Questions Further Paper 1 (256 questions)

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AQA Further Paper 1 2020 June Q11
11 marks Standard +0.8
11 The lines \(l _ { 1 } , l _ { 2 }\) and \(l _ { 3 }\) are defined as follows. $$\begin{aligned} & l _ { 1 } : \left( \mathbf { r } - \left[ \begin{array} { c } 1 \\ 5 \\ - 1 \end{array} \right] \right) \times \left[ \begin{array} { c } - 2 \\ 1 \\ - 3 \end{array} \right] = \mathbf { 0 } \\ & l _ { 2 } : \left( \mathbf { r } - \left[ \begin{array} { c } - 3 \\ 2 \\ 7 \end{array} \right] \right) \times \left[ \begin{array} { c } 2 \\ - 1 \\ 3 \end{array} \right] = \mathbf { 0 } \\ & l _ { 3 } : \left( \mathbf { r } - \left[ \begin{array} { c } - 5 \\ 12 \\ - 4 \end{array} \right] \right) \times \left[ \begin{array} { l } 4 \\ 0 \\ 9 \end{array} \right] = \mathbf { 0 } \end{aligned}$$ 11
    1. Explain how you know that two of the lines are parallel.
      11
      1. (ii)
      Show that the perpendicular distance between these two parallel lines is 7.95 units, correct to three significant figures.
      [5 marks] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)
      11
  2. Show that the lines \(l _ { 1 }\) and \(l _ { 3 }\) meet, and find the coordinates of their point of intersection. \includegraphics[max width=\textwidth, alt={}, center]{44e22a98-6424-4fb1-8a37-c965773cb7b6-23_2488_1716_219_153}
AQA Further Paper 1 2020 June Q12
8 marks Standard +0.8
12
  1. Use the definition of the cosh function to prove that $$\cosh ^ { - 1 } \left( \frac { x } { a } \right) = \ln \left( \frac { x + \sqrt { x ^ { 2 } - a ^ { 2 } } } { a } \right) \quad \text { for } a > 0$$ [6 marks]
    12
  2. The formulae booklet gives the integral of \(\frac { 1 } { \sqrt { x ^ { 2 } - a ^ { 2 } } }\) as $$\cosh ^ { - 1 } \left( \frac { x } { a } \right) \text { or } \ln \left( x + \sqrt { x ^ { 2 } - a ^ { 2 } } \right) + c$$ Ronald says that this contradicts the result given in part (a).
    Explain why Ronald is wrong.
AQA Further Paper 1 2020 June Q13
12 marks Standard +0.8
13 Two light elastic strings each have one end attached to a particle \(B\) of mass \(3 c \mathrm {~kg}\), which rests on a smooth horizontal table. The other ends of the strings are attached to the fixed points \(A\) and \(C\), which are 8 metres apart. \(A B C\) is a horizontal line. \includegraphics[max width=\textwidth, alt={}, center]{44e22a98-6424-4fb1-8a37-c965773cb7b6-26_92_910_635_566} String \(A B\) has a natural length of 4 metres and a stiffness of \(5 c\) newtons per metre.
String \(B C\) has a natural length of 1 metre and a stiffness of \(c\) newtons per metre.
The particle is pulled a distance of \(\frac { 1 } { 3 }\) metre from its equilibrium position towards \(A\), and released from rest. 13
  1. Show that the particle moves with simple harmonic motion.
    13
  2. Find the speed of the particle when it is at a point \(P\), a distance \(\frac { 1 } { 4 }\) metre from the equilibrium position. Give your answer to two significant figures.
    [0pt] [4 marks]
AQA Further Paper 1 2020 June Q14
6 marks Challenging +1.2
14
  1. Given that $$\sinh ( A + B ) = \sinh A \cosh B + \cosh A \sinh B$$ express \(\sinh ( m + 1 ) x\) and \(\sinh ( m - 1 ) x\) in terms of \(\sinh m x , \cosh m x , \sinh x\) and \(\cosh x\) 14
  2. Hence find the sum of the series $$C _ { n } = \cosh x + \cosh 2 x + \cdots + \cosh n x$$ in terms of \(\sinh x , \sinh n x\) and \(\sinh ( n + 1 ) x\) Do not write \includegraphics[max width=\textwidth, alt={}, center]{44e22a98-6424-4fb1-8a37-c965773cb7b6-30_2491_1736_219_139}
