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AQA AS Paper 1 2021 June Q14
6 marks Moderate -0.3
14 A particle, P , is moving along a straight line such that its acceleration \(a \mathrm {~ms} ^ { - 2 }\), at any time, \(t\) seconds, may be modelled by $$a = 3 + 0.2 t$$ When \(t = 2\), the velocity of P is \(k \mathrm {~m} \mathrm {~s} ^ { - 1 }\) 14
  1. Show that the initial velocity of P is given by the expression \(( k - 6.4 ) \mathrm { ms } ^ { - 1 }\) [0pt] [4 marks]
    14
  2. The initial velocity of P is one fifth of the velocity when \(t = 2\) Find the value of \(k\).
AQA AS Paper 1 2021 June Q15
10 marks Moderate -0.5
15 In this question, use \(g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) A box, B, of mass 4 kg lies at rest on a fixed rough horizontal shelf.
One end of a light string is connected to B .
The string passes over a smooth peg, attached to the end of the shelf.
The other end of the string is connected to particle, P , of mass 1 kg , which hangs freely below the shelf as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{1f887565-4587-4520-99d4-f3635b015525-22_778_910_760_566} B is initially held at rest with the string taut.
B is then released.
B and P both move with constant acceleration \(a \mathrm {~ms} ^ { - 2 }\) As B moves across the shelf it experiences a total resistance force of 5 N
15
  1. State one type of force that would be included in the total resistance force. 15
  2. Show that \(a = 1\) 15
  3. When B has moved forward exactly 20 cm the string breaks.
    Find how much further B travels before coming to rest.
    15
  4. State one assumption you have made when finding your solutions in parts (b) or (c). [1 mark]
AQA AS Paper 1 2022 June Q1
1 marks Easy -1.8
1 Express as a single logarithm $$\log _ { 10 } 2 - \log _ { 10 } x$$ Circle your answer.
[0pt] [1 mark] \(\log _ { 10 } ( 2 + x ) \quad \log _ { 10 } ( 2 - x ) \quad \log _ { 10 } ( 2 x ) \quad \log _ { 10 } \left( \frac { 2 } { x } \right)\)
AQA AS Paper 1 2022 June Q2
1 marks Easy -1.3
2 The graph of the function \(y = \cos \frac { 1 } { 2 } x\) for \(0 ^ { \circ } \leq x \leq 360 ^ { \circ }\) is one of the graphs shown below. Identify the correct graph.
Tick ( \(\checkmark\) ) one box.
[0pt] [1 mark] \includegraphics[max width=\textwidth, alt={}, center]{46d846f7-dbc6-4fd5-8e1f-bcc50cad3418-03_373_634_671_502}
□ \includegraphics[max width=\textwidth, alt={}, center]{46d846f7-dbc6-4fd5-8e1f-bcc50cad3418-03_387_634_1133_502} \includegraphics[max width=\textwidth, alt={}, center]{46d846f7-dbc6-4fd5-8e1f-bcc50cad3418-03_113_111_1265_1306} \includegraphics[max width=\textwidth, alt={}, center]{46d846f7-dbc6-4fd5-8e1f-bcc50cad3418-03_366_629_1610_502}
□ \includegraphics[max width=\textwidth, alt={}, center]{46d846f7-dbc6-4fd5-8e1f-bcc50cad3418-03_368_629_2074_502}
□
AQA AS Paper 1 2022 June Q3
3 marks Easy -1.2
3 Find the coefficient of the \(x ^ { 3 }\) term in the expansion of \(\left( 3 x + \frac { 1 } { 2 } \right) ^ { 4 }\)
AQA AS Paper 1 2022 June Q4
5 marks Moderate -0.3
4 Find all the solutions of the equation $$\cos ^ { 2 } \theta = 10 \sin \theta + 4$$ for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\), giving your answers to the nearest degree.
Fully justify your answer.
AQA AS Paper 1 2022 June Q5
3 marks Easy -1.2
5 Express \(3 x ^ { 3 } + 5 x ^ { 2 } - 27 x + 10\) in the form \(( x - 2 ) \left( a x ^ { 2 } + b x + c \right)\), where \(a , b\) and \(c\) are integers.
