Questions — CAIE (7646 questions)

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CAIE Further Paper 1 2023 November Q2
6 marks Challenging +1.2
2 Prove by mathematical induction that, for all positive integers \(n\), $$\frac { d ^ { n } } { d x ^ { n } } \left( x ^ { 2 } e ^ { x } \right) = \left( x ^ { 2 } + 2 n x + n ( n - 1 ) \right) e ^ { x }$$
CAIE Further Paper 1 2023 November Q3
8 marks Standard +0.3
3 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { l l } k & 0 \\ 0 & 1 \end{array} \right) \left( \begin{array} { l l } 1 & 0 \\ 1 & 1 \end{array} \right)\), where \(k\) is a constant and \(k \neq 0\) and \(k \neq 1\).
  1. The matrix \(\mathbf { M }\) represents a sequence of two geometrical transformations. State the type of each transformation, and make clear the order in which they are applied.
    The unit square in the \(x - y\) plane is transformed by \(\mathbf { M }\) onto parallelogram \(O P Q R\).
  2. Find, in terms of \(k\), the area of parallelogram \(O P Q R\) and the matrix which transforms \(O P Q R\) onto the unit square.
  3. Show that the line through the origin with gradient \(\frac { 1 } { k - 1 }\) is invariant under the transformation in the \(x - y\) plane represented by \(\mathbf { M }\).
CAIE Further Paper 1 2023 November Q4
10 marks Challenging +1.2
4 The cubic equation \(27 x ^ { 3 } + 18 x ^ { 2 } + 6 x - 1 = 0\) has roots \(\alpha , \beta , \gamma\).
  1. Show that a cubic equation with roots \(3 \alpha + 1,3 \beta + 1,3 \gamma + 1\) is $$y ^ { 3 } - y ^ { 2 } + y - 2 = 0$$ The sum \(( 3 \alpha + 1 ) ^ { n } + ( 3 \beta + 1 ) ^ { n } + ( 3 \gamma + 1 ) ^ { n }\) is denoted by \(\mathrm { S } _ { \mathrm { n } }\).
  2. Find the values of \(S _ { 2 }\) and \(S _ { 3 }\).
  3. Find the values of \(S _ { - 1 }\) and \(S _ { - 2 }\).
CAIE Further Paper 1 2023 November Q5
13 marks Standard +0.8
5 The plane \(\Pi _ { 1 }\) has equation \(\mathbf { r } = \mathbf { i } - \mathbf { j } - 2 \mathbf { k } + \lambda ( \mathbf { i } - 2 \mathbf { j } - 3 \mathbf { k } ) + \mu ( 3 \mathbf { i } - \mathbf { k } )\).
  1. Find an equation for \(\Pi _ { 1 }\) in the form \(\mathrm { ax } + \mathrm { by } + \mathrm { cz } = \mathrm { d }\).
    The line \(l\), which does not lie in \(\Pi _ { 1 }\), has equation \(\mathbf { r } = - 3 \mathbf { i } + \mathbf { k } + t ( \mathbf { i } + \mathbf { j } + \mathbf { k } )\).
  2. Show that \(l\) is parallel to \(\Pi _ { 1 }\).
  3. Find the distance between \(l\) and \(\Pi _ { 1 }\).
  4. The plane \(\Pi _ { 2 }\) has equation \(3 x + 3 y + 2 z = 1\). Find a vector equation of the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).
CAIE Further Paper 1 2023 November Q6
13 marks Challenging +1.8
6 The curve \(C\) has polar equation \(r = \mathrm { e } ^ { - \theta } - \mathrm { e } ^ { - \frac { 1 } { 2 } \pi }\), where \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
  1. Sketch \(C\) and state, in exact form, the greatest distance of a point on \(C\) from the pole.
  2. Find the exact value of the area of the region bounded by \(C\) and the initial line.
  3. Show that, at the point on \(C\) furthest from the initial line, $$1 - e ^ { \theta - \frac { 1 } { 2 } \pi } - \tan \theta = 0$$ and verify that this equation has a root between 0.56 and 0.57 .
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 P2 2019 June Q1
3 marks Moderate -0.8
1 Show that \(\ln \left( x ^ { 3 } - 4 x \right) - \ln \left( x ^ { 2 } - 2 x \right) \equiv \ln ( x + 2 )\).
CAIE P2 2019 June Q2
6 marks Standard +0.3
2
  1. Solve the inequality \(| 3 x - 5 | < | x + 3 |\).
  2. Hence find the greatest integer \(n\) satisfying the inequality \(\left| 3 ^ { 0.1 n + 1 } - 5 \right| < \left| 3 ^ { 0.1 n } + 3 \right|\).
CAIE P2 2019 June Q3
7 marks Standard +0.3
3 Find the equation of the normal to the curve $$x ^ { 2 } \ln y + 2 x + 5 y = 11$$ at the point \(( 3,1 )\).
CAIE P2 2019 June Q4
7 marks Moderate -0.3
4
  1. Find \(\int \tan ^ { 2 } 3 x \mathrm {~d} x\).
  2. Find the exact value of \(\int _ { 0 } ^ { 1 } \frac { \mathrm { e } ^ { 3 x } + 4 } { \mathrm { e } ^ { x } } \mathrm {~d} x\). Show all necessary working.
CAIE P2 2019 June Q5
8 marks Moderate -0.8
5 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 5 x ^ { 3 } + a x ^ { 2 } + b x - 16$$ where \(a\) and \(b\) are constants. It is given that \(( x - 2 )\) is a factor of \(\mathrm { p } ( x )\) and that the remainder is 27 when \(\mathrm { p } ( x )\) is divided by \(( x + 1 )\).
  1. Find the values of \(a\) and \(b\).
  2. Hence factorise \(\mathrm { p } ( x )\) completely.