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CAIE FP1 2019 November Q1
6 marks Standard +0.8
The curve \(C\) has equation \(y = x^a\) for \(0 \leqslant x \leqslant 1\), where \(a\) is a positive constant. Find, in terms of \(a\), the coordinates of the centroid of the region enclosed by \(C\), the line \(x = 1\) and the \(x\)-axis. [6]
CAIE FP1 2019 November Q2
6 marks Standard +0.8
It is given that \(y = \ln(ax + 1)\), where \(a\) is a positive constant. Prove by mathematical induction that, for every positive integer \(n\), $$\frac{d^n y}{dx^n} = (-1)^{n-1} \frac{(n-1)!a^n}{(ax+1)^n}.$$ [6]
CAIE FP1 2019 November Q3
7 marks Challenging +1.8
The integral \(I_n\), where \(n\) is a positive integer, is defined by $$I_n = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} x^{-n} \sin \pi x \, dx.$$
  1. Show that $$n(n+1)I_{n+2} = 2^{n+1} n + \pi - \pi^2 I_n.$$ [5]
  2. Find \(I_5\) in terms of \(\pi\) and \(I_1\). [2]
CAIE FP1 2019 November Q4
7 marks Standard +0.3
The line \(y = 2x + 1\) is an asymptote of the curve \(C\) with equation $$y = \frac{x^2 + 1}{ax + b}.$$
  1. Find the values of the constants \(a\) and \(b\). [3]
  2. State the equation of the other asymptote of \(C\). [1]
  3. Sketch \(C\). [Your sketch should indicate the coordinates of any points of intersection with the \(y\)-axis. You do not need to find the coordinates of any stationary points.] [3]
CAIE FP1 2019 November Q5
9 marks Standard +0.8
Let \(S_N = \sum_{r=1}^{N} (5r + 1)(5r + 6)\) and \(T_N = \sum_{r=1}^{N} \frac{1}{(5r + 1)(5r + 6)}\).
  1. Use standard results from the List of Formulae (MF10) to show that $$S_N = \frac{1}{3}N(25N^2 + 90N + 83).$$ [3]
  2. Use the method of differences to express \(T_N\) in terms of \(N\). [4]
  3. Find \(\lim_{N \to \infty} (N^{-3} S_N T_N)\). [2]
CAIE FP1 2019 November Q6
9 marks Challenging +1.2
With \(O\) as the origin, the points \(A\), \(B\), \(C\) have position vectors $$\mathbf{i} - \mathbf{j}, \quad 2\mathbf{i} + \mathbf{j} + 7\mathbf{k}, \quad \mathbf{i} - \mathbf{j} + \mathbf{k}$$ respectively.
  1. Find the shortest distance between the lines \(OC\) and \(AB\). [5]
  2. Find the cartesian equation of the plane containing the line \(OC\) and the common perpendicular of the lines \(OC\) and \(AB\). [4]
CAIE FP1 2019 November Q7
9 marks Challenging +1.3
The equation \(x^3 + 2x^2 + x + 7 = 0\) has roots \(\alpha\), \(\beta\), \(\gamma\).
  1. Use the relation \(x^2 = -7y\) to show that the equation $$49y^3 + 14y^2 - 27y + 7 = 0$$ has roots \(\frac{\alpha}{\beta \gamma}\), \(\frac{\beta}{\gamma \alpha}\), \(\frac{\gamma}{\alpha \beta}\). [4]
  2. Show that \(\frac{\alpha^2}{\beta^2 \gamma^2} + \frac{\beta^2}{\gamma^2 \alpha^2} + \frac{\gamma^2}{\alpha^2 \beta^2} = \frac{58}{49}\). [3]
  3. Find the exact value of \(\frac{\alpha^2}{\beta^3 \gamma^3} + \frac{\beta^2}{\gamma^3 \alpha^3} + \frac{\gamma^2}{\alpha^3 \beta^3}\). [2]
CAIE FP1 2019 November Q8
10 marks Challenging +1.2
The matrix \(\mathbf{M}\) is defined by $$\mathbf{M} = \begin{pmatrix} 2 & m & 1 \\ 0 & m & 7 \\ 0 & 0 & 1 \end{pmatrix},$$ where \(m \neq 0, 1, 2\).
