Questions FP1 (1491 questions)

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CAIE FP1 2015 November Q5
8 marks Standard +0.3
The cubic equation \(x^3 + px^2 + qx + r = 0\), where \(p\), \(q\) and \(r\) are integers, has roots \(\alpha\), \(\beta\) and \(\gamma\), such that $$\alpha + \beta + \gamma = 15,$$ $$\alpha^2 + \beta^2 + \gamma^2 = 83.$$ Write down the value of \(p\) and find the value of \(q\). [3] Given that \(\alpha\), \(\beta\) and \(\gamma\) are all real and that \(\alpha\beta + \alpha\gamma = 36\), find \(\alpha\) and hence find the value of \(r\). [5]
CAIE FP1 2015 November Q6
10 marks Standard +0.3
The matrix \(\mathbf{A}\), where $$\mathbf{A} = \begin{pmatrix} 1 & 0 & 0 \\ 10 & -7 & 10 \\ 7 & -5 & 8 \end{pmatrix},$$ has eigenvalues 1 and 3. Find corresponding eigenvectors. [3] It is given that \(\begin{pmatrix} 0 \\ 2 \\ 1 \end{pmatrix}\) is an eigenvector of \(\mathbf{A}\). Find the corresponding eigenvalue. [2] Find a diagonal matrix \(\mathbf{D}\) and matrices \(\mathbf{P}\) and \(\mathbf{P}^{-1}\) such that \(\mathbf{P}^{-1}\mathbf{A}\mathbf{P} = \mathbf{D}\). [5]
CAIE FP1 2015 November Q7
10 marks Challenging +1.3
The linear transformation \(\mathrm{T} : \mathbb{R}^4 \to \mathbb{R}^4\) is represented by the matrix \(\mathbf{M}\), where $$\mathbf{M} = \begin{pmatrix} 1 & -2 & -3 & 1 \\ 3 & -5 & -7 & 7 \\ 5 & -9 & -13 & 9 \\ 7 & -13 & -19 & 11 \end{pmatrix}.$$ Find the rank of \(\mathbf{M}\) and a basis for the null space of \(\mathrm{T}\). [6] The vector \(\begin{pmatrix} 1 \\ 2 \\ 3 \\ 4 \end{pmatrix}\) is denoted by \(\mathbf{e}\). Show that there is a solution of the equation \(\mathbf{M}\mathbf{x} = \mathbf{M}\mathbf{e}\) of the form $$\mathbf{x} = \begin{pmatrix} a \\ b \\ -1 \\ -1 \end{pmatrix}, \text{ where the constants } a \text{ and } b \text{ are to be found.}$$ [4]
CAIE FP1 2015 November Q8
11 marks Standard +0.8
The curve \(C\) has equation \(y = \frac{2x^2 + kx}{x + 1}\), where \(k\) is a constant. Find the set of values of \(k\) for which \(C\) has no stationary points. [5] For the case \(k = 4\), find the equations of the asymptotes of \(C\) and sketch \(C\), indicating the coordinates of the points where \(C\) intersects the coordinate axes. [6]
CAIE FP1 2015 November Q9
12 marks Challenging +1.3
It is given that \(I_n = \int_{1}^{e} (\ln x)^n \mathrm{d}x\) for \(n \geqslant 0\). Show that $$I_n = (n - 1)[I_{n-2} - I_{n-1}] \text{ for } n \geqslant 2.$$ [6] Hence find, in an exact form, the mean value of \((\ln x)^3\) with respect to \(x\) over the interval \(1 \leqslant x \leqslant \mathrm{e}\). [6]
CAIE FP1 2015 November Q10
12 marks Challenging +1.3
Using de Moivre's theorem, show that $$\tan 5\theta = \frac{5\tan\theta - 10\tan^3\theta + \tan^5\theta}{1 - 10\tan^2\theta + 5\tan^4\theta}.$$ [5] Hence show that the equation \(x^2 - 10x + 5 = 0\) has roots \(\tan^2\left(\frac{1}{5}\pi\right)\) and \(\tan^2\left(\frac{2}{5}\pi\right)\). [4] Deduce a quadratic equation, with integer coefficients, having roots \(\sec^2\left(\frac{1}{5}\pi\right)\) and \(\sec^2\left(\frac{2}{5}\pi\right)\). [3]
