Questions — AQA Paper 1 (122 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure 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 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 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 Stats 1 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
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 2019 June Q15
15
  1. At time \(t\) hours after a high tide, the height, \(h\) metres, of the tide and the velocity, \(v\) knots, of the tidal flow can be modelled using the parametric equations $$\begin{aligned} & v = 4 - \left( \frac { 2 t } { 3 } - 2 \right) ^ { 2 }
    & h = 3 - 2 \sqrt [ 3 ] { t - 3 } \end{aligned}$$ High tides and low tides occur alternately when the velocity of the tidal flow is zero.
    A high tide occurs at 2 am.
    15
    1. Use the model to find the height of this high tide.
      15
  3. (ii) Find the time of the first low tide after 2 am.
    15
  4. (iii) Find the height of this low tide.
    15
  5. Use the model to find the height of the tide when it is flowing with maximum velocity.
    15
  6. Comment on the validity of the model.
AQA Paper 1 2019 June Q16
16
  1. \(\quad y = \mathrm { e } ^ { - x } ( \sin x + \cos x )\) Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
    Simplify your answer.
    16
  2. Hence, show that $$\int \mathrm { e } ^ { - x } \sin x \mathrm {~d} x = a \mathrm { e } ^ { - x } ( \sin x + \cos x ) + c$$ where \(a\) is a rational number.
    16
  3. A sketch of the graph of \(y = \mathrm { e } ^ { - x } \sin x\) for \(x \geq 0\) is shown below. \(A _ { 1 } , A _ { 2 } , \ldots , A _ { n } , \ldots\)
    The areas of the finite regions bounded by the curve and the \(x\)-axis are denoted by
    \includegraphics[max width=\textwidth, alt={}, center]{6b1312f4-9a5c-4465-8129-7d37e99efefe-27_974_1507_502_262} 16
    1. Find the exact value of the area \(A _ { 1 }\)
      16
  5. (ii) Show that $$\frac { A _ { 2 } } { A _ { 1 } } = \mathrm { e } ^ { - \pi }$$ 16
  6. (iii) Given that $$\frac { A _ { n + 1 } } { A _ { n } } = \mathrm { e } ^ { - \pi }$$ show that the exact value of the total area enclosed between the curve and the \(x\)-axis is $$\frac { 1 + \mathrm { e } ^ { \pi } } { 2 \left( \mathrm { e } ^ { \pi } - 1 \right) }$$ \includegraphics[max width=\textwidth, alt={}, center]{6b1312f4-9a5c-4465-8129-7d37e99efefe-30_2488_1719_219_150}
    \includegraphics[max width=\textwidth, alt={}, center]{6b1312f4-9a5c-4465-8129-7d37e99efefe-31_2488_1719_219_150}
    \includegraphics[max width=\textwidth, alt={}, center]{6b1312f4-9a5c-4465-8129-7d37e99efefe-32_2496_1721_214_148}
AQA Paper 1 2020 June Q1
1 marks
1 The first three terms, in ascending powers of \(x\), of the binomial expansion of \(( 9 + 2 x ) ^ { \frac { 1 } { 2 } }\) are given by $$( 9 + 2 x ) ^ { \frac { 1 } { 2 } } \approx a + \frac { x } { 3 } - \frac { x ^ { 2 } } { 54 }$$ where \(a\) is a constant. 1
  1. State the range of values of \(x\) for which this expansion is valid.
    Circle your answer.
    \(| x | < \frac { 2 } { 9 }\)
    \(| x | < \frac { 2 } { 3 }\)
    \(| x | < 1\)
    \(| x | < \frac { 9 } { 2 }\) 1
  2. Find the value of \(a\).
    Circle your answer.
    [0pt] [1 mark]
    1239
AQA Paper 1 2020 June Q2
2 A student is searching for a solution to the equation \(\mathrm { f } ( x ) = 0\) He correctly evaluates $$f ( - 1 ) = - 1 \text { and } f ( 1 ) = 1$$ and concludes that there must be a root between - 1 and 1 due to the change of sign.
