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AQA Further AS Paper 1 2020 June Q11
3 marks Challenging +1.2
Sketch the polar graph of $$r = \sinh \theta + \cosh \theta$$ for \(0 \leq \theta \leq 2\pi\) \includegraphics{figure_11} [3 marks]
AQA Further AS Paper 1 2020 June Q12
2 marks Standard +0.8
The mean value of the function \(\mathbf{f}\) over the interval \(1 \leq x \leq 5\) is \(m\). The graph of \(y = \mathbf{g}(x)\) is a reflection in the \(x\)-axis of \(y = \mathbf{f}(x)\). The graph of \(y = \mathbf{h}(x)\) is a translation of \(y = \mathbf{g}(x)\) by \(\begin{bmatrix} 3 \\ 7 \end{bmatrix}\) Determine, in terms of \(m\), the mean value of the function \(\mathbf{h}\) over the interval \(4 \leq x \leq 8\) [2 marks]
AQA Further AS Paper 1 2020 June Q13
9 marks Standard +0.8
Line \(l_1\) has equation $$\frac{x - 2}{3} = \frac{1 - 2y}{4} = -z$$ and line \(l_2\) has equation $$\mathbf{r} = \begin{bmatrix} -7 \\ 4 \\ -2 \end{bmatrix} + \mu \begin{bmatrix} 12 \\ a + 3 \\ 2b \end{bmatrix}$$
  1. In the case when \(l_1\) and \(l_2\) are parallel, show that \(a = -11\) and find the value of \(b\). [4 marks]
  2. In a different case, the lines \(l_1\) and \(l_2\) intersect at exactly one point, and the value of \(b\) is 3 Find the value of \(a\). [5 marks]
AQA Further AS Paper 1 2020 June Q14
7 marks Standard +0.8
  1. Given $$\frac{x + 7}{x + 1} \leq x + 1$$ show that $$\frac{(x + a)(x + b)}{x + c} \geq 0$$ where \(a\), \(b\), and \(c\) are integers to be found. [4 marks]
  2. Briefly explain why this statement is incorrect. $$\frac{(x + p)(x + q)}{x + r} \geq 0 \Leftrightarrow (x + p)(x + q)(x + r) \geq 0$$ [1 mark]
  3. Solve $$\frac{x + 7}{x + 1} \leq x + 1$$ [2 marks]
AQA Further AS Paper 1 2020 June Q15
4 marks Standard +0.8
A segment of the line \(y = kx\) is rotated about the \(x\)-axis to generate a cone with vertex \(O\). The distance of \(O\) from the centre of the base of the cone is \(h\). The radius of the base of the cone is \(r\). \includegraphics{figure_15}
  1. Find \(k\) in terms of \(r\) and \(h\). [1 mark]
  2. Use calculus to prove that the volume of the cone is $$\frac{1}{3}\pi r^2 h$$ [3 marks]
AQA Further AS Paper 1 2020 June Q16
4 marks Moderate -0.8
\(\mathbf{A}\) and \(\mathbf{B}\) are non-singular square matrices.
  1. Write down the product \(\mathbf{AA}^{-1}\) as a single matrix. [1 mark]
  2. \(\mathbf{M}\) is a matrix such that \(\mathbf{M} = \mathbf{AB}\). Prove that \(\mathbf{M}^{-1} = \mathbf{B}^{-1}\mathbf{A}^{-1}\) [3 marks]
AQA Further AS Paper 1 2020 June Q17
4 marks Standard +0.8
The polar equation of the circle \(C\) is $$r = a(\cos \theta + \sin \theta)$$ Find, in terms of \(a\), the radius of \(C\). Fully justify your answer. [4 marks]
AQA Further AS Paper 1 2020 June Q18
5 marks Standard +0.8
The locus of points \(L_1\) satisfies the equation \(|z| = 2\) The locus of points \(L_2\) satisfies the equation \(\arg(z + 4) = \frac{\pi}{4}\)
  1. Sketch \(L_1\) on the Argand diagram below. \includegraphics{figure_18} [1 mark]
  2. Sketch \(L_2\) on the Argand diagram above. [1 mark]
  3. The complex number \(a + ib\), where \(a\) and \(b\) are real, lies on \(L_1\) The complex number \(c + id\), where \(c\) and \(d\) are real, lies on \(L_2\) Calculate the least possible value of the expression $$(c - a)^2 + (d - b)^2$$ [3 marks]
AQA Further AS Paper 2 Statistics 2020 June Q1
1 marks Easy -1.8
The discrete random variable \(X\) has the following probability distribution function. $$\mathrm{P}(X = x) = \begin{cases} 0.2 & x = 1 \\ 0.3 & x = 2 \\ 0.1 & x = 3, 4 \\ 0.25 & x = 5 \\ 0.05 & x = 6 \\ 0 & \text{otherwise} \end{cases}$$ Find the mode of \(X\). Circle your answer. [1 mark] 0.1 \quad 0.25 \quad 2 \quad 3
AQA Further AS Paper 2 Statistics 2020 June Q2
1 marks Moderate -0.8
A \(\chi^2\) test is carried out in a school to test for association between the class a student belongs to and the number of times they are late to school in a week. The contingency table below gives the expected values for the test.
