AQA Paper 1 (Paper 1) 2023 June

1 Find the coefficient of \(x ^ { 7 }\) in the expansion of \(( 2 x - 3 ) ^ { 7 }\)
Circle your answer.
-2187-128 2128

2 Given that \(y = 2 x ^ { 3 }\) find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
Circle your answer.
[0pt] [1 mark]
\(\frac { \mathrm { d } y } { \mathrm {~d} x } = 5 x ^ { 2 }\)
\(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x ^ { 2 }\)
\(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { x ^ { 4 } } { 2 }\)
\(\frac { \mathrm { d } y } { \mathrm {~d} x } = 6 x ^ { 3 }\)

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.
\(y = \frac { 1 } { 2 } \ln x \quad y = 2 \ln x \quad y = \ln \frac { x } { 2 } \quad y = \ln 2 x\)

4 Given that \(\theta\) is a small angle, find an approximation for \(\cos 2 \theta\) Circle your answer.
\(1 - \frac { \theta ^ { 2 } } { 2 }\)
\(2 - 2 \theta ^ { 2 }\)
\(1 - 2 \theta ^ { 2 }\)
\(1 - \theta ^ { 2 }\)

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 \(\int _ { 1 } ^ { 4 } \frac { 20 } { \mathrm { e } ^ { x } - 1 } \mathrm {~d} x\)

6 Show that the equation
$$\begin{aligned} & \qquad 2 \log _ { 10 } x = \log _ { 10 } 4 + \log _ { 10 } ( x + 8 )
& \text { has exactly one solution. }
& \text { Fully justify your answer. } \end{aligned}$$

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 \(n\)

8 Show that $$\int _ { 0 } ^ { \frac { \pi } { 2 } } ( x \sin 4 x ) \mathrm { d } x = - \frac { \pi } { 8 }$$

9 The points \(P\) and \(Q\) have coordinates ( \(- 6,15\) ) and (12, 19) respectively. 9

1. 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
3. 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}

10

1. Point \(A\) on the curve has coordinates ( \(a , 0.5\) )
   10
2. 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
5. 1. Find the exact value of \(\sin \left( b ^ { \circ } - 180 ^ { \circ } \right)\)
      10
6. (ii) Find the exact value of \(\cos b ^ { \circ }\)

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
3. 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
6. 1. Write down an equation for \(L\)
      11
7. (ii) Find the value of \(L\)

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}

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
2. 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}

14

1. 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
4. 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 \(n \rightarrow \infty\)

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]

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
3. 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}