CAIE FP1 2015 November Q11
28 marks Challenging +1.8
Answer only one of the following two alternatives. EITHER The points \(A\), \(B\) and \(C\) have position vectors \(\mathbf{i}\), \(2\mathbf{j}\) and \(4\mathbf{k}\) respectively, relative to an origin \(O\). The point \(N\) is the foot of the perpendicular from \(O\) to the plane \(ABC\). The point \(P\) on the line-segment \(ON\) is such that \(OP = \frac{3}{4}ON\). The line \(AP\) meets the plane \(OBC\) at \(Q\). Find a vector perpendicular to the plane \(ABC\) and show that the length of \(ON\) is \(\frac{1}{\sqrt{(21)}}\). [4] Find the position vector of the point \(Q\). [5] Show that the acute angle between the planes \(ABC\) and \(ABQ\) is \(\cos^{-1}\left(\frac{4}{5}\right)\). [5] OR The curve \(C\) has polar equation \(r = a(1 - \cos\theta)\) for \(0 \leqslant \theta < 2\pi\). Sketch \(C\). [2] Find the area of the region enclosed by the arc of \(C\) for which \(\frac{1}{3}\pi \leqslant \theta \leqslant \frac{2}{3}\pi\), the half-line \(\theta = \frac{1}{3}\pi\) and the half-line \(\theta = \frac{2}{3}\pi\). [5] Show that $$\left(\frac{\mathrm{d}s}{\mathrm{d}\theta}\right)^2 = 4a^2\sin^2\left(\frac{1}{2}\theta\right),$$ where \(s\) denotes arc length, and find the length of the arc of \(C\) for which \(\frac{1}{3}\pi \leqslant \theta \leqslant \frac{2}{3}\pi\). [7]
CAIE FP1 2018 November Q1
5 marks Moderate -0.3
The vectors \(\mathbf{a}\), \(\mathbf{b}\), \(\mathbf{c}\) and \(\mathbf{d}\) in \(\mathbb{R}^3\) are given by $$\mathbf{a} = \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} 2 \\ 9 \\ 0 \end{pmatrix}, \quad \mathbf{c} = \begin{pmatrix} 3 \\ 5 \\ 4 \end{pmatrix} \quad \text{and} \quad \mathbf{d} = \begin{pmatrix} 0 \\ -8 \\ -3 \end{pmatrix}.$$
  1. Show that \(\{\mathbf{a}, \mathbf{b}, \mathbf{c}\}\) is a basis for \(\mathbb{R}^3\). [3]
  2. Express \(\mathbf{d}\) in terms of \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\). [2]
CAIE FP1 2018 November Q2
6 marks Standard +0.8
The roots of the equation $$x^3 + px^2 + qx + r = 0$$ are \(\alpha\), \(2\alpha\), \(4\alpha\), where \(p\), \(q\), \(r\) and \(\alpha\) are non-zero real constants.
  1. Show that $$2p\alpha + q = 0.$$ [4]
  2. Show that $$p^3 r - q^3 = 0.$$ [2]
CAIE FP1 2018 November Q3
8 marks Challenging +1.2
The sequence of positive numbers \(u_1\), \(u_2\), \(u_3\), \(\ldots\) is such that \(u_1 < 3\) and, for \(n \geqslant 1\), $$u_{n+1} = \frac{4u_n + 9}{u_n + 4}.$$
  1. By considering \(3 - u_{n+1}\), or otherwise, prove by mathematical induction that \(u_n < 3\) for all positive integers \(n\). [5]
  2. Show that \(u_{n+1} > u_n\) for \(n \geqslant 1\). [3]
CAIE FP1 2018 November Q4
8 marks Challenging +1.3
A curve is defined parametrically by $$x = t - \frac{1}{2}\sin 2t \quad \text{and} \quad y = \sin^2 t.$$ The arc of the curve joining the point where \(t = 0\) to the point where \(t = \pi\) is rotated through one complete revolution about the \(x\)-axis. The area of the surface generated is denoted by \(S\).
