Questions - CAIE Further Paper 1

Find the equations of the asymptotes of $C$.
Find the coordinates of any stationary points on $C$.
Sketch $C$.
Find the coordinates of any stationary points on the curve with equation $\mathrm { y } = \frac { 1 } { \mathrm { f } ( \mathrm { x } ) }$.
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

1 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 $( k , 1 )$.

Show that $\mathbf { M } = \left( \begin{array} { c c } k & k
0 & 1 \end{array} \right)$.
The transformation represented by $\mathbf { M }$ has a line of invariant points. Find, in terms of $k$, the equation of this line.
\includegraphics[max width=\textwidth, alt={}, center]{99ac7fe2-184b-4a72-89f6-03fe4b2af138-02_2722_43_107_2005} The unit square $S$ in the $x - y$ plane is transformed by $\mathbf { M }$ onto the parallelogram $P$.
Find, in terms of $k$, a matrix which transforms $P$ onto $S$.
Given that the area of $P$ is $3 k ^ { 2 }$ units ${ } ^ { 2 }$, find the possible values of $k$.

CAIE Further Paper 1 2024 November Q2

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 $n - 1$.
\includegraphics[max width=\textwidth, alt={}, center]{99ac7fe2-184b-4a72-89f6-03fe4b2af138-05_2726_35_97_20}

CAIE Further Paper 1 2024 November Q3

3 The quartic equation $x ^ { 4 } + 2 x ^ { 3 } - 1 = 0$ has roots $\alpha , \beta , \gamma , \delta$.

Find a quartic equation whose roots are $\alpha ^ { 4 } , \beta ^ { 4 } , \gamma ^ { 4 } , \delta ^ { 4 }$ and state the value of $\alpha ^ { 4 } + \beta ^ { 4 } + \gamma ^ { 4 } + \delta ^ { 4 }$.
Find the value of $\alpha ^ { 5 } + \beta ^ { 5 } + \gamma ^ { 5 } + \delta ^ { 5 }$.
Find the value of $\alpha ^ { 8 } + \beta ^ { 8 } + \gamma ^ { 8 } + \delta ^ { 8 }$.

CAIE Further Paper 1 2024 November Q4

Use the method of differences to find $\sum _ { r = 1 } ^ { n } \frac { 5 k } { ( 5 r + k ) ( 5 r + 5 + k ) }$ in terms of $n$ and $k$, where $k$ is a positive constant.
\includegraphics[max width=\textwidth, alt={}, center]{99ac7fe2-184b-4a72-89f6-03fe4b2af138-08_2720_35_109_2010}
\includegraphics[max width=\textwidth, alt={}, center]{99ac7fe2-184b-4a72-89f6-03fe4b2af138-09_2723_35_101_20} It is given that $\sum _ { r = 1 } ^ { \infty } \frac { 5 k } { ( 5 r + k ) ( 5 r + 5 + k ) } = \frac { 1 } { 3 }$.
Find the value of $k$.
Hence find $\sum _ { r = n } ^ { n ^ { 2 } } \frac { 5 k } { ( 5 r + k ) ( 5 r + 5 + k ) }$ in terms of $n$.
$$\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 6 x y$$ has polar equation $r ^ { 2 } = 3 \sin 2 \theta$.
The curve $C$ has polar equation $r ^ { 2 } = 3 \sin 2 \theta$, for $0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi$.
Sketch $C$ and state the maximum distance of a point on $C$ from the pole.
\includegraphics[max width=\textwidth, alt={}, center]{99ac7fe2-184b-4a72-89f6-03fe4b2af138-10_2716_35_108_2012}
Find the area of the region enclosed by $C$.
Find the maximum distance of a point on $C$ from the initial line.