  1. Find a matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that \(\mathbf{M} = \mathbf{PDP}^{-1}\). [7]
  2. Find \(\mathbf{M}^T \mathbf{P}\). [3]
CAIE FP1 2019 November Q9
11 marks Challenging +1.8
  1. Use de Moivre's theorem to show that $$\sec 6\theta = \frac{\sec^6 \theta}{32 - 48 \sec^2 \theta + 18 \sec^4 \theta - \sec^6 \theta}.$$ [6]
  2. Hence obtain the roots of the equation $$3x^6 - 36x^4 + 96x^2 - 64 = 0$$ in the form \(\sec q\pi\), where \(q\) is rational. [5]
CAIE FP1 2019 November Q10
12 marks Standard +0.8
The matrix \(\mathbf{A}\) is defined by $$\mathbf{A} = \begin{pmatrix} 1 & 5 & 1 \\ 1 & -2 & -2 \\ 2 & 3 & \theta \end{pmatrix}.$$
  1. Find the rank of \(\mathbf{A}\) when \(\theta \neq -1\). [3]
  2. Find the rank of \(\mathbf{A}\) when \(\theta = -1\). [1]
Consider the system of equations \begin{align} x + 5y + z &= -1,
x - 2y - 2z &= 0,
2x + 3y + \theta z &= \theta. \end{align}
  1. Solve the system of equations when \(\theta \neq -1\). [3]
  2. Find the general solution when \(\theta = -1\). [3]
  3. Show that if \(\theta = -1\) and \(\phi \neq -1\) then \(\mathbf{A}\mathbf{x} = \begin{pmatrix} -1 \\ 0 \\ \phi \end{pmatrix}\) has no solution. [2]
CAIE FP1 2019 November Q11
28 marks Challenging +1.8
Answer only one of the following two alternatives. **EITHER** It is given that \(w = \cos y\) and $$\tan y \frac{d^2 y}{dx^2} + \left( \frac{dy}{dx} \right)^2 + 2 \tan y \frac{dy}{dx} = 1 + e^{-2x} \sec y.$$
  1. Show that $$\frac{d^2 w}{dx^2} + 2 \frac{dw}{dx} + w = -e^{-2x}.$$ [4]
  2. Find the particular solution for \(y\) in terms of \(x\), given that when \(x = 0\), \(y = \frac{1}{4}\pi\) and \(\frac{dy}{dx} = \frac{1}{\sqrt{3}}\). [10]
**OR** The curves \(C_1\) and \(C_2\) have polar equations, for \(0 \leqslant \theta \leqslant \frac{1}{2}\pi\), as follows: \begin{align} C_1 : r &= 2(e^\theta + e^{-\theta}),
C_2 : r &= e^{2\theta} - e^{-2\theta}. \end{align} The curves intersect at the point \(P\) where \(\theta = \alpha\).
  1. Show that \(e^{2\alpha} - 2e^\alpha - 1 = 0\). Hence find the exact value of \(\alpha\) and show that the value of \(r\) at \(P\) is \(4\sqrt{2}\). [6]
  2. Sketch \(C_1\) and \(C_2\) on the same diagram. [3]
  3. Find the area of the region enclosed by \(C_1\), \(C_2\) and the initial line, giving your answer correct to 3 significant figures. [5]
CAIE FP1 2019 November Q1
6 marks Standard +0.8
The curve \(C\) has equation \(y = x^a\) for \(0 \leq x \leq 1\), where \(a\) is a positive constant. Find, in terms of \(a\), the coordinates of the centroid of the region enclosed by \(C\), the line \(x = 1\) and the \(x\)-axis. [6]
CAIE FP1 2019 November Q2
6 marks Standard +0.8
It is given that \(y = \ln(ax + 1)\), where \(a\) is a positive constant. Prove by mathematical induction that, for every positive integer \(n\), $$\frac{d^n y}{dx^n} = (-1)^{n-1} \frac{(n-1)! a^n}{(ax+1)^n}.$$ [6]
CAIE FP1 2019 November Q3
7 marks Challenging +1.3
The integral \(I_n\), where \(n\) is a positive integer, is defined by $$I_n = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} x^n \sin \pi x \, dx.$$
  1. Show that $$n(n+1)I_{n+2} = 2^{n+1}n + \pi - \pi^2 I_n.$$ [5]
  2. Find \(I_5\) in terms of \(\pi\) and \(I_1\). [2]
CAIE FP1 2019 November Q7
9 marks Challenging +1.8
The equation \(x^3 + 2x^2 + x + 7 = 0\) has roots \(\alpha\), \(\beta\), \(\gamma\).
