Questions - SPS - A-Level Maths

SPS SPS ASFM 2020 May Q1

1. You are given that $z = 3 - 4 \mathrm { i }$.

Find
- $| z |$,
- $\arg ( z )$,
- $z ^ { * }$.
On an Argand diagram the complex number $w$ is represented by the point $A$ and $w ^ { * }$ is represented by the point $B$.
Describe the geometrical relationship between the points $A$ and $B$.

SPS SPS ASFM 2020 May Q2

2. The position vector of point $A$ is $\mathbf { a } = - 9 \mathbf { i } + 2 \mathbf { j } + 6 \mathbf { k }$.
The line $l$ passes through $A$ and is perpendicular to $\mathbf { a }$.

Determine the shortest distance between the origin, $O$, and $l$.
$l$ is also perpendicular to the vector $\mathbf { b }$ where $\mathbf { b } = - 2 \mathbf { i } + \mathbf { j } + \mathbf { k }$.
Find a vector which is perpendicular to both $\mathbf { a }$ and $\mathbf { b }$.
Write down an equation of $l$ in vector form.
$P$ is a point on $l$ such that $P A = 2 O A$.
Find angle $P O A$ giving your answer to 3 significant figures.
$C$ is a point whose position vector, $\mathbf { c }$, is given by $\mathbf { c } = p \mathbf { a }$ for some constant $p$. The line $m$ passes through $C$ and has equation $\mathbf { r } = \mathbf { c } + \mu \mathbf { b }$. The point with position vector $9 \mathbf { i } + 8 \mathbf { j } - 12 \mathbf { k }$ lies on $m$.
Find the value of $p$.

SPS SPS ASFM 2020 May Q3

3. \section*{In this question you must show detailed reasoning.} You are given that $\mathrm { f } ( z ) = 4 z ^ { 4 } - 12 z ^ { 3 } + 41 z ^ { 2 } - 128 z + 185$ and that $2 + \mathrm { i }$ is a root of the equation $\mathrm { f } ( z ) = 0$.

Express $\mathrm { f } ( z )$ as the product of two quadratic factors with integer coefficients.
Solve $\mathrm { f } ( z ) = 0$. Two loci on an Argand diagram are defined by $C _ { 1 } = \left\{ z : | z | = r _ { 1 } \right\}$ and $C _ { 2 } = \left\{ z : | z | = r _ { 2 } \right\}$ where $r _ { 1 } > r _ { 2 }$. You are given that two of the points representing the roots of $\mathrm { f } ( z ) = 0$ are on $C _ { 1 }$ and two are on $C _ { 2 } . R$ is the region on the Argand diagram between $C _ { 1 }$ and $C _ { 2 }$.
Find the exact area of $R$.
$\omega$ is the sum of all the roots of $\mathrm { f } ( z ) = 0$. Determine whether or not the point on the Argand diagram which represents $\omega$ lies in $R$.

SPS SPS ASFM 2020 May Q4

4. \section*{In this question you must show detailed reasoning.} You are given that $\alpha , \beta$ and $\gamma$ are the roots of the equation $5 x ^ { 3 } - 2 x ^ { 2 } + 3 x + 1 = 0$.

Find the value of $\alpha ^ { 2 } \beta ^ { 2 } + \beta ^ { 2 } \gamma ^ { 2 } + \gamma ^ { 2 } \alpha ^ { 2 }$.
Find a cubic equation whose roots are $\alpha ^ { 2 } , \beta ^ { 2 }$ and $\gamma ^ { 2 }$ giving your answer in the form $a x ^ { 3 } + b x ^ { 2 } + c x + d = 0$ where $a , b , c$ and $d$ are integers.

SPS SPS ASFM 2020 May Q5

5. A transformation T is represented by the matrix $\mathbf { T }$ where $\mathbf { T } = \left( \begin{array} { c c } x ^ { 2 } + 1 & - 4
3 - 2 x ^ { 2 } & x ^ { 2 } + 5 \end{array} \right)$.
A quadrilateral $Q$, whose area is 12 units, is transformed by T to $Q ^ { \prime }$. Find the smallest possible value of the area of $Q ^ { \prime }$.

