Questions - CAIE Further Paper 1

Sketch $C$.
Show that the area of the region bounded by the half-line $\theta = \frac { 1 } { 2 } \pi$ and $C$ is $\frac { 3 - 4 \ln 2 } { 4 \pi }$.

CAIE Further Paper 1 2021 June Q6

6 The lines $l _ { 1 }$ and $l _ { 2 }$ have equations $\mathbf { r } = - \mathbf { i } - 2 \mathbf { j } + \mathbf { k } + s ( 2 \mathbf { i } - 3 \mathbf { j } )$ and $\mathbf { r } = 3 \mathbf { i } - 2 \mathbf { k } + t ( 3 \mathbf { i } - \mathbf { j } + 3 \mathbf { k } )$ respectively. The plane $\Pi _ { 1 }$ contains $l _ { 1 }$ and the point $P$ with position vector $- 2 \mathbf { i } - 2 \mathbf { j } + 4 \mathbf { k }$.

Find an equation of $\Pi _ { 1 }$, giving your answer in the form $\mathbf { r } = \mathbf { a } + \lambda \mathbf { b } + \mu \mathbf { c }$.
The plane $\Pi _ { 2 }$ contains $l _ { 2 }$ and is parallel to $l _ { 1 }$.
Find an equation of $\Pi _ { 2 }$, giving your answer in the form $\mathrm { ax } + \mathrm { by } + \mathrm { cz } = \mathrm { d }$.
Find the acute angle between $\Pi _ { 1 }$ and $\Pi _ { 2 }$.
The point $Q$ is such that $\overrightarrow { \mathrm { OQ } } = - 5 \overrightarrow { \mathrm { OP } }$. Find the position vector of the foot of the perpendicular from the point $Q$ to $\Pi _ { 2 }$.

CAIE Further Paper 1 2021 June Q7

7 The curve $C$ has equation $y = \frac { x ^ { 2 } - x - 3 } { 1 + x - x ^ { 2 } }$.

Find the equations of the asymptotes of $C$.
Find the coordinates of any stationary points on $C$.
Sketch $C$, stating the coordinates of the intersections with the axes.
Sketch the curve with equation $y = \left| \frac { x ^ { 2 } - x - 3 } { 1 + x - x ^ { 2 } } \right|$ and find in exact form the set of values of $x$ for which $\left| \frac { x ^ { 2 } - x - 3 } { 1 + x - x ^ { 2 } } \right| < 3$.
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 June Q1

1 Let $a$ be a positive constant.

Use the method of differences to find $\sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } \frac { 1 } { ( \mathrm { ar } + 1 ) ( \mathrm { ar } + \mathrm { a } + 1 ) }$ in terms of $n$ and $a$.
Find the value of $a$ for which $\sum _ { r = 1 } ^ { \infty } \frac { 1 } { ( a r + 1 ) ( a r + a + 1 ) } = \frac { 1 } { 6 }$.

CAIE Further Paper 1 2022 June Q2

2 The points $A , B , C$ have position vectors $$4 \mathbf { i } - 4 \mathbf { j } + \mathbf { k } , \quad - 4 \mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k } , \quad 4 \mathbf { i } - \mathbf { j } - 2 \mathbf { k } ,$$ respectively, relative to the origin $O$.

Find the equation of the plane $A B C$, giving your answer in the form $a x + b y + c z = d$.
Find the perpendicular distance from $O$ to the plane $A B C$.
The point $D$ has position vector $2 \mathbf { i } + 3 \mathbf { j } - 3 \mathbf { k }$. Find the coordinates of the point of intersection of the line $O D$ with the plane $A B C$.

CAIE Further Paper 1 2022 June Q3

3 The sequence of positive numbers $u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots$ is such that $u _ { 1 } > 4$ and, for $n \geqslant 1$, $$u _ { n + 1 } = \frac { u _ { n } ^ { 2 } + u _ { n } + 12 } { 2 u _ { n } }$$

By considering $\mathrm { u } _ { \mathrm { n } + 1 } - 4$, or otherwise, prove by mathematical induction that $\mathrm { u } _ { \mathrm { n } } > 4$ for all positive integers $n$.
Show that $u _ { n + 1 } < u _ { n }$ for $n \geqslant 1$.

CAIE Further Paper 1 2022 June Q4

4 The cubic equation $2 x ^ { 3 } + 5 x ^ { 2 } - 6 = 0$ has roots $\alpha , \beta , \gamma$.

Find a cubic equation whose roots are $\frac { 1 } { \alpha ^ { 3 } } , \frac { 1 } { \beta ^ { 3 } } , \frac { 1 } { \gamma ^ { 3 } }$.
Find the value of $\frac { 1 } { \alpha ^ { 6 } } + \frac { 1 } { \beta ^ { 6 } } + \frac { 1 } { \gamma ^ { 6 } }$.
Find also the value of $\frac { 1 } { \alpha ^ { 9 } } + \frac { 1 } { \beta ^ { 9 } } + \frac { 1 } { \gamma ^ { 9 } }$.