  1. Show that $$S = a\pi \int_0^\pi \sin^3 t \, dt,$$ where the constant \(a\) is to be found. [5]
  2. Using the result \(\sin 3t = 3\sin t - 4\sin^3 t\), find the exact value of \(S\). [3]
CAIE FP1 2018 November Q5
8 marks Standard +0.8
It is given that \(\lambda\) is an eigenvalue of the matrix \(\mathbf{A}\) with \(\mathbf{e}\) as a corresponding eigenvector, and \(\mu\) is an eigenvalue of the matrix \(\mathbf{B}\) for which \(\mathbf{e}\) is also a corresponding eigenvector.
  1. Show that \(\lambda + \mu\) is an eigenvalue of the matrix \(\mathbf{A} + \mathbf{B}\) with \(\mathbf{e}\) as a corresponding eigenvector. [2]
The matrix \(\mathbf{A}\), given by $$\mathbf{A} = \begin{pmatrix} 2 & 0 & 1 \\ -1 & 2 & 3 \\ 1 & 0 & 2 \end{pmatrix}$$ has \(\begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}\), \(\begin{pmatrix} 1 \\ 4 \\ -1 \end{pmatrix}\) and \(\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}\) as eigenvectors.
  1. Find the corresponding eigenvalues. [3]
The matrix \(\mathbf{B}\) has eigenvalues \(4\), \(5\) and \(1\) with corresponding eigenvectors \(\begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}\), \(\begin{pmatrix} 1 \\ 4 \\ -1 \end{pmatrix}\) and \(\begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}\) respectively.
  1. Find a matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that \((\mathbf{A} + \mathbf{B})^3 = \mathbf{P}\mathbf{D}\mathbf{P}^{-1}\). [3]
CAIE FP1 2018 November Q6
9 marks Standard +0.8
The curve \(C\) has equation $$y = \frac{x^2 + ax - 1}{x + 1},$$ where \(a\) is constant and \(a > 1\).
  1. Find the equations of the asymptotes of \(C\). [3]
  2. Show that \(C\) intersects the \(x\)-axis twice. [1]
  3. Justifying your answer, find the number of stationary points on \(C\). [2]
  4. Sketch \(C\), stating the coordinates of its point of intersection with the \(y\)-axis. [3]
CAIE FP1 2018 November Q7
10 marks Challenging +1.2
  1. Use de Moivre's theorem to show that $$\sin 8\theta = 8\sin \theta \cos \theta(1 - 10\sin^2 \theta + 24\sin^4 \theta - 16\sin^6 \theta).$$ [6]
  2. Use the equation \(\frac{\sin 8\theta}{\sin 2\theta} = 0\) to find the roots of $$16x^6 - 24x^4 + 10x^2 - 1 = 0$$ in the form \(\sin k\pi\), where \(k\) is rational. [4]
CAIE FP1 2018 November Q8
10 marks Standard +0.3
The plane \(\Pi_1\) has equation $$\mathbf{r} = \begin{pmatrix} 5 \\ 1 \\ 0 \end{pmatrix} + s\begin{pmatrix} -4 \\ 1 \\ 3 \end{pmatrix} + t\begin{pmatrix} 0 \\ 1 \\ 2 \end{pmatrix}.$$
  1. Find a cartesian equation of \(\Pi_1\). [3]
The plane \(\Pi_2\) has equation \(3x + y - z = 3\).