SPS SPS ASFM 2020 May Q6

6. In this question you must show detailed reasoning.
$\mathbf { M }$ is the matrix $\left( \begin{array} { l l } 1 & 6
0 & 2 \end{array} \right)$.
Prove that $\mathbf { M } ^ { n } = \left( \begin{array} { c c } 1 & 3 \left( 2 ^ { n + 1 } - 2 \right)
0 & 2 ^ { n } \end{array} \right)$, for any positive integer $n$.

SPS SPS ASFM 2020 May Q7

7.

\includegraphics[max width=\textwidth, alt={}]{4718e448-71e5-452a-a905-608331f743a5-5_520_410_429_228}

A smooth wire is shaped into a circle of radius 2.5 m which is fixed in a vertical plane with its centre at a point $O$. A small bead $B$ is threaded onto the wire. $B$ is held with $O B$ vertical and is then projected horizontally with an initial speed of $8.4 \mathrm {~ms} ^ { - 1 }$ (see diagram).

Find the speed of $B$ at the instant when $O B$ makes an angle of 0.8 radians with the downward vertical through $O$.
Determine whether $B$ has sufficient energy to reach the point on the wire vertically above $O$.
\includegraphics[max width=\textwidth, alt={}, center]{4718e448-71e5-452a-a905-608331f743a5-6_675_412_296_221} As shown in the diagram, $A B$ is a long thin rod which is fixed vertically with $A$ above $B$. One end of a light inextensible string of length 1 m is attached to $A$ and the other end is attached to a particle $P$ of mass $m _ { 1 } \mathrm {~kg}$. One end of another light inextensible string of length 1 m is also attached to $P$. Its other end is attached to a small smooth ring $R$, of mass $m _ { 2 } \mathrm {~kg}$, which is free to move on $A B$. Initially, $P$ moves in a horizontal circle of radius 0.6 m with constant angular velocity $\omega \mathrm { rad } \mathrm { s } ^ { - 1 }$. The magnitude of the tension in string $A P$ is denoted by $T _ { 1 } \mathrm {~N}$ while that in string $P R$ is denoted by $T _ { 2 } \mathrm {~N}$.
By considering forces on $R$, express $T _ { 2 }$ in terms of $m _ { 2 }$.
Show that
1. $T _ { 1 } = \frac { 49 } { 4 } \left( m _ { 1 } + m _ { 2 } \right)$,
2. $\omega ^ { 2 } = \frac { 49 \left( m _ { 1 } + 2 m _ { 2 } \right) } { 4 m _ { 1 } }$.
Deduce that, in the case where $m _ { 1 }$ is much bigger than $m _ { 2 } , \omega \approx 3.5$. In a different case, where $m _ { 1 } = 2.5$ and $m _ { 2 } = 2.8 , P$ slows down. Eventually the system comes to rest with $P$ and $R$ hanging in equilibrium.
Find the total energy lost by $P$ and $R$ as the angular velocity of $P$ changes from the initial value of $\omega \mathrm { rads } ^ { - 1 }$ to zero. Three particles, $P , Q$ and $R$, are at rest on a smooth horizontal plane. The particles lie along a straight line with $Q$ between $P$ and $R$. The particles $Q$ and $R$ have masses $m$ and $k m$ respectively, where $k$ is a constant. Particle $Q$ is projected towards $R$ with speed $u$ and the particles collide directly.
The coefficient of restitution between each pair of particles is $e$.
Find, in terms of $e$, the range of values of $k$ for which there is a second collision. Given that the mass of $P$ is $k m$ and that there is a second collision,
write down, in terms of $u , k$ and $e$, the speed of $Q$ after this second collision.

SPS SPS ASFM 2020 May Q10

10. On any day, the number of orders received in one randomly chosen hour by an online supplier can be modelled by the distribution $\operatorname { Po } ( 120 )$.

