Questions Further AS Paper 1 (126 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 PURE S1 S2 S3 S4 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
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AQA Further AS Paper 1 2018 June Q1
1 marks Easy -1.8
\(z = 3 - i\) Determine the value of \(zz*\) Circle your answer. [1 mark] \(10\) \(\qquad\) \(\sqrt{10}\) \(\qquad\) \(10 - 2i\) \(\qquad\) \(10 + 2i\)
AQA Further AS Paper 1 2018 June Q2
1 marks Easy -1.8
Three matrices \(\mathbf{A}\), \(\mathbf{B}\) and \(\mathbf{C}\) are given by $$\mathbf{A} = \begin{pmatrix} 5 & 2 & -3 \\ 0 & 7 & 6 \\ 4 & 1 & 0 \end{pmatrix}, \quad \mathbf{B} = \begin{pmatrix} 1 & 0 \\ 3 & -5 \\ -2 & 6 \end{pmatrix} \quad \text{and } \mathbf{C} = \begin{pmatrix} 6 & 4 & 3 \\ 1 & 2 & 0 \end{pmatrix}$$ Which of the following **cannot** be calculated? Circle your answer. [1 mark] \(\mathbf{AB}\) \(\qquad\) \(\mathbf{AC}\) \(\qquad\) \(\mathbf{BC}\) \(\qquad\) \(\mathbf{A}^2\)
AQA Further AS Paper 1 2018 June Q3
1 marks Moderate -0.8
Which of the following functions has the fourth term \(-\frac{1}{720}x^6\) in its Maclaurin series expansion? Circle your answer. [1 mark] \(\sin x\) \(\qquad\) \(\cos x\) \(\qquad\) \(e^x\) \(\qquad\) \(\ln(1 + x)\)
AQA Further AS Paper 1 2018 June Q4
2 marks Standard +0.3
Sketch the graph given by the polar equation $$r = \frac{a}{\cos \theta}$$ where \(a\) is a positive constant. [2 marks] \includegraphics{figure_4}
AQA Further AS Paper 1 2018 June Q5
3 marks Standard +0.3
Describe fully the transformation given by the matrix \(\begin{pmatrix} -\frac{1}{2} & -\frac{\sqrt{3}}{2} & 0 \\ \frac{\sqrt{3}}{2} & -\frac{1}{2} & 0 \\ 0 & 0 & 1 \end{pmatrix}\) [3 marks]
AQA Further AS Paper 1 2018 June Q6
3 marks Standard +0.8
  1. Matthew is finding a formula for the inverse function \(\text{arsinh } x\). He writes his steps as follows: Let \(y = \sinh x\) \(y = \frac{1}{2}(e^x - e^{-x})\) \(2y = e^x - e^{-x}\) \(0 = e^x - 2y - e^{-x}\) \(0 = (e^x)^2 - 2ye^x - 1\) \(0 = (e^x - y)^2 - y^2 - 1\) \(y^2 + 1 = (e^x - y)^2\) \(\pm \sqrt{y^2 + 1} = e^x - y\) \(y + \sqrt{y^2 + 1} = e^x\) To find the inverse function, swap \(x\) and \(y\): \(x + \sqrt{x^2 + 1} = e^y\) \(\ln\left(x + \sqrt{x^2 + 1}\right) = y\) \(\text{arsinh } x = \ln\left(x + \sqrt{x^2 + 1}\right)\) Identify, and explain, the error in Matthew's proof. [2 marks]
  2. Solve \(\ln\left(x + \sqrt{x^2 + 1}\right) = 3\) [1 mark]
AQA Further AS Paper 1 2018 June Q7
2 marks Moderate -0.3
Find two invariant points under the transformation given by \(\begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix}\) [2 marks]
AQA Further AS Paper 1 2018 June Q8
5 marks Standard +0.8
\(2 - 3i\) is one root of the equation $$z^3 + mz + 52 = 0$$ where \(m\) is real.
