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AQA Paper 1 2021 June Q15
10 marks Challenging +1.2
15
  1. Show that $$\sin x - \sin x \cos 2 x \approx 2 x ^ { 3 }$$ for small values of \(x\).
    15
  2. Hence, show that the area between the graph with equation $$y = \sqrt { 8 ( \sin x - \sin x \cos 2 x ) }$$ the positive \(x\)-axis and the line \(x = 0.25\) can be approximated by $$\text { Area } \approx 2 ^ { m } \times 5 ^ { n }$$ where \(m\) and \(n\) are integers to be found.
    15
  3. (i) Explain why $$\int _ { 6.3 } ^ { 6.4 } 2 x ^ { 3 } \mathrm {~d} x$$ is not a suitable approximation for $$\int _ { 6.3 } ^ { 6.4 } ( \sin x - \sin x \cos 2 x ) d x$$ Question 15 continues on the next page 15 (c) (ii) Explain how $$\int _ { 6.3 } ^ { 6.4 } ( \sin x - \sin x \cos 2 x ) d x$$ may be approximated by $$\int _ { a } ^ { b } 2 x ^ { 3 } \mathrm {~d} x$$ for suitable values of \(a\) and \(b\). \includegraphics[max width=\textwidth, alt={}, center]{042e248a-9efa-4844-957d-f05715900ffc-31_2492_1721_217_150}
    \includegraphics[max width=\textwidth, alt={}]{042e248a-9efa-4844-957d-f05715900ffc-36_2486_1719_221_150}
AQA Paper 1 2022 June Q1
1 marks Easy -2.0
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. $$\frac { y } { x } = \tan \theta \quad x ^ { 2 } + y ^ { 2 } = 1 \quad x ^ { 2 } - y ^ { 2 } = 1 \quad x ^ { 2 } y ^ { 2 } = 1$$
AQA Paper 1 2022 June Q2
1 marks Easy -2.5
2 A periodic sequence is defined by $$U _ { n } = ( - 1 ) ^ { n }$$ State the period of the sequence.
Circle your answer.
AQA Paper 1 2022 June Q3
1 marks Easy -1.2
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] $$y = \frac { 1 } { 2 } \log _ { 4 } x \quad y = 2 \log _ { 4 } x \quad y = \log _ { 4 } 2 x \quad y = \log _ { 8 } x$$
\includegraphics[max width=\textwidth, alt={}]{22ff390e-1360-43bd-8c7f-3d2b58627e91-03_2492_1722_217_150}
AQA Paper 1 2022 June Q4
1 marks Easy -1.2
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]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{22ff390e-1360-43bd-8c7f-3d2b58627e91-04_618_634_810_703}
\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}
□ \includegraphics[max width=\textwidth, alt={}, center]{22ff390e-1360-43bd-8c7f-3d2b58627e91-05_117_117_1151_1133}
□
□
AQA Paper 1 2022 June Q5
3 marks Easy -1.2
5 Find an equation of the tangent to the curve $$y = ( x - 2 ) ^ { 4 }$$ at the point where \(x = 0\)
AQA Paper 1 2022 June Q6
6 marks Standard +0.8
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.
AQA Paper 1 2022 June Q7
3 marks Standard +0.3
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}
AQA Paper 1 2022 June Q8
11 marks Standard +0.3
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. Find the coordinates of \(Q\).
      8
      1. (ii) Hence show that \(P Q = k \sqrt { 34 }\), where \(k\) is an integer to be found. 8
    2. A circle, \(C\), has centre ( \(a , - 17\) ). \(L _ { 1 }\) and \(L _ { 2 }\) are both tangents to \(C\).
      8
      1. Find \(a\).
        8
    4. (ii) Find the equation of \(C\). \includegraphics[max width=\textwidth, alt={}, center]{22ff390e-1360-43bd-8c7f-3d2b58627e91-13_2493_1732_214_139}
AQA Paper 1 2022 June Q9
9 marks Moderate -0.3
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. (i) Write down the value of the first term of the sequence.
    9 (b) (ii) Find the value of the common difference of the sequence.
    9
  3. 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]
AQA Paper 1 2022 June Q10
12 marks Standard +0.8
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
  3. The Newton-Raphson method is used to find an approximate solution to the equation
    4. \(\theta = \sin 2 \theta\)
    10 (c) (i) 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 (c) (ii) Explain how a more accurate approximation for \(\theta\) can be found using the Newton-Raphson method.
    10 (c) (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 \(\theta\).
AQA Paper 1 2022 June Q11
10 marks Moderate -0.3
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.
11
  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 (b) (i) 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 (b) (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.
AQA Paper 1 2022 June Q12
8 marks Standard +0.8
12
  1. A geometric sequence has first term 1 and common ratio \(\frac { 1 } { 2 }\) 12
    1. (i) Find the sum to infinity of the sequence.
      12
    2. (ii) Hence, or otherwise, evaluate $$\sum _ { n = 1 } ^ { \infty } \left( \sin 30 ^ { \circ } \right) ^ { n }$$ 12
    3. Find the smallest positive exact value of \(\theta\), in radians, which satisfies the equation $$\sum _ { n = 0 } ^ { \infty } ( \cos \theta ) ^ { n } = 2 - \sqrt { 2 }$$
AQA Paper 1 2022 June Q13
9 marks Moderate -0.8
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]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{22ff390e-1360-43bd-8c7f-3d2b58627e91-24_396_991_584_529}
\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}
AQA Paper 1 2022 June Q14
9 marks Standard +0.8
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.