Find the probability that at least 28 orders are received in a randomly chosen 10 -minute period.
Find the probability that in a randomly chosen 10 -minute period on one day and a randomly chosen 10-minute period on the next day a total of at least 56 orders are received.
State a necessary assumption for the validity of your calculation in part (b).

SPS SPS ASFM 2020 May Q11

11. The members of a team stand in a random order in a straight line for a photograph. There are four men and six women.

Find the probability that all the men are next to each other.
Find the probability that no two men are next to one another.

SPS SPS ASFM 2020 May Q12

12. Alex claims that he can read people's minds. A volunteer, Jane, arranges the integers 1 to $n$ in an order of Jane's own choice and Alex tells Jane what order he believes was chosen. They agree that Alex's claim will be accepted if he gets the order completely correct or if he gets the order correct apart from two numbers which are the wrong way round. They use a value of $n$ such that, if Alex chooses the order of the integers at random, the probability that Alex's claim will be accepted is less than $1 \%$. Determine the smallest possible value of $n$.

SPS SPS FM 2019 Q2

Find the coefficient of the $x ^ { 4 }$ term in $( 2 - 3 x ) ^ { 6 }$.
A sequence $u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots$ is defined by $u _ { n } = 3 n - 1$, for $n \geq 1$.
1. Find the values of $u _ { 1 } , u _ { 2 } , u _ { 3 }$.
2. Find
$$\sum _ { n = 1 } ^ { 40 } u _ { n }$$

SPS SPS FM 2019 Q4

Show that

$$\log _ { a } \left( x ^ { 10 } \right) - 2 \log _ { a } \left( \frac { x ^ { 3 } } { 4 } \right) = 4 \log _ { a } ( 2 x )$$

SPS SPS FM 2019 Q5

Solve the following inequalities giving your answer in set notation:
1. $\quad | 4 x + 3 | < | x - 8 |$
2. $\quad \frac { x } { x ^ { 2 } + 1 } < \frac { 1 } { 2 }$
3. If $a$ and $b$ are odd integers such that 4 is a factor of ( $a - b$ ), prove by contradiction that 4 cannot be a factor of $( a + b )$.
\includegraphics[max width=\textwidth, alt={}]{e41e06f6-7d0a-496a-aa1e-b2dcd787d72c-2_568_608_1749_683}
The diagram shows a circle which passes through the points $A ( 2,9 )$ and $B ( 10,3 )$. $A B$ is a diameter of the circle.
The tangent to the circle at the point $B$ cuts the $x$-axis at $C$. Find the exact coordinates of $C$.
Find the exact area of the triangle formed by $B , C$ and the centre of the circle

SPS SPS FM 2019 Q8

8. Sketch the curve $y = 2 ^ { 2 x + 3 }$, stating the coordinates of any points of intersection with the axes. The point $P$ on the curve $y = 3 ^ { 3 x + 2 }$ has $y$-coordinate equal to 180 .
Use logarithms to find the $x$-coordinate of $P$ correct to 3 significant figures. The curves $y = 2 ^ { 2 x + 3 }$ and $y = 3 ^ { 3 x + 2 }$ intersect at the point $Q$.
Show that the $x$-coordinate of $Q$ can be written as $$x = \frac { 3 \log _ { 3 } 2 - 2 } { 3 - 2 \log _ { 3 } 2 }$$

SPS SPS FM 2019 Q9

(a) Given that $u _ { n + 1 } = 5 u _ { n } + 4 , u _ { 1 } = 4$, prove by induction that $u _ { n } = 5 ^ { n } - 1$.
(b) For all positive integers, $n \geq 2$, prove by induction that

$$\sum _ { r = 2 } ^ { n } r ^ { 2 } ( r - 1 ) = \frac { 1 } { 12 } n ( n - 1 ) ( n + 1 ) ( 3 n + 2 )$$

SPS SPS FM 2019 Q10

Show that, for any value of the real constant $b$, the equation $x ^ { 3 } - ( b + 1 ) x + b = 0$ has $x = 1$ as a solution.

