AQA Paper 2 (Paper 2) Specimen

State the values of \(|x|\) for which the binomial expansion of \((3 + 2x)^{-4}\) is valid. Circle your answer. [1 mark] \(|x| < \frac{2}{3}\) \(\quad\) \(|x| < 1\) \(\quad\) \(|x| < \frac{3}{2}\) \(\quad\) \(|x| < 3\)

A zoologist is investigating the growth of a population of red squirrels in a forest. She uses the equation \(N = \frac{200}{1 + 9e^{-\frac{t}{5}}}\) as a model to predict the number of squirrels, \(N\), in the population \(t\) weeks after the start of the investigation. What is the size of the squirrel population at the start of the investigation? Circle your answer. [1 mark] \(5\) \(\quad\) \(20\) \(\quad\) \(40\) \(\quad\) \(200\)

A curve is defined by the parametric equations $$x = t^3 + 2, \quad y = t^2 - 1$$

1. Find the gradient of the curve at the point where \(t = -2\) [4 marks]
2. Find a Cartesian equation of the curve. [2 marks]

The equation \(x^3 - 3x + 1 = 0\) has three real roots.

1. Show that one of the roots lies between \(-2\) and \(-1\) [2 marks]
2. Taking \(x_1 = -2\) as the first approximation to one of the roots, use the Newton-Raphson method to find \(x_2\), the second approximation. [3 marks]
3. Explain why the Newton-Raphson method fails in the case when the first approximation is \(x_1 = -1\) [1 mark]

1. Determine a sequence of transformations which maps the graph of \(y = \cos \theta\) onto the graph of \(y = 3\cos \theta + 3\sin \theta\) Fully justify your answer. [6 marks]
2. Hence or otherwise find the least value and greatest value of $$4 + (3\cos \theta + 3\sin \theta)^2$$ Fully justify your answer. [3 marks]

A curve \(C\) has equation \(y = x^2 - 4x + k\), where \(k\) is a constant. It crosses the \(x\)-axis at the points \((2 + \sqrt{5}, 0)\) and \((2 - \sqrt{5}, 0)\)

1. Find the value of \(k\). [2 marks]
2. Sketch the curve \(C\), labelling the exact values of all intersections with the axes. [3 marks]

A student notices that when he adds two consecutive odd numbers together the answer always seems to be the difference between two square numbers. He claims that this will always be true. He attempts to prove his claim as follows: Step 1: Check first few cases \(3 + 5 = 8\) and \(8 = 3^2 - 1^2\) \(5 + 7 = 12\) and \(12 = 4^2 - 2^2\) \(7 + 9 = 16\) and \(16 = 5^2 - 3^2\) Step 2: Use pattern to predict and check a large example \(101 + 103 = 204\) subtract 1 and divide by 2 for the first number Add 1 and divide by two for the second number \(52^2 - 50^2 = 204\) it works! Step 3: Conclusion The first few cases work and there is a pattern, which can be used to predict larger numbers. Therefore, it must be true for all consecutive odd numbers.

1. Explain what is wrong with the student's "proof". [1 mark]
2. Prove that the student's claim is correct. [3 marks]

A curve has equation \(y = 2x \cos 3x + (3x^2 - 4) \sin 3x\)

1. Find \(\frac{dy}{dx}\), giving your answer in the form \((mx^2 + n) \cos 3x\), where \(m\) and \(n\) are integers. [4 marks]
2. Show that the \(x\)-coordinates of the points of inflection of the curve satisfy the equation $$\cot 3x = \frac{9x^2 - 10}{6x}$$ [4 marks]

1. Three consecutive terms in an arithmetic sequence are \(3e^{-q}\), \(5\), \(3e^q\) Find the possible values of \(p\). Give your answers in an exact form. [6 marks]
2. Prove that there is no possible value of \(q\) for which \(3e^{-q}\), \(5\), \(3e^q\) are consecutive terms of a geometric sequence. [4 marks]

A single force of magnitude 4 newtons acts on a particle of mass 50 grams. Find the magnitude of the acceleration of the particle. Circle your answer. [1 mark] \(12.5 \text{ m s}^{-2}\) \(\quad\) \(0.08 \text{ m s}^{-2}\) \(\quad\) \(0.0125 \text{ m s}^{-2}\) \(\quad\) \(80 \text{ m s}^{-2}\)

