Questions Further Paper 1 (256 questions)

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AQA Further Paper 1 2024 June Q3
1 marks Easy -1.2
The function f is defined by $$f(x) = x^2 \quad (x \in \mathbb{R})$$ Find the mean value of \(f(x)\) between \(x = 0\) and \(x = 2\) Circle your answer. [1 mark] \(\frac{2}{3}\) \(\frac{4}{3}\) \(\frac{8}{3}\) \(\frac{16}{3}\)
AQA Further Paper 1 2024 June Q4
1 marks Moderate -0.5
Which one of the following statements is correct? Tick (\(\checkmark\)) one box. [1 mark] \(\lim_{x \to 0}(x^2 \ln x) = 0\) \(\square\) \(\lim_{x \to 0}(x^2 \ln x) = 1\) \(\square\) \(\lim_{x \to 0}(x^2 \ln x) = 2\) \(\square\) \(\lim_{x \to 0}(x^2 \ln x)\) is not defined. \(\square\)
AQA Further Paper 1 2024 June Q5
5 marks Standard +0.3
The points \(A\), \(B\) and \(C\) have coordinates \(A(5, 3, 4)\), \(B(8, -1, 9)\) and \(C(12, 5, 10)\) The points \(A\), \(B\) and \(C\) lie in the plane \(\Pi\)
  1. Find a vector that is normal to the plane \(\Pi\) [3 marks]
  2. Find a Cartesian equation of the plane \(\Pi\) [2 marks]
AQA Further Paper 1 2024 June Q6
4 marks Moderate -0.3
The sequence \(u_1, u_2, u_3, \ldots\) is defined by $$u_1 = 1$$ $$u_{n+1} = u_n + 3n$$ Prove by induction that for all integers \(n \geq 1\) $$u_n = \frac{1}{2}n^2 - \frac{3}{2}n + 1$$ [4 marks]
AQA Further Paper 1 2024 June Q7
5 marks Standard +0.3
The complex numbers \(z\) and \(w\) satisfy the simultaneous equations $$z + w^* = 5$$ $$3z^* - w = 6 + 4i$$ Find \(z\) and \(w\) [5 marks]
AQA Further Paper 1 2024 June Q8
4 marks Standard +0.8
The ellipse \(E\) has equation $$x^2 + \frac{y^2}{9} = 1$$ The line with equation \(y = mx + 4\) is a tangent to \(E\) Without using differentiation show that \(m = \pm\sqrt{7}\) [4 marks]
AQA Further Paper 1 2024 June Q9
8 marks Standard +0.8
  1. It is given that $$p = \ln\left(r + \sqrt{r^2 + 1}\right)$$ Starting from the exponential definition of the sinh function, show that \(\sinh p = r\) [4 marks]
  2. Solve the equation $$\cosh^2 x = 2\sinh x + 16$$ Give your answers in logarithmic form. [4 marks]
AQA Further Paper 1 2024 June Q10
6 marks Standard +0.8
The complex numbers \(z\) and \(w\) are defined by $$z = \cos\frac{\pi}{4} + i\sin\frac{\pi}{4}$$ and $$w = \cos\frac{\pi}{6} + i\sin\frac{\pi}{6}$$ By evaluating the product \(zw\), show that $$\tan\frac{5\pi}{12} = 2 + \sqrt{3}$$ [6 marks]
AQA Further Paper 1 2024 June Q11
5 marks Standard +0.3
  1. Find \(\frac{d}{dx}(x^2\tan^{-1} x)\) [1 mark]
  2. Hence find \(\int 2x \tan^{-1} x \, dx\) [4 marks]
AQA Further Paper 1 2024 June Q12
10 marks Challenging +1.8
The line \(L_1\) has equation $$\mathbf{r} = \begin{pmatrix} 4 \\ 2 \\ 1 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ 3 \\ -1 \end{pmatrix}$$ The transformation T is represented by the matrix $$\begin{pmatrix} 2 & 1 & 0 \\ 3 & 4 & 6 \\ -5 & 2 & -3 \end{pmatrix}$$ The transformation T transforms the line \(L_1\) to the line \(L_2\)
