Questions — CAIE Further Paper 2 (195 questions)

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CAIE Further Paper 2 2021 November Q6
11 marks Challenging +1.2
The matrix \(\mathbf{P}\) is given by $$\mathbf{P} = \begin{pmatrix} 1 & 6 & 6 \\ 0 & 2 & 6 \\ 0 & 0 & -3 \end{pmatrix}.$$
  1. Use the characteristic equation of \(\mathbf{P}\) to find \(\mathbf{P}^{-1}\). [5]
  2. Find the matrix \(\mathbf{A}\) such that $$\mathbf{P}^{-1}\mathbf{A}\mathbf{P} = \begin{pmatrix} 4 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 6 \end{pmatrix}.$$ [4]
  3. State the eigenvalues and corresponding eigenvectors of \(\mathbf{A}^3\). [2]
CAIE Further Paper 2 2021 November Q7
11 marks Challenging +1.8
It is given that \(y = x^2w\) and $$x^2\frac{d^2w}{dx^2} + 4x(x + 1)\frac{dw}{dx} + (5x^2 + 8x + 2)w = 5x^2 + 4x + 2.$$
  1. Show that $$\frac{d^2y}{dx^2} + 4\frac{dy}{dx} + 5y = 5x^2 + 4x + 2.$$ [4]
  2. Find the general solution for \(w\) in terms of \(x\). [7]
CAIE Further Paper 2 2021 November Q8
13 marks Challenging +1.2
  1. Starting from the definitions of tanh and sech in terms of exponentials, prove that $$1 - \tanh^2 x = \sech^2 x.$$ [3]
  2. Using the substitution \(u = \tanh x\), or otherwise, find \(\int \sech^2 x \tanh^2 x \, dx\). [2]
  3. It is given that, for \(n \geq 0\), \(I_n = \int_0^{\ln 3} \sech^n x \tanh^2 x \, dx\). Show that, for \(n \geq 2\), $$(n + 1)I_n = \left(\frac{4}{3}\right)^{\frac{3}{n-2}} + (n - 2)I_{n-2}.$$ [You may use the result that \(\frac{d}{dx}(\sech x) = -\tanh x \sech x\).] [5]
  4. Find the value of \(I_4\). [3]
CAIE Further Paper 2 2023 November Q1
4 marks Standard +0.8
Show that the system of equations $$14x - 4y + 6z = 5,$$ $$x + y + kz = 3,$$ $$-21x + 6y - 9z = 14,$$ where \(k\) is a constant, does not have a unique solution and interpret this situation geometrically. [4]
CAIE Further Paper 2 2023 November Q2
5 marks Standard +0.3
Find the roots of the equation \((z + 5i)^3 = 4 + 4\sqrt{3}i\), giving your answers in the form \(r\cos\theta + ir\sin\theta - 5)\), where \(r > 0\) and \(0 \leq \theta < 2\pi\). [5]
CAIE Further Paper 2 2023 November Q3
6 marks Challenging +1.2
Find the first three terms in the Maclaurin's series for \(\tanh^{-1}\left(\frac{1}{2}e^t\right)\) in the form \(\frac{1}{2}\ln a + bx + cx^2\), giving the exact values of the constants \(a\), \(b\) and \(c\). [6]
CAIE Further Paper 2 2023 November Q4
10 marks Standard +0.3
Find the particular solution of the differential equation $$\frac{d^2y}{dx^2} + 2\frac{dy}{dx} + 3y = 27x^2,$$ given that, when \(x = 0\), \(y = 2\) and \(\frac{dy}{dx} = -8\). [10]
CAIE Further Paper 2 2023 November Q5
10 marks Challenging +1.2
The curve C has parametric equations $$x = \frac{5}{3}t^{\frac{3}{2}} - 2t^{\frac{1}{2}}, \quad y = 2t + 5, \quad \text{for } 0 < t \leq 3.$$
  1. Find the exact length of C. [5]
  2. Find the set of values of \(t\) for which \(\frac{d^2y}{dx^2} > 0\). [5]
CAIE Further Paper 2 2023 November Q6
14 marks Standard +0.8
  1. Starting from the definitions of cosh and sinh in terms of exponentials, prove that $$\sinh 2x = 2\sinh x\cosh x.$$ [3]
  2. Using the substitution \(u = \sinh x\), find \(\int \sinh^2 2x\cosh x\,dx\). [4]
  3. Find the particular solution of the differential equation $$\frac{dy}{dx} + y\tanh x = \sinh^2 2x,$$ given that \(y = 4\) when \(x = 0\). Give your answer in the form \(y = f(x)\). [7]
CAIE Further Paper 2 2023 November Q7
11 marks Challenging +1.2
The matrix A is given by $$\mathbf{A} = \begin{pmatrix} -6 & 2 & 13 \\ 0 & -2 & 5 \\ 0 & 0 & 8 \end{pmatrix}.$$
  1. Find a matrix P and a diagonal matrix D such that \(\mathbf{A}^{-1} = \mathbf{PDP}^{-1}\). [7]
  2. Use the characteristic equation of A to find \(\mathbf{A}^{-1}\). [4]
CAIE Further Paper 2 2023 November Q8
15 marks Challenging +1.8
  1. State the sum of the series \(1 + z + z^2 + \ldots + z^{n-1}\), for \(z \neq 1\). [1]
  2. By letting \(z = \cos\theta + i\sin\theta\), where \(\cos\theta \neq 1\), show that $$1 + \cos\theta + \cos 2\theta + \ldots + \cos(n-1)\theta = \frac{1}{2}\left(1 - \cos n\theta + \frac{\sin n\theta \sin\theta}{1 - \cos\theta}\right).$$ [7]
\includegraphics{figure_8} The diagram shows the curve with equation \(y = \cos x\) for \(0 \leq x \leq 1\), together with a set of \(n\) rectangles of width \(\frac{1}{n}\).