[0pt] [3 marks] \includegraphics[max width=\textwidth, alt={}, center]{46d846f7-dbc6-4fd5-8e1f-bcc50cad3418-07_2488_1716_219_153} \(6 \quad A B\) is a diameter of a circle where \(A\) is \(( 1,4 )\) and \(B\) is \(( 7 , - 2 )\)
AQA AS Paper 1 2022 June Q6
9 marks Moderate -0.8
6
  1. Find the coordinates of the midpoint of \(A B\). 6
  2. Show that the equation of the circle may be written as $$x ^ { 2 } + y ^ { 2 } - 8 x - 2 y = 1$$ 6
  3. \(\quad\) The circle has centre \(C\) and crosses the \(x\)-axis at points \(D\) and \(E\). Find the exact area of triangle \(D E C\). 6
  4. The circle has centre \(C\) and crosses the \(x\)-axis at points \(D\) and \(E\).
    The area enclosed between the curve and the \(x\)-axis is 36 units.
    Find the value of \(a\).
    Fully justify your answer.
    [0pt] [6 marks] \(7 \quad\) A curve has equation \(y = a ^ { 2 } - x ^ { 2 }\), where \(a > 0\)
AQA AS Paper 1 2022 June Q8
11 marks Standard +0.3
8 A curve has equation $$y = x ^ { 3 } - 6 x + \frac { 9 } { x }$$ 8
  1. Show that the \(x\) coordinates of the stationary points of the curve satisfy the equation $$x ^ { 4 } - 2 x ^ { 2 } - 3 = 0$$ 8
  2. Deduce that the curve has exactly two stationary points.
    8
  3. Find the coordinates and nature of the two stationary points. Fully justify your answer.
    [0pt] [4 marks]
    8
  4. Write down the equation of a line which is a tangent to the curve in two places.
AQA AS Paper 1 2022 June Q10
9 marks Standard +0.8
10 Curve \(C\) has equation \(y = \frac { \sqrt { 2 } } { x ^ { 2 } }\) 10
  1. Find an equation of the tangent to \(C\) at the point \(\left( 2 , \frac { \sqrt { 2 } } { 4 } \right)\) 10
  2. Show that the tangent to \(C\) at the point \(\left( 2 , \frac { \sqrt { 2 } } { 4 } \right)\) is also a normal to the curve at a different point. \includegraphics[max width=\textwidth, alt={}, center]{46d846f7-dbc6-4fd5-8e1f-bcc50cad3418-16_588_978_1969_532}
AQA AS Paper 1 2022 June Q11
1 marks Easy -1.8
11 A car, initially at rest, moves with constant acceleration along a straight horizontal road. One of the graphs below shows how the car's velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), changes over time, \(t\) seconds. Identify the correct graph.
Tick ( \(\checkmark\) ) one box.
[0pt] [1 mark] \includegraphics[max width=\textwidth, alt={}, center]{46d846f7-dbc6-4fd5-8e1f-bcc50cad3418-18_271_296_1219_495} \includegraphics[max width=\textwidth, alt={}, center]{46d846f7-dbc6-4fd5-8e1f-bcc50cad3418-18_122_140_1290_982} \includegraphics[max width=\textwidth, alt={}, center]{46d846f7-dbc6-4fd5-8e1f-bcc50cad3418-18_270_298_1583_495} □ \includegraphics[max width=\textwidth, alt={}, center]{46d846f7-dbc6-4fd5-8e1f-bcc50cad3418-18_277_305_1946_488} □ \includegraphics[max width=\textwidth, alt={}, center]{46d846f7-dbc6-4fd5-8e1f-bcc50cad3418-18_280_305_2311_493} □
AQA AS Paper 1 2022 June Q12
1 marks Easy -1.8
12 A horizontal force of 30 N causes a crate to travel with an acceleration of \(2 \mathrm {~ms} ^ { - 2 }\), in a straight line, on a smooth horizontal surface. Find the weight of the crate.
Circle your answer. 15 kg \(15 g \mathrm {~N} 15 \mathrm {~N} 15 g\) kg
AQA AS Paper 1 2022 June Q13
3 marks Moderate -0.8
13 Two points \(A\) and \(B\) lie in a horizontal plane and have coordinates ( \(- 2,7\) ) and ( 3,19 ) respectively. A particle moves in a straight line from \(A\) to \(B\) under the action of a constant resultant force of magnitude 6.5 N Express the resultant force in vector form.