  1. Use the relation \(x^2 = -7y\) to show that the equation $$49y^3 + 14y^2 - 27y + 7 = 0$$ has roots \(\frac{\alpha}{\beta\gamma}\), \(\frac{\beta}{\gamma\alpha}\), \(\frac{\gamma}{\alpha\beta}\). [4]
  2. Show that \(\frac{\alpha^2}{\beta^2\gamma^2} + \frac{\beta^2}{\gamma^2\alpha^2} + \frac{\gamma^2}{\alpha^2\beta^2} = \frac{58}{49}\). [3]
  3. Find the exact value of \(\frac{\alpha^3}{\beta^3\gamma^3} + \frac{\beta^3}{\gamma^3\alpha^3} + \frac{\gamma^3}{\alpha^3\beta^3}\). [2]
CAIE FP1 2019 November Q8
10 marks Standard +0.8
The matrix \(\mathbf{M}\) is defined by $$\mathbf{M} = \begin{pmatrix} 2 & m & 1 \\ 0 & m & 7 \\ 0 & 0 & 1 \end{pmatrix},$$ where \(m \neq 0, 1, 2\).
  1. Find a matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that \(\mathbf{M} = \mathbf{PDP}^{-1}\). [7]
  2. Find \(\mathbf{M}^T\mathbf{P}\). [3]
CAIE FP1 2019 November Q9
11 marks Challenging +1.8
  1. Use de Moivre's theorem to show that $$\sec 6\theta = \frac{\sec^6 \theta}{32 - 48\sec^2 \theta + 18\sec^4 \theta - \sec^6 \theta}.$$ [6]
  2. Hence obtain the roots of the equation $$3t^6 - 36t^4 + 96t^2 - 64 = 0$$ in the form \(\sec q\pi\), where \(q\) is rational. [5]
CAIE FP1 2019 November Q10
12 marks Standard +0.8
The matrix \(\mathbf{A}\) is defined by $$\mathbf{A} = \begin{pmatrix} 1 & 5 & 1 \\ 1 & -2 & -2 \\ 2 & 3 & \theta \end{pmatrix}.$$
  1. Find the rank of \(\mathbf{A}\) when \(\theta \neq -1\). [3]
  2. Find the rank of \(\mathbf{A}\) when \(\theta = -1\). [1]
Consider the system of equations \begin{align} x + 5y + z &= -1,
x - 2y - 2z &= 0,
2x + 3y + \theta z &= \theta. \end{align}
  1. Solve the system of equations when \(\theta \neq -1\). [3]
  2. Find the general solution when \(\theta = -1\). [3]
  3. Show that if \(\theta = -1\) and \(\phi \neq -1\) then \(\mathbf{A}\mathbf{x} = \begin{pmatrix} -1 \\ 0 \\ \phi \end{pmatrix}\) has no solution. [2]
CAIE FP1 2019 November Q11
28 marks Challenging +1.8
Answer only one of the following two alternatives. **EITHER** It is given that \(w = \cos y\) and $$\tan y \frac{d^2 y}{dx^2} + \left(\frac{dy}{dx}\right)^2 + 2\tan y \frac{dy}{dx} = 1 + e^{-2x} \sec y.$$
  1. Show that $$\frac{d^2 w}{dx^2} + 2\frac{dw}{dx} + w = -e^{-2x}.$$ [4]
  2. Find the particular solution for \(y\) in terms of \(x\), given that when \(x = 0\), \(y = \frac{1}{4}\pi\) and \(\frac{dy}{dx} = \frac{1}{\sqrt{3}}\). [10]
**OR** The curves \(C_1\) and \(C_2\) have polar equations, for \(0 \leq \theta \leq \frac{1}{2}\pi\), as follows: \begin{align} C_1: r &= 2(e^\theta + e^{-\theta}),
C_2: r &= e^{2\theta} - e^{-2\theta}. \end{align} The curves intersect at the point \(P\) where \(\theta = \alpha\).