  1. Show that the angle between \(L_1\) and \(L_2\) is 0.701 radians, correct to three decimal places. [4 marks]
  2. Find the shortest distance between \(L_1\) and \(L_2\) Give your answer in an exact form. [6 marks]
AQA Further Paper 1 2024 June Q13
9 marks Standard +0.3
  1. Use de Moivre's theorem to show that $$\cos 3\theta = 4\cos^3 \theta - 3\cos \theta$$ [3 marks]
  2. Use de Moivre's theorem to express \(\sin 3\theta\) in terms of \(\sin \theta\) [2 marks]
  3. Hence show that $$\cot 3\theta = \frac{\cot^3 \theta - 3\cot \theta}{3\cot^2 \theta - 1}$$ [4 marks]
AQA Further Paper 1 2024 June Q14
7 marks Challenging +1.2
Solve the differential equation $$\frac{dy}{dx} + y\tanh x = \sinh^3 x$$ given that \(y = 3\) when \(x = \ln 2\) Give your answer in an exact form. [7 marks]
AQA Further Paper 1 2024 June Q15
5 marks Challenging +1.2
A curve is defined parametrically by the equations $$x = \frac{3}{2}t^3 + 5$$ $$y = t^{\frac{3}{2}} \quad (t \geq 0)$$ Show that the arc length of the curve from \(t = 0\) to \(t = 2\) is equal to 26 units. [5 marks]
AQA Further Paper 1 2024 June Q16
9 marks Challenging +1.8
The curve \(C\) has polar equation \(r = 2 + \tan \theta\) The curve \(C\) meets the line \(\theta = \frac{\pi}{4}\) at the point \(A\) The point \(B\) has polar coordinates \((4, 0)\) The diagram shows part of the curve \(C\), and the points \(A\) and \(B\) \includegraphics{figure_16}
  1. Show that the area of triangle \(OAB\) is \(3\sqrt{2}\) units. [2 marks]
  2. Find the area of the shaded region. Give your answer in an exact form. [7 marks]
AQA Further Paper 1 2024 June Q17
7 marks Challenging +1.8
By making a suitable substitution, show that $$\int_{-2}^{1} \sqrt{x^2 + 6x + 8} \, dx = 2\sqrt{15} - \frac{1}{2}\cosh^{-1}(4)$$ [7 marks]
AQA Further Paper 1 2024 June Q18
12 marks Challenging +1.2
In this question use \(g = 9.8\) m s\(^{-2}\) Two light elastic strings each have one end attached to a small ball \(B\) of mass 0.5 kg The other ends of the strings are attached to the fixed points \(A\) and \(C\), which are 8 metres apart with \(A\) vertically above \(C\) The whole system is in a thin tube of oil, as shown in the diagram below. \includegraphics{figure_18} The string connecting \(A\) and \(B\) has natural length 2 metres, and the tension in this string is \(7e\) newtons when the extension is \(e\) metres. The string connecting \(B\) and \(C\) has natural length 3 metres, and the tension in this string is \(3e\) newtons when the extension is \(e\) metres.
  1. Find the extension of each string when the system is in equilibrium. [3 marks]
  2. It is known that in a large bath of oil, the oil causes a resistive force of magnitude \(4.5v\) newtons to act on the ball, where \(v\) m s\(^{-1}\) is the speed of the ball. Use this model to answer part (b)(i) and part (b)(ii).