  1. Find the acute angle between \(\Pi_1\) and \(\Pi_2\), giving your answer in degrees. [2]
  2. Find an equation of the line of intersection of \(\Pi_1\) and \(\Pi_2\), giving your answer in the form \(\mathbf{r} = \mathbf{a} + \lambda\mathbf{b}\). [5]
CAIE FP1 2018 November Q9
10 marks Standard +0.8
The curve \(C\) has polar equation $$r = 5\sqrt{\cot \theta},$$ where \(0.01 \leqslant \theta \leqslant \frac{1}{2}\pi\).
  1. Find the area of the finite region bounded by \(C\) and the line \(\theta = 0.01\), showing full working. Give your answer correct to \(1\) decimal place. [3]
Let \(P\) be the point on \(C\) where \(\theta = 0.01\).
  1. Find the distance of \(P\) from the initial line, giving your answer correct to \(1\) decimal place. [2]
  2. Find the maximum distance of \(C\) from the initial line. [3]
  3. Sketch \(C\). [2]
CAIE FP1 2018 November Q10
13 marks Challenging +1.2
  1. Find the particular solution of the differential equation $$\frac{d^2x}{dt^2} + 2\frac{dx}{dt} + 10x = 37\sin 3t,$$ given that \(x = 3\) and \(\frac{dx}{dt} = 0\) when \(t = 0\). [10]
  2. Show that, for large positive values of \(t\) and for any initial conditions, $$x \approx \sqrt{(37)}\sin(3t - \phi),$$ where the constant \(\phi\) is such that \(\tan \phi = 6\). [3]
CAIE FP1 2018 November Q11
26 marks Challenging +1.2
Answer only one of the following two alternatives. EITHER
  1. By considering \((2r + 1)^2 - (2r - 1)^2\), use the method of differences to prove that $$\sum_{r=1}^n r = \frac{1}{2}n(n + 1).$$ [3]
  2. By considering \((2r + 1)^4 - (2r - 1)^4\), use the method of differences and the result given in part (i) to prove that $$\sum_{r=1}^n r^3 = \frac{1}{4}n^2(n + 1)^2.$$ [5]
The sums \(S\) and \(T\) are defined as follows: $$S = 1^3 + 2^3 + 3^3 + 4^3 + \ldots + (2N)^3 + (2N + 1)^3,$$ $$T = 1^3 + 3^3 + 5^3 + 7^3 + \ldots + (2N - 1)^3 + (2N + 1)^3.$$
  1. Use the result given in part (ii) to show that \(S = (2N + 1)^2(N + 1)^2\). [1]
  2. Hence, or otherwise, find an expression in terms of \(N\) for \(T\), factorising your answer as far as possible. [2]
  3. Deduce the value of \(\frac{S}{T}\) as \(N \to \infty\). [2]
OR The curve \(C\) has equation $$x^2 + 2xy = y^3 - 2.$$
  1. Show that \(A(-1, 1)\) is the only point on \(C\) with \(x\)-coordinate equal to \(-1\). [2]
For \(n \geqslant 1\), let \(A_n\) denote the value of \(\frac{d^n y}{dx^n}\) at the point \(A(-1, 1)\).
  1. Show that \(A_1 = 0\). [3]
  2. Show that \(A_2 = \frac{2}{5}\). [3]
Let \(I_n = \int_{-1}^0 x^n \frac{d^n y}{dx^n} dx\).
  1. Show that for \(n \geqslant 2\), $$I_n = (-1)^{n+1} A_{n-1} - nI_{n-1}.$$ [3]
  2. Deduce the value of \(I_3\) in terms of \(I_1\). [2]
CAIE FP1 2018 November Q1
5 marks Standard +0.3
The roots of the cubic equation $$x^3 - 5x^2 + 13x - 4 = 0$$ are \(\alpha, \beta, \gamma\).
