Questions — AQA Paper 1 (122 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
AQA Paper 1 2023 June Q4
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
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
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
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
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
4 marks
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
  2. (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
  4. (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}
AQA Paper 1 2023 June Q10
10
  1. Point \(A\) on the curve has coordinates ( \(a , 0.5\) )
    10
    1. Find the value of \(a\)
      10
  3. (ii) State the value of \(\sin \left( 180 ^ { \circ } - a ^ { \circ } \right)\)
    10
  4. Point \(B\) on the curve has coordinates \(\left( b , - \frac { 3 } { 7 } \right)\)
    10
    1. Find the exact value of \(\sin \left( b ^ { \circ } - 180 ^ { \circ } \right)\)
      10
  6. (ii) Find the exact value of \(\cos b ^ { \circ }\)
AQA Paper 1 2023 June Q11
11 The \(n\)th term of a sequence is \(u _ { n }\)
The sequence is defined by $$u _ { n + 1 } = p u _ { n } + 70$$ where \(u _ { 1 } = 400\) and \(p\) is a constant.
11
  1. Find an expression, in terms of \(p\), for \(u _ { 2 }\) 11
  2. It is given that \(u _ { 3 } = 382\)
    11
    1. Show that \(p\) satisfies the equation $$200 p ^ { 2 } + 35 p - 156 = 0$$ 11
  4. (ii) It is given that the sequence is a decreasing sequence. Find the value of \(u _ { 4 }\) and the value of \(u _ { 5 }\)
    11
  5. The limit of \(u _ { n }\) as \(n\) tends to infinity is \(L\)
    11
    1. Write down an equation for \(L\)
      11
  7. (ii) Find the value of \(L\)
AQA Paper 1 2023 June Q12
1 marks
12 One of the rides at a theme park is a room where the floor and ceiling both move up and down for \(10 \pi\) seconds. At time \(t\) seconds after the ride begins, the distance \(f\) metres of the floor above the ground is $$f = 1 - \cos t$$ At time \(t\) seconds after the ride begins, the distance \(c\) metres of the ceiling above the ground is $$c = 8 - 4 \sin t$$ The ride is shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{6a03a035-ff32-4734-864b-a076aa9cbec0-16_448_766_932_635} 12
  1. Show that the initial distance between the floor and ceiling is 8 metres.
    [0pt] [1 mark]
    \includegraphics[max width=\textwidth, alt={}]{6a03a035-ff32-4734-864b-a076aa9cbec0-17_2500_1721_214_148}
AQA Paper 1 2023 June Q13
13 The function f is defined by $$\mathrm { f } ( x ) = \arccos x \text { for } 0 \leq x \leq a$$ The curve with equation \(y = \mathrm { f } ( x )\) is shown below.
\includegraphics[max width=\textwidth, alt={}, center]{6a03a035-ff32-4734-864b-a076aa9cbec0-18_842_837_550_603} 13
  1. State the value of \(a\) 13
    1. On the diagram above, sketch the curve with equation $$y = \cos x \text { for } 0 \leq x \leq \frac { \pi } { 2 }$$ and
      sketch the line with equation $$y = x \text { for } 0 \leq x \leq \frac { \pi } { 2 }$$ 13
  3. (ii) Explain why the solution to the equation $$x - \cos x = 0$$ must also be a solution to the equation $$\cos x = \arccos x$$ Question 13 continues on the next page 13
  4. Use the Newton-Raphson method with \(x _ { 0 } = 0\) to find an approximate solution, \(x _ { 3 }\), to the equation $$x - \cos x = 0$$ Give your answer to four decimal places.
    \includegraphics[max width=\textwidth, alt={}, center]{6a03a035-ff32-4734-864b-a076aa9cbec0-21_2491_1716_219_153}
AQA Paper 1 2023 June Q14
3 marks
14
    1. Given that $$y = 2 ^ { x }$$ write down \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) 14
  2. (ii) Hence find $$\int 2 ^ { x } \mathrm {~d} x$$ 14
  3. The area, \(A\), bounded by the curve with equation \(y = 2 ^ { x }\), the \(x\)-axis, the \(y\)-axis and the line \(x = - 4\) is approximated using eight rectangles of equal width as shown in the diagram below.
    \includegraphics[max width=\textwidth, alt={}, center]{6a03a035-ff32-4734-864b-a076aa9cbec0-23_1319_978_450_532} 14
    1. Show that the exact area of the largest rectangle is \(\frac { \sqrt { 2 } } { 4 }\)
      14
  5. (ii) The areas of these rectangles form a geometric sequence with common ratio \(\frac { \sqrt { 2 } } { 2 }\)
    Find the exact value of the total area of the eight rectangles.