Find all values of $b$ for which this equation has exactly two real solutions \section*{11. In the question you must show detailed reasoning} Given that the coefficients of $x , x ^ { 2 }$ and $x ^ { 4 }$ in the expansion of $( 1 + k x ) ^ { n }$ are the consecutive terms of a geometric series, where $n \geq 4$ and $k$ is a positive constant

Show that $k = \frac { 6 ( n - 1 ) } { ( n - 2 ) ( n - 3 ) }$
For the case when $k = \frac { 7 } { 5 }$, find the value of $n$.
Given that $= \frac { 7 } { 5 } , n$ is a positive integer, and that the first term of the geometric series is the coefficient of $x$, find the number of terms required for the sum of the geometric series to exceed $1.12 \times 10 ^ { 12 }$. \section*{12. In the question you must show detailed reasoning} Given that $\log _ { a } x = \frac { \log _ { b } x } { \log _ { b } a }$ show that the sum of the infinite series, where $n = 0,1,2 \ldots$, $$\log _ { 2 } e - \log _ { 4 } e + \log _ { 16 } e - \cdots + ( - 1 ) ^ { n } \log _ { 2 ^ { 2 } } n e + \cdots$$ is $$\frac { 1 } { \ln ( 2 \sqrt { 2 } ) }$$ \section*{Advanced GCE (H245)} \section*{Further Mathematics A} \section*{Formulae Booklet} \section*{Pure Mathematics} \section*{Arithmetic series} $S _ { n } = \frac { 1 } { 2 } n ( a + l ) = \frac { 1 } { 2 } n \{ 2 a + ( n - 1 ) d \}$ \section*{Geometric series} $S _ { n } = \frac { a \left( 1 - r ^ { n } \right) } { 1 - r }$
$S _ { \infty } = \frac { a } { 1 - r }$ for $| r | < 1$ \section*{Binomial series} $( a + b ) ^ { n } = a ^ { n } + { } ^ { n } \mathrm { C } _ { 1 } a ^ { n - 1 } b + { } ^ { n } \mathrm { C } _ { 2 } a ^ { n - 2 } b ^ { 2 } + \ldots + { } ^ { n } \mathrm { C } _ { r } a ^ { n - r } b ^ { r } + \ldots + b ^ { n } \quad ( n \in \mathbb { N } )$,
where ${ } ^ { n } \mathrm { C } _ { r } = { } _ { n } \mathrm { C } _ { r } = \binom { n } { r } = \frac { n ! } { r ! ( n - r ) ! }$ $$( 1 + x ) ^ { n } = 1 + n x + \frac { n ( n - 1 ) } { 2 ! } x ^ { 2 } + \ldots + \frac { n ( n - 1 ) \ldots ( n - r + 1 ) } { r ! } x ^ { r } + \ldots \quad ( | x | < 1 , n \in \mathbb { R } )$$

SPS SPS FM 2020 December Q1

Solve $2 \sin x = \tan x$ exactly, where $- \frac { \pi } { 2 } < x < \frac { \pi } { 2 }$.
Let $a , b$ satisfy $0 < a < b$.
i. Find, in terms of $a$ and $b$, the value of

$$\int _ { a } ^ { b } \frac { 81 } { x ^ { 4 } } d x$$ ii. Explaining clearly any limiting processes used, find the value of $a$, given that $$\int _ { a } ^ { \infty } \frac { 81 } { x ^ { 4 } } d x = \frac { 216 } { 125 }$$

SPS SPS FM 2020 December Q3

i. Sketch the graph of $y = | 3 x - 1 |$.
ii. Hence, solve $5 x + 3 < | 3 x - 1 |$.
The following diagram shows the curve $y = a \sin ( b ( x + c ) ) + d$, where $a , b , c$ and $d$ are all positive constants and $x$ is measured in radians. The curve has a maximum point at $( 1,3.5 )$ and a minimum point at $( 2,0.5 )$.
\includegraphics[max width=\textwidth, alt={}, center]{85570d22-3596-4795-93ed-bfe8d094e8b4-05_826_1109_269_532}
i) Write down the value of $a$ and the value of $d$.
ii) Find the value of $b$.
iii) Find the smallest possible value of $c$, given that $c > 0$.
The $2 \times 2$ matrix $\mathbf { A }$ represents a rotation by $90 ^ { \circ }$ anticlockwise about the origin. The $2 \times 2$ matrix $\mathbf { B }$ represents a reflection in the line $y = - x$.
The matrix $\mathbf { B }$ is given by