    1. The ball is pulled a distance of 0.6 metres downwards from its equilibrium position towards \(C\), and released from rest. Show that during the subsequent motion the particle satisfies the differential equation $$\frac{d^2x}{dt^2} + 9\frac{dx}{dt} + 20x = 0$$ where \(x\) metres is the displacement of the particle below the equilibrium position at time \(t\) seconds after the particle is released. [3 marks]
    2. Find \(x\) in terms of \(t\) [5 marks]
  3. State one limitation of the model used in part (b) [1 mark]
AQA Further Paper 1 Specimen Q1
1 marks Easy -1.8
A vector is given by \(\mathbf{a} = \begin{bmatrix} 2 \\ -1 \\ -3 \end{bmatrix}\) Which vector is not perpendicular to \(\mathbf{a}\)? Circle your answer. \(\begin{bmatrix} 1 \\ -1 \\ 1 \end{bmatrix}\) \quad \(\begin{bmatrix} 3 \\ 0 \\ 2 \end{bmatrix}\) \quad \(\begin{bmatrix} 5 \\ -1 \\ 3 \end{bmatrix}\) \quad \(\begin{bmatrix} 2 \\ 1 \\ 1 \end{bmatrix}\) [1 mark]
AQA Further Paper 1 Specimen Q2
2 marks Moderate -0.8
Use the definitions of \(\cosh x\) and \(\sinh x\) in terms of \(e^x\) and \(e^{-x}\) to show that \(\cosh^2 x - \sinh^2 x = 1\) [2 marks]
AQA Further Paper 1 Specimen Q3
6 marks Standard +0.8
\begin{enumerate}[label=(\alph*)] \item Given that $$\frac{2}{(r + 1)(r + 2)(r + 3)} \equiv \frac{A}{(r + 1)(r + 2)} + \frac{B}{(r + 2)(r + 3)}$$ find the values of the integers \(A\) and \(B\) [2 marks] \item Use the method of differences to show clearly that $$\sum_{r=4}^{97} \frac{1}{(r + 1)(r + 2)(r + 3)} = \frac{89}{19800}$$ [4 marks]
AQA Further Paper 1 Specimen Q4
2 marks Standard +0.8
A student states that \(\int_0^{\pi/2} \frac{\cos x + \sin x}{\cos x - \sin x} \, dx\) is not an improper integral because \(\frac{\cos x + \sin x}{\cos x - \sin x}\) is defined at both \(x = 0\) and \(x = \frac{\pi}{2}\) Assess the validity of the student's argument. [2 marks]
AQA Further Paper 1 Specimen Q5
6 marks Standard +0.8
\(p(z) = z^4 + 3z^2 + az + b\), \(a \in \mathbb{R}\), \(b \in \mathbb{R}\) \(2 - 3i\) is a root of the equation \(p(z) = 0\)
  1. Express \(p(z)\) as a product of quadratic factors with real coefficients. [5 marks]
  2. Solve the equation \(p(z) = 0\). [1 mark]
AQA Further Paper 1 Specimen Q6
7 marks Standard +0.8
  1. Obtain the general solution of the differential equation $$\tan x \frac{dy}{dx} + y = \sin x \tan x$$ where \(0 < x < \frac{\pi}{2}\) [5 marks]
  2. Hence find the particular solution of this differential equation, given that \(y = \frac{1}{2\sqrt{2}}\) when \(x = \frac{\pi}{4}\) [2 marks]
AQA Further Paper 1 Specimen Q7
11 marks Challenging +1.8
Three planes have equations, $$x - y + kz = 3$$ $$kx - 3y + 5z = -1$$ $$x - 2y + 3z = -4$$ Where \(k\) is a real constant. The planes do not meet at a unique point.
  1. Find the possible values of \(k\) [3 marks]
  2. There are two possible geometric configurations of the given planes. Identify each possible configurations, stating the corresponding value of \(k\) Fully justify your answer. [5 marks]
  3. Given further that the equations of the planes form a consistent system, find the solution of the system of equations. [3 marks]
AQA Further Paper 1 Specimen Q8
5 marks Standard +0.3
A curve has equation $$y = \frac{5 - 4x}{1 + x}$$
  1. Sketch the curve. [4 marks]
  2. Hence sketch the graph of \(y = \left|\frac{5 - 4x}{1 + x}\right|\). [1 mark]
AQA Further Paper 1 Specimen Q9
13 marks Challenging +1.3
A line has Cartesian equations \(x - p = \frac{y + 2}{q} = 3 - z\) and a plane has equation \(\mathbf{r} \cdot \begin{bmatrix} 1 \\ -1 \\ -2 \end{bmatrix} = -3\)
  1. In the case where the plane fully contains the line, find the values of \(p\) and \(q\). [3 marks]
  2. In the case where the line intersects the plane at a single point, find the range of values of \(p\) and \(q\). [3 marks]
  3. In the case where the angle \(\theta\) between the line and the plane satisfies \(\sin \theta = \frac{1}{\sqrt{6}}\) and the line intersects the plane at \(z = 0\)
    1. Find the value of \(q\). [4 marks]
    2. Find the value of \(p\). [3 marks]