Questions — CAIE Further Paper 1 (151 questions)

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CAIE Further Paper 1 2022 November Q3
3
  1. By considering \(( 2 r + 1 ) ^ { 3 } - ( 2 r - 1 ) ^ { 3 }\), use the method of differences to prove that $$\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )$$ Let \(S _ { n } = 1 ^ { 2 } + 3 \times 2 ^ { 2 } + 3 ^ { 2 } + 3 \times 4 ^ { 2 } + 5 ^ { 2 } + 3 \times 6 ^ { 2 } + \ldots + \left( 2 + ( - 1 ) ^ { n } \right) n ^ { 2 }\).
  2. Show that \(\mathrm { S } _ { 2 \mathrm { n } } = \frac { 1 } { 3 } \mathrm { n } ( 2 \mathrm { n } + 1 ) ( \mathrm { an } + \mathrm { b } )\), where \(a\) and \(b\) are integers to be determined.
  3. State the value of \(\lim _ { n \rightarrow \infty } \frac { S _ { 2 n } } { n ^ { 3 } }\).
CAIE Further Paper 1 2022 November Q4
4 The plane \(\Pi\) contains the lines \(\mathbf { r } = 3 \mathbf { i } - 2 \mathbf { j } + \mathbf { k } + \lambda ( - \mathbf { i } + 2 \mathbf { j } + \mathbf { k } )\) and \(\mathbf { r } = 4 \mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k } + \mu ( 3 \mathbf { i } + 2 \mathbf { j } - \mathbf { k } )\).
  1. Find a Cartesian equation of \(\Pi\), giving your answer in the form \(a x + b y + c z = d\).
    The line \(l\) passes through the point \(P\) with position vector \(2 \mathbf { i } + 3 \mathbf { j } + \mathbf { k }\) and is parallel to the vector \(\mathbf { j } + \mathbf { k }\).
  2. Find the acute angle between \(I\) and \(\Pi\).
  3. Find the position vector of the foot of the perpendicular from \(P\) to \(\Pi\).
CAIE Further Paper 1 2022 November Q5
5 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { r r } \frac { 1 } { 2 } \sqrt { 2 } & - \frac { 1 } { 2 } \sqrt { 2 }
\frac { 1 } { 2 } \sqrt { 2 } & \frac { 1 } { 2 } \sqrt { 2 } \end{array} \right) \left( \begin{array} { c c } 1 & k
0 & 1 \end{array} \right)\), where \(k\) is a constant.
  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. The triangle \(A B C\) in the \(x - y\) plane is transformed by \(\mathbf { M }\) onto triangle \(D E F\). Find, in terms of \(k\), the single matrix which transforms triangle \(D E F\) onto triangle \(A B C\).
  3. Find the set of values of \(k\) for which the transformation represented by \(\mathbf { M }\) has no invariant lines through the origin.
CAIE Further Paper 1 2022 November Q6
6
  1. Show that the curve with Cartesian equation $$\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 36 \left( x ^ { 2 } - y ^ { 2 } \right)$$ has polar equation \(r ^ { 2 } = 36 \cos 2 \theta\).
    The curve \(C\) has polar equation \(r ^ { 2 } = 36 \cos 2 \theta\), for \(- \frac { 1 } { 4 } \pi \leqslant \theta \leqslant \frac { 1 } { 4 } \pi\).
  2. Sketch \(C\) and state the maximum distance of a point on \(C\) from the pole.
  3. Find the area of the region enclosed by \(C\).
  4. Find the maximum distance of a point on \(C\) from the initial line, giving the answer in exact form.
CAIE Further Paper 1 2022 November Q7
7 The curve \(C\) has equation \(y = \frac { 5 x ^ { 2 } } { 5 x - 2 }\).
  1. Find the equations of the asymptotes of \(C\).
  2. Find the coordinates of the stationary points on \(C\).
  3. Sketch \(C\).
  4. Sketch the curve with equation \(y = \left| \frac { 5 x ^ { 2 } } { 5 x - 2 } \right|\) and find in exact form the set of values of \(x\) for which \(\left| \frac { 5 x ^ { 2 } } { 5 x - 2 } \right| < 2\).
