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AQA Paper 1 2020 June Q12
13 marks Standard +0.3
12 A curve \(C\) has equation $$x ^ { 3 } \sin y + \cos y = A x$$ where \(A\) is a constant. \(C\) passes through the point \(P \left( \sqrt { 3 } , \frac { \pi } { 6 } \right)\) 12
  1. Show that \(A = 2\) 12
    1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 2 - 3 x ^ { 2 } \sin y } { x ^ { 3 } \cos y - \sin y }\) 12
  3. (ii) Hence, find the gradient of the curve at \(P\).
    12
  4. (iii) The tangent to \(C\) at \(P\) intersects the \(x\)-axis at \(Q\).
    Find the exact \(x\)-coordinate of \(Q\).
AQA Paper 1 2020 June Q13
15 marks Standard +0.3
13 The function f is defined by $$\mathrm { f } ( x ) = \frac { 2 x + 3 } { x - 2 } \quad x \in \mathbb { R } , x \neq 2$$ 13
    1. Find f-1
      13
  2. (ii) Write down an expression for \(\mathrm { ff } ( x )\).
    13
  3. The function g is defined by $$g ( x ) = \frac { 2 x ^ { 2 } - 5 x } { 2 } \quad x \in \mathbb { R } , 0 \leq x \leq 4$$ 13
    1. Find the range of g .
      13
  5. (ii) Determine whether g has an inverse.
    Fully justify your answer.
  6. Show that $$g f ( x ) = \frac { 48 + 29 x - 2 x ^ { 2 } } { 2 x ^ { 2 } - 8 x + 8 }$$ 13
  7. It can be shown that fg is given by $$f g ( x ) = \frac { 4 x ^ { 2 } - 10 x + 6 } { 2 x ^ { 2 } - 5 x - 4 }$$ with domain \(\{ x \in \mathbb { R } : 0 \leq x \leq 4 , x \neq a \}\) Find the value of \(a\).
    Fully justify your answer.
AQA Paper 1 2020 June Q14
9 marks Standard +0.3
14 The function f is defined by $$f ( x ) = 3 ^ { x } \sqrt { x } - 1 \quad \text { where } x \geq 0$$ 14
  1. \(\quad \mathrm { f } ( x ) = 0\) has a single solution at the point \(x = \alpha\) By considering a suitable change of sign, show that \(\alpha\) lies between 0 and 1
    14
    1. Show that $$\mathrm { f } ^ { \prime } ( x ) = \frac { 3 ^ { x } ( 1 + x \ln 9 ) } { 2 \sqrt { x } }$$
      14
    2. (ii) Use the Newton-Raphson method with \(x _ { 1 } = 1\) to find \(x _ { 3 }\), an approximation for \(\alpha\).
      3. Give your answer to five decimal places.
      [2 marks] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)
      14
    3. (iii) Explain why the Newton-Raphson method fails to find \(\alpha\) with \(x _ { 1 } = 0\)
      4. [2 marks] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)
AQA Paper 1 2020 June Q15
10 marks Standard +0.3
15 The region enclosed between the curves \(y = \mathrm { e } ^ { x } , y = 6 - \mathrm { e } ^ { \overline { 2 } }\) and the line \(x = 0\) is shown shaded in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{08e1f291-7052-40a5-b7b2-13fd1d0137c2-28_1155_1009_424_513} Show that the exact area of the shaded region is $$6 \ln 4 - 5$$ Fully justify your answer. \includegraphics[max width=\textwidth, alt={}, center]{08e1f291-7052-40a5-b7b2-13fd1d0137c2-30_2491_1736_219_139}
AQA Paper 1 2021 June Q1
1 marks Easy -1.8
1 State the set of values of \(x\) which satisfies the inequality $$( x - 3 ) ( 2 x + 7 ) > 0$$ Tick ( \(\checkmark\) ) one box. $$\begin{aligned} & \left\{ x : - \frac { 7 } { 2 } < x < 3 \right\} \\ & \left\{ x : x < - 3 \text { or } x > \frac { 7 } { 2 } \right\} \\ & \left\{ x : x < - \frac { 7 } { 2 } \text { or } x > 3 \right\} \\ & \left\{ x : - 3 < x < \frac { 7 } { 2 } \right\} \end{aligned}$$
AQA Paper 1 2021 June Q2
1 marks Easy -1.8
2 Given that \(y = \ln ( 5 x )\) find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) Circle your answer. $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { x } \quad \frac { \mathrm {~d} y } { \mathrm {~d} x } = \frac { 1 } { 5 x } \quad \frac { \mathrm {~d} y } { \mathrm {~d} x } = \frac { 5 } { x } \quad \frac { \mathrm {~d} y } { \mathrm {~d} x } = \ln 5$$
AQA Paper 1 2021 June Q3
1 marks Moderate -0.8
3 A geometric sequence has a sum to infinity of - 3 A second sequence is formed by multiplying each term of the original sequence by - 2 What is the sum to infinity of the new sequence? Circle your answer. The sum to infinity does not
AQA Paper 1 2021 June Q4
1 marks Easy -1.8
4 Millie is attempting to use proof by contradiction to show that the result of multiplying an irrational number by a non-zero rational number is always an irrational number. Select the assumption she should make to start her proof.
