 } , \frac { 2 \pi } { 5 } \right]\)
   

   10 The Newton-Raphson method is used to find an approximate solution to the equation

   \(\theta = \sin 2 \theta\)

   10 Using \(\theta _ { 1 } = \frac { \pi } { 5 }\) as a first approximation for \(\theta\) apply the Newton-Raphson method twice

   to find the value of \(\theta _ { 3 }\) Give your answer to three decimal places.
   10

(ii) Explain how a more accurate approximation for \(\theta\) can be found using the Newton-Raphson method.
10

(iii) Explain why using \(\theta _ { 1 } = \frac { \pi } { 6 }\) as a first approximation in the Newton-Raphson method
[0pt] [2 marks] does not lead to a solution for \(\theta\).

11 The polynomial \(\mathrm { p } ( x )\) is given by $$\mathrm { p } ( x ) = x ^ { 3 } + ( b + 2 ) x ^ { 2 } + 2 ( b + 2 ) x + 8$$ where \(b\) is a constant.
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1. Use the factor theorem to prove that \(( x + 2 )\) is a factor of \(\mathrm { p } ( x )\) for all values of \(b\).
   11
2. The graph of \(y = \mathrm { p } ( x )\) meets the \(x\)-axis at exactly two points.
   11
3. 1. Sketch a possible graph of \(y = \mathrm { p } ( x )\)
      \includegraphics[max width=\textwidth, alt={}, center]{22ff390e-1360-43bd-8c7f-3d2b58627e91-20_1084_965_1619_532} 11
4. (ii) Given \(\mathrm { p } ( x )\) can be written as $$\mathrm { p } ( x ) = ( x + 2 ) \left( x ^ { 2 } + b x + 4 \right)$$ find the value of \(b\). Fully justify your answer.

12

1. A geometric sequence has first term 1 and common ratio \(\frac { 1 } { 2 }\)
   12
2. 1. Find the sum to infinity of the sequence.
      12
3. (ii) Hence, or otherwise, evaluate $$\sum _ { n = 1 } ^ { \infty } \left( \sin 30 ^ { \circ } \right) ^ { n }$$ 12
4. Find the smallest positive exact value of \(\theta\), in radians, which satisfies the equation $$\sum _ { n = 0 } ^ { \infty } ( \cos \theta ) ^ { n } = 2 - \sqrt { 2 }$$

13 Figure 2 shows the approximate shape of the vertical cross section of the entrance to a cave. The cave has a horizontal floor. The entrance to the cave joins the floor at the points \(O\) and \(P\). \begin{figure}[h] \end{figure} Garry models the shape of the cross section of the entrance to the cave using the equation $$x ^ { 2 } + y ^ { 2 } = a \sqrt { x } - y$$ where \(a\) is a constant, and \(x\) and \(y\) are the horizontal and vertical distances respectively, in metres, measured from \(O\). 13

1. The distance \(O P\) is 16 metres.
   Find the value of \(a\) that Garry should use in the model.
   \includegraphics[max width=\textwidth, alt={}, center]{22ff390e-1360-43bd-8c7f-3d2b58627e91-25_2518_1723_196_148}

14 The region bounded by the curve $$y = ( 2 x - 8 ) \ln x$$ and the \(x\)-axis is shaded in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{22ff390e-1360-43bd-8c7f-3d2b58627e91-26_867_908_543_566} 14

1. Use the trapezium rule with 5 ordinates to find an estimate for the area of the shaded region. Give your answer correct to three significant figures.
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2. Show that the exact area is given by $$32 \ln 2 - \frac { 33 } { 2 }$$ Fully justify your answer.

15

1. Given that $$y = \operatorname { cosec } \theta$$ 15
2. 1. Express \(y\) in terms of \(\sin \theta\). 15
3. (ii) Hence, prove that $$\frac { \mathrm { d } y } { \mathrm {~d} \theta } = - \operatorname { cosec } \theta \cot \theta$$ 15
4. (iii) Show that $$\frac { \sqrt { y ^ { 2 } - 1 } } { y } = \cos \theta \quad \text { for } 0 < \theta < \frac { \pi } { 2 }$$ 15
5. 1. Use the substitution $$x = 2 \operatorname { cosec } u$$ to show that $$\int \frac { 1 } { x ^ { 2 } \sqrt { x ^ { 2 } - 4 } } \mathrm {~d} x \quad \text { for } x > 2$$ can be written as $$k \int \sin u \mathrm {~d} u$$ where \(k\) is a constant to be found.
      15
6. (ii) Hence, show $$\int \frac { 1 } { x ^ { 2 } \sqrt { x ^ { 2 } - 4 } } \mathrm {~d} x = \frac { \sqrt { x ^ { 2 } - 4 } } { 4 x } + c \quad \text { for } x > 2$$ where \(c\) is a constant.
   \includegraphics[max width=\textwidth, alt={}, center]{22ff390e-1360-43bd-8c7f-3d2b58627e91-32_2492_1721_217_150}
   \includegraphics[max width=\textwidth, alt={}]{22ff390e-1360-43bd-8c7f-3d2b58627e91-36_2496_1721_214_148}