CAIE Further Paper 1 2023 June Q1
Standard +0.3
1 Let \(\mathbf { A } = \left( \begin{array} { l l } 3 & 0 \\ 1 & 1 \end{array} \right)\).
  1. Prove by mathematical induction that, for all positive integers \(n\), $$2 \mathbf { A } ^ { n } = \left( \begin{array} { l l } 2 \times 3 ^ { n } & 0 \\ 3 ^ { n } - 1 & 2 \end{array} \right)$$
  2. Find, in terms of \(n\), the inverse of \(\mathbf { A } ^ { n }\).
CAIE Further Paper 1 2023 June Q2
Standard +0.8
2 The cubic equation \(x ^ { 3 } + 4 x ^ { 2 } + 6 x + 1 = 0\) has roots \(\alpha , \beta , \gamma\).
  1. Find the value of \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\).
  2. Use standard results from the list of formulae (MF19) to show that $$\sum _ { r = 1 } ^ { n } \left( ( \alpha + r ) ^ { 2 } + ( \beta + r ) ^ { 2 } + ( \gamma + r ) ^ { 2 } \right) = n \left( n ^ { 2 } + a n + b \right)$$ where \(a\) and \(b\) are constants to be determined.
CAIE Further Paper 1 2024 November Q1
10 marks Standard +0.3
The matrix \(\mathbf{M}\) represents the sequence of two transformations in the \(x\)-\(y\) plane given by a stretch parallel to the \(x\)-axis, scale factor \(k\) (\(k \neq 0\)), followed by a shear, \(x\)-axis fixed, with \((0, 1)\) mapped to \((k, 1)\).
  1. Show that \(\mathbf{M} = \begin{pmatrix} k & k \\ 0 & 1 \end{pmatrix}\). [4]
  2. The transformation represented by \(\mathbf{M}\) has a line of invariant points. Find, in terms of \(k\), the equation of this line. [3]
The unit square \(S\) in the \(x\)-\(y\) plane is transformed by \(\mathbf{M}\) onto the parallelogram \(P\).
  1. Find, in terms of \(k\), a matrix which transforms \(P\) onto \(S\). [1]
  2. Given that the area of \(P\) is \(3k^2\) units\(^2\), find the possible values of \(k\). [2]
CAIE Further Paper 1 2024 November Q2
6 marks Challenging +1.2
Prove by mathematical induction that, for all positive integers \(n\), $$\frac{\mathrm{d}^n}{\mathrm{d}x^n}\left(\tan^{-1}x\right) = P_n(x)\left(1+x^2\right)^{-n},$$ where \(P_n(x)\) is a polynomial of degree \(n-1\). [6]
CAIE Further Paper 1 2024 November Q3
10 marks Challenging +1.8
The quartic equation \(x^4 + 2x^3 - 1 = 0\) has roots \(\alpha, \beta, \gamma, \delta\).
  1. Find a quartic equation whose roots are \(\alpha^4, \beta^4, \gamma^4, \delta^4\) and state the value of \(\alpha^4 + \beta^4 + \gamma^4 + \delta^4\). [5]
  2. Find the value of \(\alpha^5 + \beta^5 + \gamma^5 + \delta^5\). [3]
  3. Find the value of \(\alpha^8 + \beta^8 + \gamma^8 + \delta^8\). [2]
CAIE Further Paper 1 2024 November Q4
8 marks Challenging +1.2
  1. Use the method of differences to find \(\sum_{r=1}^{n} \frac{5k}{(5r+k)(5r+5+k)}\) in terms of \(n\) and \(k\), where \(k\) is a positive constant. [4]
It is given that \(\sum_{r=1}^{\infty} \frac{5k}{(5r+k)(5r+5+k)} = \frac{1}{3}\).