Tick ( \(\checkmark\) ) one box. Every irrational multiplied by a non-zero rational is irrational. □ Every irrational multiplied by a non-zero rational is rational. □ There exists a non-zero rational and an irrational whose product is irrational. □ There exists a non-zero rational and an irrational whose product is rational. □
AQA Paper 1 2021 June Q5
6 marks Moderate -0.3
5
  1. Find the equation of the line perpendicular to \(L\) which passes through \(P\). 5 The line \(L\) has equation 5
  2. Hence, find the shortest distance from \(P\) to \(L\). \includegraphics[max width=\textwidth, alt={}, center]{042e248a-9efa-4844-957d-f05715900ffc-05_2488_1716_219_153}
AQA Paper 1 2021 June Q6
7 marks Standard +0.3
6
  1. The ninth term of an arithmetic series is 3 The sum of the first \(n\) terms of the series is \(S _ { n }\) and \(S _ { 21 } = 42\) Find the first term and common difference of the series.
    [0pt] [4 marks]
    6
  2. A second arithmetic series has first term - 18 and common difference \(\frac { 3 } { 4 }\) The sum of the first \(n\) terms of this series is \(T _ { n }\) Find the value of \(n\) such that \(T _ { n } = S _ { n }\) [0pt] [3 marks]
AQA Paper 1 2021 June Q7
7 marks Moderate -0.3
7 The equation \(x ^ { 2 } = x ^ { 3 } + x - 3\) has a single solution, \(x = \alpha\) 7
  1. By considering a suitable change of sign, show that \(\alpha\) lies between 1.5 and 1.6
    [0pt] [2 marks]
    7
  2. Show that the equation \(x ^ { 2 } = x ^ { 3 } + x - 3\) can be rearranged into the form $$x ^ { 2 } = x - 1 + \frac { 3 } { x }$$ 7
  3. Use the iterative formula $$x _ { n + 1 } = \sqrt { x _ { n } - 1 + \frac { 3 } { x _ { n } } }$$ with \(x _ { 1 } = 1.5\), to find \(x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\), giving your answers to four decimal places.
    7
  4. Hence, deduce an interval of width 0.001 in which \(\alpha\) lies.
AQA Paper 1 2021 June Q8
9 marks Standard +0.8
8
  1. Given that $$9 \sin ^ { 2 } \theta + \sin 2 \theta = 8$$ show that $$8 \cot ^ { 2 } \theta - 2 \cot \theta - 1 = 0$$ 8
  2. Hence, solve $$9 \sin ^ { 2 } \theta + \sin 2 \theta = 8$$ in the interval \(0 < \theta < 2 \pi\) Give your answers to two decimal places.
    8
  3. Solve $$9 \sin ^ { 2 } \left( 2 x - \frac { \pi } { 4 } \right) + \sin \left( 4 x - \frac { \pi } { 2 } \right) = 8$$ in the interval \(0 < x < \frac { \pi } { 2 }\) Give your answers to one decimal place.
AQA Paper 1 2021 June Q9
15 marks Moderate -0.3
9 The table below shows the annual global production of plastics, \(P\), measured in millions of tonnes per year, for six selected years.
Year198019851990199520002005
\(\boldsymbol { P }\)7594120156206260
It is thought that \(P\) can be modelled by $$P = A \times 10 ^ { k t }$$ where \(t\) is the number of years after 1980 and \(A\) and \(k\) are constants.
