AQA AS Paper 1 (AS Paper 1) 2023 June

1 At a point \(P\) on a curve, the gradient of the tangent to the curve is 10 State the gradient of the normal to the curve at \(P\) Circle your answer.
-10
-0.1
0.1
10

2 Identify the expression below which is equivalent to \(\left( \frac { 2 x } { 5 } \right) ^ { - 3 }\)
Circle your answer.
[0pt] [1 mark]
\(\frac { 8 x ^ { 3 } } { 125 }\)
\(\frac { 125 x ^ { 3 } } { 8 }\)
\(\frac { 125 } { 8 x ^ { 3 } }\)
\(\frac { 8 } { 125 x ^ { 3 } }\) Find the two possible values of \(a\) The coefficient of \(x ^ { 2 }\) in the binomial expansion of \(( 1 + a x ) ^ { 6 }\) is \(\frac { 20 } { 3 }\)

3 The coefficient of \(x ^ { 2 }\) in the binomial expansion of \(( 1 + a x ) ^ { 6 }\) is \(\frac { 20 } { 3 }\)

4

1. Find the possible values of \(\tan \theta\), giving your answers in exact form.
   4
2. Hence, or otherwise, solve the equation $$5 \cos ^ { 2 } \theta - 4 \sin ^ { 2 } \theta = 0$$ giving all solutions of \(\theta\) to the nearest \(0.1 ^ { \circ }\) in the interval \(0 ^ { \circ } \leq \theta \leq 360 ^ { \circ }\)
   \begin{center} \begin{tabular}{|l|l|} \hline

5 (b) & 5 (a) Given that \(y = x \sqrt { x }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
\hline \end{tabular} \end{center}

6

1. The curve \(C _ { 1 }\) has equation \(y = 2 x ^ { 2 } - 20 x + 42\) Express the equation of \(C _ { 1 }\) in the form $$y = a ( x - b ) ^ { 2 } + c$$ where \(a , b\) and \(c\) are integers.
   6
2. Write down the coordinates of the minimum point of \(C _ { 1 }\)
   6
3. The curve \(C _ { 1 }\) is mapped onto the curve \(C _ { 2 }\) by a stretch in the \(y\)-direction.
   The minimum point of \(C _ { 2 }\) is at \(( 5 , - 4 )\)
   Find the equation of \(C _ { 2 }\)
   \(7 \quad\) Points \(P\) and \(Q\) lie on the curve with equation \(y = x ^ { 4 }\) The \(x\)-coordinate of \(P\) is \(x\)
   The \(x\)-coordinate of \(Q\) is \(x + h\)

7

1. \(\quad\) Expand \(( x + h ) ^ { 4 }\)
   7
2. Hence, find an expression, in terms of \(x\) and \(h\), for the gradient of the line \(P Q\)
   7
3. Explain how to use the answer from part (b) to obtain the gradient function of \(y = x ^ { 4 }\)
   [0pt] [2 marks]

8

1. Show that $$\int _ { 1 } ^ { a } \left( 6 - \frac { 12 } { \sqrt { x } } \right) \mathrm { d } x = 6 a - 24 \sqrt { a } + 18$$ 8
2. The curve \(y = 6 - \frac { 12 } { \sqrt { x } }\), the line \(x = 1\) and the line \(x = a\) are shown in the diagram below. The shaded region \(R _ { 1 }\) is bounded by the curve, the line \(x = 1\) and the \(x\)-axis.
   The shaded region \(R _ { 2 }\) is bounded by the curve, the line \(x = a\) and the \(x\)-axis.
   \includegraphics[max width=\textwidth, alt={}, center]{9cd7f38d-a2a1-4fd3-9ed9-cb389e8ee3b6-09_705_931_632_648} It is given that the areas of \(R _ { 1 }\) and \(R _ { 2 }\) are equal.
   Find the value of \(a\)
   Fully justify your answer.

9 A continuous curve has equation \(y = \mathrm { f } ( x )\) The curve passes through the points \(A ( 2,1 ) , B ( 4,5 )\) and \(C ( 6,1 )\)
It is given that \(f ^ { \prime } ( 4 ) = 0\)
Jasmin made two statements about the nature of the curve \(y = \mathrm { f } ( x )\) at the point \(B\) :
Statement 1: There is a turning point at \(B\)
Statement 2: There is a maximum point at \(B\)
9

1. Draw a sketch of the curve \(y = \mathrm { f } ( x )\) such that Statement 1 is correct and Statement 2 is correct.
   [0pt] [1 mark]
   \includegraphics[max width=\textwidth, alt={}, center]{9cd7f38d-a2a1-4fd3-9ed9-cb389e8ee3b6-10_593_588_1043_817} 9
2. Draw a sketch of the curve \(y = \mathrm { f } ( x )\) such that Statement 1 is correct and Statement 2 is not correct.
   \includegraphics[max width=\textwidth, alt={}, center]{9cd7f38d-a2a1-4fd3-9ed9-cb389e8ee3b6-11_607_597_497_813} 9
3. Draw a sketch of the curve \(y = \mathrm { f } ( x )\) such that Statement 1 is not correct and Statement 2 is not correct.
   [0pt] [1 mark]
   \includegraphics[max width=\textwidth, alt={}, center]{9cd7f38d-a2a1-4fd3-9ed9-cb389e8ee3b6-11_605_597_1541_813} 1 Charlie buys a car for \(\pounds 18000\) on 1 January 2016.
   The value of the car decreases exponentially.
   The car has a value of \(\pounds 12000\) on 1 January 2018.