  1. Find the other roots. [3 marks]
  2. Determine the value of \(m\). [2 marks]
AQA Further AS Paper 1 2018 June Q9
6 marks Standard +0.3
  1. Sketch the graph of \(y^2 = 4x\) [1 mark] \includegraphics{figure_9a}
  2. Ben is using a 3D printer to make a plastic bowl which holds exactly \(1000\text{cm}^3\) of water. Ben models the bowl as a region which is rotated through \(2\pi\) radians about the \(x\)-axis. He uses the finite region enclosed by the lines \(x = d\) and \(y = 0\) and the curve with equation \(y^2 = 4x\) for \(y \geq 0\)
    1. Find the depth of the bowl to the nearest millimetre. [4 marks]
    2. What assumption has Ben made about the bowl? [1 mark]
AQA Further AS Paper 1 2018 June Q10
8 marks Standard +0.8
  1. Prove by induction that, for all integers \(n \geq 1\), $$\sum_{r=1}^{n} r^3 = \frac{1}{4}n^2(n + 1)^2$$ [4 marks]
  2. Hence show that $$\sum_{r=1}^{2n} r(r - 1)(r + 1) = n(n + 1)(2n - 1)(2n + 1)$$ [4 marks]
AQA Further AS Paper 1 2018 June Q11
3 marks Challenging +1.2
Four finite regions \(A\), \(B\), \(C\) and \(D\) are enclosed by the curve with equation $$y = x^3 - 7x^2 + 11x + 6$$ and the lines \(y = k\), \(x = 1\) and \(x = 4\), as shown in the diagram below. \includegraphics{figure_11} The areas of \(B\) and \(C\) are equal. Find the value of \(k\). [3 marks]
AQA Further AS Paper 1 2018 June Q12
6 marks Standard +0.3
  1. Show that the matrix \(\begin{pmatrix} 5 - k & 2 \\ k^3 + 1 & k \end{pmatrix}\) is singular when \(k = 1\). [1 mark]
  2. Find the values of \(k\) for which the matrix \(\begin{pmatrix} 5 - k & 2 \\ k^3 + 1 & k \end{pmatrix}\) has a negative determinant. Fully justify your answer. [5 marks]
AQA Further AS Paper 1 2018 June Q13
9 marks Challenging +1.2
The graph of the rational function \(y = f(x)\) intersects the \(x\)-axis exactly once at \((-3, 0)\) The graph has exactly two asymptotes, \(y = 2\) and \(x = -1\)
  1. Find \(f(x)\) [2 marks]
  2. Sketch the graph of the function. [3 marks] \includegraphics{figure_13b}
  3. Find the range of values of \(x\) for which \(f(x) \leq 5\) [4 marks]
AQA Further AS Paper 1 2018 June Q14
7 marks Challenging +1.2
  1. Sketch, on the Argand diagram below, the locus of points satisfying the equation $$|z - 3| = 2$$ [1 mark] \includegraphics{figure_14a}
  2. There is a unique complex number \(w\) that satisfies both $$|w - 3| = 2 \quad \text{and} \quad \arg(w + 1) = \alpha$$ where \(\alpha\) is a constant such that \(0 < \alpha < \pi\)
    1. Find the value of \(\alpha\). [2 marks]
    2. Express \(w\) in the form \(r(\cos \theta + i \sin \theta)\). Give each of \(r\) and \(\theta\) to two significant figures. [4 marks]
AQA Further AS Paper 1 2018 June Q15
4 marks Standard +0.3
  1. Show that $$\frac{1}{r + 2} - \frac{1}{r + 3} = \frac{1}{(r + 2)(r + 3)}$$ [1 mark]
  2. Use the method of differences to show that $$\sum_{r=1}^{n} \frac{1}{(r + 2)(r + 3)} = \frac{n}{3(n + 3)}$$ [3 marks]
AQA Further AS Paper 1 2018 June Q16
3 marks Standard +0.8
Two matrices \(\mathbf{A}\) and \(\mathbf{B}\) satisfy the equation $$\mathbf{AB} = \mathbf{I} + 2\mathbf{A}$$ where \(\mathbf{I}\) is the identity matrix and \(\mathbf{B} = \begin{pmatrix} 3 & -2 \\ -4 & 8 \end{pmatrix}\) Find \(\mathbf{A}\). [3 marks]
