AQA Paper 1 (Paper 1) 2019 June

1 Given that \(a > 0\), determine which of these expressions is not equivalent to the others. Circle your answer.
[0pt] [1 mark] $$- 2 \log _ { 10 } \left( \frac { 1 } { a } \right) \quad 2 \log _ { 10 } ( a ) \quad \log _ { 10 } \left( a ^ { 2 } \right) \quad - 4 \log _ { 10 } ( \sqrt { a } )$$

2 Given \(y = \mathrm { e } ^ { k x }\), where \(k\) is a constant, find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
Circle your answer. $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \mathrm { e } ^ { k x } \quad \frac { \mathrm {~d} y } { \mathrm {~d} x } = k \mathrm { e } ^ { k x } \quad \frac { \mathrm {~d} y } { \mathrm {~d} x } = k x \mathrm { e } ^ { k x - 1 } \quad \frac { \mathrm {~d} y } { \mathrm {~d} x } = \frac { \mathrm { e } ^ { k x } } { k }$$

3 The diagram below shows a sector of a circle.
\includegraphics[max width=\textwidth, alt={}, center]{6b1312f4-9a5c-4465-8129-7d37e99efefe-02_375_406_1647_817} The radius of the circle is 4 cm and \(\theta = 0.8\) radians. Find the area of the sector. Circle your answer.
[0pt] [1 mark]
\(1.28 \mathrm {~cm} ^ { 2 }\)
\(3.2 \mathrm {~cm} ^ { 2 }\)
\(6.4 \mathrm {~cm} ^ { 2 }\)
\(12.8 \mathrm {~cm} ^ { 2 }\)

4 The point \(A\) has coordinates \(( - 1 , a )\) and the point \(B\) has coordinates \(( 3 , b )\) The line \(A B\) has equation \(5 x + 4 y = 17\)
Find the equation of the perpendicular bisector of the points \(A\) and \(B\).

5 An arithmetic sequence has first term \(a\) and common difference \(d\). The sum of the first 16 terms of the sequence is 260 5

1. Show that \(4 a + 30 d = 65\)
   5
2. Given that the sum of the first 60 terms is 315 , find the sum of the first 41 terms.
   5
3. \(\quad S _ { n }\) is the sum of the first \(n\) terms of the sequence. Explain why the value you found in part (b) is the maximum value of \(S _ { n }\)

6 The function f is defined by $$\mathrm { f } ( x ) = \frac { 1 } { 2 } \left( x ^ { 2 } + 1 \right) , x \geq 0$$ 6

1. Find the range of f . 6
2. 1. Find \(\mathrm { f } ^ { - 1 } ( x )\)
      6
3. (ii) State the range of \(\mathrm { f } ^ { - 1 } ( x )\)
   6
4. State the transformation which maps the graph of \(y = \mathrm { f } ( x )\) onto the graph of \(y = \mathrm { f } ^ { - 1 } ( x )\) 6
5. Find the coordinates of the point of intersection of the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\)

7

1. By sketching the graphs of \(y = \frac { 1 } { x }\) and \(y = \sec 2 x\) on the axes below, show that the equation $$\frac { 1 } { x } = \sec 2 x$$ has exactly one solution for \(x > 0\)
   \includegraphics[max width=\textwidth, alt={}, center]{6b1312f4-9a5c-4465-8129-7d37e99efefe-08_675_771_689_639} 7
2. By considering a suitable change of sign, show that the solution to the equation lies between 0.4 and 0.6
   7
3. Show that the equation can be rearranged to give $$x = \frac { 1 } { 2 } \cos ^ { - 1 } x$$ 7
4. 1. Use the iterative formula $$x _ { n + 1 } = \frac { 1 } { 2 } \cos ^ { - 1 } x _ { n }$$ with \(x _ { 1 } = 0.4\), to find \(x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\), giving your answers to four decimal places.
      7
5. (ii) On the graph below, draw a cobweb or staircase diagram to show how convergence takes place, indicating the positions of \(x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\).
   [0pt] [2 marks]
   \includegraphics[max width=\textwidth, alt={}, center]{6b1312f4-9a5c-4465-8129-7d37e99efefe-09_954_1600_1717_223}

