Questions — CAIE FP1 (594 questions)

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CAIE FP1 2015 November Q10
Challenging +1.2
10 Using de Moivre's theorem, show that $$\tan 5 \theta = \frac { 5 \tan \theta - 10 \tan ^ { 3 } \theta + \tan ^ { 5 } \theta } { 1 - 10 \tan ^ { 2 } \theta + 5 \tan ^ { 4 } \theta }$$ Hence show that the equation \(x ^ { 2 } - 10 x + 5 = 0\) has roots \(\tan ^ { 2 } \left( \frac { 1 } { 5 } \pi \right)\) and \(\tan ^ { 2 } \left( \frac { 2 } { 5 } \pi \right)\). Deduce a quadratic equation, with integer coefficients, having roots \(\sec ^ { 2 } \left( \frac { 1 } { 5 } \pi \right)\) and \(\sec ^ { 2 } \left( \frac { 2 } { 5 } \pi \right)\).
[0pt] [Question 11 is printed on the next page.]
CAIE FP1 2015 November Q11 EITHER
Challenging +1.8
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 \(A B C\). The point \(P\) on the line-segment \(O N\) is such that \(O P = \frac { 3 } { 4 } O N\). The line \(A P\) meets the plane \(O B C\) at \(Q\). Find a vector perpendicular to the plane \(A B C\) and show that the length of \(O N\) is \(\frac { 4 } { \sqrt { } ( 21 ) }\). Find the position vector of the point \(Q\). Show that the acute angle between the planes \(A B C\) and \(A B Q\) is \(\cos ^ { - 1 } \left( \frac { 2 } { 3 } \right)\).
CAIE FP1 2015 November Q11 OR
Standard +0.8
The curve \(C\) has polar equation \(r = a ( 1 - \cos \theta )\) for \(0 \leqslant \theta < 2 \pi\). Sketch \(C\). Find the area of the region enclosed by the arc of \(C\) for which \(\frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 3 } { 2 } \pi\), the half-line \(\theta = \frac { 1 } { 2 } \pi\) and the half-line \(\theta = \frac { 3 } { 2 } \pi\). Show that $$\left( \frac { \mathrm { d } s } { \mathrm {~d} \theta } \right) ^ { 2 } = 4 a ^ { 2 } \sin ^ { 2 } \left( \frac { 1 } { 2 } \theta \right) ,$$ where \(s\) denotes arc length, and find the length of the arc of \(C\) for which \(\frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 3 } { 2 } \pi\). {www.cie.org.uk} after the live examination series.
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CAIE FP1 2018 November Q1
Moderate -0.3
1 The vectors \(\mathbf { a } , \mathbf { b } , \mathbf { c }\) and \(\mathbf { d }\) in \(\mathbb { R } ^ { 3 }\) are given by $$\mathbf { a } = \left( \begin{array} { l } 1 \\ 2 \\ 1 \end{array} \right) , \quad \mathbf { b } = \left( \begin{array} { l } 2 \\ 9 \\ 0 \end{array} \right) , \quad \mathbf { c } = \left( \begin{array} { l } 3 \\ 3 \\ 4 \end{array} \right) \quad \text { and } \quad \mathbf { d } = \left( \begin{array} { r } 0 \\ - 8 \\ 3 \end{array} \right) .$$
  1. Show that \(\{ \mathbf { a } , \mathbf { b } , \mathbf { c } \}\) is a basis for \(\mathbb { R } ^ { 3 }\).
  2. Express \(\mathbf { d }\) in terms of \(\mathbf { a } , \mathbf { b }\) and \(\mathbf { c }\).
CAIE FP1 2018 November Q2
Standard +0.8
2 The roots of the equation $$x ^ { 3 } + p x ^ { 2 } + q x + r = 0$$ are \(\alpha , 2 \alpha , 4 \alpha\), where \(p , q , r\) and \(\alpha\) are non-zero real constants.
  1. Show that $$2 p \alpha + q = 0$$
  2. Show that $$p ^ { 3 } r - q ^ { 3 } = 0$$
CAIE FP1 2018 November Q3
Challenging +1.2
3 The sequence of positive numbers \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is such that \(u _ { 1 } < 3\) and, for \(n \geqslant 1\), $$u _ { n + 1 } = \frac { 4 u _ { n } + 9 } { u _ { n } + 4 }$$
  1. By considering \(3 - u _ { n + 1 }\), or otherwise, prove by mathematical induction that \(u _ { n } < 3\) for all positive integers \(n\).
