Questions — CAIE Further Paper 1 (151 questions)

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CAIE Further Paper 1 2023 November Q7
16 marks Challenging +1.2
7 The curve \(C\) has equation \(y = f ( x )\), where \(f ( x ) = \frac { x ^ { 2 } } { x + 1 }\).
  1. Find the equations of the asymptotes of \(C\).
  2. Find the coordinates of any stationary points on \(C\).
  3. Sketch \(C\).
  4. Find the coordinates of any stationary points on the curve with equation \(\mathrm { y } = \frac { 1 } { \mathrm { f } ( \mathrm { x } ) }\).
  5. Sketch the curve with equation \(y = \frac { 1 } { f ( x ) }\) and find, in exact form, the set of values for which $$\frac { 1 } { \mathrm { f } ( x ) } > \mathrm { f } ( x ) .$$ If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE Further Paper 1 2024 November Q1
5 marks Moderate -0.3
1 The sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is such that \(u _ { 1 } = 4\) and \(u _ { n + 1 } = 3 u _ { n } - 2\) for \(n \geqslant 1\).
Prove by induction that \(u _ { n } = 3 ^ { n } + 1\) for all positive integers \(n\).
CAIE Further Paper 1 2024 November Q2
7 marks Standard +0.3
2 The line \(l _ { 1 }\) has equation \(\mathbf { r } = \mathbf { i } + 3 \mathbf { j } - \mathbf { k } + \lambda ( \mathbf { i } - \mathbf { j } - 4 \mathbf { k } )\).
The plane \(\Pi\) contains \(l _ { 1 }\) and is parallel to the vector \(2 \mathbf { i } + 5 \mathbf { j } - 4 \mathbf { k }\).
  1. Find the equation of \(\Pi\), giving your answer in the form \(a x + b y + c z = d\). \includegraphics[max width=\textwidth, alt={}, center]{beb9c1f1-1676-4432-a42a-c418ff9f45d8-05_2723_33_99_22} The line \(l _ { 2 }\) is parallel to the vector \(5 \mathbf { i } - 5 \mathbf { j } - 2 \mathbf { k }\).
  2. Find the acute angle between \(l _ { 2 }\) and \(\Pi\).
CAIE Further Paper 1 2024 November Q3
10 marks Standard +0.8
3 It is given that $$\begin{aligned} & \alpha + \beta + \gamma + \delta = 2 \\ & \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 } = 3 \\ & \alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } + \delta ^ { 3 } = 4 \end{aligned}$$
  1. Find the value of \(\alpha \beta + \alpha \gamma + \alpha \delta + \beta \gamma + \beta \delta + \gamma \delta\).
  2. Find the value of \(\alpha ^ { 2 } \beta + \alpha ^ { 2 } \gamma + \alpha ^ { 2 } \delta + \beta ^ { 2 } \alpha + \beta ^ { 2 } \gamma + \beta ^ { 2 } \delta + \gamma ^ { 2 } \alpha + \gamma ^ { 2 } \beta + \gamma ^ { 2 } \delta + \delta ^ { 2 } \alpha + \delta ^ { 2 } \beta + \delta ^ { 2 } \gamma\). \includegraphics[max width=\textwidth, alt={}, center]{beb9c1f1-1676-4432-a42a-c418ff9f45d8-06_2717_33_109_2014} \includegraphics[max width=\textwidth, alt={}, center]{beb9c1f1-1676-4432-a42a-c418ff9f45d8-07_2723_33_99_22}
  3. It is given that \(\alpha , \beta , \gamma , \delta\) are the roots of the equation $$6 x ^ { 4 } - 12 x ^ { 3 } + 3 x ^ { 2 } + 2 x + 6 = 0 .$$
    1. Find the value of \(\alpha ^ { 4 } + \beta ^ { 4 } + \gamma ^ { 4 } + \delta ^ { 4 }\).
    2. Find the value of \(\alpha ^ { 5 } + \beta ^ { 5 } + \gamma ^ { 5 } + \delta ^ { 5 }\).
CAIE Further Paper 1 2024 November Q4
13 marks Standard +0.8
4 The matrices \(\mathbf { A } , \mathbf { B }\) and \(\mathbf { C }\) are given by $$\mathbf { A } = \left( \begin{array} { l l l } 1 & 2 & 3 \\ 2 & 1 & 3 \\ 3 & 2 & 5 \end{array} \right) , \mathbf { B } = \left( \begin{array} { r r } 0 & - 2 \\ - 1 & 3 \\ 0 & 0 \end{array} \right) \text { and } \mathbf { C } = \left( \begin{array} { r r r } - 2 & - 1 & 1 \\ 1 & 1 & 3 \end{array} \right)$$
  1. Show that \(\mathbf { C A B } = \left( \begin{array} { r r } 3 & - 7 \\ - 9 & 3 \end{array} \right)\).
  2. Find the equations of the invariant lines, through the origin, of the transformation in the \(x - y\) plane represented by \(\mathbf { C A B }\). \includegraphics[max width=\textwidth, alt={}, center]{beb9c1f1-1676-4432-a42a-c418ff9f45d8-08_2715_31_106_2016} Let \(\mathbf { M } = \left( \begin{array} { l l } 3 & 0 \\ 0 & 1 \end{array} \right)\).
