Questions AS Paper 1 (363 questions)

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AQA AS Paper 1 2019 June Q2
1 marks
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]
\(n = 3\)
\(n = 4\)
\(n = 5\)
\(n = 6\)
AQA AS Paper 1 2019 June Q3
3 marks
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.}
AQA AS Paper 1 2019 June Q5
5
  1. Sketch the curve \(y = \mathrm { g } ( x )\) where $$g ( x ) = ( x + 2 ) ( x - 1 ) ^ { 2 }$$ 5
  2. Hence, solve \(\mathrm { g } ( x ) \leq 0\)
AQA AS Paper 1 2019 June Q6
6
    1. Show that \(\cos \theta = \frac { 1 } { 2 }\) is one solution of the equation $$6 \sin ^ { 2 } \theta + 5 \cos \theta = 7$$ 6
  1. (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
  2. 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.
AQA AS Paper 1 2019 June Q7
6 marks
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]
AQA AS Paper 1 2019 June Q8
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.
AQA AS Paper 1 2019 June Q9
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 )
AQA AS Paper 1 2019 June Q10
1 marks
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. State the value that Jude should use for \(A\).
      10
  1. (ii) State the value that Jude should use for \(B\).
    10
  2. (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
  3. (iv) Using Jude's model, find how many days during 2019 will have at least 17.4 hours of daylight in Inverness.
    10
  4. (v) Explain why Jude's model will become inaccurate for 2020 and future years.
    10
  5. 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.
AQA AS Paper 1 2019 June Q11
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 \(m\) □
\includegraphics[max width=\textwidth, alt={}, center]{9f84ae5b-15d9-40e7-bdc2-a7a8715082b4-15_2488_1716_219_153}
AQA AS Paper 1 2019 June Q13
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.
AQA AS Paper 1 2019 June Q14
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
14

  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
    1. Show that the acceleration of \(B\) towards \(A\) is \(16.25 \mathrm {~ms} ^ { - 2 }\) 14
  3. (ii) Hence, find the time taken for \(B\) to reach \(A\).
    Give your answer to two significant figures.
AQA AS Paper 1 2019 June Q15
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 \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
15
  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 \(9 \mathrm {~km} \mathrm {~h} ^ { - 1 }\)
    \includegraphics[max width=\textwidth, alt={}, center]{9f84ae5b-15d9-40e7-bdc2-a7a8715082b4-22_2488_1719_219_150}
    \includegraphics[max width=\textwidth, alt={}, center]{9f84ae5b-15d9-40e7-bdc2-a7a8715082b4-23_2488_1719_219_150}
    \includegraphics[max width=\textwidth, alt={}, center]{9f84ae5b-15d9-40e7-bdc2-a7a8715082b4-24_2498_1721_213_148}
AQA AS Paper 1 2020 June Q1
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 □
AQA AS Paper 1 2020 June Q2
2 Given that \(\mathrm { f } ( x ) = 10\) when \(x = 4\), which statement below must be correct?
Tick \(( \checkmark )\) one box. $$\begin{aligned} & \mathrm { f } ( 2 x ) = 5 \text { when } x = 4
& \mathrm { f } ( 2 x ) = 10 \text { when } x = 2
& \mathrm { f } ( 2 x ) = 10 \text { when } x = 8
& \mathrm { f } ( 2 x ) = 20 \text { when } x = 4 \end{aligned}$$ □



AQA AS Paper 1 2020 June Q3
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 \(- 180 ^ { \circ } \leq \theta \leq 180 ^ { \circ }\)
AQA AS Paper 1 2020 June Q4
4 In the binomial expansion of \(( \sqrt { } 3 + \sqrt { } 2 ) ^ { 4 }\) there are two irrational terms. Find the difference between these two terms.
AQA AS Paper 1 2020 June Q5
5 Differentiate from first principles $$y = 4 x ^ { 2 } + x$$
AQA AS Paper 1 2020 June Q6
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 \(L _ { 1 }\)
AQA AS Paper 1 2020 June Q8
3 marks
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]
AQA AS Paper 1 2020 June Q9
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.
AQA AS Paper 1 2020 June Q10
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.
    \(\boldsymbol { C }\)0.601.151.50
    \(\boldsymbol { P }\)49512001720
    10
    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
  3. (ii) Explain which feature of the plot suggests that Raj's belief may be correct.
    10
  4. (iii) Using the graph on page 15 , estimate the value of \(a\) and the value of \(n\). 10
  5. Explain the significance of \(a\) in this context.
    10
  6. Raj wants to buy a ring with a brilliant-cut diamond of weight 2 carats. Estimate the price of such a ring.
AQA AS Paper 1 2020 June Q11
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
AQA AS Paper 1 2020 June Q12
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. $$\begin{array} { l l } \sqrt { 13 ^ { 2 } + 4 ^ { 2 } } & \square
\sqrt { 13 ^ { 2 } + 10 ^ { 2 } } & \square
\sqrt { 23 ^ { 2 } + 4 ^ { 2 } } & \square
\sqrt { 23 ^ { 2 } + 10 ^ { 2 } } & \square \end{array}$$
AQA AS Paper 1 2020 June Q13
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}
AQA AS Paper 1 2020 June Q15
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}