  2. Show that \(u _ { n + 1 } > u _ { n }\) for \(n \geqslant 1\).
CAIE FP1 2018 November Q4
Challenging +1.2
4 A curve is defined parametrically by $$x = t - \frac { 1 } { 2 } \sin 2 t \quad \text { and } \quad y = \sin ^ { 2 } t$$ 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 $$S = a \pi \int _ { 0 } ^ { \pi } \sin ^ { 3 } t \mathrm {~d} t$$ where the constant \(a\) is to be found.
  2. Using the result \(\sin 3 t = 3 \sin t - 4 \sin ^ { 3 } t\), find the exact value of \(S\).
CAIE FP1 2018 November Q5
Standard +0.3
5 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.
    The matrix \(\mathbf { A }\), given by $$\mathbf { A } = \left( \begin{array} { r r r } 2 & 0 & 1 \\ - 1 & 2 & 3 \\ 1 & 0 & 2 \end{array} \right)$$ has \(\left( \begin{array} { l } 1 \\ 2 \\ 1 \end{array} \right) , \left( \begin{array} { r } 1 \\ 4 \\ - 1 \end{array} \right)\) and \(\left( \begin{array} { l } 0 \\ 1 \\ 0 \end{array} \right)\) as eigenvectors.
  2. Find the corresponding eigenvalues.
    The matrix \(\mathbf { B }\) has eigenvalues 4, 5 and 1 with corresponding eigenvectors \(\left( \begin{array} { l } 1 \\ 2 \\ 1 \end{array} \right) , \left( \begin{array} { r } 1 \\ 4 \\ - 1 \end{array} \right)\) and \(\left( \begin{array} { l } 0 \\ 1 \\ 0 \end{array} \right)\) respectively.
  3. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(( \mathbf { A } + \mathbf { B } ) ^ { 3 } = \mathbf { P D P } ^ { - 1 }\).
CAIE FP1 2018 November Q6
Standard +0.8
6 The curve \(C\) has equation $$y = \frac { x ^ { 2 } + a x - 1 } { x + 1 }$$ where \(a\) is constant and \(a > 1\).
  1. Find the equations of the asymptotes of \(C\).
  2. Show that \(C\) intersects the \(x\)-axis twice.
  3. Justifying your answer, find the number of stationary points on \(C\).
  4. Sketch \(C\), stating the coordinates of its point of intersection with the \(y\)-axis.
CAIE FP1 2018 November Q7
Challenging +1.2
7
  1. Use de Moivre's theorem to show that $$\sin 8 \theta = 8 \sin \theta \cos \theta \left( 1 - 10 \sin ^ { 2 } \theta + 24 \sin ^ { 4 } \theta - 16 \sin ^ { 6 } \theta \right) .$$
  2. Use the equation \(\frac { \sin 8 \theta } { \sin 2 \theta } = 0\) to find the roots of $$16 x ^ { 6 } - 24 x ^ { 4 } + 10 x ^ { 2 } - 1 = 0$$ in the form \(\sin k \pi\), where \(k\) is rational.
CAIE FP1 2018 November Q8
5 marks Standard +0.3
8 The plane \(\Pi _ { 1 }\) has equation $$\mathbf { r } = \left( \begin{array} { l } 5 \\ 1 \\ 0 \end{array} \right) + s \left( \begin{array} { r } - 4 \\ 1 \\ 3 \end{array} \right) + t \left( \begin{array} { l } 0 \\ 1 \\ 2 \end{array} \right)$$
  1. Find a cartesian equation of \(\Pi _ { 1 }\).
    The plane \(\Pi _ { 2 }\) has equation \(3 x + y - z = 3\).
  2. Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), giving your answer in degrees.
  3. 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
Challenging +1.2
9 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.
    Let \(P\) be the point on \(C\) where \(\theta = 0.01\).
  2. Find the distance of \(P\) from the initial line, giving your answer correct to 1 decimal place.
  3. Find the maximum distance of \(C\) from the initial line.
  4. Sketch \(C\).
CAIE FP1 2018 November Q10
Challenging +1.3
10
  1. Find the particular solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 10 x = 37 \sin 3 t$$ given that \(x = 3\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 0\) when \(t = 0\).
  2. Show that, for large positive values of \(t\) and for any initial conditions, $$x \approx \sqrt { } ( 37 ) \sin ( 3 t - \phi ) ,$$ where the constant \(\phi\) is such that \(\tan \phi = 6\).
