AQA AS Paper 1 (AS Paper 1) 2024 June

It is given that \(\tan \theta^\circ = k\), where \(k\) is a constant. Find \(\tan (\theta + 180)^\circ\) Circle your answer. [1 mark] \(-k\) \qquad \(-\frac{1}{k}\) \qquad \(\frac{1}{k}\) \qquad \(k\)

Curve \(C\) has equation \(y = \frac{1}{(x-1)^2}\) State the equations of the asymptotes to curve \(C\) Tick (\(\checkmark\)) one box. [1 mark] \(x = 0\) and \(y = 0\) \qquad \(\square\) \(x = 0\) and \(y = 1\) \qquad \(\square\) \(x = 1\) and \(y = 0\) \qquad \(\square\) \(x = 1\) and \(y = 1\) \qquad \(\square\)

Express \(\frac{\sqrt{3} + 3\sqrt{5}}{\sqrt{5} - \sqrt{3}}\) in the form \(a + b\sqrt{c}\), where \(a\) and \(b\) are integers. Fully justify your answer. [4 marks]

1. 1. By using a suitable trigonometric identity, show that the equation $$\sin \theta \tan \theta = 4 \cos \theta$$ can be written as $$\tan^2 \theta = 4$$ [1 mark]
   2. Hence solve the equation $$\sin \theta \tan \theta = 4 \cos \theta$$ where \(0^\circ < \theta < 360^\circ\) Give your answers to the nearest degree. [3 marks]
2. Deduce all solutions of the equation $$\sin 3\alpha \tan 3\alpha = 4 \cos 3\alpha$$ where \(0^\circ < \alpha < 180^\circ\) Give your answers to the nearest degree. [3 marks]

A student is looking for factors of the polynomial \(f(x)\) They suggest that \((x - 2)\) is a factor of \(f(x)\) The method they use to check this suggestion is to calculate \(f(-2)\) They correctly calculate that \(f(-2) = 0\) They conclude that their suggestion is correct.

1. Make one comment about the student's method. [1 mark]
2. Make two comments about the student's conclusion. [2 marks] 1 2

Determine the set of values of \(x\) which satisfy the inequality $$3x^2 + 3x > x + 6$$ Give your answer in exact form using set notation. [4 marks]

A triangular field of grass, \(ABC\), has boundaries with lengths as follows: $$AB = 234 \text{ m} \qquad BC = 225 \text{ m} \qquad AC = 310 \text{ m}$$ The field is shown in the diagram below. \includegraphics{figure_7}

1. Find angle \(A\) [2 marks]
2. Farmers calculate the number of sheep they can keep in a field, by allowing one sheep for every \(1200 \text{ m}^2\) of grass. Find the maximum number of sheep which can be kept in the field \(ABC\) [3 marks]

It is given that $$\ln x - \ln y = 3$$

1. Express \(x\) in terms of \(y\) in a form not involving logarithms. [3 marks]
2. Given also that $$x + y = 10$$ find the exact value of \(y\) and the exact value of \(x\) [3 marks]

A curve has equation \(y = f(x)\) where $$f(x) = x(6 - x)$$

1. Find \(f'(x)\) [2 marks]
2. The diagram below shows the graph of \(y = f(x)\) On the same diagram sketch the gradient function for this curve, stating the coordinates of any points where the gradient function cuts the axes. [3 marks] \includegraphics{figure_9}

It is given that $$\frac{\mathrm{d}y}{\mathrm{d}x} = (x + 2)(2x - 1)^2$$ and when \(x = 6\), \(y = 900\) Find \(y\) in terms of \(x\) [6 marks]

It is given that for the continuous function \(g\) • \(g'(1) = 0\) • \(g'(4) = 0\) • \(g''(x) = 2x - 5\)

1. Determine the nature of each of the turning points of \(g\) Fully justify your answer. [3 marks]
2. Find the set of values of \(x\) for which \(g\) is an increasing function. [2 marks]

The monthly mean temperature of a city, \(T\) degrees Celsius, may be modelled by the equation $$T = 15 + 8 \sin (30m - 120)^\circ$$ where \(m\) is the month number, counting January = 1, February = 2, through to December = 12