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE Further Paper 1 2022 November Q1
1
  1. Use the list of formulae (MF19) to find \(\sum _ { r = 1 } ^ { n } r ( r + 2 )\) in terms of \(n\), simplifying your answer.
  2. Express \(\frac { 1 } { r ( r + 2 ) }\) in partial fractions and hence find \(\sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } \frac { 1 } { \mathrm { r } ( \mathrm { r } + 2 ) }\) in terms of \(n\).
  3. Deduce the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { r ( r + 2 ) }\).
CAIE Further Paper 1 2022 November Q2
2 The equation \(x ^ { 4 } + 3 x ^ { 2 } + 2 x + 6 = 0\) has roots \(\alpha , \beta , \gamma , \delta\).
  1. Find a quartic equation whose roots are \(\frac { 1 } { \alpha ^ { 2 } } , \frac { 1 } { \beta ^ { 2 } } , \frac { 1 } { \gamma ^ { 2 } } , \frac { 1 } { \delta ^ { 2 } }\) and state the value of \(\frac { 1 } { \alpha ^ { 2 } } + \frac { 1 } { \beta ^ { 2 } } + \frac { 1 } { \gamma ^ { 2 } } + \frac { 1 } { \delta ^ { 2 } }\).
  2. Find the value of \(\beta ^ { 2 } \gamma ^ { 2 } \delta ^ { 2 } + \alpha ^ { 2 } \gamma ^ { 2 } \delta ^ { 2 } + \alpha ^ { 2 } \beta ^ { 2 } \delta ^ { 2 } + \alpha ^ { 2 } \beta ^ { 2 } \gamma ^ { 2 }\).
  3. Find the value of \(\frac { 1 } { \alpha ^ { 4 } } + \frac { 1 } { \beta ^ { 4 } } + \frac { 1 } { \gamma ^ { 4 } } + \frac { 1 } { \delta ^ { 4 } }\).
CAIE Further Paper 1 2022 November Q3
4 marks
3 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { c c } 1 & 0
0 & k \end{array} \right) \left( \begin{array} { c c } 1 & 0
k & 1 \end{array} \right)\), where \(k\) is a constant and \(k \neq 0\) or 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.
  2. Write \(\mathbf { M } ^ { - 1 }\) as the product of two matrices, neither of which is \(\mathbf { I }\).
  3. Show that the invariant points of the transformation represented by \(\mathbf { M }\) lie on the line \(\mathrm { y } = \frac { \mathrm { k } ^ { 2 } } { 1 - \mathrm { k } } \mathrm { x }\). [4]
  4. The triangle \(A B C\) in the \(x - y\) plane is transformed by \(\mathbf { M }\) onto triangle \(D E F\). Find the value of \(k\) for which the area of triangle \(D E F\) is equal to the area of triangle \(A B C\).
CAIE Further Paper 1 2022 November Q4
4 The function f is such that \(\mathrm { f } ^ { \prime \prime } ( x ) = \mathrm { f } ( x )\).
Prove by mathematical induction that, for every positive integer \(n\), $$\frac { d ^ { 2 n - 1 } } { d x ^ { 2 n - 1 } } ( x f ( x ) ) = x f ^ { \prime } ( x ) + ( 2 n - 1 ) f ( x )$$
CAIE Further Paper 1 2022 November Q5
5 The curve \(C\) has polar equation \(r = \operatorname { asec } ^ { 2 } \theta\), where \(a\) is a positive constant and \(0 \leqslant \theta \leqslant \frac { 1 } { 4 } \pi\).
  1. Sketch \(C\), stating the polar coordinates of the point of intersection of \(C\) with the initial line and also with the half-line \(\theta = \frac { 1 } { 4 } \pi\).