CAIE Further Paper 1 2024 November Q6

6 The curve $C$ has equation $y = \frac { 4 x ^ { 2 } + x + 1 } { 2 x ^ { 2 } - 7 x + 3 }$.

Find the equations of the asymptotes of $C$.
Find the coordinates of any stationary points on $C$.
\includegraphics[max width=\textwidth, alt={}, center]{99ac7fe2-184b-4a72-89f6-03fe4b2af138-13_2720_40_106_18}
Sketch $C$, stating the coordinates of any intersections with the axes.
Sketch the curve with equation $y = \left| \frac { 4 x ^ { 2 } + x + 1 } { 2 x ^ { 2 } - 7 x + 3 } \right|$ and state the set of values of $k$ for which $\left| \frac { 4 x ^ { 2 } + x + 1 } { 2 x ^ { 2 } - 7 x + 3 } \right| = k$ has 4 distinct real solutions.

CAIE Further Paper 1 2024 November Q7

7 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 $l _ { 2 }$.

Find the equation of $\Pi _ { 1 }$, giving your answer in the form $a x + b y + c z = d$.
The plane $\Pi _ { 2 }$ contains $l _ { 2 }$ and the point with coordinates $( 2 , - 1,7 )$.
Find the acute angle between $\Pi _ { 1 }$ and $\Pi _ { 2 }$.
\includegraphics[max width=\textwidth, alt={}, center]{99ac7fe2-184b-4a72-89f6-03fe4b2af138-15_2723_35_101_20} 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 $l _ { 2 }$.
Find a vector equation for $P Q$.
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

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

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 $2 \mathbf { i } + 5 \mathbf { j } - 4 \mathbf { k }$.

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 $5 \mathbf { i } - 5 \mathbf { j } - 2 \mathbf { k }$.
Find the acute angle between $l _ { 2 }$ and $\Pi$.

CAIE Further Paper 1 2024 November Q3

3 It is given that $$\begin{aligned} & \alpha + \beta + \gamma + \delta = 2
& \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 } = 3
& \alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } + \delta ^ { 3 } = 4 \end{aligned}$$

Find the value of $\alpha \beta + \alpha \gamma + \alpha \delta + \beta \gamma + \beta \delta + \gamma \delta$.
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}
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

4 The matrices $\mathbf { A } , \mathbf { B }$ and $\mathbf { C }$ are given by $$\mathbf { A } = \left( \begin{array} { l l l } 1 & 2 & 3
2 & 1 & 3
3 & 2 & 5 \end{array} \right) , \mathbf { B } = \left( \begin{array} { r r } 0 & - 2
- 1 & 3
0 & 0 \end{array} \right) \text { and } \mathbf { C } = \left( \begin{array} { r r r } - 2 & - 1 & 1
1 & 1 & 3 \end{array} \right)$$

Show that $\mathbf { C A B } = \left( \begin{array} { r r } 3 & - 7
- 9 & 3 \end{array} \right)$.
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 $\mathbf { M } = \left( \begin{array} { l l } 3 & 0
0 & 1 \end{array} \right)$.
Give full details of the transformation represented by $\mathbf { M }$.
Find the matrix $\mathbf { N }$ such that $\mathbf { N M } = \mathbf { C A B }$.

CAIE Further Paper 1 2024 November Q5

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 $x > 0$.

Find $S _ { n }$ in terms of $n , x$ and the function f .
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}
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

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

Show that $C$ has no vertical asymptotes and state the equation of the horizontal asymptote.
Show that $1 < y \leqslant 3$ for all real values of $x$.
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}
Sketch $C$, stating the coordinates of any intersections with the axes and labelling the asymptote.
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

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.

Find a Cartesian equation for $C _ { 1 }$ and show that it represents a circle, stating its radius and the Cartesian coordinates of its centre.
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 $( a \phi , \phi )$.
Verify that $1.25 < \phi < 1.26$.
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 2024 November Q1

Show that $\mathbf { M } = \left( \begin{array} { c c } k & k
0 & 1 \end{array} \right)$.
The transformation represented by $\mathbf { M }$ has a line of invariant points. Find, in terms of $k$, the equation of this line.
The unit square $S$ in the $x - y$ plane is transformed by $\mathbf { M }$ onto the parallelogram $P$.
Find, in terms of $k$, a matrix which transforms $P$ onto $S$.
Given that the area of $P$ is $3 k ^ { 2 }$ units ${ } ^ { 2 }$, find the possible values of $k$.