AQA Further AS Paper 1 2018 June Q17
4 marks Standard +0.8
Find the exact solution to the equation $$\sinh \theta(\sinh \theta + \cosh \theta) = 1$$ [4 marks]
AQA Further AS Paper 1 2018 June Q18
4 marks Challenging +1.8
\(\alpha\), \(\beta\) and \(\gamma\) are the real roots of the cubic equation $$x^3 + mx^2 + nx + 2 = 0$$ By considering \((\alpha - \beta)^2 + (\gamma - \alpha)^2 + (\beta - \gamma)^2\), prove that $$m^2 \geq 3n$$ [4 marks]
AQA Further AS Paper 1 2018 June Q19
8 marks Challenging +1.2
A theme park has two zip wires. Sarah models the two zip wires as straight lines using coordinates in metres. The ends of one wire are located at \((0, 0, 0)\) and \((0, 100, -20)\) The ends of the other wire are located at \((10, 0, 20)\) and \((-10, 100, -5)\)
  1. Use Sarah's model to find the shortest distance between the zip wires. [7 marks]
  2. State one way in which Sarah's model could be refined. [1 mark]
AQA Further AS Paper 1 2019 June Q1
1 marks Easy -2.5
Which of the following matrices is an identity matrix? Circle your answer. [1 mark] \(\begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}\) \quad \(\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\) \quad \(\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\) \quad \(\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}\)
AQA Further AS Paper 1 2019 June Q2
1 marks Easy -1.8
Which of the following expressions is the determinant of the matrix \(\begin{bmatrix} a & 2 \\ b & 5 \end{bmatrix}\)? Circle your answer. [1 mark] \(5a - 2b\) \quad \(2a - 5b\) \quad \(5b - 2a\) \quad \(2b - 5a\)
AQA Further AS Paper 1 2019 June Q3
1 marks Easy -1.8
Point \(P\) has polar coordinates \(\left(2, \frac{2\pi}{3}\right)\). Which of the following are the Cartesian coordinates of \(P\)? Circle your answer. [1 mark] \((1, -\sqrt{3})\) \quad \((-\sqrt{3}, 1)\) \quad \((\sqrt{3}, -1)\) \quad \((-1, \sqrt{3})\)
AQA Further AS Paper 1 2019 June Q4
2 marks Moderate -0.3
The line \(L\) has polar equation $$r = \frac{k}{\sin \theta}$$ where \(k\) is a positive constant.
  1. Sketch \(L\). [1 mark]
  2. State the minimum distance between \(L\) and the point \(O\). [1 mark]
AQA Further AS Paper 1 2019 June Q5
8 marks Challenging +1.2
A hyperbola \(H\) has the equation $$\frac{x^2}{a^2} - \frac{y^2}{4a^2} = 1$$ where \(a\) is a positive constant.
  1. Write down the equations of the asymptotes of \(H\). [1 mark]
  2. Sketch the hyperbola \(H\) on the axes below, indicating the coordinates of any points of intersection with the coordinate axes. The asymptotes have already been drawn. [2 marks]
  3. The finite region bounded by \(H\), the positive \(x\)-axis, the positive \(y\)-axis and the line \(y = a\) is rotated through \(360°\) about the \(y\)-axis. Show that the volume of the solid generated is \(ma^3\), where \(m = 3.40\) correct to three significant figures. [5 marks]
AQA Further AS Paper 1 2019 June Q6
5 marks Standard +0.3
  1. On the axes provided, sketch the graph of $$x = \cosh(y + b)$$ where \(b\) is a positive constant. [4 marks]
  2. Determine the minimum distance between the graph of \(x = \cosh(y + b)\) and the \(y\)-axis. [1 mark]