  2. Find the maximum distance of a point of \(C\) from the initial line.
  3. Find the area of the region enclosed by \(C\), the initial line and the half-line \(\theta = \frac { 1 } { 4 } \pi\).
  4. Find, in the form \(y = f ( x )\), the Cartesian equation of \(C\).
CAIE Further Paper 1 2022 November Q6
6 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations \(\mathbf { r } = 2 \mathbf { i } + \mathbf { k } + \lambda ( \mathbf { i } - \mathbf { j } + 2 \mathbf { k } )\) and \(\mathbf { r } = 2 \mathbf { j } + 6 \mathbf { k } + \mu ( \mathbf { i } + 2 \mathbf { j } - 2 \mathbf { k } )\) respectively. The point \(P\) on \(l _ { 1 }\) and the point \(Q\) on \(l _ { 2 }\) are such that \(P Q\) is perpendicular to both \(l _ { 1 }\) and \(l _ { 2 }\).
  1. Find the length \(P Q\).
    The plane \(\Pi _ { 1 }\) contains \(P Q\) and \(l _ { 1 }\).
    The plane \(\Pi _ { 2 }\) contains \(P Q\) and \(l _ { 2 }\).
    1. Write down an equation of \(\Pi _ { 1 }\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + \mathbf { s b } + \mathbf { t c }\).
    2. Find an equation of \(\Pi _ { 2 }\), giving your answer in the form \(a x + b y + c z = d\).
  3. Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).
CAIE Further Paper 1 2022 November Q7
7 The curve \(C\) has equation \(\mathrm { y } = \frac { \mathrm { x } ^ { 2 } - \mathrm { x } } { \mathrm { x } + 1 }\).
  1. Find the equations of the asymptotes of \(C\).
  2. Find the exact coordinates of the stationary points on \(C\).
  3. Sketch \(C\), stating the coordinates of any intersections with the axes.
  4. Sketch the curve with equation \(y = \left| \frac { x ^ { 2 } - x } { x + 1 } \right|\) and find in exact form the set of values of \(x\) for which \(\left| \frac { x ^ { 2 } - x } { x + 1 } \right| < 6\).
    If you use the following page to complete the answer to any question, the question number must be clearly shown.
CAIE Further Paper 1 2023 November Q1
1
  1. By considering \(( r + 1 ) ^ { 2 } - r ^ { 2 }\), use the method of differences to prove that $$\sum _ { r = 1 } ^ { n } r = \frac { 1 } { 2 } n ( n + 1 )$$
  2. Given that \(\sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } ( \mathrm { r } + \mathrm { a } ) = \mathrm { n }\), find \(a\) in terms of \(n\).
CAIE Further Paper 1 2023 November Q2
2 Prove by mathematical induction that, for all positive integers \(n\), $$1 + 2 x + 3 x ^ { 2 } + \ldots + n x ^ { n - 1 } = \frac { 1 - ( n + 1 ) x ^ { n } + n x ^ { n + 1 } } { ( 1 - x ) ^ { 2 } }$$
CAIE Further Paper 1 2023 November Q3
3 The quartic equation \(\mathrm { x } ^ { 4 } + \mathrm { bx } ^ { 3 } + \mathrm { cx } ^ { 2 } + \mathrm { dx } - 2 = 0\) has roots \(\alpha , \beta , \gamma , \delta\). It is given that $$\alpha + \beta + \gamma + \delta = 3 , \quad \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 } = 5 , \quad \alpha ^ { - 1 } + \beta ^ { - 1 } + \gamma ^ { - 1 } + \delta ^ { - 1 } = 6$$
  1. Find the values of \(b , c\) and \(d\).
  2. Given also that \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } + \delta ^ { 3 } = - 27\), find the value of \(\alpha ^ { 4 } + \beta ^ { 4 } + \gamma ^ { 4 } + \delta ^ { 4 }\).
CAIE Further Paper 1 2023 November Q4
4 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations $$\mathbf { r } = - 2 \mathbf { i } - 3 \mathbf { j } - 5 \mathbf { k } + \lambda ( - 4 \mathbf { i } + 3 \mathbf { j } + 5 \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = 2 \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k } + \mu ( 2 \mathbf { i } - 3 \mathbf { j } + \mathbf { k } )$$ respectively.