9
  1. Show algebraically that the graph of \(\log _ { 10 } P\) against \(t\) should be linear.
    9
    1. Complete the table below.
      \(\boldsymbol { t }\)0510152025
      \(\boldsymbol { \operatorname { l o g } } _ { \mathbf { 1 0 } } \boldsymbol { P }\)1.881.972.082.31
      9
  3. (ii) Plot \(\log _ { 10 } P\) against \(t\), and draw a line of best fit for the data. \includegraphics[max width=\textwidth, alt={}, center]{042e248a-9efa-4844-957d-f05715900ffc-13_1203_1308_360_367} 9
    1. Hence, show that \(k\) is approximately 0.02
      9
  5. (ii) Find the value of \(A\).
    9
  6. Using the model with \(k = 0.02\) predict the number of tonnes of annual global production of plastics in 2030. 9
  7. Using the model with \(k = 0.02\) predict the year in which \(P\) first exceeds 8000
    9
  8. Give a reason why it may be inappropriate to use the model to make predictions about future annual global production of plastics. \includegraphics[max width=\textwidth, alt={}, center]{042e248a-9efa-4844-957d-f05715900ffc-15_2488_1716_219_153}
AQA Paper 1 2021 June Q10
8 marks Standard +0.3
10
  1. Given that $$y = \tan x$$ use the quotient rule to show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \sec ^ { 2 } x$$ 10
  2. The region enclosed by the curve \(y = \tan ^ { 2 } x\) and the horizontal line, which intersects the curve at \(x = - \frac { \pi } { 4 }\) and \(x = \frac { \pi } { 4 }\), is shaded in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{042e248a-9efa-4844-957d-f05715900ffc-17_1059_967_461_539} Show that the area of the shaded region is $$\pi - 2$$ Fully justify your answer.
    Do not write outside the box \includegraphics[max width=\textwidth, alt={}, center]{042e248a-9efa-4844-957d-f05715900ffc-19_2488_1716_219_153}
AQA Paper 1 2021 June Q11
8 marks Challenging +1.8
11 A curve, \(C\), passes through the point with coordinates \(( 1,6 )\) The gradient of \(C\) is given by $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 6 } ( x y ) ^ { 2 }$$ Show that \(C\) intersects the coordinate axes at exactly one point and state the coordinates of this point. Fully justify your answer.
AQA Paper 1 2021 June Q12
8 marks Moderate -0.3
12 The equation of a curve is $$( x + y ) ^ { 2 } = 4 y + 2 x + 8$$ The curve intersects the positive \(x\)-axis at the point \(P\).
12
  1. Show that the gradient of the curve at \(P\) is \(- \frac { 3 } { 2 }\)
    12
  		2. Find the equation of the normal to the curve at \(P\), giving your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers.
    [2 marks]
    \(\_\_\_\_\)
AQA Paper 1 2021 June Q13
8 marks Standard +0.3
13
  1. Given that $$P ( x ) = 125 x ^ { 3 } + 150 x ^ { 2 } + 55 x + 6$$ use the factor theorem to prove that ( \(5 x + 1\) ) is a factor of \(\mathrm { P } ( x )\).
    [0pt] [2 marks]
    13
  2. Factorise \(\mathrm { P } ( x )\) completely.
    13
  3. Hence, prove that \(250 n ^ { 3 } + 300 n ^ { 2 } + 110 n + 12\) is a multiple of 12 when \(n\) is a positive whole number.
AQA Paper 1 2021 June Q14
10 marks Standard +0.3
14 The curve \(C\) is defined for \(t \geq 0\) by the parametric equations $$x = t ^ { 2 } + t \quad \text { and } \quad y = 4 t ^ { 2 } - t ^ { 3 }$$ \(C\) is shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{042e248a-9efa-4844-957d-f05715900ffc-26_691_608_541_717} 14
  1. Find the gradient of \(C\) at the point where it intersects the positive \(x\)-axis.
    14
    1. The area \(A\) enclosed between \(C\) and the \(x\)-axis is given by $$A = \int _ { 0 } ^ { b } y \mathrm {~d} x$$ Find the value of \(b\).
      14
  3. (ii) Use the substitution \(y = 4 t ^ { 2 } - t ^ { 3 }\) to show that $$A = \int _ { 0 } ^ { 4 } \left( 4 t ^ { 2 } + 7 t ^ { 3 } - 2 t ^ { 4 } \right) \mathrm { d } t$$ 14
  4. (iii) Find the value of \(A\).
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
    1. 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
  4. (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.