  3. Give full details of the transformation represented by \(\mathbf { M }\).
  4. Find the matrix \(\mathbf { N }\) such that \(\mathbf { N M } = \mathbf { C A B }\).
CAIE Further Paper 1 2024 November Q5
9 marks Challenging +1.2
5 It is given that \(S _ { n } = \sum _ { r = 1 } ^ { n } u _ { r }\), where \(u _ { r } = x ^ { \mathrm { f } ( r ) } - x ^ { \mathrm { f } ( r + 1 ) }\) and \(x > 0\).
  1. Find \(S _ { n }\) in terms of \(n , x\) and the function f .
  2. Given that \(\mathrm { f } ( r ) = \ln r\), find the set of values of \(x\) for which the infinite series $$u _ { 1 } + u _ { 2 } + u _ { 3 } + \ldots$$ is convergent and give the sum to infinity when this exists. \includegraphics[max width=\textwidth, alt={}, center]{beb9c1f1-1676-4432-a42a-c418ff9f45d8-10_2716_31_106_2016} \includegraphics[max width=\textwidth, alt={}, center]{beb9c1f1-1676-4432-a42a-c418ff9f45d8-11_2723_35_101_20}
  3. Given instead that \(\mathrm { f } ( r ) = 2 \log _ { x } r\) where \(x \neq 1\), use standard results from the List of formulae (MF19) to find \(\sum _ { n = 1 } ^ { N } S _ { n }\) in terms of \(N\). Fully factorise your answer.
CAIE Further Paper 1 2024 November Q6
15 marks Standard +0.8
6 The curve \(C\) has equation \(y = \frac { x ^ { 2 } + 3 } { x ^ { 2 } + 1 }\).
  1. Show that \(C\) has no vertical asymptotes and state the equation of the horizontal asymptote.
  2. Show that \(1 < y \leqslant 3\) for all real values of \(x\).
  3. Find the coordinates of any stationary points on \(C\). \includegraphics[max width=\textwidth, alt={}, center]{beb9c1f1-1676-4432-a42a-c418ff9f45d8-12_2718_42_107_2007} \includegraphics[max width=\textwidth, alt={}, center]{beb9c1f1-1676-4432-a42a-c418ff9f45d8-13_2720_40_106_18}
  4. Sketch \(C\), stating the coordinates of any intersections with the axes and labelling the asymptote.
  5. Sketch the curve with equation \(y = \frac { x ^ { 2 } + 1 } { x ^ { 2 } + 3 }\) and find the set of values of \(x\) for which \(\frac { x ^ { 2 } + 1 } { x ^ { 2 } + 3 } < \frac { 1 } { 2 }\).
CAIE Further Paper 1 2024 November Q7
16 marks Challenging +1.2
7 The curve \(C _ { 1 }\) has polar equation \(r = a ( \cos \theta + \sin \theta )\) for \(- \frac { 1 } { 4 } \pi \leqslant \theta \leqslant \frac { 3 } { 4 } \pi\), where \(a\) is a positive constant.
  1. Find a Cartesian equation for \(C _ { 1 }\) and show that it represents a circle, stating its radius and the Cartesian coordinates of its centre.
  2. Sketch \(C _ { 1 }\) and state the greatest distance of a point on \(C _ { 1 }\) from the pole. \includegraphics[max width=\textwidth, alt={}, center]{beb9c1f1-1676-4432-a42a-c418ff9f45d8-14_2721_40_107_2010} The curve \(C _ { 2 }\) with polar equation \(r = a \theta\) intersects \(C _ { 1 }\) at the pole and the point with polar coordinates \(( a \phi , \phi )\).
  3. Verify that \(1.25 < \phi < 1.26\).
  4. Show that the area of the smaller region enclosed by \(C _ { 1 }\) and \(C _ { 2 }\) is equal to $$\frac { 1 } { 2 } a ^ { 2 } \left( \frac { 3 } { 4 } \pi + \frac { 1 } { 3 } \phi ^ { 3 } - \phi + \frac { 1 } { 2 } \cos 2 \phi \right)$$ and deduce, in terms of \(a\) and \(\phi\), the area of the larger region enclosed by \(C _ { 1 }\) and \(C _ { 2 }\).
