AQA AS Paper 1 (AS Paper 1) 2020 June

1 At the point ( 1,0 ) on the curve \(y = \ln x\), which statement below is correct? Tick ( \(\checkmark\) ) one box. The gradient is negative and decreasing □ The gradient is negative and increasing
\includegraphics[max width=\textwidth, alt={}, center]{091aecd0-d812-4a8f-8596-a1c91f3bae1c-02_109_109_995_1306} The gradient is positive and decreasing □ The gradient is positive and increasing □

2 Given that \(\mathrm { f } ( x ) = 10\) when \(x = 4\), which statement below must be correct?
Tick \(( \checkmark )\) one box. $$\begin{aligned} & \mathrm { f } ( 2 x ) = 5 \text { when } x = 4
& \mathrm { f } ( 2 x ) = 10 \text { when } x = 2
& \mathrm { f } ( 2 x ) = 10 \text { when } x = 8
& \mathrm { f } ( 2 x ) = 20 \text { when } x = 4 \end{aligned}$$ □
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3 Jia has to solve the equation $$2 - 2 \sin ^ { 2 } \theta = \cos \theta$$ where \(- 180 ^ { \circ } \leq \theta \leq 180 ^ { \circ }\)
Jia's working is as follows: $$\begin{gathered} 2 - 2 \left( 1 - \cos ^ { 2 } \theta \right) = \cos \theta
2 - 2 + 2 \cos ^ { 2 } \theta = \cos \theta
2 \cos ^ { 2 } \theta = \cos \theta
2 \cos \theta = 1
\cos \theta = 0.5
\theta = 60 ^ { \circ } \end{gathered}$$ Jia's teacher tells her that her solution is incomplete.
3

1. Explain the two errors that Jia has made.
   3
2. Write down all the values of \(\theta\) that satisfy the equation $$2 - 2 \sin ^ { 2 } \theta = \cos \theta$$ where \(- 180 ^ { \circ } \leq \theta \leq 180 ^ { \circ }\)

4 In the binomial expansion of \(( \sqrt { } 3 + \sqrt { } 2 ) ^ { 4 }\) there are two irrational terms. Find the difference between these two terms.

5 Differentiate from first principles $$y = 4 x ^ { 2 } + x$$

6

1. It is given that $$f ( x ) = x ^ { 3 } - x ^ { 2 } + x - 6$$ Use the factor theorem to show that \(( x - 2 )\) is a factor of \(\mathrm { f } ( x )\).
   6
2. Find the quadratic factor of \(\mathrm { f } ( x )\).
   6
3. Hence, show that there is only one real solution to \(\mathrm { f } ( x ) = 0\)
   6
4. Find the exact value of \(x\) that solves $$\mathrm { e } ^ { 3 x } - \mathrm { e } ^ { 2 x } + \mathrm { e } ^ { x } - 6 = 0$$ \(7 \quad\) Curve \(C\) has equation \(y = x ^ { 2 }\)
   \(C\) is translated by vector \(\left[ \begin{array} { l } 3
   0 \end{array} \right]\) to give curve \(C _ { 1 }\)
   Line \(L\) has equation \(y = x\)
   \(L\) is stretched by scale factor 2 parallel to the \(x\)-axis to give line \(L _ { 1 }\)
   Find the exact distance between the two intersection points of \(C _ { 1 }\) and \(L _ { 1 }\)

8

1. Find the equation of the tangent to the curve \(y = \mathrm { e } ^ { 4 x }\) at the point ( \(a , \mathrm { e } ^ { 4 a }\) ).
   8
2. Find the value of \(a\) for which this tangent passes through the origin.
   8
3. Hence, find the set of values of \(m\) for which the equation
   has no real solutions. $$\mathrm { e } ^ { 4 x } = m x$$ has no real solutions.
   [0pt] [3 marks]

9 The diagram below shows a circle and four triangles.
\includegraphics[max width=\textwidth, alt={}, center]{091aecd0-d812-4a8f-8596-a1c91f3bae1c-12_974_739_356_648}
\(A B\) is a diameter of the circle. \(C\) is a point on the circumference of the circle.
Triangles \(A B K , B C L\) and \(C A M\) are equilateral.
Prove that the area of triangle \(A B K\) is equal to the sum of the areas of triangle \(B C L\) and triangle CAM.

