Questions - SPS SPS FM Pure

Find the matrix which represents the stretch that maps triangle $T _ { 1 }$ onto triangle $T _ { 2 }$.
The triangle $T _ { 2 }$ is reflected in the line $y = \sqrt { 3 } x$ to give a third triangle, $T _ { 3 }$. Find, using surd forms where appropriate:
1. the matrix which represents the reflection that maps triangle $T _ { 2 }$ onto triangle $T _ { 3 }$;
2. the matrix which represents the combined transformation that maps triangle $T _ { 1 }$ onto triangle $T _ { 3 }$.
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SPS SPS FM Pure 2022 February Q1

(a) Express $\frac { 1 } { ( 2 r - 1 ) ( 2 r + 1 ) }$ in partial fractions.
(b) Hence find $\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 2 r - 1 ) ( 2 r + 1 ) }$, expressing the result as a single fraction.
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$$\mathbf { A } = \left( \begin{array} { r r } 4 & - 2
5 & 3 \end{array} \right)$$ The matrix $\mathbf { A }$ represents the linear transformation $M$.
Prove that, for the linear transformation $M$, there are no invariant lines.
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SPS SPS FM Pure 2022 February Q3

3. The line $l _ { 1 }$ has equation $\mathbf { r } = \left( \begin{array} { r } 1
- 3
3 \end{array} \right) + \lambda \left( \begin{array} { r } 3
2
- 2 \end{array} \right)$.
The plane $\Pi$ has equation $\mathbf { r } . \left( \begin{array} { r } 2
- 5
- 3 \end{array} \right) = 4$.

Find the position vector of the point of intersection of $l _ { 1 }$ and $\Pi$.
Find the acute angle between $l _ { 1 }$ and $\Pi$.
$A$ is the point on $l _ { 1 }$ where $\lambda = 1$.
$l _ { 2 }$ is the line with the following properties.
- $l _ { 2 }$ passes through $A$
- $l _ { 2 }$ is perpendicular to $l _ { 1 }$
- $l _ { 2 }$ is parallel to $\Pi$
- Find, in vector form, the equation of $l _ { 2 }$.
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SPS SPS FM Pure 2022 February Q4

1. $\mathbf { A }$ is a 2 by 2 matrix and $\mathbf { B }$ is a 2 by 3 matrix.

Giving a reason for your answer, explain whether it is possible to evaluate

$\mathbf { A B }$
$\mathbf { A } + \mathbf { B }$
(ii) Given that $$\left( \begin{array} { r r r } - 5 & 3 & 1
a & 0 & 0
b & a & b \end{array} \right) \left( \begin{array} { r r r } 0 & 5 & 0
2 & 12 & - 1
- 1 & - 11 & 3 \end{array} \right) = \lambda \mathbf { I }$$ where $a , b$ and $\lambda$ are constants,
determine
- the value of $\lambda$
- the value of $a$
- the value of $b$
- Hence deduce the inverse of the matrix $\left( \begin{array} { r r r } - 5 & 3 & 1
  a & 0 & 0
  b & a & b \end{array} \right)$
  (iii) Given that
$$\mathbf { M } = \left( \begin{array} { c c c } 1 & 1 & 1
0 & \sin \theta & \cos \theta
0 & \cos 2 \theta & \sin 2 \theta \end{array} \right) \quad \text { where } 0 \leqslant \theta < \pi$$ determine the values of $\theta$ for which the matrix $\mathbf { M }$ is singular.
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SPS SPS FM Pure 2022 February Q5

5. Points $A , B$ and $C$ have coordinates $( 4,2,0 ) , ( 1,5,3 )$ and $( 1,4 , - 2 )$ respectively. The line $l$ passes through $A$ and $B$.

Find a cartesian equation for $l$.
$M$ is the point on $l$ that is closest to $C$.
Find the coordinates of $M$.
Find the exact area of the triangle $A B C$.
[0pt] [BLANK PAGE] \section*{6. The curve $C$ has equation} $$r = a ( p + 2 \cos \theta ) \quad 0 \leqslant \theta < 2 \pi$$ where $a$ and $p$ are positive constants and $p > 2$
There are exactly four points on $C$ where the tangent is perpendicular to the initial line.
Show that the range possible values for $p$ is
Sketch the curve with equation $$r = a ( 3 + 2 \cos \theta ) \quad 0 \leqslant \theta < 2 \pi \quad \text { where } a > 0$$ John digs a hole in his garden in order to make a pond.
The pond has a uniform horizontal cross section that is modelled by the curve with equation $$r = 20 ( 3 + 2 \cos \theta ) \quad 0 \leqslant \theta < 2 \pi$$ where $r$ is measured in centimetres. The depth of the pond is 90 centimetres.
Water flows through a hosepipe into the pond at a rate of 50 litres per minute.
Given that the pond is initially empty,
determine how long it will take to completely fill the pond with water using the hosepipe, according to the model. Give your answer to the nearest minute.
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SPS SPS FM Pure 2022 February Q7