Number of times late
01234
A8.121415.12144.76
Class B8.9915.516.7415.55.27
C11.8920.522.1420.56.97
Find a possible value for the degrees of freedom for the test. Circle your answer. [1 mark] 6 \quad 8 \quad 12 \quad 15
AQA Further AS Paper 2 Statistics 2020 June Q3
5 marks Moderate -0.8
The random variable \(X\) represents the value on the upper face of an eight-sided dice after it has been rolled. The faces are numbered 1 to 8 The random variable \(X\) is modelled by a discrete uniform distribution with \(n = 8\)
  1. Find E\((X)\) [1 mark]
  2. Find Var\((X)\) [1 mark]
  3. Find P\((X \geq 6)\) [1 mark]
  4. The dice was rolled 800 times and the results below were obtained.
    \(x\)12345678
    Frequency1036384110744185240
    State, with a reason, how you would refine the model for the random variable \(X\). [2 marks]
AQA Further AS Paper 2 Statistics 2020 June Q4
3 marks Moderate -0.8
Murni is investigating the annual salary of people from a particular town. She takes a random sample of 200 people from the town and records their annual salary. The mean annual salary is £28 500 and the standard deviation is £5100 Calculate a 97% confidence interval for the population mean of annual salaries for the people who live in the town, giving your values to the nearest pound. [3 marks]
AQA Further AS Paper 2 Statistics 2020 June Q5
7 marks Moderate -0.3
The discrete random variable \(X\) has the following probability distribution.
\(x\)2469
P\((X = x)\)0.20.60.10.1
  1. Find P\((X \leq 6)\) [1 mark]
  2. Let \(Y = 3X + 2\) Show that Var\((Y) = 32.49\) [5 marks]
  3. The continuous random variable \(T\) is independent of \(Y\). Given that Var\((T) = 5\), find Var\((T + Y)\) [1 mark]
AQA Further AS Paper 2 Statistics 2020 June Q6
8 marks Standard +0.3
The continuous random variable \(X\) has probability density function $$f(x) = \begin{cases} \frac{4}{45}(x^3 - 10x^2 + 29x - 20) & 1 \leq x \leq 4 \\ 0 & \text{otherwise} \end{cases}$$
  1. Find P\((X < 2)\) [2 marks]
  2. Verify that the median of \(X\) is 2.3, correct to two significant figures. [4 marks]
  3. Find the mean of \(X\). [2 marks]
AQA Further AS Paper 2 Statistics 2020 June Q7
6 marks Moderate -0.8
A restaurant has asked Sylvia to conduct a \(\chi^2\) test for association between meal ordered and age of customer.
  1. State the hypotheses that Sylvia should use for her test. [1 mark]
  2. Sylvia correctly calculates her value of the test statistic to be 44.1 She uses a 5% level of significance and the degrees of freedom for the test is 30 Sylvia accepts the null hypothesis. Explain whether or not Sylvia was correct to accept the null hypothesis. [4 marks]
  3. State in context the correct conclusion to Sylvia's test. [1 mark]
AQA Further AS Paper 2 Statistics 2020 June Q8
9 marks Challenging +1.2
There are two hospitals in a city. Over a period of time, the first hospital recorded an average of 20 births a day. Over the same period of time, the second hospital recorded an average of 5 births a day. Stuart claims that birth rates in the hospitals have changed over time. On a randomly chosen day, he records a total of 16 births from the two hospitals.