  1. Find the value of \(k\). [2]
  2. Hence find \(\sum_{r=1}^{n-1} \frac{5k}{(5r+k)(5r+5+k)}\) in terms of \(n\). [2]
CAIE Further Paper 1 2024 November Q5
13 marks Challenging +1.2
  1. Show that the curve with Cartesian equation $$\left(x^2+y^2\right)^2 = 6xy$$ has polar equation \(r^2 = 3\sin 2\theta\). [2]
The curve \(C\) has polar equation \(r^2 = 3\sin 2\theta\), for \(0 \leqslant \theta \leqslant \frac{1}{2}\pi\).
  1. Sketch \(C\) and state the maximum distance of a point on \(C\) from the pole. [3]
  2. Find the area of the region enclosed by \(C\). [2]
  3. Find the maximum distance of a point on \(C\) from the initial line. [6]
CAIE Further Paper 1 2024 November Q6
13 marks Challenging +1.2
The curve \(C\) has equation \(y = \frac{4x^2 + x + 1}{2x^2 - 7x + 3}\).
  1. Find the equations of the asymptotes of \(C\). [2]
  2. Find the coordinates of any stationary points on \(C\). [4]
  3. Sketch \(C\), stating the coordinates of any intersections with the axes. [5]
  4. Sketch the curve with equation \(y = \left|\frac{4x^2 + x + 1}{2x^2 - 7x + 3}\right|\) and state the set of values of \(k\) for which \(\left|\frac{4x^2 + x + 1}{2x^2 - 7x + 3}\right| = k\) has 4 distinct real solutions. [2]
CAIE Further Paper 1 2024 November Q7
15 marks Challenging +1.3
The lines \(l_1\) and \(l_2\) have equations \(\mathbf{r} = \mathbf{i} + 3\mathbf{j} - 2\mathbf{k} + \lambda(2\mathbf{i} + \mathbf{j} + \mathbf{k})\) and \(\mathbf{r} = \mathbf{i} - 2\mathbf{j} + 9\mathbf{k} + \mu(\mathbf{i} - 4\mathbf{j} + 2\mathbf{k})\) respectively. The plane \(\Pi_1\) contains \(l_1\) and is parallel to \(l_2\).
  1. Find the equation of \(\Pi_1\), giving your answer in the form \(ax + by + cz = d\). [4]
The plane \(\Pi_2\) contains \(l_2\) and the point with coordinates \((2, -1, 7)\).
  1. Find the acute angle between \(\Pi_1\) and \(\Pi_2\). [4]
The point \(P\) on \(l_1\) and the point \(Q\) on \(l_2\) are such that \(PQ\) is perpendicular to both \(l_1\) and \(l_2\).
  1. Find a vector equation for \(PQ\). [7]
CAIE Further Paper 1 2024 November Q4
8 marks Challenging +1.2
  1. Use the method of differences to find \(\sum_{r=1}^n \frac{5k}{(5r+k)(5r+5+k)}\) in terms of \(n\) and \(k\), where \(k\) is a positive constant. [4]
It is given that \(\sum_{r=1}^{\infty} \frac{5k}{(5r+k)(5r+5+k)} = \frac{1}{3}\).
  1. Find the value of \(k\). [2]
  2. Hence find \(\sum_{r=7}^{n+5} \frac{5k}{(5r+k)(5r+5+k)}\) in terms of \(n\). [2]
CAIE Further Paper 1 2024 November Q5
13 marks Challenging +1.2
  1. Show that the curve with Cartesian equation \(\left(x^2 + y^2\right)^2 = 6xy\) has polar equation \(r^2 = 3\sin 2\theta\). [2]
The curve \(C\) has polar equation \(r^2 = 3\sin 2\theta\), for \(0 \leqslant \theta \leqslant \frac{1}{2}\pi\).