7. The matrix $\mathbf { M }$ is defined by $\mathbf { M } = \left[ \begin{array} { c c c } 3 & 2 & - 2
0 & 1 & 0
0 & 0 & 1 \end{array} \right]$
Prove by induction that $\mathbf { M } ^ { n } = \left[ \begin{array} { c c c } 3 ^ { n } & 3 ^ { n } - 1 & - 3 ^ { n } + 1
0 & 1 & 0
0 & 0 & 1 \end{array} \right]$ for all integers $n \geq 1$
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SPS SPS FM Pure 2022 February Q8

8. The complex number $z$ satisfies the equations $$\left| z ^ { * } - 1 - 2 i \right| = | z - 3 |$$ and $$| z - a | = 3$$ where $a$ is real.
Show that $a$ must lie in the interval $[ 1 - s \sqrt { t } , 1 + s \sqrt { t } ]$, where $s$ and $t$ are prime numbers.
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SPS SPS FM Pure 2022 February Q9

9. The equation $4 x ^ { 4 } - 4 x ^ { 3 } + p x ^ { 2 } + q x - 9 = 0$, where $p$ and $q$ are constants, has roots $\alpha , - \alpha , \beta$ and $\frac { 1 } { \beta }$.

Determine the exact roots of the equation.
Determine the values of $p$ and $q$.
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SPS SPS FM Pure 2022 February Q10

10. You are given that $\mathrm { f } ( x ) = 4 \sinh x + 3 \cosh x$.

Show that the curve $y = \mathrm { f } ( x )$ has no turning points.
Determine the exact solution of the equation $\mathrm { f } ( x ) = 5$.
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SPS SPS FM Pure 2022 February Q11

11. A particle $P$ of mass 2 kg can only move along the straight line segment $O A$, where $O A$ is on a rough horizontal surface. The particle is initially at rest at $O$ and the distance $O A$ is 0.9 m . When the time is $t$ seconds the displacement of $P$ from $O$ is $x \mathrm {~m}$ and the velocity of $P$ is $v \mathrm {~ms} ^ { - 1 }$. $P$ is subject to a force of magnitude $4 \mathrm { e } ^ { - 2 t } \mathrm {~N}$ in the direction of $A$ for any $t \geqslant 0$. The resistance to the motion of $P$ is modelled as being proportional to $v$. At the instant when $t = \ln 2 , v = 0.5$ and the resultant force on $P$ is 0 N .

Show that, according to the model, $\frac { \mathrm { d } v } { \mathrm {~d} t } + v = 2 \mathrm { e } ^ { - 2 t }$.
Find an expression for $v$ in terms of $t$ for $t \geqslant 0$.
By considering the behaviour of $v$ as $t$ becomes large explain why, according to the model, $P$ 's speed must reach a maximum value for some $t > 0$.
Determine the maximum speed considered in part (c).
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SPS SPS FM Pure 2022 February Q12

12. In this question you must show detailed reasoning.

By using an appropriate Maclaurin series prove that if $x > 0$ then $\mathrm { e } ^ { x } > 1 + x$.
Hence, by using a suitable substitution, deduce that $\mathrm { e } ^ { t } > \mathrm { e } t$ for $t > 1$.
Using the inequality in part (b), and by making a suitable choice for $t$, determine which is greater, $\mathrm { e } ^ { \pi }$ or $\pi ^ { \mathrm { e } }$.
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SPS SPS FM Pure 2023 November Q1

(i) Solve the equation

$$x \sqrt { 2 } - \sqrt { 18 } = x$$ writing the answer as a surd in simplest form.
(ii) Solve the equation $$4 ^ { 3 x - 2 } = \frac { 1 } { 2 \sqrt { 2 } }$$

SPS SPS FM Pure 2023 November Q2

2. $$\mathrm { f } ( x ) = x ^ { 3 } + 4 x ^ { 2 } + x - 6$$

Use the factor theorem to show that $( x + 2 )$ is a factor of $\mathrm { f } ( x )$.
Factorise f(x) completely.
Write down all the solutions to the equation $$x ^ { 3 } + 4 x ^ { 2 } + x - 6 = 0$$ [BLANK PAGE]

SPS SPS FM Pure 2023 November Q3

3. The curve $C$ has equation $$y = \frac { 1 } { 2 } x ^ { 3 } - 9 x ^ { \frac { 3 } { 2 } } + \frac { 8 } { x } + 30 , \quad x > 0$$

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
Show that the point $P ( 4 , - 8 )$ lies on $C$.
Find an equation of the normal to $C$ at the point $P$, giving your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers.
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SPS SPS FM Pure 2023 November Q4