1. Using this model, calculate the monthly mean temperature of the city for May, the fifth month. [2 marks]
2. Using this model, find the month with the highest mean temperature. [2 marks]
3. Climate change may affect the parameters, 8, 30, 120 and 15, used in this model.
   1. State, with a reason, which parameter would be increased because of an overall rise in temperatures. [1 mark]
   2. State, with a reason, which parameter would be increased because of the occurrence of more extreme temperatures. [1 mark]

A particle is moving in a straight line with constant acceleration \(a\) m s\(^{-2}\) The particle's velocity, \(v\) m s\(^{-1}\), varies with time, \(t\) seconds, so that $$v = 3 - 4t$$ Deduce the value of \(a\) Circle your answer. [1 mark] \(-4\) \qquad \(-1\) \qquad \(3\) \qquad \(4\)

Two forces, \(\mathbf{F}_1 = 3\mathbf{i} + 2\mathbf{j}\) newtons and \(\mathbf{F}_2 = \mathbf{i} - 3\mathbf{j}\) newtons, are added together to find a resultant force, \(\mathbf{R}\) newtons. This vector addition can be represented using a diagram. Identify the diagram below which correctly represents this vector addition. Tick (\(\checkmark\)) one box. [1 mark] \includegraphics{figure_14}

A graph indicating how the velocity, \(v\) m s\(^{-1}\), of a particle changes with respect to time, \(t\) seconds, is shown below. \includegraphics{figure_15}

1. Find the total distance travelled by the particle over the 8 second period shown. [3 marks]
2. A student claims that "The displacement of the particle is less than the distance travelled." State the range of values of \(t\) for which this claim is true. [1 mark]

In this question use \(g = 9.8\) m s\(^{-2}\) A ball is launched vertically upwards from the Earth's surface with velocity \(u\) m s\(^{-1}\) The ball reaches a maximum height of 15 metres. You may assume that air resistance can be ignored. Find the value of \(u\) [3 marks]

A particle moves in a straight line with acceleration \(a\) m s\(^{-2}\), at time \(t\) seconds, where $$a = 10 - 6t$$ The particle's velocity, \(v\) m s\(^{-1}\), and displacement, \(r\) metres, are both initially zero. Show that $$r = t^2(5 - t)$$ Fully justify your answer. [4 marks]

It is given that two points \(A\) and \(B\) have position vectors $$\overrightarrow{OA} = \begin{bmatrix} 5 \\ -1 \end{bmatrix} \text{ metres} \quad \text{and} \quad \overrightarrow{OB} = \begin{bmatrix} 13 \\ 5 \end{bmatrix} \text{ metres.}$$

1. Show that the distance from \(A\) to \(B\) is 10 metres. [3 marks]
2. A constant resultant force, of magnitude \(R\) newtons, acts on a particle so that it moves in a straight line passing through the same two points \(A\) and \(B\) At \(A\), the speed of the particle is 3 m s\(^{-1}\) in the direction from \(A\) to \(B\) The particle takes 2 seconds to travel from \(A\) to \(B\) The mass of the particle is 150 grams. Find the value of \(R\) [3 marks]

Two objects, \(M\) and \(N\), are connected by a light inextensible string that passes over a smooth peg. \(M\) has a mass of 0.6 kilograms. \(N\) has a mass of 0.5 kilograms. \(M\) and \(N\) are initially held at rest, with the string taut, as shown in the diagram below. \includegraphics{figure_19} \(M\) and \(N\) are released at the same instant and begin to move vertically. You may assume that air resistance can be ignored.

1. It is given that \(M\) and \(N\) move with acceleration \(a\) m s\(^{-2}\) By forming two equations of motion show that $$a = \frac{1}{11}g$$ [5 marks]
2. The speed of \(N\), 0.5 seconds after its release, is \(\frac{g}{k}\) m s\(^{-1}\) where \(k\) is a constant. Find the value of \(k\) [2 marks]
3. State one assumption that must be made for the answer in part (b) to be valid. [1 mark]