AQA Paper 2 (Paper 2) 2020 June

Which one of these functions is decreasing for all real values of \(x\)? Circle your answer. \(f(x) = e^x\) \quad \(f(x) = -e^{1-x}\) \quad \(f(x) = -e^{x-1}\) \quad \(f(x) = -e^{-x}\) [1 mark]

Which one of the following equations has no real solutions? Tick (\(\checkmark\)) one box. \(\cot x = 0\) \(\ln x = 0\) \(|x + 1| = 0\) \(\sec x = 0\) [1 mark]

Find the coefficient of \(x^2\) in the binomial expansion of \(\left(2x - \frac{3}{x}\right)^8\) [3 marks]

Using small angle approximations, show that for small, non-zero, values of \(x\) $$\frac{x \tan 5x}{\cos 4x - 1} \approx A$$ where \(A\) is a constant to be determined. [4 marks]

Use integration by substitution to show that $$\int_{-\frac{3}{4}}^6 x\sqrt{4x + 1} \, dx = \frac{875}{12}$$ Fully justify your answer. [6 marks]

The line \(L\) has equation $$5y + 12x = 298$$ A circle, \(C\), has centre \((7, 9)\) \(L\) is a tangent to \(C\).

1. Find the coordinates of the point of intersection of \(L\) and \(C\). Fully justify your answer. [5 marks]
2. Find the equation of \(C\). [3 marks]

\(a\) and \(b\) are two positive irrational numbers. The sum of \(a\) and \(b\) is rational. The product of \(a\) and \(b\) is rational. Caroline is trying to prove \(\frac{1}{a} + \frac{1}{b}\) is rational. Here is her proof: Step 1 \quad \(\frac{1}{a} + \frac{1}{b} = \frac{2}{a + b}\) Step 2 \quad \(2\) is rational and \(a + b\) is non-zero and rational. Step 3 \quad Therefore \(\frac{2}{a + b}\) is rational. Step 4 \quad Hence \(\frac{1}{a} + \frac{1}{b}\) is rational.

1. 1. Identify Caroline's mistake. [1 mark]
   2. Write down a correct version of the proof. [2 marks]
2. Prove by contradiction that the difference of any rational number and any irrational number is irrational. [4 marks]

The curve defined by the parametric equations $$x = t^2 \text{ and } y = 2t \quad -\sqrt{2} \leq t \leq \sqrt{2}$$ is shown in Figure 1 below. \includegraphics{figure_1}

1. Find a Cartesian equation of the curve in the form \(y^2 = f(x)\) [2 marks]
2. The point \(A\) lies on the curve where \(t = a\) The tangent to the curve at \(A\) is at an angle \(\theta\) to a line through \(A\) parallel to the \(x\)-axis. The point \(B\) has coordinates \((1, 0)\) The line \(AB\) is at an angle \(\phi\) to the \(x\)-axis. \includegraphics{figure_1_extended}
   1. By considering the gradient of the curve, show that $$\tan \theta = \frac{1}{a}$$ [3 marks]
   2. Find \(\tan \phi\) in terms of \(a\). [2 marks]
   3. Show that \(\tan 2\theta = \tan \phi\) [3 marks]

A cylinder is to be cut out of the circular face of a solid hemisphere. The cylinder and the hemisphere have the same axis of symmetry. The cylinder has height \(h\) and the hemisphere has a radius of \(R\). \includegraphics{figure_9}

1. Show that the volume, \(V\), of the cylinder is given by $$V = \pi R^2 h - \pi h^3$$ [3 marks]
2. Find the maximum volume of the cylinder in terms of \(R\). Fully justify your answer. [7 marks]

A vehicle is driven at a constant speed of \(12\text{ ms}^{-1}\) along a straight horizontal road. Only one of the statements below is correct. Identify the correct statement. Tick (\(\checkmark\)) one box. The vehicle is accelerating The vehicle's driving force exceeds the total force resisting its motion The resultant force acting on the vehicle is zero The resultant force acting on the vehicle is dependent on its mass [1 mark]

A number of forces act on a particle such that the resultant force is \(\begin{pmatrix} 6 \\ -3 \end{pmatrix}\) N One of the forces acting on the particle is \(\begin{pmatrix} 8 \\ -5 \end{pmatrix}\) N Calculate the total of the other forces acting on the particle. Circle your answer. \(\begin{pmatrix} 2 \\ -2 \end{pmatrix}\) N \quad \(\begin{pmatrix} 14 \\ -8 \end{pmatrix}\) N \quad \(\begin{pmatrix} -2 \\ 2 \end{pmatrix}\) N \quad \(\begin{pmatrix} -14 \\ 8 \end{pmatrix}\) N [1 mark]