CAIE Further Paper 1 2024 November Q2

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 $n - 1$.
\includegraphics[max width=\textwidth, alt={}, center]{beb69d0f-3f83-49bf-b9a2-329ddc7243fa-04_2718_42_107_2007}
\includegraphics[max width=\textwidth, alt={}, center]{beb69d0f-3f83-49bf-b9a2-329ddc7243fa-05_2726_35_97_20}

CAIE Further Paper 1 2024 November Q4

Use the method of differences to find $\sum _ { r = 1 } ^ { n } \frac { 5 k } { ( 5 r + k ) ( 5 r + 5 + k ) }$ in terms of $n$ and $k$, where $k$ is a positive constant.
\includegraphics[max width=\textwidth, alt={}, center]{beb69d0f-3f83-49bf-b9a2-329ddc7243fa-08_2715_35_110_2012}
\includegraphics[max width=\textwidth, alt={}, center]{beb69d0f-3f83-49bf-b9a2-329ddc7243fa-09_2723_35_101_20} It is given that $\sum _ { r = 1 } ^ { \infty } \frac { 5 k } { ( 5 r + k ) ( 5 r + 5 + k ) } = \frac { 1 } { 3 }$.
Find the value of $k$.
Hence find $\sum _ { r = n } ^ { n ^ { 2 } } \frac { 5 k } { ( 5 r + k ) ( 5 r + 5 + k ) }$ in terms of $n$.
$$\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 6 x y$$ has polar equation $r ^ { 2 } = 3 \sin 2 \theta$.
The curve $C$ has polar equation $r ^ { 2 } = 3 \sin 2 \theta$, for $0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi$.
Sketch $C$ and state the maximum distance of a point on $C$ from the pole.
\includegraphics[max width=\textwidth, alt={}, center]{beb69d0f-3f83-49bf-b9a2-329ddc7243fa-10_2716_35_108_2012}
Find the area of the region enclosed by $C$.
Find the maximum distance of a point on $C$ from the initial line.

CAIE Further Paper 1 2024 November Q6

6 The curve $C$ has equation $y = \frac { 4 x ^ { 2 } + x + 1 } { 2 x ^ { 2 } - 7 x + 3 }$.

Find the equations of the asymptotes of $C$.
Find the coordinates of any stationary points on $C$.
\includegraphics[max width=\textwidth, alt={}, center]{beb69d0f-3f83-49bf-b9a2-329ddc7243fa-13_2720_40_106_18}
Sketch $C$, stating the coordinates of any intersections with the axes.
Sketch the curve with equation $y = \left| \frac { 4 x ^ { 2 } + x + 1 } { 2 x ^ { 2 } - 7 x + 3 } \right|$ and state the set of values of $k$ for which $\left| \frac { 4 x ^ { 2 } + x + 1 } { 2 x ^ { 2 } - 7 x + 3 } \right| = k$ has 4 distinct real solutions.

CAIE Further Paper 1 2024 November Q7

Find the equation of $\Pi _ { 1 }$, giving your answer in the form $a x + b y + c z = d$.
The plane $\Pi _ { 2 }$ contains $l _ { 2 }$ and the point with coordinates $( 2 , - 1,7 )$.
Find the acute angle between $\Pi _ { 1 }$ and $\Pi _ { 2 }$.
\includegraphics[max width=\textwidth, alt={}, center]{beb69d0f-3f83-49bf-b9a2-329ddc7243fa-15_2723_35_101_20} 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 $l _ { 2 }$.
Find a vector equation for $P Q$.
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

Gie it h $\mathrm { f } ( r ) = \frac { 1 } { ( r + 1 ) ( r + 2 ) } , \mathrm { s }$ th t $$\mathrm { f } \left( r - 1 \quad \mathrm { f } ( r ) = \frac { 2 } { r ( r + 1 ) ( r + 2 ) } \right.$$
Hen e fid $\sum _ { r = 1 } ^ { n } \frac { 1 } { r ( r + 1 ) ( r + 2 ) }$.
Ded e th le $6 \sum _ { r = 1 } ^ { \infty } \frac { 1 } { r ( r + 1 ) ( r + 2 ) }$.

