AQA Paper 1 (Paper 1) 2021 June

1 State the set of values of \(x\) which satisfies the inequality $$( x - 3 ) ( 2 x + 7 ) > 0$$ Tick ( \(\checkmark\) ) one box. $$\begin{aligned} & \left\{ x : - \frac { 7 } { 2 } < x < 3 \right\}
& \left\{ x : x < - 3 \text { or } x > \frac { 7 } { 2 } \right\}
& \left\{ x : x < - \frac { 7 } { 2 } \text { or } x > 3 \right\}
& \left\{ x : - 3 < x < \frac { 7 } { 2 } \right\} \end{aligned}$$

2 Given that \(y = \ln ( 5 x )\)
find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
Circle your answer. $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { x } \quad \frac { \mathrm {~d} y } { \mathrm {~d} x } = \frac { 1 } { 5 x } \quad \frac { \mathrm {~d} y } { \mathrm {~d} x } = \frac { 5 } { x } \quad \frac { \mathrm {~d} y } { \mathrm {~d} x } = \ln 5$$

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

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. □

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}

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]

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.

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.

9 The table below shows the annual global production of plastics, \(P\), measured in millions of tonnes per year, for six selected years. 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
2. 1. Complete the table below.

      \(\boldsymbol { t }\)	0	5	10	15	20	25
      \(\boldsymbol { \operatorname { l o g } } _ { \mathbf { 1 0 } } \boldsymbol { P }\)	1.88	1.97	2.08		2.31

      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
4. 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}

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}

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.

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	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]
   	\(\_\_\_\_\)

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.

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
2. 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 \(A\).

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. 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}