    Give your answer in the form \(k ( 1 + \sqrt { 2 } )\) where \(k\) is a rational number.
    [0pt] [3 marks]
    14
  6. (iii) More accurate approximations for \(A\) can be found by increasing the number, \(n\), of rectangles used. Find the exact value of the limit of the approximations for \(A\) as \(n \rightarrow \infty\)
AQA Paper 1 2023 June Q15
2 marks
15 The curve with equation $$x ^ { 2 } + 2 y ^ { 3 } - 4 x y = 0$$ has a single stationary point at \(P\) as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{6a03a035-ff32-4734-864b-a076aa9cbec0-26_656_1138_548_450} 15
  1. Show that the \(y\)-coordinate of \(P\) satisfies the equation $$y ^ { 2 } ( y - 2 ) = 0$$ 15
  2. Hence, find the coordinates of \(P\)
    [0pt] [2 marks]
AQA Paper 1 2023 June Q16
16
  1. Given that $$\frac { 1 } { 16 - 9 x ^ { 2 } } \equiv \frac { A } { 4 - 3 x } + \frac { B } { 4 + 3 x }$$ find the values of \(A\) and \(B\)
    16
  2. An empty container, in the shape of a cuboid, has length 1.6 metres, width 1.25 metres and depth 0.5 metres, as shown in the diagram below.
    \includegraphics[max width=\textwidth, alt={}, center]{6a03a035-ff32-4734-864b-a076aa9cbec0-29_469_812_404_616} The container has a small hole in the bottom. Water is poured into the container at a rate of 0.16 cubic metres per minute.
    At time \(t\) minutes after the container starts to be filled, the depth of water is \(d\) metres and water leaks out at a rate of \(0.36 d ^ { 2 }\) cubic metres per minute. At time \(t\) minutes after the container starts to be filled, the volume of water in the container is \(V\) cubic metres. 16
    1. Show that $$\frac { \mathrm { d } V } { \mathrm {~d} t } = \frac { 16 - 9 V ^ { 2 } } { 100 }$$ \includegraphics[max width=\textwidth, alt={}, center]{6a03a035-ff32-4734-864b-a076aa9cbec0-30_2493_1721_214_150}
      \includegraphics[max width=\textwidth, alt={}, center]{6a03a035-ff32-4734-864b-a076aa9cbec0-31_2492_1721_217_150} Question number Additional page, if required.
      Write the question numbers in the left-hand margin. Question number Additional page, if required.
      Write the question numbers in the left-hand margin. Question number Additional page, if required.
      Write the question numbers in the left-hand margin.
      \includegraphics[max width=\textwidth, alt={}, center]{6a03a035-ff32-4734-864b-a076aa9cbec0-36_2498_1723_213_148}
AQA Paper 1 2024 June Q1
1 marks
1 Find the coefficient of \(x\) in the expansion of $$\left( 4 x ^ { 3 } - 5 x ^ { 2 } + 3 x - 2 \right) \left( x ^ { 5 } + 4 x + 1 \right)$$ Circle your answer.
[0pt] [1 mark]
\(\begin{array} { l l l l } - 5 & - 2 & 7 & 11 \end{array}\)
AQA Paper 1 2024 June Q2
2 The function f is defined by \(\mathrm { f } ( x ) = \mathrm { e } ^ { x } + 1\) for \(x \in \mathbb { R }\) Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\) Tick ( ✓ ) one box.
\(\mathrm { f } ^ { - 1 } ( x ) = \ln ( x - 1 )\)
\includegraphics[max width=\textwidth, alt={}, center]{0320e0a6-adc0-440a-b1da-d1a49fe06179-03_113_113_534_804}
\(\mathrm { f } ^ { - 1 } ( x ) = \ln ( x ) - 1\)
\includegraphics[max width=\textwidth, alt={}, center]{0320e0a6-adc0-440a-b1da-d1a49fe06179-03_109_113_689_804}
\(\mathrm { f } ^ { - 1 } ( x ) = \frac { 1 } { \mathrm { e } ^ { x } + 1 }\)
\includegraphics[max width=\textwidth, alt={}, center]{0320e0a6-adc0-440a-b1da-d1a49fe06179-03_113_108_840_804}
\(\mathrm { f } ^ { - 1 } ( x ) = \frac { x - 1 } { \mathrm { e } }\)
\includegraphics[max width=\textwidth, alt={}, center]{0320e0a6-adc0-440a-b1da-d1a49fe06179-03_108_109_991_808}
AQA Paper 1 2024 June Q3
1 marks
3 The expression $$\frac { 12 x ^ { 2 } + 3 x + 7 } { 3 x - 5 }$$ can be written as $$A x + B + \frac { C } { 3 x - 5 }$$ State the value of \(A\) Circle your answer.