10 Raj is investigating how the price, \(P\) pounds, of a brilliant-cut diamond ring is related to the weight, \(C\) carats, of the diamond. He believes that they are connected by a formula $$P = a C ^ { n }$$ where \(a\) and \(n\) are constants.
10

1. Express \(\ln P\) in terms of \(\ln C\).
   10
2. Raj researches the price of three brilliant-cut diamond rings on a website with the following results.

   \(\boldsymbol { C }\)	0.60	1.15	1.50
   \(\boldsymbol { P }\)	495	1200	1720

   10
3. 1. Plot \(\ln P\) against \(\ln C\) for the three rings on the grid below.
      \includegraphics[max width=\textwidth, alt={}, center]{091aecd0-d812-4a8f-8596-a1c91f3bae1c-15_1018_1467_751_283} 10
4. (ii) Explain which feature of the plot suggests that Raj's belief may be correct.
   10
5. (iii) Using the graph on page 15 , estimate the value of \(a\) and the value of \(n\). 10
6. Explain the significance of \(a\) in this context.
   10
7. Raj wants to buy a ring with a brilliant-cut diamond of weight 2 carats. Estimate the price of such a ring.

11 A go-kart and driver, of combined mass 55 kg , move forward in a straight line with a constant acceleration of \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) The total driving force is 14 N
Find the total resistance force acting on the go-kart and driver.
Circle your answer. 0 N 3 N 11 N 14 N

12 One of the following is an expression for the distance between the points represented by position vectors \(5 \mathbf { i } - 3 \mathbf { j }\) and \(18 \mathbf { i } + 7 \mathbf { j }\) Identify the correct expression.
Tick ( \(\checkmark\) ) one box. $$\begin{array} { l l } \sqrt { 13 ^ { 2 } + 4 ^ { 2 } } & \square
\sqrt { 13 ^ { 2 } + 10 ^ { 2 } } & \square
\sqrt { 23 ^ { 2 } + 4 ^ { 2 } } & \square
\sqrt { 23 ^ { 2 } + 10 ^ { 2 } } & \square \end{array}$$

13 An object is moving in a straight line, with constant acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), over a time period of \(t\) seconds. It has an initial velocity \(u\) and final velocity \(v\) as shown in the graph below.
\includegraphics[max width=\textwidth, alt={}, center]{091aecd0-d812-4a8f-8596-a1c91f3bae1c-19_606_784_493_628} Use the graph to show that $$v = u + a t$$ \includegraphics[max width=\textwidth, alt={}, center]{091aecd0-d812-4a8f-8596-a1c91f3bae1c-21_2488_1728_219_141}

15 A particle, \(P\), is moving in a straight line with acceleration \(a \mathrm {~ms} ^ { - 2 }\) at time \(t\) seconds, where $$a = 4 - 3 t ^ { 2 }$$ 15

1. Initially \(P\) is stationary.
   Find an expression for the velocity of \(P\) in terms of \(t\).
   
   \includegraphics[max width=\textwidth, alt={}]{091aecd0-d812-4a8f-8596-a1c91f3bae1c-23_2496_1723_214_148}

16 A simple lifting mechanism comprises a light inextensible wire which is passed over a smooth fixed pulley. One end of the wire is attached to a rigid triangular container of mass 2 kg , which rests on horizontal ground. A load of \(m \mathrm {~kg}\) is placed in the container.
The other end of the wire is attached to a particle of mass 5 kg , which hangs vertically downwards. The mechanism is initially held at rest as shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{091aecd0-d812-4a8f-8596-a1c91f3bae1c-24_858_677_943_683} The mechanism is released from rest, and the container begins to move upwards with acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) The wire remains taut throughout the motion. 16 (c) In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) The load reaches a height of 2 metres above the ground 1 second after it is released.
Find the mass of the load.
16 (d) Ignoring air resistance, describe one assumption you have made in your model.
[0pt] [1 mark]