CAIE Further Paper 1 2022 June Q5

5 The curve $C$ has equation $y = \frac { 2 x ^ { 2 } - x - 1 } { x ^ { 2 } + x + 1 }$.

Show that $C$ has no vertical asymptotes and state the equation of the horizontal asymptote of $C$.
Find the coordinates of the stationary points on $C$.
Sketch $C$, stating the coordinates of the intersections with the axes.
Sketch the curve with equation $y = \left| \frac { 2 x ^ { 2 } - x - 1 } { x ^ { 2 } + x + 1 } \right|$ and state the set of values of $k$ for which $\left| \frac { 2 x ^ { 2 } - x - 1 } { x ^ { 2 } + x + 1 } \right| = k$ has 4 distinct real solutions.

CAIE Further Paper 1 2022 June Q6

6 The curve $C$ has polar equation $r ^ { 2 } = \tan ^ { - 1 } \left( \frac { 1 } { 2 } \theta \right)$, where $0 \leqslant \theta \leqslant 2$.

Sketch $C$ and state, in exact form, the greatest distance of a point on $C$ from the pole.
Find the exact value of the area of the region bounded by $C$ and the half-line $\theta = 2$.
Now consider the part of $C$ where $0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi$.
Show that, at the point furthest from the half-line $\theta = \frac { 1 } { 2 } \pi$, $$\left( \theta ^ { 2 } + 4 \right) \tan ^ { - 1 } \left( \frac { 1 } { 2 } \theta \right) \sin \theta - \cos \theta = 0$$ and verify that this equation has a root between 0.6 and 0.7 .
$7 \quad$ The matrix $\mathbf { A }$ is given by $\mathbf { A } = \left( \begin{array} { l l l } 1 & 2 & 3
4 & k & 6
7 & 8 & 9 \end{array} \right)$.
Find the set of values of $k$ for which $\mathbf { A }$ is non-singular.
Given that $\mathbf { A }$ is non-singular, find, in terms of $k$, the entries in the top row of $\mathbf { A } ^ { - 1 }$.
Given that $\mathbf { B } = \left( \begin{array} { l l l } 1 & 0 & 0
0 & 1 & 0 \end{array} \right)$, give an example of a matrix $\mathbf { C }$ such that $\mathbf { B A C } = \left( \begin{array} { l l } 2 & 1
k & 4 \end{array} \right)$.
Find the set of values of $k$ for which the transformation in the $x - y$ plane represented by $\left( \begin{array} { l l } 2 & 1
k & 4 \end{array} \right)$ has two distinct invariant lines through the origin.
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 June Q1

Sketch the curve with equation $\mathrm { y } = \frac { \mathrm { x } + 1 } { \mathrm { x } - 1 }$.
Sketch the curve with equation $\mathrm { y } = \frac { | \mathrm { x } | + 1 } { | \mathrm { x } | - 1 }$ and find the set of values of x for which $\frac { | \mathrm { x } | + 1 } { | \mathrm { x } | - 1 } < - 2$.

CAIE Further Paper 1 2022 June Q2

2 The cubic equation $x ^ { 3 } + 5 x ^ { 2 } + 10 x - 2 = 0$ has roots $\alpha , \beta , \gamma$.

Find the value of $\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }$.
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Show that the matrix $\left( \begin{array} { c c c } 1 & \alpha & \beta
\alpha & 1 & \gamma
\beta & \gamma & 1 \end{array} \right)$ is singular.

CAIE Further Paper 1 2022 June Q3

3 A curve $C$ has equation $\mathrm { y } = \frac { \mathrm { a } \mathrm { x } ^ { 2 } + \mathrm { x } - 1 } { \mathrm { x } - 1 }$, where $a$ is a positive constant.

Find the equations of the asymptotes of $C$.
Show that there is no point on $C$ for which $1 < \mathrm { y } < 1 + 4 \mathrm { a }$.
Sketch C. You do not need to find the coordinates of the intersections with the axes.

CAIE Further Paper 1 2022 June Q4

4 Let $\mathrm { u } _ { \mathrm { r } } = \mathrm { e } ^ { \mathrm { rx } } \left( \mathrm { e } ^ { 2 \mathrm { x } } - 2 \mathrm { e } ^ { \mathrm { x } } + 1 \right)$.

Using the method of differences, or otherwise, find $\sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } \mathrm { u } _ { \mathrm { r } }$ in terms of $n$ and $x$.
Deduce the set of non-zero 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.
Using a standard result from the list of formulae (MF19), find $\sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } \ln \mathrm { u } _ { \mathrm { r } }$ in terms of $n$ and $x$.