[0pt] [3 marks]
AQA AS Paper 1 2022 June Q14
3 marks Moderate -0.8
14 A ball is released from rest from a height \(h\) metres above horizontal ground and falls freely downwards. When the ball reaches the ground, its speed is \(v \mathrm {~ms} ^ { - 1 }\), where \(v \leq 10\) Show that $$h \leq \frac { 50 } { g }$$ \includegraphics[max width=\textwidth, alt={}, center]{46d846f7-dbc6-4fd5-8e1f-bcc50cad3418-21_2488_1728_219_141} \begin{center} \begin{tabular}{|l|l|l|l|} \hline \multicolumn{3}{|l|}{\begin{tabular}{l}
AQA AS Paper 1 2022 June Q15
5 marks Moderate -0.8
15
15
\end{tabular}} &
Two particles, \(P\) and \(Q\), are initially at rest at the same point on a horizontal plane.
A force of \(\left[ \begin{array} { l } 4
0 \end{array} \right] \mathrm { N }\) is applied to \(P\).
A force of \(\left[ \begin{array} { c } 8
15 \end{array} \right] \mathrm { N }\) is applied to \(Q\).
Calculate, to the nearest degree, the acute angle between the two forces.
[2 marks] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

\hline & & &
\hline \end{tabular} \end{center}
\includegraphics[max width=\textwidth, alt={}]{46d846f7-dbc6-4fd5-8e1f-bcc50cad3418-23_2495_1719_219_150}
AQA AS Paper 1 2022 June Q16
6 marks Moderate -0.3
16 Jermaine and his friend Meena are walking in the same direction along a straight path. Meena is walking at a constant speed of \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Jermaine is walking \(0.2 \mathrm {~ms} ^ { - 1 }\) more slowly than Meena.
When Jermaine is \(d\) metres behind Meena he starts to run with a constant acceleration of \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), for a time of \(t\) seconds, until he reaches her. 16
  1. Show that $$d = t ^ { 2 } - 0.2 t$$ 16
  2. When Jermaine's speed is \(7.8 \mathrm {~ms} ^ { - 1 }\), he reaches Meena. Given that \(u = 1.4\) find the value of \(d\).
AQA AS Paper 1 2022 June Q17
7 marks Moderate -0.5
17 \includegraphics[max width=\textwidth, alt={}, center]{46d846f7-dbc6-4fd5-8e1f-bcc50cad3418-26_270_1036_274_552} A car and caravan, connected by a tow bar, move forward together along a horizontal road. Their velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t\) seconds, for \(0 \leq t < 20\), is given by $$v = 0.5 t + 0.01 t ^ { 2 }$$ 17
  1. Show that when \(t = 15\) their acceleration is \(0.8 \mathrm {~ms} ^ { - 2 }\) 17
  2. The car has a mass of 1500 kg
    The caravan has a mass of 850 kg
    When \(t = 15\) the tension in the tow bar is 800 N and the car experiences a resistance force of 100 N 17
    1. Find the total resistance force experienced by the caravan when \(t = 15\) 17
  4. (ii) Find the driving force being applied by the car when \(t = 15\) [0pt] [3 marks]
    17
  5. State one assumption you have made about the tow bar. \includegraphics[max width=\textwidth, alt={}, center]{46d846f7-dbc6-4fd5-8e1f-bcc50cad3418-28_2492_1721_217_150}
    \includegraphics[max width=\textwidth, alt={}]{46d846f7-dbc6-4fd5-8e1f-bcc50cad3418-32_2486_1719_221_150}
AQA AS Paper 1 2023 June Q1
1 marks Easy -1.8
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
AQA AS Paper 1 2023 June Q2
1 marks Easy -1.8
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 \(\frac { 20 } { 3 }\)
AQA AS Paper 1 2023 June Q3
3 marks Easy -1.2
3 The coefficient of \(x ^ { 2 }\) in the binomial expansion of \(( 1 + a x ) ^ { 6 }\) is \(\frac { 20 } { 3 }\)
AQA AS Paper 1 2023 June Q4
5 marks Moderate -0.3
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
AQA AS Paper 1 2023 June Q5
7 marks Easy -1.2
5 (b) & 5 (a) Given that \(y = x \sqrt { x }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) \hline \end{tabular} \end{center}
AQA AS Paper 1 2023 June Q6
6 marks Moderate -0.8
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 \(x + h\)
AQA AS Paper 1 2023 June Q7
5 marks Moderate -0.8
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]
AQA AS Paper 1 2023 June Q8
7 marks Moderate -0.3
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.