Questions — CAIE Further Paper 1 (151 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics Stats 1 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
CAIE Further Paper 1 2020 November Q7
7
  1. Show that the curve with Cartesian equation $$\left( x ^ { 2 } + y ^ { 2 } \right) ^ { \frac { 5 } { 2 } } = 4 x y \left( x ^ { 2 } - y ^ { 2 } \right)$$ has polar equation \(r = \sin 4 \theta\).
    The curve \(C\) has polar equation \(r = \sin 4 \theta\), for \(0 \leqslant \theta \leqslant \frac { 1 } { 4 } \pi\).
  2. Sketch \(C\) and state the equation of the line of symmetry.
  3. Find the exact value of the area of the region enclosed by \(C\).
  4. Using the identity \(\sin 4 \theta \equiv 4 \sin \theta \cos ^ { 3 } \theta - 4 \sin ^ { 3 } \theta \cos \theta\), find the maximum distance of \(C\) from the line \(\theta = \frac { 1 } { 2 } \pi\). Give your answer correct to 2 decimal places.
    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 Further Paper 1 2020 November Q1
1 The cubic equation \(\mathrm { x } ^ { 3 } + \mathrm { bx } ^ { 2 } + \mathrm { cx } + \mathrm { d } = 0\), where \(b , c\) and \(d\) are constants, has roots \(\alpha , \beta , \gamma\). It is given that \(\alpha \beta \gamma = - 1\).
  1. State the value of \(d\).
  2. Find a cubic equation, with coefficients in terms of \(b\) and \(c\), whose roots are \(\alpha + 1 , \beta + 1 , \gamma + 1\).
  3. Given also that \(\gamma + 1 = - \alpha - 1\), deduce that \(( \mathrm { c } - 2 \mathrm {~b} + 3 ) ( \mathrm { b } - 3 ) = \mathrm { b } - \mathrm { c }\).
CAIE Further Paper 1 2020 November Q2
2 Prove by mathematical induction that \(7 ^ { 2 n } - 1\) is divisible by 12 for every positive integer \(n\).
CAIE Further Paper 1 2020 November Q3
1 marks
3
  1. By simplifying \(\left( x ^ { n } - \sqrt { x ^ { 2 n } + 1 } \right) \left( x ^ { n } + \sqrt { x ^ { 2 n } + 1 } \right)\), show that \(\frac { 1 } { x ^ { n } - \sqrt { x ^ { 2 n } + 1 } } = - x ^ { n } - \sqrt { x ^ { 2 n } + 1 }\). [1]
    Let \(u _ { n } = x ^ { n + 1 } + \sqrt { x ^ { 2 n + 2 } + 1 } + \frac { 1 } { x ^ { n } - \sqrt { x ^ { 2 n } + 1 } }\).
  2. Use the method of differences to find \(\sum _ { \mathrm { n } = 1 } ^ { \mathrm { N } } \mathrm { u } _ { \mathrm { n } }\) in terms of \(N\) and \(x\).
  3. Deduce 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.
CAIE Further Paper 1 2020 November Q4
4 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by $$\mathbf { A } = \left( \begin{array} { l l } 0 & 1
1 & 0 \end{array} \right) \text { and } \mathbf { B } = \left( \begin{array} { c c } \frac { 1 } { 2 } & - \frac { 1 } { 2 } \sqrt { 3 }
\frac { 1 } { 2 } \sqrt { 3 } & \frac { 1 } { 2 } \end{array} \right)$$
  1. Give full details of the geometrical transformation in the \(x - y\) plane represented by \(\mathbf { A }\).
  2. Give full details of the geometrical transformation in the \(x - y\) plane represented by \(\mathbf { B }\).
    The triangle \(D E F\) in the \(x - y\) plane is transformed by \(\mathbf { A B }\) onto triangle \(P Q R\).
  3. Show that the triangles \(D E F\) and \(P Q R\) have the same area.
  4. Find the matrix which transforms triangle \(P Q R\) onto triangle \(D E F\).
  5. Find the equations of the invariant lines, through the origin, of the transformation in the \(x - y\) plane represented by \(\mathbf { A B }\).