A uniform rod, \(AB\), has length 3 metres and mass 24 kg. A particle of mass \(M\) kg is attached to the rod at \(A\). The rod is balanced in equilibrium on a support at \(C\), which is 0.8 metres from \(A\). \includegraphics{figure_11} Find the value of \(M\). [2 marks]

A particle moves on a straight line with a constant acceleration, \(a\) m s\(^{-2}\). The initial velocity of the particle is \(U\) m s\(^{-1}\). After \(T\) seconds the particle has velocity \(V\) m s\(^{-1}\). This information is shown on the velocity-time graph. \includegraphics{figure_12} The displacement, \(S\) metres, of the particle from its initial position at time \(T\) seconds is given by the formula $$S = \frac{1}{2}(U + V)T$$

1. By considering the gradient of the graph, or otherwise, write down a formula for \(a\) in terms of \(U\), \(V\) and \(T\). [1 mark]
2. Hence show that \(V^2 = U^2 + 2aS\) [3 marks]

The three forces \(\mathbf{F_1}\), \(\mathbf{F_2}\) and \(\mathbf{F_3}\) are acting on a particle. \(\mathbf{F_1} = (25\mathbf{i} + 12\mathbf{j})\) N \(\mathbf{F_2} = (-7\mathbf{i} + 5\mathbf{j})\) N \(\mathbf{F_3} = (15\mathbf{i} - 28\mathbf{j})\) N The unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal and vertical respectively. The resultant of these three forces is \(\mathbf{F}\) newtons.

The fourth force, \(\mathbf{F_4}\), is applied to the particle so that the four forces are in equilibrium. Find \(\mathbf{F_4}\), giving your answer in terms of \(\mathbf{i}\) and \(\mathbf{j}\). [1 mark]

The graph below models the velocity of a small train as it moves on a straight track for 20 seconds. The front of the train is at the point \(A\) when \(t = 0\) The mass of the train is 800kg. \includegraphics{figure_14}

1. Find the total distance travelled in the 20 seconds. [3 marks]
2. Find the distance of the front of the train from the point \(A\) at the end of the 20 seconds. [1 mark]
3. Find the maximum magnitude of the resultant force acting on the train. [2 marks]
4. Explain why, in reality, the graph may not be an accurate model of the motion of the train. [1 mark]

At time \(t = 0\), a parachutist jumps out of an airplane that is travelling horizontally. The velocity, \(\mathbf{v}\) m s\(^{-1}\), of the parachutist at time \(t\) seconds is given by: $$\mathbf{v} = (40e^{-0.2t})\mathbf{i} + 50(e^{-0.2t} - 1)\mathbf{j}$$ The unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal and vertical respectively. Assume that the parachutist is at the origin when \(t = 0\) Model the parachutist as a particle.

1. Find an expression for the position vector of the parachutist at time \(t\). [4 marks]
2. The parachutist opens her parachute when she has travelled 100 metres horizontally. Find the vertical displacement of the parachutist from the origin when she opens her parachute. [4 marks]
3. Carefully, explaining the steps that you take, deduce the value of \(g\) used in the formulation of this model. [3 marks]

In this question use \(g = 9.8\) m s\(^{-2}\). The diagram shows a box, of mass 8.0 kg, being pulled by a string so that the box moves at a constant speed along a rough horizontal wooden board. The string is at an angle of 40° to the horizontal. The tension in the string is 50 newtons. \includegraphics{figure_16a} The coefficient of friction between the box and the board is \(\mu\) Model the box as a particle.

1. Show that \(\mu = 0.83\) [4 marks]
2. One end of the board is lifted up so that the board is now inclined at an angle of 5° to the horizontal. The box is pulled up the inclined board. The string remains at an angle of 40° to the board. The tension in the string is increased so that the box accelerates up the board at 3 m s\(^{-2}\) \includegraphics{figure_16b}
   1. Draw a diagram to show the forces acting on the box as it moves. [1 mark]
   2. Find the tension in the string as the box accelerates up the slope at 3 m s\(^{-2}\). [7 marks]

In this question use \(g = 9.81\) m s\(^{-2}\). A ball is projected from the origin. After 2.5 seconds, the ball lands at the point with position vector \((40\mathbf{i} - 10\mathbf{j})\) metres. The unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal and vertical respectively. Assume that there are no resistance forces acting on the ball.

1. Find the speed of the ball when it is at a height of 3 metres above its initial position. [6 marks]
2. State the speed of the ball when it is at its maximum height. [1 mark]
3. Explain why the answer you found in part (b) may not be the actual speed of the ball when it is at its maximum height. [1 mark]