A particle, \(P\), is moving with constant velocity \(8\mathbf{i} - 12\mathbf{j}\) A second particle, \(Q\), is moving with constant velocity \(a\mathbf{i} + 9\mathbf{j}\) \(Q\) travels in a direction which is parallel to the motion of \(P\). Find \(a\). Circle your answer. \(-6\) \quad \(-5\) \quad \(5\) \quad \(6\) [1 mark]

A uniform rod, \(AB\), has length \(7\) metres and mass \(4\) kilograms. The rod rests on a single fixed pivot point, \(C\), where \(AC = 2\) metres. A particle of weight \(W\) newtons is fixed at \(A\), as shown in the diagram. \includegraphics{figure_13} The system is in equilibrium with the rod resting horizontally.

1. Find \(W\), giving your answer in terms of \(g\). [2 marks]
2. Explain how you have used the fact that the rod is uniform in part (a). [1 mark]

At time \(t\) seconds a particle, \(P\), has position vector \(\mathbf{r}\) metres, with respect to a fixed origin, such that $$\mathbf{r} = (t^3 - 5t^2)\mathbf{i} + (8t - t^2)\mathbf{j}$$

1. Find the exact speed of \(P\) when \(t = 2\) [4 marks]
2. Bella claims that the magnitude of acceleration of \(P\) will never be zero. Determine whether Bella's claim is correct. Fully justify your answer. [3 marks]

A particle is moving in a straight line with velocity \(v\text{ ms}^{-1}\) at time \(t\) seconds as shown by the graph below. \includegraphics{figure_15}

1. Use the trapezium rule with four strips to estimate the distance travelled by the particle during the time period \(20 \leq t \leq 100\) [4 marks]
2. Over the same time period, the curve can be very closely modelled by a particular quadratic. Explain how you could find an alternative estimate using this quadratic. [1 mark]

Two particles \(A\) and \(B\) are released from rest from different starting points above a horizontal surface. \(A\) is released from a height of \(h\) metres. \(B\) is released at a time \(t\) seconds after \(A\) from a height of \(kh\) metres, where \(0 < k < 1\) Both particles land on the surface \(5\) seconds after \(A\) was released. Assuming any resistance forces may be ignored, prove that $$t = 5(1 - \sqrt{k})$$ Fully justify your answer. [5 marks]

A ball is projected forward from a fixed point, \(P\), on a horizontal surface with an initial speed \(u\text{ ms}^{-1}\), at an acute angle \(\theta\) above the horizontal. The ball needs to first land at a point at least \(d\) metres away from \(P\). You may assume the ball may be modelled as a particle and that air resistance may be ignored. Show that $$\sin 2\theta \geq \frac{dg}{u^2}$$ [6 marks]

Block \(A\), of mass \(0.2\) kg, lies at rest on a rough plane. The plane is inclined at an angle \(\theta\) to the horizontal, such that \(\tan \theta = \frac{7}{24}\) A light inextensible string is attached to \(A\) and runs parallel to the line of greatest slope until it passes over a smooth fixed pulley at the top of the slope. The other end of this string is attached to particle \(B\), of mass \(2\) kg, which is held at rest so that the string is taut, as shown in the diagram below. \includegraphics{figure_18}

1. \(B\) is released from rest so that it begins to move vertically downwards with an acceleration of \(\frac{543}{625}\) g ms\(^{-2}\) Show that the coefficient of friction between \(A\) and the surface of the inclined plane is \(0.17\) [8 marks]
2. In this question use \(g = 9.81\text{ ms}^{-2}\) When \(A\) reaches a speed of \(0.5\text{ ms}^{-1}\) the string breaks.
   1. Find the distance travelled by \(A\) after the string breaks until first coming to rest. [4 marks]
   2. State an assumption that could affect the validity of your answer to part (b)(i). [1 mark]

A particle moves so that its acceleration, \(a\text{ ms}^{-2}\), at time \(t\) seconds may be modelled in terms of its velocity, \(v\text{ ms}^{-1}\), as $$a = -0.1v^2$$ The initial velocity of the particle is \(4\text{ ms}^{-1}\)

1. By first forming a suitable differential equation, show that $$v = \frac{20}{5 + 2t}$$ [6 marks]
2. Find the acceleration of the particle when \(t = 5.5\) [2 marks]