CAIE Further Paper 1 2020 November Q5
5 The curve \(C\) has polar equation \(r = \ln ( 1 + \pi - \theta )\), for \(0 \leqslant \theta \leqslant \pi\).
  1. Sketch \(C\) and state the polar coordinates of the point of \(C\) furthest from the pole.
  2. Using the substitution \(u = 1 + \pi - \theta\), or otherwise, show that the area of the region enclosed by \(C\) and the initial line is $$\frac { 1 } { 2 } ( 1 + \pi ) \ln ( 1 + \pi ) ( \ln ( 1 + \pi ) - 2 ) + \pi$$
  3. Show that, at the point of \(C\) furthest from the initial line, $$( 1 + \pi - \theta ) \ln ( 1 + \pi - \theta ) - \tan \theta = 0$$ and verify that this equation has a root between 1.2 and 1.3.
CAIE Further Paper 1 2020 November Q6
6 Let \(a\) be a positive constant.
  1. The curve \(C _ { 1 }\) has equation \(\mathrm { y } = \frac { \mathrm { x } - \mathrm { a } } { \mathrm { x } - 2 \mathrm { a } }\). Sketch \(C _ { 1 }\). The curve \(C _ { 2 }\) has equation \(\mathrm { y } = \left( \frac { \mathrm { x } - \mathrm { a } } { \mathrm { x } - 2 \mathrm { a } } \right) ^ { 2 }\). The curve \(C _ { 3 }\) has equation \(\mathrm { y } = \left| \frac { \mathrm { x } - \mathrm { a } } { \mathrm { x } - 2 \mathrm { a } } \right|\).
    1. Find the coordinates of any stationary points of \(C _ { 2 }\).
    2. Find also the coordinates of any points of intersection of \(C _ { 2 }\) and \(C _ { 3 }\).
  3. Sketch \(C _ { 2 }\) and \(C _ { 3 }\) on a single diagram, clearly identifying each curve. Hence find the set of values of \(x\) for which \(\left( \frac { x - a } { x - 2 a } \right) ^ { 2 } \leqslant \left| \frac { x - a } { x - 2 a } \right|\).
CAIE Further Paper 1 2020 November Q7
7 The points \(A , B , C\) have position vectors $$- 2 \mathbf { i } + 2 \mathbf { j } - \mathbf { k } , \quad - 2 \mathbf { i } + \mathbf { j } + 2 \mathbf { k } , \quad - 2 \mathbf { j } + \mathbf { k } ,$$ respectively, relative to the origin \(O\).
  1. Find the equation of the plane \(A B C\), giving your answer in the form \(a x + b y + c z = d\).
  2. Find the acute angle between the planes \(O B C\) and \(A B C\).
    The point \(D\) has position vector \(t \mathbf { i } - \mathbf { j }\).
  3. Given that the shortest distance between the lines \(A B\) and \(C D\) is \(\sqrt { \mathbf { 1 0 } }\), find the value of \(t\).
    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 Further Paper 1 2021 November Q1
1 It is given that $$\alpha + \beta + \gamma = 3 , \quad \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 5 , \quad \alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } = 6 .$$ The cubic equation \(\mathrm { x } ^ { 3 } + \mathrm { bx } ^ { 2 } + \mathrm { cx } + \mathrm { d } = 0\) has roots \(\alpha , \beta , \gamma\).
Find the values of \(b , c\) and \(d\).
CAIE Further Paper 1 2021 November Q2
2
  1. Use standard results from the list of formulae (MF19) to find \(\sum _ { r = 1 } ^ { n } r ( r + 1 ) ( r + 2 )\) in terms of \(n\),
    fully factorising your answer. fully factorising your answer.
  2. Express \(\frac { 1 } { r ( r + 1 ) ( r + 2 ) }\) in partial fractions and hence use the method of differences to 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 2021 November Q3
3 The sequence of real numbers \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is such that \(a _ { 1 } = 1\) and $$a _ { n + 1 } = \left( a _ { n } + \frac { 1 } { a _ { n } } \right) ^ { 3 }$$
  1. Prove by mathematical induction that \(\ln a _ { n } \geqslant 3 ^ { n - 1 } \ln 2\) for all integers \(n \geqslant 2\).
    [0pt] [You may use the fact that \(\ln \left( x + \frac { 1 } { x } \right) > \ln x\) for \(x > 0\).]
  2. Show that \(\ln \mathrm { a } _ { \mathrm { n } + 1 } - \ln \mathrm { a } _ { \mathrm { n } } > 3 ^ { \mathrm { n } - 1 } \ln 4\) for \(n \geqslant 2\).
CAIE Further Paper 1 2021 November Q4
4 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { c c } \cos \theta & - \sin \theta
\sin \theta & \cos \theta \end{array} \right) \left( \begin{array} { l l } 3 & 0
0 & 1 \end{array} \right)\).