  1. Investigate Stuart's claim, using a suitable test at the 5% level of significance. [6 marks]
  2. For a test of the type carried out in part (a), find the probability of making a Type I error, giving your answer to two significant figures. [3 marks]
AQA Further AS Paper 2 Mechanics 2019 June Q1
1 marks Easy -1.8
A turntable rotates at a constant speed of \(33\frac{1}{3}\) revolutions per minute. Find the angular speed in radians per second. Circle your answer. [1 mark] \(\frac{5\pi}{9}\) \quad \(\frac{10\pi}{9}\) \quad \(\frac{5\pi}{3}\) \quad \(\frac{20\pi}{9}\)
AQA Further AS Paper 2 Mechanics 2019 June Q2
1 marks Easy -1.2
The graph shows the resistance force experienced by a cyclist over the first 20 metres of a bicycle ride. \includegraphics{figure_2} Find the work done by the resistance force over the 20 metres of the bicycle ride. Circle your answer. [1 mark] 1600 J \quad 3000 J \quad 3200 J \quad 4000 J
AQA Further AS Paper 2 Mechanics 2019 June Q3
3 marks Moderate -0.5
A formula for the elastic potential energy, \(E\), stored in a stretched spring is given by $$E = \frac{kx^2}{2}$$ where \(x\) is the extension of the spring and \(k\) is a constant. Use dimensional analysis to find the dimensions of \(k\). [3 marks]
AQA Further AS Paper 2 Mechanics 2019 June Q4
7 marks Standard +0.3
In this question use \(g = 9.8\,\text{m}\,\text{s}^{-2}\) A ride in a fairground consists of a hollow vertical cylinder of radius 4.6 metres with a horizontal floor. Stephi, who has mass 50 kilograms, stands inside the cylinder with her back against the curved surface. The cylinder begins to rotate about a vertical axis through the centre of the cylinder. When the cylinder is rotating at a constant angular speed of \(\omega\) radians per second, the magnitude of the normal reaction between Stephi and the curved surface is 980 newtons. The floor is lowered and Stephi remains against the curved surface with her feet above the floor, as shown in the diagram. \includegraphics{figure_4}
  1. Explain, with the aid of a force diagram, why the magnitude of the frictional force acting on Stephi is 490 newtons. [2 marks]
  2. Find \(\omega\) [3 marks]
  3. State one modelling assumption that you have used in this question. Explain the effect of this assumption. [2 marks]
AQA Further AS Paper 2 Mechanics 2019 June Q5
7 marks Standard +0.8
A car of mass 1000 kg has a maximum speed of \(40\,\text{m}\,\text{s}^{-1}\) when travelling on a straight horizontal race track. The maximum power output of the car's engine is 48 kW The total resistance force experienced by the car can be modelled as being proportional to the car's speed. Find the maximum possible acceleration of the car when it is travelling at \(25\,\text{m}\,\text{s}^{-1}\) on the straight horizontal race track. Fully justify your answer. [7 marks]
AQA Further AS Paper 2 Mechanics 2019 June Q6
9 marks Standard +0.3
In this question use \(g = 9.8\,\text{m}\,\text{s}^{-2}\) Martin, who is of mass 40 kg, is using a slide. The slide is made of two straight sections \(AB\) and \(BC\). The section \(AB\) has length 15 metres and is at an angle of \(50°\) to the horizontal. The section \(BC\) has length 2 metres and is horizontal. \includegraphics{figure_6} Martin pushes himself from \(A\) down the slide with initial speed \(1\,\text{m}\,\text{s}^{-1}\) He reaches \(B\) with speed \(5\,\text{m}\,\text{s}^{-1}\) Model Martin as a particle.
  1. Find the energy lost as Martin slides from \(A\) to \(B\). [4 marks]
  2. Assume that a resistance force of constant magnitude acts on Martin while he is moving on the slide.
    1. Show that the magnitude of this resistance force is approximately 270 N [2 marks]
    2. Determine if Martin reaches the point \(C\). [3 marks]
AQA Further AS Paper 2 Mechanics 2019 June Q7
12 marks Standard +0.3
Two smooth spheres, \(P\) and \(Q\), of equal radius are free to move on a smooth horizontal surface. The masses of \(P\) and \(Q\) are \(3m\) and \(m\) respectively. \(P\) is set in motion with speed \(u\) directly towards \(Q\), which is initially at rest. \(P\) subsequently collides with \(Q\). \includegraphics{figure_7} Immediately after the collision, \(P\) moves with speed \(v\) and \(Q\) moves with speed \(w\). The coefficient of restitution between the spheres is \(e\).
    1. Show that $$v = \frac{u(3-e)}{4}$$ [4 marks]
    2. Find \(w\), in terms of \(e\) and \(u\), simplifying your answer. [2 marks]
  2. Deduce that $$\frac{u}{2} \leq v \leq \frac{3u}{4}$$ [2 marks]
    1. Find, in terms of \(m\) and \(u\), the maximum magnitude of the impulse that \(P\) exerts on \(Q\). [3 marks]
    2. Describe the impulse that \(Q\) exerts on \(P\). [1 mark]
AQA Further AS Paper 2 Mechanics 2021 June Q1
1 marks Easy -1.8
A light spring of natural length 0.6 metres is compressed to a length of 0.4 metres by a force of 20 newtons. The stiffness of the spring is \(k\) N m\(^{-1}\) Find \(k\) Circle your answer. [1 mark] 20 50 100 200
AQA Further AS Paper 2 Mechanics 2021 June Q2
1 marks Easy -2.0
State the dimensions of force. Circle your answer. [1 mark] \(MLT\) \(ML^2T\) \(MLT^{-1}\) \(MLT^{-2}\)