CAIE Further Paper 1 2020 Specimen Q2

2 It is g $n$th $\mathrm { t } \phi ( n ) = 5 ^ { n } ( 4 n + 1 )$, ff $\quad \mathbf { o } \quad n = , \mathbb { B } .$.
Pro tyn ath matical id tin th $t \phi ( n )$ is $\dot { \mathbf { d } } \dot { \mathbf { v } }$ sib eff $\quad \mathbf { o }$ eyp itie in eg $r n$.

CAIE Further Paper 1 2020 Specimen Q3

3 Th cn $C \mathbf { h }$ sp areq tin $r = 2 + 2$ co $\theta$, fo $0 \leqslant \theta \leqslant \pi$.

Sk tch $C$.
Fid $b$ area 6 th reg œ $n$ lo edy $C$ ad $b$ in tial lie .
Sh the the Cartesiare q tim $C$ carb essed $\mathrm { s } \left( 4 x ^ { 2 } + y ^ { 2 } \right) = \left( x ^ { 2 } + y ^ { 2 } - 2 x \right) ^ { 2 }$. $\quad [ \beta$

CAIE Further Paper 1 2020 Specimen Q4

4 Th cb c ę tin $$z ^ { 3 } - z ^ { 2 } - z - 5 = 0$$ h s ro $\mathrm { s } \alpha , \beta$ ad $\gamma$.

Sth th t th le $6 \alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 }$ is 9
Fid he le $6 \alpha ^ { 4 } + \beta ^ { 4 } + \gamma ^ { 4 }$.
Fird cb ceq tin $N$ ith o s $\alpha + 1 \beta + 1 \mathrm {~d} \gamma + \underset { \text { g } } { \text { vg } } \quad$ as wer in th fo m $$p x ^ { 3 } + q x ^ { 2 } + r x + s = 0$$ we re $p , q , r$ ad $s$ are co tan s to $\mathbf { b } \quad \mathbf { d }$ termin d

CAIE Further Paper 1 2020 Specimen Q5

5 Th matrix $\mathbf { A }$ is g it $$\mathbf { A } = \left( \begin{array} { r r } 5 & k
- 3 & - 4 \end{array} \right)$$

Fid b le $6 k$ fo wh ch $\mathbf { A }$ is sig ar. It is $\mathbf { M } \quad \mathbf { g }$ vert h $\mathrm { t } k = 6$ d $\mathbf { h } \mathrm { t } \mathbf { A } = \left( \begin{array} { r r } 5 & 6
- 3 & - 4 \end{array} \right)$.
Fid th eq tim 6 th in rian lies s, th g th o ign 6 th tras fo matin in th $x - y \mathrm { p }$ ae rep esen edy $\mathbf { A }$.
Th triag e $D E F$ in b $x - y$ p ae is tras fo medy An a riag e $P Q R$.
1. Gie it $\mathbf { h }$ th area 6 triag e $D E F$ is $\mathbb { I } \mathrm { cm } ^ { 2 }$, f id $\mathbf { b }$ area 6 triag e $P Q R$.
2. Find b matrixw h cht ras fo ms triag e $P Q R$ b d riag e $D E F$.

CAIE Further Paper 1 2020 Specimen Q6

6 Th p itim ctosg th $\dot { \mathrm { p } }$ ns $A , B , C , D$ are $$2 \mathbf { i } + 4 \mathbf { j } - 3 \mathbf { k } , \quad - 2 \mathbf { i } + 5 \mathbf { j } - 4 \mathbf { k } , \quad \mathbf { i } + 4 \mathbf { j } + \mathbf { k } , \quad \mathbf { i } + 5 \mathbf { j } + m \mathbf { k }$$ resp ctively, wh re $m$ is an in eg r. It is g n th the sh test d stan e b tween th lie th g $\quad A$ ad $B$ ad lie th g $\quad C$ ad $D$ is 3

Shat that to b sib ey le $6 m$ is 2
Fid b sh test il stan e $6 D$ frm th lie thrg $\mathrm { h } A$ ad $C$.
Swat the actu eas eb tweert b p an $\mathrm { s } A C D$ ad $B C D$ is co ${ } ^ { - 1 } \left( \frac { 1 } { \sqrt { 3 } } \right)$.

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