    14
  2. Show that the exact area is given by $$32 \ln 2 - \frac { 33 } { 2 }$$ Fully justify your answer.
AQA Paper 1 2022 June Q15
16 marks Challenging +1.2
15
  1. Given that $$y = \operatorname { cosec } \theta$$ 15
      1. Express \(y\) in terms of \(\sin \theta\). 15
      2. Hence, prove that $$\frac { \mathrm { d } y } { \mathrm {~d} \theta } = - \operatorname { cosec } \theta \cot \theta$$ 15
      3. Show that $$\frac { \sqrt { y ^ { 2 } - 1 } } { y } = \cos \theta \quad \text { for } 0 < \theta < \frac { \pi } { 2 }$$ 15
        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
    2. (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}
AQA Paper 1 2023 June Q1
1 marks Easy -1.2
1 Find the coefficient of \(x ^ { 7 }\) in the expansion of \(( 2 x - 3 ) ^ { 7 }\) Circle your answer.
-2187-128 2128
AQA Paper 1 2023 June Q2
1 marks Easy -2.5
2 Given that \(y = 2 x ^ { 3 }\) find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) Circle your answer.
[0pt] [1 mark] \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 5 x ^ { 2 }\) \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x ^ { 2 }\) \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x ^ { 4 } } { 2 }\) \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x ^ { 3 }\)
AQA Paper 1 2023 June Q3
1 marks Moderate -0.8
3 The curve with equation \(y = \ln x\) is transformed by a stretch parallel to the \(x\)-axis with scale factor 2 Find the equation of the transformed curve.
Circle your answer. \(y = \frac { 1 } { 2 } \ln x \quad y = 2 \ln x \quad y = \ln \frac { x } { 2 } \quad y = \ln 2 x\)
AQA Paper 1 2023 June Q4
1 marks Moderate -0.8
4 Given that \(\theta\) is a small angle, find an approximation for \(\cos 2 \theta\) Circle your answer. \(1 - \frac { \theta ^ { 2 } } { 2 }\) \(2 - 2 \theta ^ { 2 }\) \(1 - 2 \theta ^ { 2 }\) \(1 - \theta ^ { 2 }\)
AQA Paper 1 2023 June Q5
4 marks Moderate -0.3
5
  1. Use the trapezium rule with 6 ordinates ( 5 strips) to find an approximate value for the shaded area. Give your answer to four decimal places.
    5
  2. Using your answer to part (a) deduce an estimate for \(\int _ { 1 } ^ { 4 } \frac { 20 } { \mathrm { e } ^ { x } - 1 } \mathrm {~d} x\)
AQA Paper 1 2023 June Q6
5 marks Standard +0.8
6 Show that the equation $$\begin{aligned} & \qquad 2 \log _ { 10 } x = \log _ { 10 } 4 + \log _ { 10 } ( x + 8 ) \\ & \text { has exactly one solution. } \\ & \text { Fully justify your answer. } \end{aligned}$$
AQA Paper 1 2023 June Q7
4 marks Moderate -0.3
7
  1. Given that \(n\) is a positive integer, express $$\frac { 7 } { 3 + 5 \sqrt { n } } - \frac { 7 } { 5 \sqrt { n } - 3 }$$ as a single fraction not involving surds.
    7
  2. Hence, deduce that $$\frac { 7 } { 3 + 5 \sqrt { n } } - \frac { 7 } { 5 \sqrt { n } - 3 }$$ is a rational number for all positive integer values of \(n\)
AQA Paper 1 2023 June Q8
6 marks Moderate -0.3
8 Show that $$\int _ { 0 } ^ { \frac { \pi } { 2 } } ( x \sin 4 x ) \mathrm { d } x = - \frac { \pi } { 8 }$$
\includegraphics[max width=\textwidth, alt={}]{6a03a035-ff32-4734-864b-a076aa9cbec0-09_2491_1716_219_153}
AQA Paper 1 2023 June Q9
9 marks Moderate -0.3
9 The points \(P\) and \(Q\) have coordinates ( \(- 6,15\) ) and (12, 19) respectively. 9
    1. Find the coordinates of the midpoint of \(P Q\) 9
      1. (ii) Find the equation of the perpendicular bisector of \(P Q\) Give your answer in the form \(a x + b y = c\) where \(a , b\) and \(c\) are integers.
        [0pt] [4 marks]
        9
      1. A circle passes through the points \(P\) and \(Q\) The centre of the circle lies on the line with equation \(2 x - 5 y = - 30\) Find the equation of the circle. 9
    3. (ii) The circle intersects the coordinate axes at \(n\) points.
      State the value of \(n\) $$y = \sin x ^ { \circ }$$ for \(- 360 \leq x \leq 360\) is shown below. \includegraphics[max width=\textwidth, alt={}, center]{6a03a035-ff32-4734-864b-a076aa9cbec0-12_613_1552_532_246}