AQA AS Paper 1 (AS Paper 1) 2022 June

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

The graph of the function \(y = \cos \frac{1}{2}x\) for \(0° \leq x \leq 360°\) is one of the graphs shown below. Identify the correct graph. Tick (✓) one box. [1 mark] \includegraphics{figure_2}

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

Find all the solutions of the equation $$\cos^2 \theta = 10 \sin \theta + 4$$ for \(0° < \theta < 360°\), giving your answers to the nearest degree. Fully justify your answer. [5 marks]

Express \(3x^3 + 5x^2 - 27x + 10\) in the form \((x - 2)(ax^2 + bx + c)\), where \(a\), \(b\) and \(c\) are integers. [3 marks]

\(AB\) is a diameter of a circle where \(A\) is \((1, 4)\) and \(B\) is \((7, -2)\)

1. Find the coordinates of the midpoint of \(AB\). [1 mark]
2. Show that the equation of the circle may be written as $$x^2 + y^2 - 8x - 2y = 1$$ [4 marks]
3. The circle has centre \(C\) and crosses the \(x\)-axis at points \(D\) and \(E\). Find the exact area of triangle \(DEC\). [4 marks]

A curve has equation \(y = a^2 - x^2\), where \(a > 0\) The area enclosed between the curve and the \(x\)-axis is 36 units. Find the value of \(a\). Fully justify your answer. [6 marks]

A curve has equation $$y = x^3 - 6x + \frac{9}{x}$$

1. Show that the \(x\) coordinates of the stationary points of the curve satisfy the equation $$x^4 - 2x^2 - 3 = 0$$ [3 marks]
2. Deduce that the curve has exactly two stationary points. [3 marks]
3. Find the coordinates and nature of the two stationary points. Fully justify your answer. [4 marks]
4. Write down the equation of a line which is a tangent to the curve in two places. [1 mark]

Integers \(m\) and \(n\) are both odd. Prove that \(m^2 + n^2\) is a multiple of 2 but not a multiple of 4 [5 marks]

Curve \(C\) has equation \(y = \frac{\sqrt{2}}{x^2}\)

1. Find an equation of the tangent to \(C\) at the point \(\left(2, \frac{\sqrt{2}}{4}\right)\) [4 marks]
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{figure_10} [5 marks]

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, \(v\) m s\(^{-1}\), changes over time, \(t\) seconds. Identify the correct graph. Tick (✓) one box. [1 mark] \includegraphics{figure_11}

A horizontal force of 30 N causes a crate to travel with an acceleration of 2 m s\(^{-2}\), in a straight line, on a smooth horizontal surface. Find the weight of the crate. Circle your answer. [1 mark] 15 kg \quad 15g N \quad 15 N \quad 15g kg

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. [3 marks]

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 \(v\) m s\(^{-1}\), where \(v \leq 10\) Show that $$h \leq \frac{50}{g}$$ [3 marks]

Two particles, \(P\) and \(Q\), are initially at rest at the same point on a horizontal plane. A force of \(\begin{bmatrix} 4 \\ 0 \end{bmatrix}\) N is applied to \(P\). A force of \(\begin{bmatrix} 8 \\ 15 \end{bmatrix}\) N is applied to \(Q\).

1. Calculate, to the nearest degree, the acute angle between the two forces. [2 marks]
2. The particles begin to move under the action of the respective forces. \(P\) and \(Q\) have the same mass. \(P\) has an acceleration of magnitude 5 m s\(^{-2}\) Find the magnitude of the acceleration of \(Q\). [3 marks]

Jermaine and his friend Meena are walking in the same direction along a straight path. Meena is walking at a constant speed of \(u\) m s\(^{-1}\) Jermaine is walking 0.2 m s\(^{-1}\) more slowly than Meena. When Jermaine is \(d\) metres behind Meena he starts to run with a constant acceleration of 2 m s\(^{-2}\), for a time of \(t\) seconds, until he reaches her.

1. Show that $$d = t^2 - 0.2t$$ [4 marks]
2. When Jermaine's speed is 7.8 m s\(^{-1}\), he reaches Meena. Given that \(u = 1.4\) find the value of \(d\). [2 marks]

\includegraphics{figure_17} A car and caravan, connected by a tow bar, move forward together along a horizontal road. Their velocity \(v\) m s\(^{-1}\) at time \(t\) seconds, for \(0 \leq t < 20\), is given by $$v = 0.5t + 0.01t^2$$

1. Show that when \(t = 15\) their acceleration is 0.8 m s\(^{-2}\) [2 marks]
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
   1. Find the total resistance force experienced by the caravan when \(t = 15\) [2 marks]
   2. Find the driving force being applied by the car when \(t = 15\) [3 marks]
3. State one assumption you have made about the tow bar. [1 mark]