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE Further Paper 1 2020 Specimen Q1
6 marks Standard +0.3
1
  1. Given that \(\mathrm { f } ( r ) = \frac { 1 } { ( r + 1 ) ( r + 2 ) }\), show that $$\mathrm { f } ( r - 1 ) - \mathrm { f } ( r ) = \frac { 2 } { r ( r + 1 ) ( r + 2 ) } .$$
  2. Hence find \(\sum _ { r = 1 } ^ { n } \frac { 1 } { r ( r + 1 ) ( r + 2 ) }\).
  3. Deduce the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { r ( r + 1 ) ( r + 2 ) }\).
CAIE Further Paper 1 2020 Specimen Q2
7 marks Standard +0.3
2 It is given that \(\phi ( n ) = 5 ^ { n } ( 4 n + 1 ) - 1\), for \(n = 1,2,3 , \ldots\).
Prove, by mathematical induction, that \(\phi ( n )\) is divisible by 8 for every positive integer \(n\).
CAIE Further Paper 1 2020 Specimen Q3
10 marks Standard +0.8
3 The curve \(C\) has polar equation \(r = 2 + 2 \cos \theta\), for \(0 \leqslant \theta \leqslant \pi\).
  1. Sketch \(C\).
  2. Find the area of the region enclosed by \(C\) and the initial line.
  3. Show that the Cartesian equation of \(C\) can be expressed as \(4 \left( x ^ { 2 } + y ^ { 2 } \right) = \left( x ^ { 2 } + y ^ { 2 } - 2 x \right) ^ { 2 }\).
CAIE Further Paper 1 2020 Specimen Q4
9 marks Standard +0.8
4 The cubic equation $$z ^ { 3 } - z ^ { 2 } - z - 5 = 0$$ has roots \(\alpha , \beta\) and \(\gamma\).
  1. Show that the value of \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 }\) is 19 .
  2. Find the value of \(\alpha ^ { 4 } + \beta ^ { 4 } + \gamma ^ { 4 }\).
  3. Find a cubic equation with roots \(\alpha + 1 , \beta + 1\) and \(\gamma + 1\), giving your answer in the form $$p x ^ { 3 } + q x ^ { 2 } + r x + s = 0 ,$$ where \(p , q , r\) and \(s\) are constants to be determined.
CAIE Further Paper 1 2020 Specimen Q5
12 marks Standard +0.3
5 The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r }
CAIE Further Paper 1 2020 Specimen Q6
14 marks Challenging +1.8
6 The position vectors of the points \(A , B , C , D\) are $$2 \mathbf { i } + 4 \mathbf { j } - 3 \mathbf { k } , \quad - 2 \mathbf { i } + 5 \mathbf { j } - 4 \mathbf { k } , \quad \mathbf { i } + 4 \mathbf { j } + \mathbf { k } , \quad \mathbf { i } + 5 \mathbf { j } + m \mathbf { k } ,$$ respectively, where \(m\) is an integer. It is given that the shortest distance between the line through \(A\) and \(B\) and the line through \(C\) and \(D\) is 3 .
  1. Show that the only possible value of \(m\) is 2 .
  2. Find the shortest distance of \(D\) from the line through \(A\) and \(C\).
  3. Show that the acute angle between the planes \(A C D\) and \(B C D\) is \(\cos ^ { - 1 } \left( \frac { 1 } { \sqrt { 3 } } \right)\).
CAIE Further Paper 1 2020 Specimen Q7
17 marks Challenging +1.2
7 The curve \(C\) has equation \(y = \frac { 2 x ^ { 2 } - 3 x - 2 } { x ^ { 2 } - 2 x + 1 }\).
  1. State the equations of the asymptotes of \(C\).
  2. Show that \(y \leqslant \frac { 25 } { 12 }\) at all points on \(C\).
  3. Find the coordinates of any stationary points of \(C\).
  4. Sketch \(C\), stating the coordinates of any intersections of \(C\) with the coordinate axes and the asymptotes.
  5. Sketch the curve with equation \(y = \left| \frac { 2 x ^ { 2 } - 3 x - 2 } { x ^ { 2 } - 2 x + 1 } \right|\) and find the set of values of \(x\) for which \(\left| \frac { 2 x ^ { 2 } - 3 x - 2 } { x ^ { 2 } - 2 x + 1 } \right| < 2\).
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]