AQA AS Paper 1 (AS Paper 1) 2019 June

1 State the number of solutions to the equation \(\tan 4 \theta = 1\) for \(0 ^ { \circ } < \theta < 180 ^ { \circ }\)
Circle your answer. 1288

2 Dan believes that for every positive integer \(n\), at least one of \(2 ^ { n } - 1\) and \(2 ^ { n } + 1\) is prime. Which value of \(n\) shown below is a counter example to Dan's belief?
Circle your answer.
[0pt] [1 mark]
\(n = 3\)
\(n = 4\)
\(n = 5\)
\(n = 6\)

3 It is given that \(( x + 1 )\) and \(( x - 3 )\) are two factors of \(\mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = p x ^ { 3 } - 3 x ^ { 2 } - 8 x + q$$ 3

1. Find the values of \(p\) and \(q\).
   [0pt] [3 marks]
   3
2. Fully factorise f (x).
   \section*{Fully justify your answer.}

5

1. Sketch the curve \(y = \mathrm { g } ( x )\) where $$g ( x ) = ( x + 2 ) ( x - 1 ) ^ { 2 }$$ 5
2. Hence, solve \(\mathrm { g } ( x ) \leq 0\)

6

1. 1. Show that \(\cos \theta = \frac { 1 } { 2 }\) is one solution of the equation $$6 \sin ^ { 2 } \theta + 5 \cos \theta = 7$$ 6
2. (ii) Find all the values of \(\theta\) that solve the equation $$6 \sin ^ { 2 } \theta + 5 \cos \theta = 7$$ for \(0 ^ { \circ } \leq \theta \leq 360 ^ { \circ }\)
   Give your answers to the nearest degree.
   6
3. Hence, find all the solutions of the equation $$6 \sin ^ { 2 } 2 \theta + 5 \cos 2 \theta = 7$$ for \(0 ^ { \circ } \leq \theta \leq 360 ^ { \circ }\)
   Give your answers to the nearest degree.

7 Given that \(y \in \mathbb { R }\), prove that $$( 2 + 3 y ) ^ { 4 } + ( 2 - 3 y ) ^ { 4 } \geq 32$$ Fully justify your answer.
[0pt] [6 marks]

8 Prove that the curve with equation $$y = 2 x ^ { 5 } + 5 x ^ { 4 } + 10 x ^ { 3 } - 8$$ has only one stationary point, stating its coordinates.

9 A curve cuts the \(x\)-axis at ( 2,0 ) and has gradient function $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 24 } { x ^ { 3 } }$$ 9

1. Find the equation of the curve.
2. Show that the perpendicular bisector of the line joining \(A ( - 2,8 )\) to \(B ( - 6 , - 4 )\) is the 9
3. Snormal to the curve at ( 2,0 )

10 On 18 March 2019 there were 12 hours of daylight in Inverness.
On 16 June 2019, 90 days later, there will be 18 hours of daylight in Inverness.
Jude decides to model the number of hours of daylight in Inverness, \(N\), by the formula $$N = A + B \sin t ^ { \circ }$$ where \(t\) is the number of days after 18 March 2019.
10

1. 1. State the value that Jude should use for \(A\).
      10
2. (ii) State the value that Jude should use for \(B\).
   10
3. (iii) Using Jude's model, calculate the number of hours of daylight in Inverness on 15 May 2019, 58 days after 18 March 2019.
   [0pt] [1 mark]
   10
4. (iv) Using Jude's model, find how many days during 2019 will have at least 17.4 hours of daylight in Inverness.
   10
5. (v) Explain why Jude's model will become inaccurate for 2020 and future years.
   10
6. Anisa decides to model the number of hours of daylight in Inverness with the formula $$N = A + B \sin \left( \frac { 360 } { 365 } t \right) \circ$$ Explain why Anisa's model is better than Jude's model.

11 A ball moves in a straight line and passes through two fixed points, \(A\) and \(B\), which are 0.5 m apart. The ball is moving with a constant acceleration of \(0.39 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) in the direction \(A B\).
The speed of the ball at \(A\) is \(1.9 \mathrm {~ms} ^ { - 1 }\)
Find the speed of the ball at \(B\).
Circle your answer.
\(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
\(3.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
\(3.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
\(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) A particle \(P\), of mass \(m\) kilograms, is attached to one end of a light inextensible string.
The other end of this string is held at a fixed position, \(O\).
\(P\) hangs freely, in equilibrium, vertically below \(O\).
Identify the statement below that correctly describes the tension, \(T\) newtons, in the string as \(m\) varies. Tick \(( \checkmark )\) one box.
\(T\) varies along the string, with its greatest value at \(O\) □
\(T\) varies along the string, with its greatest value at \(P\) □
\(T = 0\) because the system is in equilibrium □
\(T\) is directly proportional to \(m\) □
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13 A car, starting from rest, is driven along a horizontal track. The velocity of the car, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), at time \(t\) seconds, is modelled by the equation $$v = 0.48 t ^ { 2 } - 0.024 t ^ { 3 } \text { for } 0 \leq t \leq 15$$ 13

1. Find the distance the car travels during the first 10 seconds of its journey.
   13
2. Find the maximum speed of the car.
   Give your answer to three significant figures.
   13
3. Deduce the range of values of \(t\) for which the car is modelled as decelerating.

14
Two particles, \(A\) and \(B\), lie at rest on a smooth horizontal plane.
\(A\) has position vector \(\mathbf { r } _ { A } = ( 13 \mathbf { i } - 22 \mathbf { j } )\) metres
\(B\) has position vector \(\mathbf { r } _ { B } = ( 3 \mathbf { i } + 2 \mathbf { j } )\) metres
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1. 
   Calculate the distance between \(A\) and \(B\).
   \end{tabular}
   \hline \end{tabular} \end{center} 14
2. A force of \(( 5 \mathbf { i } - 12 \mathbf { j } )\) newtons, is applied to \(B\), so that \(B\) moves, from rest, in a straight line towards \(A\).
   \(B\) has a mass of 0.8 kg
   14
3. 1. Show that the acceleration of \(B\) towards \(A\) is \(16.25 \mathrm {~ms} ^ { - 2 }\) 14
4. (ii) Hence, find the time taken for \(B\) to reach \(A\).
   Give your answer to two significant figures.

15 A tractor and its driver have a combined mass of \(m\) kilograms.
The tractor is towing a trailer of mass \(4 m\) kilograms in a straight line along a horizontal road. The tractor and trailer are connected by a horizontal tow bar, modelled as a light rigid rod. A driving force of 11080 N and a total resistance force of 160 N act on the tractor.
A total resistance force of 600 N acts on the trailer.
The tractor and the trailer have an acceleration of \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
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1. Find \(m\).
   15
2. Find the tension in the tow bar.
   15
3. At the instant the speed of the tractor reaches \(18 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) the tow bar breaks. The total resistance force acting on the trailer remains constant. Starting from the instant the tow bar breaks, calculate the time taken until the speed of the trailer reduces to \(9 \mathrm {~km} \mathrm {~h} ^ { - 1 }\)
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