  1. Find the value of \(\alpha^2 + \beta^2 + \gamma^2\). [3]
  2. Find the value of \(\alpha^3 + \beta^3 + \gamma^3\). [2]
CAIE FP1 2018 November Q2
6 marks Standard +0.3
It is given that $$\mathbf{A} = \begin{pmatrix} 2 & 3 & 1 \\ 0 & -2 & 1 \\ 0 & 0 & 1 \end{pmatrix}$$
  1. Find the eigenvalue of \(\mathbf{A}\) corresponding to the eigenvector \(\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}\). [1]
  2. Write down the negative eigenvalue of \(\mathbf{A}\) and find a corresponding eigenvector. [3]
  3. Find an eigenvalue and a corresponding eigenvector of the matrix \(\mathbf{A} + \mathbf{A}^6\). [2]
CAIE FP1 2018 November Q3
8 marks Standard +0.8
The curve \(C\) has polar equation \(r = a \cos 3\theta\), for \(-\frac{1}{6}\pi \leqslant \theta \leqslant \frac{1}{6}\pi\), where \(a\) is a positive constant.
  1. Sketch \(C\). [2]
  2. Find the area of the region enclosed by \(C\), showing full working. [3]
  3. Using the identity \(\cos 3\theta \equiv 4\cos^3 \theta - 3\cos \theta\), find a cartesian equation of \(C\). [3]
CAIE FP1 2018 November Q4
8 marks Standard +0.8
  1. Find the general solution of the differential equation $$\frac{\mathrm{d}^2 x}{\mathrm{d}t^2} + 2\frac{\mathrm{d}x}{\mathrm{d}t} + x = 4\sin t.$$ [7]
  2. State an approximate solution for large positive values of \(t\). [1]
CAIE FP1 2018 November Q5
9 marks Challenging +1.2
The linear transformation \(\mathrm{T} : \mathbb{R}^4 \to \mathbb{R}^4\) is represented by the matrix \(\mathbf{M}\), where $$\mathbf{M} = \begin{pmatrix} 3 & 2 & 0 & 1 \\ 6 & 5 & -1 & 3 \\ 9 & 8 & -2 & 5 \\ -3 & -2 & 0 & -1 \end{pmatrix}.$$
  1. Find the rank of \(\mathbf{M}\). [3]
Let \(K\) be the null space of \(\mathrm{T}\).
  1. Find a basis for \(K\). [3]
  2. Find the general solution of $$\mathbf{M}\mathbf{x} = \begin{pmatrix} 2 \\ 5 \\ 8 \\ -2 \end{pmatrix}.$$ [3]
CAIE FP1 2018 November Q6
8 marks Challenging +1.3
It is given that \(y = e^x u\), where \(u\) is a function of \(x\). The \(r\)th derivatives \(\frac{\mathrm{d}^r y}{\mathrm{d}x^r}\) and \(\frac{\mathrm{d}^r u}{\mathrm{d}x^r}\) are denoted by \(y^{(r)}\) and \(u^{(r)}\) respectively. Prove by mathematical induction that, for all positive integers \(n\), $$y^{(n)} = e^x\left[\binom{n}{0}u + \binom{n}{1}u^{(1)} + \binom{n}{2}u^{(2)} + \ldots + \binom{n}{r}u^{(r)} + \ldots + \binom{n}{n}u^{(n)}\right].$$ [8] [You may use without proof the result \(\binom{k}{r} + \binom{k}{r-1} = \binom{k+1}{r}\).]
CAIE FP1 2018 November Q7
10 marks Standard +0.8
Let $$S_N = \sum_{r=1}^{N}(3r + 1)(3r + 4) \quad \text{and} \quad T_N = \sum_{r=1}^{N}\frac{1}{(3r + 1)(3r + 4)}.$$
  1. Use standard results from the List of Formulae (MF10) to show that $$S_N = N(3N^2 + 12N + 13).$$ [3]
  2. Use the method of differences to show that $$T_N = \frac{1}{12} - \frac{1}{3(3N + 4)}.$$ [3]
  3. Deduce that \(\frac{S_N}{T_N}\) is an integer. [2]
  4. Find \(\lim_{N \to \infty} \frac{S_N}{N^3 T_N}\). [2]