[0pt] [1 mark] $$\begin{array} { l l l l } 3 & 4 & 7 & 9 \end{array}$$
AQA Paper 1 2024 June Q4
1 marks
4 One of the diagrams below shows the graph of \(y = \arccos x\) Identify the graph of \(y = \arccos x\) Tick ( ✓ ) one box.
[0pt] [1 mark]
\includegraphics[max width=\textwidth, alt={}, center]{0320e0a6-adc0-440a-b1da-d1a49fe06179-05_1339_1545_555_315}
AQA Paper 1 2024 June Q5
3 marks
5 Solve the equation $$\sin ^ { 2 } x = 1$$ for \(0 ^ { \circ } < x < 360 ^ { \circ }\)
[0pt] [3 marks]
AQA Paper 1 2024 June Q6
6 Use the chain rule to find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) when \(y = \left( x ^ { 3 } + 5 x \right) ^ { 7 }\)
AQA Paper 1 2024 June Q7
4 marks
7 Show that $$\frac { 3 + \sqrt { 8 n } } { 1 + \sqrt { 2 n } }$$ can be written as $$\frac { 4 n - 3 + \sqrt { 2 n } } { 2 n - 1 }$$ where \(n\) is a positive integer.
[0pt] [4 marks]
AQA Paper 1 2024 June Q8
3 marks
8
  1. Find the first three terms, in ascending powers of \(x\), in the expansion of $$( 2 + k x ) ^ { 5 }$$ where \(k\) is a positive constant.
    [0pt] [3 marks]
    8
  2. Hence, given that the coefficient of \(x\) is four times the coefficient of \(x ^ { 2 }\), find the value of \(k\)
AQA Paper 1 2024 June Q9
5 marks
9
  1. Show that, for small values of \(\theta\) measured in radians $$\cos 4 \theta + 2 \sin 3 \theta - \tan 2 \theta \approx A + B \theta + C \theta ^ { 2 }$$ where \(A , B\) and \(C\) are constants to be found.
    [0pt] [3 marks]
    9
  2. Use your answer to part (a) to find an approximation for $$\cos 0.28 + 2 \sin 0.21 - \tan 0.14$$ Give your answer to three decimal places.
    [0pt] [2 marks]
AQA Paper 1 2024 June Q10
2 marks
10
  1. An arithmetic sequence has 300 terms. The first term of the sequence is - 7 and the last term is 32 Find the sum of the 300 terms.
    [0pt] [2 marks]
    10
  2. A school holds a raffle at its summer fair. There are nine prizes.
    The total value of the prizes is \(\pounds 1260\)
    The values of the prizes form an arithmetic sequence.
    The top prize has the highest value, and the bottom prize has the least value.
    The value of the top prize is six times the value of the bottom prize.
    Find the value of the top prize.
AQA Paper 1 2024 June Q11
11 It is given that $$f ( x ) = x ( x - a ) ( x - 6 )$$ where \(0 < a < 6\) 11
  1. Sketch the graph of \(y = \mathrm { f } ( x )\) on the axes below.
    \includegraphics[max width=\textwidth, alt={}, center]{0320e0a6-adc0-440a-b1da-d1a49fe06179-14_1207_1105_733_447} 11
  2. Sketch the graph of \(y = \mathrm { f } ( - 2 x )\) on the axes below.
    \includegraphics[max width=\textwidth, alt={}, center]{0320e0a6-adc0-440a-b1da-d1a49fe06179-15_1207_1107_413_445}
AQA Paper 1 2024 June Q12
4 marks
12 The terms, \(u _ { n }\), of a periodic sequence are defined by $$u _ { 1 } = 3 \text { and } u _ { n + 1 } = \frac { - 6 } { u _ { n } }$$ 12
  1. \(\quad\) Find \(u _ { 2 } , u _ { 3 }\) and \(u _ { 4 }\)
    [0pt] [2 marks]
    12
  2. State the period of the sequence.
    12
  3. Find the value of \(\sum _ { n = 1 } ^ { 101 } u _ { n }\)
    [0pt] [2 marks]