  1. By considering the sum of the areas of these rectangles, show that $$\int_0^1 \cos x\,dx < \frac{1}{2n}\left(1 - \cos 1 + \frac{\sin 1\sin\frac{1}{n}}{1 - \cos\frac{1}{n}}\right).$$ [4]
  2. Use a similar method to find, in terms of \(n\), a lower bound for \(\int_0^1 \cos x\,dx\). [3]
CAIE Further Paper 2 2024 November Q1
4 marks Standard +0.8
Find the value of \(\int_6^7 \frac{1}{\sqrt{(x-5)^2-1}} \, dx\), giving your answer in the form \(\ln(a + \sqrt{b})\), where \(a\) and \(b\) are integers to be determined. [4]
CAIE Further Paper 2 2024 November Q2
7 marks Standard +0.3
The curve \(C\) has equation $$4y^2 + 4\ln(xy) = 1.$$
  1. Show that, at the point \(\left(2, \frac{1}{2}\right)\) on \(C\), \(\frac{dy}{dx} = -\frac{1}{6}\). [3]
  2. Find the value of \(\frac{d^2y}{dx^2}\) at the point \(\left(2, \frac{1}{2}\right)\). [4]
CAIE Further Paper 2 2024 November Q3
7 marks Challenging +1.8
The curve \(C\) has parametric equations $$x = \frac{1}{2}e^{2t} - \frac{1}{3}t^3 - \frac{1}{2}, \quad y = 2e^t(t-1), \quad \text{for } 0 \leqslant t \leqslant 1.$$ Find the exact length of \(C\). [7]
CAIE Further Paper 2 2024 November Q4
10 marks Challenging +1.8
  1. Use de Moivre's theorem to show that $$\cot 6\theta = \frac{\cot^4 \theta - 15\cot^4 \theta + 15\cot^2 \theta - 1}{6\cot^5 \theta - 20\cot^3 \theta + 6\cot \theta}.$$ [6]
  2. Hence obtain the roots of the equation $$x^6 - 6x^5 - 15x^4 + 20x^3 + 15x^2 - 6x - 1 = 0$$ in the form \(\cot(q\pi)\), where \(q\) is a rational number. [4]
CAIE Further Paper 2 2024 November Q5
10 marks Standard +0.8
Find the particular solution of the differential equation $$3\frac{d^2y}{dx^2} + 2\frac{dy}{dx} + y = x^2,$$ given that, when \(x = 0\), \(y = \frac{dy}{dx} = 0\). [10]
CAIE Further Paper 2 2024 November Q6
13 marks Challenging +1.2
\includegraphics{figure_6} The diagram shows the curve with equation \(y = e^{1-x}\) for \(0 \leqslant x \leqslant 1\), together with a set of \(n\) rectangles of width \(\frac{1}{n}\).
  1. By considering the sum of the areas of these rectangles, show that \(\int_0^1 e^{1-x} \, dx < U_n\), where $$U_n = \frac{e-1}{n(1-e^{-1})}.$$ [4]
  2. Use a similar method to find, in terms of \(n\), a lower bound \(L_n\) for \(\int_0^1 e^{1-x} \, dx\). [4]
  3. Show that \(\lim_{n \to \infty}(U_n - L_n) = 0\). [2]
  4. Use the Maclaurin's series for \(e^x\) given in the list of formulae (MF19) to find the first three terms of the series expansion of \(z(1-e^{-z})\), in ascending powers of \(z\), and deduce the value of \(\lim_{n \to \infty}(U_n)\). [3]
CAIE Further Paper 2 2024 November Q7
10 marks Challenging +1.2
  1. Show that \(\frac{d}{dx}(\ln(\tanh x)) = 2\cosh 2x\). [3]
  2. Find the solution of the differential equation $$\sinh 2x \frac{dy}{dx} + 2y = \sinh 2x$$ for which \(y = 5\) when \(x = \ln 2\). Give your answer in an exact form. [7]
CAIE Further Paper 2 2024 November Q8
14 marks Challenging +1.3
The matrix \(\mathbf{A}\) is given by $$\mathbf{A} = \begin{pmatrix} -2 & 0 & 0 \\ 0 & 7 & 9 \\ 4 & 1 & 7 \end{pmatrix}.$$
  1. Show that the characteristic equation of \(\mathbf{A}\) is \(\lambda^3 - 12\lambda^2 + 124 + 80 = 0\) and find the eigenvalues of \(\mathbf{A}\). [4]
  2. Use the characteristic equation of \(\mathbf{A}\) to show that $$\mathbf{A}^4 = p\mathbf{A}^2 + q\mathbf{A} + r\mathbf{I},$$ where \(p\), \(q\) and \(r\) are integers to be determined. [4]
  3. Find a matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that \((\mathbf{A} - 3\mathbf{I})^4 = \mathbf{PDP}^{-1}\). [6]
CAIE Further Paper 2 2020 Specimen Q0
Standard +0.3
0 & 2 & 2
- 1 & 1 & 3 \end{array} \right) .$$
  1. Find the eigenvalues of \(\mathbf { A }\).
  2. Use the characteristic equation of \(\mathbf { A }\) to find \(\mathbf { A } ^ { - 1 }\).