CAIE Further Paper 1 2022 June Q5

5 Let $\mathbf { A } = \left( \begin{array} { l l } 1 & a
0 & 1 \end{array} \right)$, where $a$ is a positive constant.

State the type of the geometrical transformation in the $x - y$ plane represented by $\mathbf { A }$.
Prove by mathematical induction that, for all positive integers $n$, $$\mathbf { A } ^ { \mathrm { n } } = \left( \begin{array} { c c } 1 & \mathrm { na }
0 & 1 \end{array} \right)$$ Let $\mathbf { B } = \left( \begin{array} { c c } b & b
a ^ { - 1 } & a ^ { - 1 } \end{array} \right)$, where $b$ is a positive constant.
Find the equations of the invariant lines, through the origin, of the transformation in the $x - y$ plane represented by $\mathbf { A } ^ { n } \mathbf { B }$.

CAIE Further Paper 1 2022 June Q6

6 The curve $C$ has Cartesian equation $x ^ { 2 } + x y + y ^ { 2 } = a$, where $a$ is a positive constant.

Show that the polar equation of $C$ is $r ^ { 2 } = \frac { 2 a } { 2 + \sin 2 \theta }$.
Sketch the part of $C$ for $0 \leqslant \theta \leqslant \frac { 1 } { 4 } \pi$. The region $R$ is enclosed by this part of $C$, the initial line and the half-line $\theta = \frac { 1 } { 4 } \pi$.
It is given that $\sin 2 \theta$ may be expressed as $\frac { 2 \tan \theta } { 1 + \tan ^ { 2 } \theta }$. Use this result to show that the area of $R$ is $$\frac { 1 } { 2 } a \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \frac { 1 + \tan ^ { 2 } \theta } { 1 + \tan \theta + \tan ^ { 2 } \theta } \mathrm {~d} \theta$$ and use the substitution $t = \tan \theta$ to find the exact value of this area.

CAIE Further Paper 1 2022 June Q7

7 The position vectors of the points $A , B , C , D$ are $$7 \mathbf { i } + 4 \mathbf { j } - \mathbf { k } , \quad 11 \mathbf { i } + 3 \mathbf { j } , \quad 2 \mathbf { i } + 6 \mathbf { j } + 3 \mathbf { k } , \quad 2 \mathbf { i } + 7 \mathbf { j } + \lambda \mathbf { k }$$ respectively.

Given that the shortest distance between the line $A B$ and the line $C D$ is 3 , show that $\lambda ^ { 2 } - 5 \lambda + 4 = 0$.
Let $\Pi _ { 1 }$ be the plane $A B D$ when $\lambda = 1$.
Let $\Pi _ { 2 }$ be the plane $A B D$ when $\lambda = 4$.
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 $a x + b y + c z = d$.
Find the acute angle between $\Pi _ { 1 }$ and $\Pi _ { 2 }$.
If you use the following page to complete the answer to any question, the question number must be clearly shown.

CAIE Further Paper 1 2023 June Q1

1 Let $\mathbf { A } = \left( \begin{array} { l l } 3 & 0
1 & 1 \end{array} \right)$.

Prove by mathematical induction that, for all positive integers $n$, $$2 \mathbf { A } ^ { n } = \left( \begin{array} { l l }

CAIE Further Paper 1 2023 June Q2

2 \times 3 ^ { n } & 0
3 ^ { n } - 1 & 2 \end{array} \right)$$ (b) Find, in terms of $n$, the inverse of $\mathbf { A } ^ { n }$.
2 The cubic equation $x ^ { 3 } + 4 x ^ { 2 } + 6 x + 1 = 0$ has roots $\alpha , \beta , \gamma$.
(a) Find the value of $\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }$.
(b) Use standard results from the list of formulae (MF19) to show that $$\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 2023 June Q3

Use the method of differences to find $\sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } \frac { 1 } { ( \mathrm { kr } + 1 ) ( \mathrm { kr } - \mathrm { k } + 1 ) }$ in terms of $n$ and $k$, where $k$ is a positive constant.
Deduce the value of $\sum _ { \mathrm { r } = 1 } ^ { \infty } \frac { 1 } { ( \mathrm { kr } + 1 ) ( \mathrm { kr } - \mathrm { k } + 1 ) }$.
Find also $\sum _ { \mathrm { r } = \mathrm { n } } ^ { \mathrm { n } ^ { 2 } } \frac { 1 } { ( \mathrm { kr } + 1 ) ( \mathrm { kr } - \mathrm { k } + 1 ) }$ in terms of $n$ and $k$.