  1. Show that \(e^{2\alpha} - 2e^\alpha - 1 = 0\). Hence find the exact value of \(\alpha\) and show that the value of \(r\) at \(P\) is \(4\sqrt{2}\). [6]
  2. Sketch \(C_1\) and \(C_2\) on the same diagram. [3]
  3. Find the area of the region enclosed by \(C_1\), \(C_2\) and the initial line, giving your answer correct to 3 significant figures. [5]
CAIE Further Paper 2 2020 June Q1
6 marks Standard +0.3
Find the general solution of the differential equation $$\frac{d^2x}{dt^2} - 8\frac{dx}{dt} - 9x = 9e^{8t}.$$ [6]
CAIE Further Paper 2 2020 June Q2
6 marks Challenging +1.2
Let \(I_n = \int_0^1 (1+3x)^n e^{-3x} dx\), where \(n\) is an integer.
  1. Show that \(3I_n = 1 - 4^n e^{-3} + 3nI_{n-1}\). [3]
  2. Find the exact value of \(I_2\). [3]
CAIE Further Paper 2 2020 June Q3
8 marks Challenging +1.2
The matrix \(\mathbf{A}\) is given by $$\mathbf{A} = \begin{pmatrix} 5 & -1 & 7 \\ 0 & 6 & 0 \\ 7 & 7 & 5 \end{pmatrix}.$$
  1. Find the eigenvalues of \(\mathbf{A}\). [4]
  2. Use the characteristic equation of \(\mathbf{A}\) to find \(\mathbf{A}^{-1}\). [4]
CAIE Further Paper 2 2020 June Q4
8 marks Challenging +1.2
\includegraphics{figure_4} The diagram shows the curve with equation \(y = \ln x\) for \(x \geqslant 1\), together with a set of \((N-1)\) rectangles of unit width.
  1. By considering the sum of the areas of these rectangles, show that $$\ln N! > N \ln N - N + 1.$$ [5]
  2. Use a similar method to find, in terms of \(N\), an upper bound for \(\ln N!\). [3]
CAIE Further Paper 2 2020 June Q5
9 marks Challenging +1.3
The curve \(C\) has parametric equations $$x = \frac{1}{2}t^2 - \ln t, \quad y = 2t + 1, \quad \text{for } \frac{1}{2} \leqslant t \leqslant 2.$$
  1. Find the exact length of \(C\). [5]
  2. Find \(\frac{d^2y}{dx^2}\) in terms of \(t\), simplifying your answer. [4]
CAIE Further Paper 2 2020 June Q6
12 marks Standard +0.8
  1. Starting from the definitions of \(\tanh\) and \(\sech\) in terms of exponentials, prove that $$1 - \tanh^2 \theta = \sech^2 \theta.$$ [3]
The variables \(x\) and \(y\) are such that \(\tanh y = \cos\left(x + \frac{1}{4}\pi\right)\), for \(-\frac{1}{4}\pi < x < \frac{3}{4}\pi\).
  1. By differentiating the equation \(\tanh y = \cos\left(x + \frac{1}{4}\pi\right)\) with respect to \(x\), show that $$\frac{dy}{dx} = -\operatorname{cosec}\left(x + \frac{1}{4}\pi\right).$$ [4]
  2. Hence find the first three terms in the Maclaurin's series for \(\tanh^{-1}\left(\cos\left(x + \frac{1}{4}\pi\right)\right)\) in the form \(\frac{1}{2}\ln a + bx + cx^2\), giving the exact values of the constants \(a\), \(b\) and \(c\). [5]