$$\mathbf { B } = \left( \begin{array} { c c } 0 & - 1
- 1 & 0 \end{array} \right)$$ i. Write down the matrix representing $\mathbf { A }$.
ii. The $2 \times 2$ matrix $\mathbf { C }$ represents a rotation by $90 ^ { \circ }$ anticlockwise about the origin, followed by a reflection in the line $y = - x$. Compute the matrix $\mathbf { C }$ and describe geometrically the single transformation represented by $\mathbf { C }$.

SPS SPS FM 2020 December Q6

6. Given that $z$ is the complex number $x + i y$ and satisfies $$| z | + z = 6 - 2 i$$ find the value of $x$ and the value of $y$.

SPS SPS FM 2020 December Q7

7. The diagram below shows part of a curve C with equation $y = 1 + 3 x - \frac { 1 } { 2 } x ^ { 2 }$.
\includegraphics[max width=\textwidth, alt={}, center]{85570d22-3596-4795-93ed-bfe8d094e8b4-08_709_898_214_603}
i. The curve crosses the $y$ axis at the point A . The straight line L is normal to the curve at A and meets the curve again at B . Find the equation of L and the $x$ coordinate of the point B .
ii. The region $R$ is bounded by the curve $C$ and the line $L$. Find the exact area of $R$.

SPS SPS FM 2020 December Q8

8.
a) The $2 \times 2$ matrix $\mathbf { A }$ is given by $$\mathbf { A } = \left( \begin{array} { l l } 7 & 3
2 & 1 \end{array} \right) .$$ The $2 \times 2$ matrix $\mathbf { B }$ satisfies $$\mathbf { B } \mathbf { A } ^ { 2 } = \mathbf { A } .$$ Find the matrix $\mathbf { B }$.
b) The $2 \times 2$ matrix $\mathbf { C }$ is given by $$\mathbf { C } = \left( \begin{array} { l l } - 2 & 4
- 1 & 2 \end{array} \right) .$$ By considering $\mathbf { C } ^ { 2 }$, show that the matrices $\mathbf { I } - \mathbf { C }$ and $\mathbf { I } + \mathbf { C }$ are inverse to each other.

SPS SPS FM 2020 December Q9

9. Sketch on an Argand diagram the locus of all points that satisfy $| z + 4 - 4 i | = 2 \sqrt { 2 }$ and hence find $\theta , \phi \in ( - \pi , \pi ]$ such that $\theta \leq \arg z \leq \phi$.

SPS SPS FM 2020 December Q10

10. The $2 \times 2$ matrix $\mathbf { M }$ is defined by $$\mathbf { M } = \left( \begin{array} { c c } 0 & 0.25
0.36 & 0 \end{array} \right)$$ Find, by calculation, the equations of the two lines that pass through the origin, that remain invariant under the transformation represented by $\mathbf { M }$.

SPS SPS FM 2020 December Q11

11. In the triangle $P Q R , P Q = 6 , P R = k , P \hat { Q } R = 30 ^ { \circ }$.
i. For the case $k = 4$, find the two possible values of $Q R$ exactly.
ii. Determine the value(s) of $k$ for which the conditions above define a unique triangle.

SPS SPS FM 2020 December Q12

12. Consider the binomial expansion of $\left( 1 + \frac { x } { 5 } \right) ^ { n }$ in ascending powers of $x$, where $n$ is a positive integer.
i. Write down the first four terms of the expansion, giving the coefficients as polynomials in $n$. The coefficients of the second, third and fourth terms of the expansion are consecutive terms of an arithmetic sequence.
ii. Show that $n ^ { 3 } - 33 n ^ { 2 } + 182 n = 0$.
iii. Hence find the possible values of $n$ and the corresponding values of the common difference.

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