AQA AS Paper 1 (AS Paper 1) 2018 June

1 Three of the following points lie on the same straight line.
Which point does not lie on this line?
Tick one box.
(-2, 14) □
(-1, 8)
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\(( 1 , - 1 )\) □
\(( 2 , - 6 )\) □

2 A circle has equation \(( x - 2 ) ^ { 2 } + ( y + 3 ) ^ { 2 } = 13\)
Find the gradient of the tangent to this circle at the origin.
Circle your answer.
[0pt] [1 mark]
\(- \frac { 3 } { 2 }\)
\(- \frac { 2 } { 3 }\)
\(\frac { 2 } { 3 }\)
\(\frac { 3 } { 2 }\)

3 State the interval for which \(\sin x\) is a decreasing function for \(0 ^ { \circ } \leq x \leq 360 ^ { \circ }\)
[0pt] [2 marks]

4

1. Find the first three terms in the expansion of \(( 1 - 3 x ) ^ { 4 }\) in ascending powers of \(x\). 4
2. Using your expansion, approximate \(( 0.994 ) ^ { 4 }\) to six decimal places.

5 Point \(C\) has coordinates \(( c , 2 )\) and point \(D\) has coordinates \(( 6 , d )\). The line \(y + 4 x = 11\) is the perpendicular bisector of \(C D\).
Find \(c\) and \(d\).
[0pt] [5 marks]
\(6 \quad A B C\) is a right-angled triangle.
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\(D\) is the point on hypotenuse \(A C\) such that \(A D = A B\).
The area of \(\triangle A B D\) is equal to half that of \(\triangle A B C\).

6

1. Show that \(\tan A = 2 \sin A\)
   6
2. 1. Show that the equation given in part (a) has two solutions for \(0 ^ { \circ } \leq A \leq 90 ^ { \circ }\)
      6
3. (ii) State the solution which is appropriate in this context.

8 Maxine measures the pressure, \(P\) kilopascals, and the volume, \(V\) litres, in a fixed quantity of gas. Maxine believes that the pressure and volume are connected by the equation $$P = c V ^ { d }$$ where \(c\) and \(d\) are constants. Using four experimental results, Maxine plots \(\log _ { 10 } P\) against \(\log _ { 10 } V\), as shown in the graph below.
\includegraphics[max width=\textwidth, alt={}, center]{c982106c-b742-444f-aeed-6f59ff3fae56-10_1386_1076_792_482} 8

1. Find the value of \(P\) and the value of \(V\) for the data point labelled \(A\) on the graph.
   8
2. Calculate the value of each of the constants \(c\) and \(d\).
   8
3. Estimate the pressure of the gas when the volume is 2 litres.

9 Craig is investigating the gradient of chords of the curve with equation \(\mathrm { f } ( x ) = x - x ^ { 2 }\) Each chord joins the point \(( 3 , - 6 )\) to the point \(( 3 + h , \mathrm { f } ( 3 + h ) )\)
The table shows some of Craig's results. 9

1. Show how the value - 5.1 has been calculated. 9
2. Complete the third row of the table above.
   

   9	State the limit suggested by Craig's investigation for the gradient of these chords as \(h\) tends to 0
   	[1 mark]
   9	Using differentiation from first principles, verify that your result in part (c) is correct. [4 marks] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

10 A curve has equation \(y = 2 x ^ { 2 } - 8 x \sqrt { x } + 8 x + 1\) for \(x \geq 0\) 10

1. Prove that the curve has a maximum point at ( 1,3 )
   Fully justify your answer.
   10
2. Find the coordinates of the other stationary point of the curve and state its nature. [2 marks]

11 In this question use \(g = 9.8 \mathrm {~ms} ^ { - 2 }\)
A ball, initially at rest, is dropped from a height of 40 m above the ground.
Calculate the speed of the ball when it reaches the ground.
Circle your answer.
\(- 28 \mathrm {~ms} ^ { - 1 }\)
\(28 \mathrm {~ms} ^ { - 1 }\)
\(- 780 \mathrm {~ms} ^ { - 1 }\)
\(780 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)

12 An object of mass 5 kg is moving in a straight line.
As a result of experiencing a forward force of \(F\) newtons and a resistant force of \(R\) newtons it accelerates at \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) Which one of the following equations is correct?
Circle your answer.
[0pt] [1 mark]
\(F - R = 0\)
\(F - R = 5\)
\(F - R = 3\)
\(F - R = 0.6\)

13 A vehicle, which begins at rest at point \(P\), is travelling in a straight line. For the first 4 seconds the vehicle moves with a constant acceleration of \(0.75 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
For the next 5 seconds the vehicle moves with a constant acceleration of \(- 1.2 \mathrm {~ms} ^ { - 2 }\) The vehicle then immediately stops accelerating, and travels a further 33 m at constant speed. 13

1. Draw a velocity-time graph for this journey on the grid below.
   [0pt] [3 marks]
   \includegraphics[max width=\textwidth, alt={}, center]{c982106c-b742-444f-aeed-6f59ff3fae56-17_739_1670_790_185} 13
2. Find the distance of the car from \(P\) after 20 seconds.

14 In this question use \(g = 9.81 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) Two particles, of mass 1.8 kg and 1.2 kg , are connected by a light, inextensible string over a smooth peg.
\includegraphics[max width=\textwidth, alt={}, center]{c982106c-b742-444f-aeed-6f59ff3fae56-18_556_680_488_680} 14

1. Initially the particles are held at rest 1.5 m above horizontal ground and the string between them is taut. The particles are released from rest.
   Find the time taken for the 1.8 kg particle to reach the ground.
   14
2. State one assumption you have made in answering part (a).

15 (b) (ii) State one assumption you have made that could affect your answer to part (b)(i).
[0pt] [1 mark] Turn over for the next question

16 A remote-controlled toy car is moving over a horizontal surface. It moves in a straight line through a point \(A\). The toy is initially at the point with displacement 3 metres from \(A\). Its velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), at time \(t\) seconds is defined by $$v = 0.06 \left( 2 + t - t ^ { 2 } \right)$$ 16

1. Find an expression for the displacement, \(r\) metres, of the toy from \(A\) at time \(t\) seconds.
   16
2. In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) At time \(t = 2\) seconds, the toy launches a ball which travels directly upwards with initial speed \(3.43 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Find the time taken for the ball to reach its highest point.
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