  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.
  2. Find the values of \(\theta\), for \(0 \leqslant \theta \leqslant \pi\), for which the transformation represented by \(\mathbf { M }\) has exactly one invariant line through the origin, giving your answers in terms of \(\pi\).
CAIE Further Paper 1 2021 November Q5
5 The plane \(\Pi\) has equation \(\mathbf { r } = - 2 \mathbf { i } + 3 \mathbf { j } + 3 \mathbf { k } + \lambda ( \mathbf { i } + \mathbf { k } ) + \mu ( 2 \mathbf { i } + 3 \mathbf { j } )\).
  1. Find a Cartesian equation of \(\Pi\), giving your answer in the form \(a x + b y + c = d\).
    The line \(l\) passes through the point \(P\) with position vector \(2 \mathbf { i } - 3 \mathbf { j } + 5 \mathbf { k }\) and is parallel to the vector \(\mathbf { k }\).
  2. Find the position vector of the point where \(l\) meets \(\Pi\).
  3. Find the acute angle between \(l\) and \(\Pi\).
  4. Find the perpendicular distance from \(P\) to \(\Pi\).
CAIE Further Paper 1 2021 November Q6
6 The curve \(C\) has polar equation \(r = 2 \cos \theta ( 1 + \sin \theta )\), for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
  1. Find the polar coordinates of the point on \(C\) that is furthest from the pole.
  2. Sketch C.
  3. Find the area of the region bounded by \(C\) and the initial line, giving your answer in exact form.
CAIE Further Paper 1 2021 November Q7
7 The curve \(C\) has equation \(y = \frac { 4 x + 5 } { 4 - 4 x ^ { 2 } }\).
  1. Find the equations of the asymptotes of \(C\).
  2. Find the coordinates of any stationary points on \(C\).
  3. Sketch \(C\), stating the coordinates of the intersections with the axes.
  4. Sketch the curve with equation \(y = \left| \frac { 4 x + 5 } { 4 - 4 x ^ { 2 } } \right|\) and find in exact form the set of values of \(x\) for which \(4 | 4 x + 5 | > 5 \left| 4 - 4 x ^ { 2 } \right|\).
    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 Further Paper 1 2021 November Q1
1
  1. Give full details of the geometrical transformation in the \(x - y\) plane represented by the matrix \(\left( \begin{array} { l l } 6 & 0
    0 & 6 \end{array} \right)\). Let \(\mathbf { A } = \left( \begin{array} { l l } 3 & 4
    2 & 2 \end{array} \right)\).
  2. The triangle \(D E F\) in the \(x - y\) plane is transformed by \(\mathbf { A }\) onto triangle \(P Q R\). Given that the area of triangle \(D E F\) is \(13 \mathrm {~cm} ^ { 2 }\), find the area of triangle \(P Q R\).
  3. Find the matrix \(\mathbf { B }\) such that \(\mathbf { A B } = \left( \begin{array} { l l } 6 & 0
    0 & 6 \end{array} \right)\).
  4. Show that the origin is the only invariant point of the transformation in the \(x - y\) plane represented by \(\mathbf { A }\).
CAIE Further Paper 1 2021 November Q2
2 It is given that \(\mathrm { y } = \mathrm { xe } ^ { \mathrm { ax } }\), where \(a\) is a constant.
Prove by mathematical induction that, for all positive integers \(n\), $$\frac { d ^ { n } y } { d x ^ { n } } = \left( a ^ { n } x + n a ^ { n - 1 } \right) e ^ { a x }$$
CAIE Further Paper 1 2021 November Q3
3 Let \(S _ { n } = \sum _ { r = 1 } ^ { n } \ln \frac { r ( r + 2 ) } { ( r + 1 ) ^ { 2 } }\).
  1. Using the method of differences, or otherwise, show that \(S _ { n } = \ln \frac { n + 2 } { 2 ( n + 1 ) }\).
    Let \(S = \sum _ { r = 1 } ^ { \infty } \ln \frac { r ( r + 2 ) } { ( r + 1 ) ^ { 2 } }\).