8 \(\quad \mathrm { P } ( n ) = \sum _ { k = 0 } ^ { n } k ^ { 3 } - \sum _ { k = 0 } ^ { n - 1 } k ^ { 3 }\) where \(n\) is a positive integer.
8

1. Find \(\mathrm { P } ( 3 )\) and \(\mathrm { P } ( 10 )\)
   8
2. Solve the equation \(\mathrm { P } ( n ) = 1.25 \times 10 ^ { 8 }\)

9 Prove that the sum of a rational number and an irrational number is always irrational.
Jodie is attempting to use differentiation from first principles to prove that the gradient of \(y = \sin x\) is zero when \(x = \frac { \pi } { 2 }\) Jodie's teacher tells her that she has made mistakes starting in Step 4 of her working. Her working is shown below.
\includegraphics[max width=\textwidth, alt={}, center]{6b1312f4-9a5c-4465-8129-7d37e99efefe-14_645_1298_584_370} Step 1 Gradient of chord \(A B = \frac { \sin \left( \frac { \pi } { 2 } + h \right) - \sin \left( \frac { \pi } { 2 } \right) } { h }\) Step 2 \(= \frac { \sin \left( \frac { \pi } { 2 } \right) \cos ( h ) + \cos \left( \frac { \pi } { 2 } \right) \sin ( h ) - \sin \left( \frac { \pi } { 2 } \right) } { h }\) Step 3 $$= \sin \left( \frac { \pi } { 2 } \right) \left( \frac { \cos ( h ) - 1 } { h } \right) + \cos \left( \frac { \pi } { 2 } \right) \frac { \sin ( h ) } { h }$$ Step 4
For gradient of curve at \(A\),
let \(h = 0\) then
\(\frac { \cos ( h ) - 1 } { h } = 0\) and \(\frac { \sin ( h ) } { h } = 0\)
Step 5
Hence the gradient of the curve at \(A\) is given by \(\sin \left( \frac { \pi } { 2 } \right) \times 0 + \cos \left( \frac { \pi } { 2 } \right) \times 0 = 0\) Complete Steps 4 and 5 of Jodie's working below, to correct her proof. Step 4 For gradient of curve at \(A\), Step 5 Hence the gradient of the curve at \(A\) is given by

12

1. Show that the equation $$2 \cot ^ { 2 } x + 2 \operatorname { cosec } ^ { 2 } x = 1 + 4 \operatorname { cosec } x$$ can be written in the form $$a \operatorname { cosec } ^ { 2 } x + b \operatorname { cosec } x + c = 0$$ 12
2. Hence, given \(x\) is obtuse and $$2 \cot ^ { 2 } x + 2 \operatorname { cosec } ^ { 2 } x = 1 + 4 \operatorname { cosec } x$$ find the exact value of \(\tan x\) Fully justify your answer.
   

   13	A curve, \(C\), has equation
   	\(y = \frac { \mathrm { e } ^ { 3 x - 5 } } { x ^ { 2 } }\)
   	Show that \(C\) has exactly one stationary point.
   	Fully justify your answer.

   \includegraphics[max width=\textwidth, alt={}]{6b1312f4-9a5c-4465-8129-7d37e99efefe-19_2488_1716_219_153}

14 The graph of \(y = \frac { 2 x ^ { 3 } } { x ^ { 2 } + 1 }\) is shown for \(0 \leq x \leq 4\)
\includegraphics[max width=\textwidth, alt={}, center]{6b1312f4-9a5c-4465-8129-7d37e99efefe-20_1022_640_411_701} Caroline is attempting to approximate the shaded area, \(A\), under the curve using the trapezium rule by splitting the area into \(n\) trapezia. 14

1. When \(n = 4\)
   14
2. 1. State the number of ordinates that Caroline uses. 14
3. (ii) Calculate the area that Caroline should obtain using this method.
   Give your answer correct to two decimal places.
   14
4. Show that the exact area of \(A\) is $$16 - \ln 17$$ Fully justify your answer.
   

   14

   Explain what would happen to Caroline's answer to part (a)(ii) as \(n \rightarrow \infty\)

   [1 mark]

   Do not write outside the box

   \includegraphics[max width=\textwidth, alt={}]{6b1312f4-9a5c-4465-8129-7d37e99efefe-23_2488_1716_219_153}

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
2. 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.

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
4. 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}