10

1. Charlie says:
   • because the car has lost \(\pounds 6000\) after two years, after another two years it will be worth £6000.
   Charlie's friend Kaya says:
   • because the car has lost one third of its value after two years, after another two years it will be worth \(\pounds 8000\).
   Explain whose statement is correct, justifying the value they have stated.
   

   10

   The value of Charlie's car, \(\pounds V , t\) years after 1 January 2016 may be modelled by the equation \(V = A \mathrm { e } ^ { - k t }\) where \(A\) and \(k\) are positive constants. Find the value of \(t\) when the car has a value of \(\pounds 10000\), giving your answer to two significant figures. [5 marks] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) [1 mark] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

11

1. A circle has equation $$x ^ { 2 } + y ^ { 2 } - 10 x - 6 = 0$$ Find the centre and the radius of the circle.
   11
2. An equilateral triangle has one vertex at the origin, and one side along the line \(x = 8\), as shown in the diagram below.
   \includegraphics[max width=\textwidth, alt={}, center]{9cd7f38d-a2a1-4fd3-9ed9-cb389e8ee3b6-14_697_750_1311_735} 11
3. 1. Show that the vertex at the origin lies inside the circle \(x ^ { 2 } + y ^ { 2 } - 10 x - 6 = 0\)
      11
4. (ii) Prove that the triangle lies completely within the circle \(x ^ { 2 } + y ^ { 2 } - 10 x - 6 = 0\)

12 A particle, initially at rest, starts to move forward in a straight line with constant acceleration, \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) After 6 seconds the particle has a velocity of \(3 \mathrm {~ms} ^ { - 1 }\)
Find the value of \(a\) Circle your answer.
\(\begin{array} { l l l l } - 2 & - 0.5 & 0.5 & 2 \end{array}\)

13 A resultant force of \(\left[ \begin{array} { c } - 2
6 \end{array} \right] \mathrm { N }\) acts on a particle.
The acceleration of the particle is \(\left[ \begin{array} { c } - 6
y \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 2 }\)
Find the value of \(y\)
Circle your answer.
[0pt] [1 mark] 231018

15 A particle is moving in a straight line such that its velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), changes with respect to time, \(t\) seconds, as shown in the graph below.
\includegraphics[max width=\textwidth, alt={}, center]{9cd7f38d-a2a1-4fd3-9ed9-cb389e8ee3b6-18_501_889_406_667} 15

1. Show that the acceleration of the particle over the first 4 seconds is \(3.5 \mathrm {~ms} ^ { - 2 }\) 15
2. The particle is initially at a fixed point \(P\)
   Show that the displacement of the particle from \(P\), when \(t = 9\), is 62 metres.

16 A toy remote control speed boat is launched from one edge of a small pond and moves in a straight line across the pond's surface. The boat's velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), is modelled in terms of time, \(t\) seconds after the boat is launched, by the expression $$v = 0.9 + 0.16 t - 0.06 t ^ { 2 }$$ 16

1. Find the acceleration of the boat when \(t = 2\)
   16
2. Find the displacement of the boat, from the point where it was launched, when \(t = 2\)

17 A particle, \(P\), is initially at rest on a smooth horizontal surface. A resultant force of \(\left[ \begin{array} { c } 12
9 \end{array} \right] \mathrm { N }\) is then applied to \(P\), so that it moves in a straight line.
17

1. Find the magnitude of the resultant force. 17
2. Two fixed points \(A\) and \(B\) have position vectors $$\overrightarrow { O A } = \left[ \begin{array} { l } 3
   7 \end{array} \right] \text { metres } \quad \text { and } \quad \overrightarrow { O B } = \left[ \begin{array} { c } k
   k - 1 \end{array} \right] \text { metres }$$ with respect to a fixed origin, \(O\)
   \(P\) moves in a straight line parallel to \(\overrightarrow { A B }\)
   17
3. 1. Find \(\overrightarrow { A B }\) in terms of \(k\) 17
4. (ii) Find the value of \(k\)

18 A rescue van is towing a broken-down car by using a tow bar. The van and the car are moving with a constant acceleration of \(0.6 \mathrm {~ms} ^ { - 2 }\) along a straight horizontal road as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{9cd7f38d-a2a1-4fd3-9ed9-cb389e8ee3b6-22_77_163_511_991} The van has a total mass of 2780 kg
The car has a total mass of 1620 kg
The van experiences a driving force of \(D\) newtons. The van experiences a total resistance force of \(R\) newtons.
The car experiences a total resistance force of \(0.6 R\) newtons. 18

1. The tension in the tow bar, \(T\) newtons, may be modelled by $$T = k D - 18$$ where \(k\) is a constant. Find \(k\)
   18
2. State one assumption that must be made in answering part (a).