CAIE FP1 2018 November Q11 EITHER
Standard +0.8
  1. By considering \(( 2 r + 1 ) ^ { 2 } - ( 2 r - 1 ) ^ { 2 }\), use the method of differences to prove that $$\sum _ { r = 1 } ^ { n } r = \frac { 1 } { 2 } n ( n + 1 )$$
  2. By considering \(( 2 r + 1 ) ^ { 4 } - ( 2 r - 1 ) ^ { 4 }\), use the method of differences and the result given in part (i) to prove that $$\sum _ { r = 1 } ^ { n } r ^ { 3 } = \frac { 1 } { 4 } n ^ { 2 } ( n + 1 ) ^ { 2 }$$ The sums \(S\) and \(T\) are defined as follows: $$\begin{aligned} & S = 1 ^ { 3 } + 2 ^ { 3 } + 3 ^ { 3 } + 4 ^ { 3 } + \ldots + ( 2 N ) ^ { 3 } + ( 2 N + 1 ) ^ { 3 } , \\ & T = 1 ^ { 3 } + 3 ^ { 3 } + 5 ^ { 3 } + 7 ^ { 3 } + \ldots + ( 2 N - 1 ) ^ { 3 } + ( 2 N + 1 ) ^ { 3 } . \end{aligned}$$
  3. Use the result given in part (ii) to show that \(S = ( 2 N + 1 ) ^ { 2 } ( N + 1 ) ^ { 2 }\).
  4. Hence, or otherwise, find an expression in terms of \(N\) for \(T\), factorising your answer as far as possible.
  5. Deduce the value of \(\frac { S } { T }\) as \(N \rightarrow \infty\).
CAIE FP1 2018 November Q11 OR
Challenging +1.8
The curve \(C\) has equation $$x ^ { 2 } + 2 x y = y ^ { 3 } - 2$$
  1. Show that \(A ( - 1,1 )\) is the only point on \(C\) with \(x\)-coordinate equal to - 1 .
    For \(n \geqslant 1\), let \(A _ { n }\) denote the value of \(\frac { \mathrm { d } ^ { n } y } { \mathrm {~d} x ^ { n } }\) at the point \(A ( - 1,1 )\).
  2. Show that \(A _ { 1 } = 0\).
  3. Show that \(A _ { 2 } = \frac { 2 } { 5 }\).
    Let \(I _ { n } = \int _ { - 1 } ^ { 0 } x ^ { n } \frac { \mathrm {~d} ^ { n } y } { \mathrm {~d} x ^ { n } } \mathrm {~d} x\).
  4. Show that for \(n \geqslant 2\), $$I _ { n } = ( - 1 ) ^ { n + 1 } A _ { n - 1 } - n I _ { n - 1 } .$$
  5. Deduce the value of \(I _ { 3 }\) in terms of \(I _ { 1 }\).
    If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
CAIE FP1 2003 November Q1
6 marks Challenging +1.2
\includegraphics{figure_1} The curve \(C\) has polar equation $$r = \theta^{\frac{1}{2}}e^{\theta/\pi},$$ where \(0 \leq \theta \leq \pi\). The area of the finite region bounded by \(C\) and the line \(\theta = \beta\) is \(\pi\) (see diagram). Show that $$\beta = (\pi \ln 3)^{\frac{1}{2}}.$$ [6]
CAIE FP1 2003 November Q2
6 marks Challenging +1.2
Given that $$u_n = \frac{1}{n^2 - n + 1} - \frac{1}{n^2 + n + 1},$$ find \(S_N = \sum_{n=N+1}^{2N} u_n\) in terms of \(N\). [3] Find a number \(M\) such that \(S_N < 10^{-20}\) for all \(N > M\). [3]
CAIE FP1 2003 November Q3
6 marks Challenging +1.2
Three \(n \times 1\) column vectors are denoted by \(\mathbf{x}_1\), \(\mathbf{x}_2\), \(\mathbf{x}_3\), and \(\mathbf{M}\) is an \(n \times n\) matrix. Show that if \(\mathbf{x}_1\), \(\mathbf{x}_2\), \(\mathbf{x}_3\) are linearly dependent then the vectors \(\mathbf{Mx}_1\), \(\mathbf{Mx}_2\), \(\mathbf{Mx}_3\) are also linearly dependent. [2] The vectors \(\mathbf{y}_1\), \(\mathbf{y}_2\), \(\mathbf{y}_3\) and the matrix \(\mathbf{P}\) are defined as follows: $$\mathbf{y}_1 = \begin{pmatrix} 1 \\ 5 \\ 7 \end{pmatrix}, \quad \mathbf{y}_2 = \begin{pmatrix} 2 \\ -3 \\ 4 \end{pmatrix}, \quad \mathbf{y}_3 = \begin{pmatrix} 5 \\ 51 \\ 55 \end{pmatrix},$$ $$\mathbf{P} = \begin{pmatrix} 1 & -4 & 3 \\ 0 & 2 & 5 \\ 0 & 0 & -7 \end{pmatrix}$$
  1. Show that \(\mathbf{y}_1\), \(\mathbf{y}_2\), \(\mathbf{y}_3\) are linearly dependent. [2]