  1. Sketch \(C\) and state the maximum distance of a point on \(C\) from the pole. [3]
  2. Find the area of the region enclosed by \(C\). [2]
  3. Find the maximum distance of a point on \(C\) from the initial line. [6]
AQA Further Paper 1 2019 June Q1
1 marks Easy -1.2
Which one of these functions has the set \(\{x : |x| < 1\}\) as its greatest possible domain? Circle your answer. [1 mark] \(\cosh x\) \quad \(\cosh^{-1} x\) \quad \(\tanh x\) \quad \(\tanh^{-1} x\)
AQA Further Paper 1 2019 June Q2
1 marks Moderate -0.5
The first two non-zero terms of the Maclaurin series expansion of \(f(x)\) are \(x\) and \(-\frac{1}{2}x^3\) Which one of the following could be \(f(x)\)? Circle your answer. [1 mark] \(xe^{\frac{1}{2}x^2}\) \quad \(\frac{1}{2}\sin 2x\) \quad \(x \cos x\) \quad \((1 + x^3)^{-\frac{1}{2}}\)
AQA Further Paper 1 2019 June Q3
1 marks Moderate -0.8
The function \(f(x) = x^2 - 1\) Find the mean value of \(f(x)\) from \(x = -0.5\) to \(x = 1.7\) Give your answer to three significant figures. Circle your answer. [1 mark] \(-0.521\) \quad \(-0.434\) \quad \(-0.237\) \quad \(0.786\)
AQA Further Paper 1 2019 June Q4
4 marks Moderate -0.5
Solve the equation \(2z - 5iz^* = 12\) [4 marks]
AQA Further Paper 1 2019 June Q5
3 marks Standard +0.3
A plane has equation \(\mathbf{r} \cdot \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} = 7\) A line has equation \(\mathbf{r} = \begin{pmatrix} 2 \\ 0 \\ 1 \end{pmatrix} + \mu \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}\) Calculate the acute angle between the line and the plane. Give your answer to the nearest \(0.1°\) [3 marks]
AQA Further Paper 1 2019 June Q6
8 marks Standard +0.8
  1. Show that $$\cosh^3 x + \sinh^3 x = \frac{1}{4}e^{mx} + \frac{3}{4}e^{nx}$$ where \(m\) and \(n\) are integers. [3 marks]
  2. Hence find \(\cosh^6 x - \sinh^6 x\) in the form $$\frac{a \cosh(kx) + b}{8}$$ where \(a\), \(b\) and \(k\) are integers. [5 marks]
AQA Further Paper 1 2019 June Q7
4 marks Challenging +1.2
Three non-singular square matrices, \(\mathbf{A}\), \(\mathbf{B}\) and \(\mathbf{R}\) are such that $$\mathbf{AR} = \mathbf{B}$$ The matrix \(\mathbf{R}\) represents a rotation about the \(z\)-axis through an angle \(\theta\) and $$\mathbf{B} = \begin{pmatrix} -\cos \theta & \sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{pmatrix}$$
  1. Show that \(\mathbf{A}\) is independent of the value of \(\theta\). [3 marks]
  2. Give a full description of the single transformation represented by the matrix \(\mathbf{A}\). [1 mark]
AQA Further Paper 1 2019 June Q8
10 marks Standard +0.8
  1. If \(z = \cos \theta + i \sin \theta\), use de Moivre's theorem to prove that $$z^n - \frac{1}{z^n} = 2i \sin n\theta$$ [3 marks]
  2. Express \(\sin^5 \theta\) in terms of \(\sin 5\theta\), \(\sin 3\theta\) and \(\sin \theta\) [4 marks]
  3. Hence show that $$\int_0^{\frac{\pi}{3}} \sin^5 \theta \, d\theta = \frac{53}{480}$$ [3 marks]
AQA Further Paper 1 2019 June Q9
9 marks Challenging +1.8
  1. Solve the equation \(z^3 = \sqrt{2} - \sqrt{6}i\), giving your answers in the form \(re^{i\theta}\) where \(r > 0\) and \(0 \leq \theta < 2\pi\) [5 marks]
  2. The transformation represented by the matrix \(\mathbf{M} = \begin{pmatrix} 5 & 1 \\ 1 & 3 \end{pmatrix}\) acts on the points on an Argand Diagram which represent the roots of the equation in part (a). Find the exact area of the shape formed by joining the transformed points. [4 marks]
AQA Further Paper 1 2019 June Q10
8 marks Standard +0.8
The points \(A(5, -4, 6)\) and \(B(6, -6, 8)\) lie on the line \(L\). The point \(C\) is \((15, -5, 9)\).
  1. \(D\) is the point on \(L\) that is closest to \(C\). Find the coordinates of \(D\). [6 marks]
  2. Hence find, in exact form, the shortest distance from \(C\) to \(L\). [2 marks]