Select the function \(\mathrm { f } ( x )\) for which the conclusion is incorrect.
Circle your answer. $$\mathrm { f } ( x ) = \frac { 1 } { x } \quad \mathrm { f } ( x ) = x \quad \mathrm { f } ( x ) = x ^ { 3 } \quad \mathrm { f } ( x ) = \frac { 2 x + 1 } { x + 2 }$$
AQA Paper 1 2020 June Q3
1 marks
3 The diagram shows a sector \(O A B\) of a circle with centre \(O\) and radius 2
\includegraphics[max width=\textwidth, alt={}, center]{08e1f291-7052-40a5-b7b2-13fd1d0137c2-03_374_455_1187_790} The angle \(A O B\) is \(\theta\) radians and the perimeter of the sector is 6
Find the value of \(\theta\) Circle your answer.
[0pt] [1 mark]
1
\(\sqrt { 3 }\)
2
3
AQA Paper 1 2020 June Q4
4
  1. Sketch the graph of
    \includegraphics[max width=\textwidth, alt={}, center]{08e1f291-7052-40a5-b7b2-13fd1d0137c2-04_933_1093_349_475} 4
  2. Solve the inequality $$4 - | 2 x - 6 | > 2$$
AQA Paper 1 2020 June Q5
5 Prove that, for integer values of \(n\) such that \(0 \leq n < 4\) $$2 ^ { n + 2 } > 3 ^ { n }$$
AQA Paper 1 2020 June Q6
6 Four students, Tom, Josh, Floella and Georgia are attempting to complete the indefinite integral $$\int \frac { 1 } { x } \mathrm {~d} x \quad \text { for } x > 0$$ Each of the students' solutions is shown below: $$\begin{array} { l l } \text { Tom } & \int \frac { 1 } { x } \mathrm {~d} x = \ln x
\text { Josh } & \int \frac { 1 } { x } \mathrm {~d} x = k \ln x
\text { Floella } & \int \frac { 1 } { x } \mathrm {~d} x = \ln A x
\text { Georgia } & \int \frac { 1 } { x } \mathrm {~d} x = \ln x + c \end{array}$$ 6
    1. Explain what is wrong with Tom's answer. 6
  2. (ii) Explain what is wrong with Josh's answer.
    6
  3. Explain why Floella and Georgia's answers are equivalent.
AQA Paper 1 2020 June Q7
7 Consecutive terms of a sequence are related by $$u _ { n + 1 } = 3 - \left( u _ { n } \right) ^ { 2 }$$ 7
  1. In the case that \(u _ { 1 } = 2\)
    7
    1. Find \(u _ { 3 }\) 7
  3. (ii) Find \(u _ { 50 }\) 7
  4. State a different value for \(u _ { 1 }\) which gives the same value for \(u _ { 50 }\) as found in part (a)(ii).
AQA Paper 1 2020 June Q8
2 marks
8 Mike, an amateur astronomer who lives in the South of England, wants to know how the number of hours of darkness changes through the year. On various days between February and September he records the length of time, \(H\) hours, of darkness along with \(t\), the number of days after 1 January. His results are shown in Figure 1 below. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{08e1f291-7052-40a5-b7b2-13fd1d0137c2-08_940_1541_696_246}
\end{figure} Mike models this data using the equation $$H = 3.87 \sin \left( \frac { 2 \pi ( t + 101.75 ) } { 365 } \right) + 11.7$$ 8
  1. Find the minimum number of hours of darkness predicted by Mike's model. Give your answer to the nearest minute.
    [0pt] [2 marks] 8
  2. Find the maximum number of consecutive days where the number of hours of darkness predicted by Mike's model exceeds 14
    8
  3. Mike's friend Sofia, who lives in Spain, also records the number of hours of darkness on various days throughout the year. Her results are shown in Figure 2 below. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{08e1f291-7052-40a5-b7b2-13fd1d0137c2-10_933_1537_561_248}
    \end{figure} Sofia attempts to model her data by refining Mike's model.