4. A sequence of numbers is defined by $$\begin{aligned} u _ { 1 } & = 2
u _ { n + 1 } & = 5 u _ { n } - 4 , \quad n \geqslant 1 . \end{aligned}$$ Prove by induction that, for $n \in \mathbb { Z } ^ { + } , u _ { n } = 5 ^ { n - 1 } + 1$.
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SPS SPS FM Pure 2023 November Q5

5. (a) Show that the equation $$4 \cos \theta - 1 = 2 \sin \theta \tan \theta$$ can be written in the form $$6 \cos ^ { 2 } \theta - \cos \theta - 2 = 0$$ (b) Hence solve, for $0 \leqslant x < 90 ^ { \circ }$ $$4 \cos 3 x - 1 = 2 \sin 3 x \tan 3 x$$ giving your answers, where appropriate, to one decimal place.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
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SPS SPS FM Pure 2023 November Q6

6. Find the values of $x$ such that $$2 \log _ { 3 } x - \log _ { 3 } ( x - 2 ) = 2$$ [BLANK PAGE]

SPS SPS FM Pure 2023 November Q7

7. $$\mathrm { f } ( x ) = 2 x ^ { 2 } + 4 x + 9 \quad x \in \mathbb { R }$$

Write $\mathrm { f } ( x )$ in the form $a ( x + b ) ^ { 2 } + c$, where $a , b$ and $c$ are integers to be found.
Sketch the curve with equation $y = \mathrm { f } ( x )$ showing any points of intersection with the coordinate axes and the coordinates of any turning point.
1. Describe fully the transformation that maps the curve with equation $y = \mathrm { f } ( x )$ onto the curve with equation $y = \mathrm { g } ( x )$ where $$\mathrm { g } ( x ) = 2 ( x - 2 ) ^ { 2 } + 4 x - 3 \quad x \in \mathbb { R }$$
2. Find the range of the function $$\mathrm { h } ( x ) = \frac { 21 } { 2 x ^ { 2 } + 4 x + 9 } \quad x \in \mathbb { R }$$ [BLANK PAGE]

SPS SPS FM Pure 2023 November Q8

8. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{3dcde139-bc6b-412d-8d1f-c45543d67430-16_703_851_150_701} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a sketch of the graph with equation $$y = 2 | x + 4 | - 5$$ The vertex of the graph is at the point $P$, shown in Figure 2.

Find the coordinates of $P$.
Solve the equation $$3 x + 40 = 2 | x + 4 | - 5$$ A line $l$ has equation $y = a x$, where $a$ is a constant.
Given that $l$ intersects $y = 2 | x + 4 | - 5$ at least once,
find the range of possible values of $a$, writing your answer in set notation.
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SPS SPS FM Pure 2023 November Q9

9. In this question you must show all stages of your working. \section*{Solutions relying entirely on calculator technology are not acceptable.} A geometric series has common ratio $r$ and first term $a$.
Given $r \neq 1$ and $a \neq 0$

prove that $$S _ { n } = \frac { a \left( 1 - r ^ { n } \right) } { 1 - r }$$ Given also that $S _ { 10 }$ is four times $S _ { 5 }$
find the exact value of $r$.
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SPS SPS FM Pure 2023 November Q10

10. (a) Find the first 4 terms, in ascending powers of $x$, in the binomial expansion of $$( 1 + k x ) ^ { 10 }$$ where $k$ is a non-zero constant. Write each coefficient as simply as possible. Given that in the expansion of $( 1 + k x ) ^ { 10 }$ the coefficient $x ^ { 3 }$ is 3 times the coefficient of $x$, (b) find the possible values of $k$.
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SPS SPS FM Pure 2023 June Q1

You are given that $g f ( x ) = | 3 x - 1 |$ for $x \in \mathbb { R }$.
1. Given that $f ( x ) = 3 x - 1$, express $g ( x )$ in terms of $x$.
2. State the range of $g f ( x )$.
3. Solve the inequality $| 3 x - 1 | > 1$.
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\section*{2. In this question you must show detailed reasoning.}
Express $8 \cos x + 5 \sin x$ in the form $R \cos ( x - \alpha )$, where $R$ and $\alpha$ are constants with $R > 0$ and $0 < \alpha < \frac { 1 } { 2 } \pi$.
Hence solve the equation $8 \cos x + 5 \sin x = 6$ for $0 \leqslant x < 2 \pi$, giving your answers correct to 4 decimal places.
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SPS SPS FM Pure 2023 June Q3

3. You are given that $f ( x ) = \ln ( 2 x - 5 ) + 2 x ^ { 2 } - 30$, for $x > 2.5$.

Show that $f ( x ) = 0$ has a root $\alpha$ in the interval [3.5, 4]. A student takes 4 as the first approximation to $\alpha$.
Given $f ( 4 ) = 3.099$ and $f ^ { \prime } ( 4 ) = 16.67$ to 4 significant figures,
apply the Newton-Raphson procedure once to obtain a second approximation for $\alpha$, giving your answer to 3 significant figures.
Show that $\alpha$ is the only root of $f ( x ) = 0$.
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