CAIE Further Paper 1 2023 June Q4

4 The matrix $\mathbf { M }$ is given by $\mathbf { M } = \left( \begin{array} { l l } \mathrm { a } & \mathrm { b } ^ { 2 }
\mathrm { c } ^ { 2 } & \mathrm { a } \end{array} \right)$, where $a , b , c$ are real constants and $b \neq 0$.

Show that $\mathbf { M }$ does not represent a rotation about the origin.
Find the equations of the invariant lines, through the origin, of the transformation in the $x - y$ plane represented by $\mathbf { M }$.
It is given that $\mathbf { M }$ represents the sequence of two transformations in the $x - y$ plane given by an enlargement, centre the origin, scale factor 5 followed by a shear, $x$-axis fixed, with $( 0,1 )$ mapped to $( 5,1 )$.
Find $\mathbf { M }$.
The triangle $D E F$ in the $x - y$ plane is transformed by $\mathbf { M }$ onto triangle $P Q R$. Given that the area of triangle $D E F$ is $12 \mathrm {~cm} ^ { 2 }$, find the area of triangle $P Q R$.

CAIE Further Paper 1 2023 June Q5

5 The curve $C$ has polar equation $r ^ { 2 } = \frac { 1 } { \theta ^ { 2 } + 1 }$, for $0 \leqslant \theta \leqslant \pi$.

Sketch $C$ and state the polar coordinates of the point of $C$ furthest from the pole.
Find the area of the region enclosed by $C$, the initial line, and the half-line $\theta = \pi$.
Show that, at the point of $C$ furthest from the initial line, $$\left( \theta + \frac { 1 } { \theta } \right) \cot \theta - 1 = 0$$ and verify that this equation has a root between 1.1 and 1.2.

CAIE Further Paper 1 2023 June Q6

6 The curve $C$ has equation $\mathrm { y } = \frac { \mathrm { x } ^ { 2 } + 2 \mathrm { x } - 15 } { \mathrm { x } - 2 }$.

Find the equations of the asymptotes of $C$.
Show that $C$ has no stationary points.
Sketch $C$, stating the coordinates of the intersections with the axes.
Sketch the curve with equation $\mathrm { y } = \left| \frac { \mathrm { x } ^ { 2 } - 2 \mathrm { x } - 15 } { \mathrm { x } - 2 } \right|$.
Find the set of values of $x$ for which $\left| \frac { 2 x ^ { 2 } + 4 x - 30 } { x - 2 } \right| < 15$.

CAIE Further Paper 1 2023 June Q7

7 The plane $\Pi _ { 1 }$ has equation $r = - 4 \mathbf { j } - 3 \mathbf { k } + \lambda ( \mathbf { i } - \mathbf { j } + \mathbf { k } ) + \mu ( \mathbf { i } + \mathbf { j } - \mathbf { k } )$.

Obtain an equation of $\Pi _ { 1 }$ in the form $\mathrm { px } + \mathrm { qy } + \mathrm { rz } = \mathrm { d }$.
The plane $\Pi _ { 2 }$ has equation $\mathbf { r } . ( - 5 \mathbf { i } + 3 \mathbf { j } + 5 \mathbf { k } ) = 4$. Find a vector equation of the line of intersection of $\Pi _ { 1 }$ and $\Pi _ { 2 }$.
The line $l$ passes through the point $A$ with position vector $a \mathbf { i } + a \mathbf { j } + ( a - 7 ) \mathbf { k }$ and is parallel to $( 1 - b ) \mathbf { i } + b \mathbf { j } + b \mathbf { k }$, where $a$ and $b$ are positive constants.
Given that the perpendicular distance from $A$ to $\Pi _ { 1 }$ is $\sqrt { 2 }$, find the value of $a$.
Given that the obtuse angle between $l$ and $\Pi _ { 1 }$ is $\frac { 3 } { 4 } \pi$, find the exact value of $b$.
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 June Q1

1 Prove by mathematical induction that, for all positive integers $n , 5 ^ { 3 n } + 32 ^ { n } - 33$ is divisible by 31 .

CAIE Further Paper 1 2023 June Q2

Use standard results from the list of formulae (MF19) to show that $$\sum _ { r = 1 } ^ { n } \left( 6 r ^ { 2 } + 6 r - 5 \right) = a n ^ { 3 } + b n ^ { 2 } + c n$$ where $a$, $b$ and $c$ are integers to be determined.
Use the method of differences to find $\sum _ { r = 1 } ^ { n } \frac { 6 r ^ { 2 } + 6 r - 5 } { r ^ { 2 } + r }$ in terms of $n$.
Find also $\sum _ { r = n + 1 } ^ { 2 n } \frac { 6 r ^ { 2 } + 6 r - 5 } { r ^ { 2 } + r }$ in terms of $n$.

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