  1. Find the shortest distance between \(l _ { 1 }\) and \(l _ { 2 }\).
    The plane \(\Pi\) contains \(l _ { 1 }\) and the point with position vector \(- \mathbf { i } - 3 \mathbf { j } - 4 \mathbf { k }\).
  2. Find an equation of \(\Pi\), giving your answer in the form \(a x + b y + c z = d\).
CAIE Further Paper 1 2023 November Q5
5 Let \(k\) be a constant. The matrices \(\mathbf { A } , \mathbf { B }\) and \(\mathbf { C }\) are given by $$\mathbf { A } = \left( \begin{array} { l l l } 1 & k & 3
2 & 1 & 3
3 & 2 & 5 \end{array} \right) , \quad \mathbf { B } = \left( \begin{array} { r r } 0 & - 2
- 1 & 3
0 & 0 \end{array} \right) \quad \text { and } \quad \mathbf { C } = \left( \begin{array} { r r r } - 2 & - 1 & 1
1 & 1 & 3 \end{array} \right)$$ It is given that \(\mathbf { A }\) is singular.
  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 \(\mathbf { C A B }\).
  3. The matrices \(\mathbf { D } , \mathbf { E }\) and \(\mathbf { F }\) represent geometrical transformations in the \(x - y\) plane.
    • D represents an enlargement, centre the origin.
    • E represents a stretch parallel to the \(x\)-axis.
    • F represents a reflection in the line \(y = x\).
    Given that \(\mathbf { C A B } = \mathbf { D } - 9 \mathbf { E F }\), find \(\mathbf { D } , \mathbf { E }\) and \(\mathbf { F }\).
CAIE Further Paper 1 2023 November Q6
6
  1. Show that the curve with Cartesian equation $$\left( x - \frac { 1 } { 2 } \right) ^ { 2 } + y ^ { 2 } = \frac { 1 } { 4 }$$ has polar equation \(r = \cos \theta\).
    The curves \(C _ { 1 }\) and \(C _ { 2 }\) have polar equations $$r = \cos \theta \quad \text { and } \quad r = \sin 2 \theta$$ respectively, where \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\). The curves \(C _ { 1 }\) and \(C _ { 2 }\) intersect at the pole and at another point \(P\).
  2. Find the polar coordinates of \(P\).
  3. In a single diagram sketch \(C _ { 1 }\) and \(C _ { 2 }\), clearly identifying each curve, and mark the point \(P\).
  4. The region \(R\) is enclosed by \(C _ { 1 }\) and \(C _ { 2 }\) and includes the line \(O P\). Find, in exact form, the area of \(R\).
CAIE Further Paper 1 2023 November Q7
7 The curve \(C\) has equation \(y = f ( x )\), where \(f ( x ) = \frac { x ^ { 2 } + 2 } { x ^ { 2 } - x - 2 }\).
  1. Find the equations of the asymptotes of \(C\).
  2. Find the coordinates of any stationary points on \(C\), giving your answers correct to 1 decimal place.
  3. Sketch \(C\), stating the coordinates of any intersections with the axes.
  4. Sketch the curve with equation \(\mathrm { y } = \frac { 1 } { \mathrm { f } ( \mathrm { x } ) }\).
  5. Find 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 2023 November Q1
1
  1. Use standard results from the list of formulae (MF19) to find \(\sum _ { r = 1 } ^ { n } \left( 3 r ^ { 2 } + 3 r + 1 \right)\) in terms of \(n\), simplifying your answer.
  2. Show that $$\frac { 1 } { r ^ { 3 } } - \frac { 1 } { ( r + 1 ) ^ { 3 } } = \frac { 3 r ^ { 2 } + 3 r + 1 } { r ^ { 3 } ( r + 1 ) ^ { 3 } }$$ and hence use the method of differences to find \(\sum _ { r = 1 } ^ { n } \frac { 3 r ^ { 2 } + 3 r + 1 } { r ^ { 3 } ( r + 1 ) ^ { 3 } }\).
  3. Deduce the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 3 r ^ { 2 } + 3 r + 1 } { r ^ { 3 } ( r + 1 ) ^ { 3 } }\).
CAIE Further Paper 1 2023 November Q2
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
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
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
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
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 .