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AQA Further AS Paper 1 2020 June Q2
1 marks Moderate -0.8
Given that \(1 - i\) is a root of the equation \(z^3 - 3z^2 + 4z - 2 = 0\), find the other two roots. Tick \((\checkmark)\) one box. [1 mark] \(-1 + i\) and \(-1\) \(1 + i\) and \(1\) \(-1 + i\) and \(1\) \(1 + i\) and \(-1\)
AQA Further AS Paper 1 2020 June Q3
1 marks Moderate -0.8
Given \((x - 1)(x - 2)(x - a) < 0\) and \(a > 2\) Find the set of possible values of \(x\). Tick \((\checkmark)\) one box. [1 mark] \(\{x : x < 1\} \cup \{x : 2 < x < a\}\) \(\{x : 1 < x < 2\} \cup \{x : x > a\}\) \(\{x : x < -a\} \cup \{x : -2 < x < -1\}\) \(\{x : -a < x < -2\} \cup \{x : x > -1\}\)
AQA Further AS Paper 1 2020 June Q4
5 marks Standard +0.3
The matrices \(\mathbf{A}\) and \(\mathbf{B}\) are such that $$\mathbf{A} = \begin{bmatrix} 2 & a & 3 \\ 0 & -2 & 1 \end{bmatrix} \quad \text{and} \quad \mathbf{B} = \begin{bmatrix} 1 & -3 \\ -2 & 4a \\ 0 & 5 \end{bmatrix}$$
  1. Find the product \(\mathbf{AB}\) in terms of \(a\). [2 marks]
  2. Find the determinant of \(\mathbf{AB}\) in terms of \(a\). [1 mark]
  3. Show that \(\mathbf{AB}\) is singular when \(a = -1\) [2 marks]
AQA Further AS Paper 1 2020 June Q5
4 marks Standard +0.3
  1. Show that $$r^2(r + 1)^2 - (r - 1)^2r^2 = pr^3$$ where \(p\) is an integer to be found. [1 mark]
  2. Hence use the method of differences to show that $$\sum_{r=1}^{n} r^3 = \frac{1}{4}n^2(n + 1)^2$$ [3 marks]
AQA Further AS Paper 1 2020 June Q6
2 marks Standard +0.3
Anna has been asked to describe the transformation given by the matrix $$\begin{bmatrix} 1 & 0 & 0 \\ 0 & -\frac{\sqrt{3}}{2} & -\frac{1}{2} \\ 0 & \frac{1}{2} & -\frac{\sqrt{3}}{2} \end{bmatrix}$$ She writes her answer as follows: The transformation is a rotation about the \(x\)-axis through an angle of \(\theta\), where $$\sin \theta = \frac{1}{2} \quad \text{and} \quad -\sin \theta = -\frac{1}{2}$$ $$\theta = 30°$$ Identify and correct the error in Anna's work. [2 marks]
AQA Further AS Paper 1 2020 June Q7
4 marks Moderate -0.3
Prove by induction that, for all integers \(n \geq 1\), the expression \(7^n - 3^n\) is divisible by 4 [4 marks]
AQA Further AS Paper 1 2020 June Q8
8 marks Standard +0.3
  1. Prove that $$\tanh^{-1} x = \frac{1}{2}\ln\left(\frac{1 + x}{1 - x}\right)$$ [5 marks]
  2. Prove that the graphs of $$y = \sinh x \quad \text{and} \quad y = \cosh x$$ do not intersect. [3 marks]
AQA Further AS Paper 1 2020 June Q9
8 marks Standard +0.3
The quadratic equation \(2x^2 + px + 3 = 0\) has two roots, \(\alpha\) and \(\beta\), where \(\alpha > \beta\).
    1. Write down the value of \(\alpha\beta\). [1 mark]
    2. Express \(\alpha + \beta\) in terms of \(p\). [1 mark]
  2. Hence find \((\alpha - \beta)^2\) in terms of \(p\). [2 marks]
  3. Hence find, in terms of \(p\), a quadratic equation with roots \(\alpha - 1\) and \(\beta + 1\) [4 marks]
AQA Further AS Paper 1 2020 June Q10
8 marks Standard +0.3
  1. Show that the equation $$y = \frac{3x - 5}{2x + 4}$$ can be written in the form $$(x + a)(y + b) = c$$ where \(a\), \(b\) and \(c\) are integers to be found. [3 marks]
  2. Write down the equations of the asymptotes of the graph of $$y = \frac{3x - 5}{2x + 4}$$ [2 marks]
  3. Sketch, on the axes provided, the graph of $$y = \frac{3x - 5}{2x + 4}$$ \includegraphics{figure_10} [3 marks]
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]