  2. Find a basis for the linear space spanned by the vectors \(\mathbf{Py}_1\), \(\mathbf{Py}_2\), \(\mathbf{Py}_3\). [2]
CAIE FP1 2003 November Q4
8 marks Challenging +1.2
Given that \(y = x \sin x\), find \(\frac{d^2y}{dx^2}\) and \(\frac{d^4y}{dx^4}\), simplifying your results as far as possible, and show that $$\frac{d^6y}{dx^6} = -x \sin x + 6 \cos x.$$ [3] Use induction to establish an expression for \(\frac{d^{2n}y}{dx^{2n}}\), where \(n\) is a positive integer. [5]
CAIE FP1 2003 November Q5
8 marks Challenging +1.8
The integral \(I_n\) is defined by $$I_n = \int_0^{\frac{1}{4}\pi} \sec^n x \, dx.$$ By considering \(\frac{d}{dx}(\tan x \sec^n x)\), or otherwise, show that $$(n + 1)I_{n+2} = 2^{\frac{4n}{n}} + nI_n.$$ [4] Find the value of \(I_6\). [4]
CAIE FP1 2003 November Q6
9 marks Challenging +1.2
Find the sum of the squares of the roots of the equation $$x^3 + x + 12 = 0,$$ and deduce that only one of the roots is real. [4] The real root of the equation is denoted by \(\alpha\). Prove that \(-3 < \alpha < -2\), and hence prove that the modulus of each of the other roots lies between 2 and \(\sqrt{6}\). [5]
CAIE FP1 2003 November Q7
9 marks Standard +0.3
Find the general solution of the differential equation $$\frac{d^2y}{dt^2} + 4\frac{dy}{dt} + 4y = e^{-\alpha t},$$ where \(\alpha\) is a constant and \(\alpha \neq 2\). [7] Show that if \(\alpha < 2\) then, whatever the initial conditions, \(ye^{\alpha t} \to \frac{1}{(2-\alpha)^2}\) as \(t \to \infty\). [2]
CAIE FP1 2003 November Q8
11 marks Challenging +1.2
Given that \(z = e^{i\theta}\) and \(n\) is a positive integer, show that $$z^n + \frac{1}{z^n} = 2 \cos n\theta \quad \text{and} \quad z^n - \frac{1}{z^n} = 2i \sin n\theta.$$ [2] Hence express \(\sin^6 \theta\) in the form $$p \cos 6\theta + q \cos 4\theta + r \cos 2\theta + s,$$ where the constants \(p\), \(q\), \(r\), \(s\) are to be determined. [4] Hence find the mean value of \(\sin^6 \theta\) with respect to \(\theta\) over the interval \(0 \leq \theta \leq \frac{1}{4}\pi\). [5]
CAIE FP1 2003 November Q9
11 marks Challenging +1.8
The line \(l_1\) passes through the point \(A\) with position vector \(\mathbf{i} - \mathbf{j} - 2\mathbf{k}\) and is parallel to the vector \(3\mathbf{i} - 4\mathbf{j} - 2\mathbf{k}\). The variable line \(l_2\) passes through the point \((1 + 5 \cos t)\mathbf{i} - (1 + 5 \sin t)\mathbf{j} - 14\mathbf{k}\), where \(0 \leq t < 2\pi\), and is parallel to the vector \(15\mathbf{i} + 8\mathbf{j} - 3\mathbf{k}\). The points \(P\) and \(Q\) are on \(l_1\) and \(l_2\) respectively, and \(PQ\) is perpendicular to both \(l_1\) and \(l_2\).
  1. Find the length of \(PQ\) in terms of \(t\). [4]
  2. Hence show that the lines \(l_1\) and \(l_2\) do not intersect, and find the maximum length of \(PQ\) as \(t\) varies. [3]
  3. The plane \(\Pi_1\) contains \(l_1\) and \(PQ\); the plane \(\Pi_2\) contains \(l_2\) and \(PQ\). Find the angle between the planes \(\Pi_1\) and \(\Pi_2\), correct to the nearest tenth of a degree. [4]
CAIE FP1 2003 November Q10
12 marks Standard +0.8
Find the eigenvalues and corresponding eigenvectors of the matrix \(\mathbf{A}\), where $$\mathbf{A} = \begin{pmatrix} 6 & 4 & 1 \\ -6 & -1 & 3 \\ 8 & 8 & 4 \end{pmatrix}.$$ [8] Hence find a non-singular matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that \(\mathbf{A} + \mathbf{A}^2 + \mathbf{A}^3 = \mathbf{PDP}^{-1}\). [4]