AQA AS Paper 1 (AS Paper 1) 2022 June

1 Express as a single logarithm $$\log _ { 10 } 2 - \log _ { 10 } x$$ Circle your answer.
[0pt] [1 mark]
\(\log _ { 10 } ( 2 + x ) \quad \log _ { 10 } ( 2 - x ) \quad \log _ { 10 } ( 2 x ) \quad \log _ { 10 } \left( \frac { 2 } { x } \right)\)

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}
□

3 Find the coefficient of the \(x ^ { 3 }\) term in the expansion of \(\left( 3 x + \frac { 1 } { 2 } \right) ^ { 4 }\)

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.

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 \(( 7 , - 2 )\)

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 \(a > 0\)

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.

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}

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} □

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

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]

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}

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}

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 \(d\).

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
3. 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}