    She decides to increase the 3.87 value, leaving everything else unchanged.
    Explain whether Sofia's refinement is appropriate.
    \includegraphics[max width=\textwidth, alt={}, center]{08e1f291-7052-40a5-b7b2-13fd1d0137c2-11_2488_1730_219_141}
    \(9 \quad\) Chloe is attempting to write \(\frac { 2 x ^ { 2 } + x } { ( x + 1 ) ( x + 2 ) ^ { 2 } }\) as partial fractions, with constant numerators. Her incorrect attempt is shown below. Step 1 $$\frac { 2 x ^ { 2 } + x } { ( x + 1 ) ( x + 2 ) ^ { 2 } } \equiv \frac { A } { x + 1 } + \frac { B } { ( x + 2 ) ^ { 2 } }$$ Step 2 $$2 x ^ { 2 } + x \equiv A ( x + 2 ) ^ { 2 } + B ( x + 1 )$$ Step 3 $$\begin{aligned} & \text { Let } x = - 1 \Rightarrow A = 1
    & \text { Let } x = - 2 \Rightarrow B = - 6 \end{aligned}$$ Answer $$\frac { 2 x ^ { 2 } + x } { ( x + 1 ) ( x + 2 ) ^ { 2 } } \equiv \frac { 1 } { x + 1 } - \frac { 6 } { ( x + 2 ) ^ { 2 } }$$
AQA Paper 1 2020 June Q9
9
    1. By using a counter example, show that the answer obtained by Chloe cannot be correct.
      9
  2. (ii) Explain her mistake in Step 1.
    9
  3. Write \(\frac { 2 x ^ { 2 } + x } { ( x + 1 ) ( x + 2 ) ^ { 2 } }\) as partial fractions, with constant numerators.
AQA Paper 1 2020 June Q10
4 marks
10
  1. An arithmetic series is given by $$\sum _ { r = 5 } ^ { 20 } ( 4 r + 1 )$$ 10
    1. Write down the first term of the series.
      10
  3. (ii) Write down the common difference of the series.
    10
  4. (iii) Find the number of terms of the series.
    10
  5. A different arithmetic series is given by \(\sum _ { r = 10 } ^ { 100 } ( b r + c )\)
    where \(b\) and \(c\) are constants.
    The sum of this series is 7735
    10
  6. (ii) The 40th term of the series is 4 times the 2nd term. Find the values of \(b\) and \(c\).
    [0pt] [4 marks]
AQA Paper 1 2020 June Q11
2 marks
11 The region \(R\) enclosed by the lines \(x = 1 , x = 6 , y = 0\) and the curve $$y = \ln ( 8 - x )$$ is shown shaded in Figure 3 below. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{08e1f291-7052-40a5-b7b2-13fd1d0137c2-17_419_869_598_587}
\end{figure} All distances are measured in centimetres.
11
  1. Use a single trapezium to find an approximate value of the area of the shaded region, giving your answer in \(\mathrm { cm } ^ { 2 }\) to two decimal places.
    [0pt] [2 marks]
    \section*{Question 11 continues on the next page} 11
  2. Shape \(B\) is made from four copies of region \(R\) as shown in Figure 4 below. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{08e1f291-7052-40a5-b7b2-13fd1d0137c2-18_707_711_438_667}
    \end{figure} Shape \(B\) is cut from metal of thickness 2 mm
    The metal has a density of \(10.5 \mathrm {~g} / \mathrm { cm } ^ { 3 }\)
    Use the trapezium rule with six ordinates to calculate an approximate value of the mass of Shape B. Give your answer to the nearest gram.
    11
  3. Without further calculation, give one reason why the mass found in part (b) may be:
    11
    1. an underestimate.
      11
  5. (ii) an overestimate.
AQA Paper 1 2020 June Q12
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
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
4 marks
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
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 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
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
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
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
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
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
2 marks
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
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.