  2. Find the least value of \(n\) such that \(\mathrm { S } _ { \mathrm { n } } - \mathrm { S } < 0.01\).
CAIE Further Paper 1 2021 November Q4
4 The cubic equation \(x ^ { 3 } + 2 x ^ { 2 } + 3 x + 3 = 0\) has roots \(\alpha , \beta , \gamma\).
  1. Find the value of \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\).
  2. Show that \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } = 1\).
  3. Use standard results from the list of formulae (MF19) to show that $$\sum _ { r = 1 } ^ { n } \left( ( \alpha + r ) ^ { 3 } + ( \beta + r ) ^ { 3 } + ( \gamma + r ) ^ { 3 } \right) = n + \frac { 1 } { 4 } n ( n + 1 ) \left( a n ^ { 2 } + b n + c \right)$$ where \(a\), \(b\) and \(c\) are constants to be determined.
CAIE Further Paper 1 2021 November Q5
5 The curve \(C\) has polar equation \(r = 3 + 2 \sin \theta\), for \(- \pi < \theta \leqslant \pi\).
  1. The diagram shows part of \(C\). Sketch the rest of \(C\) on the diagram.
    \includegraphics[max width=\textwidth, alt={}, center]{3dbf7021-79c0-4ebf-b96a-5ddeeed45011-08_865_702_408_1023} The straight line \(l\) has polar equation \(r \sin \theta = 2\).
  2. Add \(l\) to the diagram in part (a) and find the polar coordinates of the points of intersection of \(C\) and \(l\).
  3. The region \(R\) is enclosed by \(C\) and \(l\), and contains the pole. Find the area of \(R\), giving your answer in exact form.
CAIE Further Paper 1 2021 November Q6
1 marks
6 The curve \(C\) has equation \(\mathrm { y } = \frac { \mathrm { x } ^ { 2 } } { \mathrm { x } - 3 }\).
  1. Find the equations of the asymptotes of \(C\).
  2. Show that there is no point on \(C\) for which \(0 < y < 12\).
  3. Sketch C.
    1. Sketch the graphs of \(y = \left| \frac { x ^ { 2 } } { x - 3 } \right|\) and \(y = | x | - 3\) on a single diagram, stating the coordinates of the intersections with the axes.
    2. Use your sketch to find the set of values of \(c\) for which \(\left| \frac { x ^ { 2 } } { x - 3 } \right| \leqslant | x | + c\) has no solution. [1]
CAIE Further Paper 1 2021 November Q7
7 The points \(A , B , C\) have position vectors $$2 \mathbf { i } + 2 \mathbf { j } , \quad - \mathbf { j } + \mathbf { k } \quad \text { and } \quad 2 \mathbf { i } + \mathbf { j } - 7 \mathbf { k }$$ respectively, relative to the origin \(O\).
  1. Find an equation of the plane \(O A B\), giving your answer in the form \(\mathbf { r } . \mathbf { n } = p\).
    The plane \(\Pi\) has equation \(\mathrm { x } - 3 \mathrm { y } - 2 \mathrm { z } = 1\).
  2. Find the perpendicular distance of \(\Pi\) from the origin.
  3. Find the acute angle between the planes \(O A B\) and \(\Pi\).
  4. Find an equation for the common perpendicular to the lines \(O C\) and \(A B\).
    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 Further Paper 1 2021 November Q7
7 The curve \(C\) has equation \(\mathrm { y } = \frac { 4 \mathrm { x } + 5 } { 4 - 4 \mathrm { x } ^ { 2 } }\).
  1. Find the equations of the asymptotes of \(C\).
  2. Find the coordinates of any stationary points on \(C\).
  3. Sketch \(C\), stating the coordinates of the intersections with the axes.
  4. Sketch the curve with equation \(y = \left| \frac { 4 x + 5 } { 4 - 4 x ^ { 2 } } \right|\) and find in exact form the set of values of \(x\) for which \(4 | 4 x + 5 | > 5 \left| 4 - 4 x ^ { 2 } \right|\).
    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 Further Paper 1 2022 November Q1
1 The cubic equation \(x ^ { 3 } + b x ^ { 2 } + d = 0\) has roots \(\alpha , \beta , \gamma\), where \(\alpha = \beta\) and \(d \neq 0\).
  1. Show that \(4 b ^ { 3 } + 27 d = 0\).
  2. Given that \(2 \alpha ^ { 2 } + \gamma ^ { 2 } = 3 b\), find the values of \(b\) and \(d\).
CAIE Further Paper 1 2022 November Q2
6 marks
2 Prove by mathematical induction that, for all positive integers \(n , 7 ^ { 2 n } + 97 ^ { n } - 50\) is divisible by 48. [6]