Questions - SPS - A-Level Maths

Write down expressions for
1. $\mathrm { fg } ( x )$,
2. $\mathrm { gf } ( x )$.
Hence find the values of $x$ for which $\mathrm { fg } ( x ) - \mathrm { gf } ( x ) = 24$.
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SPS SPS FM 2023 October Q7

7. The seventh term of a geometric progression is equal to twice the fifth term. The sum of the first seven terms is 254 and the terms are all positive. Find the first term, showing that it can be written in the form $p + q \sqrt { r }$ where $p , q$ and $r$ are integers.
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SPS SPS FM 2023 October Q8

8. Prove that $2 ^ { 3 n } - 3 ^ { n }$ is divisible by 5 for all integers $n \geq 1$.
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SPS SPS FM 2023 October Q9

9. (a)
\includegraphics[max width=\textwidth, alt={}, center]{9c377549-1fbd-4790-b57e-37ec9707c9d8-20_540_529_157_278} The shape $A B C$ shown in the diagram is a student's design for the sail of a small boat. The curve $A C$ has equation $y = 2 \log _ { 2 } x$ and the curve $B C$ has equation $y = \log _ { 2 } \left( x - \frac { 3 } { 2 } \right) + 3$. State the $x$-coordinate of point $A$.
(b) Determine the $x$-coordinate of point $B$.
(c) By solving an equation involving logarithms, show that the $x$-coordinate of point $C$ is 2 . It is given that, correct to 3 significant figures, the area of the sail is 0.656 units $^ { 2 }$.
(d) Calculate by how much the area is over-estimated or under-estimated when the curved edges of the sail are modelled as straight lines.
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SPS SPS FM Pure 2024 January Q1

Fig. 6 shows the region enclosed by part of the curve $y = 2 x ^ { 2 }$, the straight line $x + y = 3$, and the $y$-axis. The curve and the straight line meet at $\mathrm { P } ( 1,2 )$.

\begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f4c02a9b-802e-4e51-94c2-d1c5d69855b5-04_552_806_287_625} \captionsetup{labelformat=empty} \caption{Fig. 6}

\end{figure} The shaded region is rotated through $360 ^ { \circ }$ about the $y$-axis. Find, in terms of $\pi$, the volume of the solid of revolution formed.
[0pt] [BLANK PAGE] \section*{2. a)} Find, in terms of $k$, the set of values of $x$ for which $$k - | 2 x - 3 k | > x - k$$ giving your answer in set notation.
b) Find, in terms of $k$, the coordinates of the minimum point of the graph with equation $$y = 3 - 5 \mathrm { f } \left( \frac { 1 } { 2 } x \right)$$ where $$\mathrm { f } ( x ) = k - | 2 x - 3 k |$$ [BLANK PAGE]

SPS SPS FM Pure 2024 January Q3

3. Find the value of the integral:
$\int _ { 0 } ^ { 1 } \frac { x ^ { \frac { 1 } { 3 } } + x ^ { - \frac { 1 } { 3 } } } { x } \mathrm {~d} x$.
(4 marks)
(Total 4 marks)
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SPS SPS FM Pure 2024 January Q4

4. The points $A$ and $B$ have position vectors $5 \mathbf { j } + 11 \mathbf { k }$ and $c \mathbf { i } + d \mathbf { j } + 21 \mathbf { k }$ respectively, where $c$ and $d$ are constants. The line $l$, through the points $A$ and $B$, has vector equation $\mathbf { r } = 5 \mathbf { j } + 11 \mathbf { k } + \lambda ( 2 \mathbf { i } + \mathbf { j } + 5 \mathbf { k } )$, where $\lambda$ is a parameter.

Find the value of $c$ and the value of $d$.
(3) The point $P$ lies on the line $l$, and $\overrightarrow { O P }$ is perpendicular to $l$, where $O$ is the origin.
Find the position vector of $P$.
(6)
Find the area of triangle $O A B$, giving your answer to 3 significant figures.
(4)
(Total 13 marks)
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SPS SPS FM Pure 2024 January Q5

5. Let $$f ( x ) = \frac { 27 x ^ { 2 } + 32 x + 16 } { ( 3 x + 2 ) ^ { 2 } ( 1 - x ) }$$

Express $f ( x )$ in terms of partial fractions
Hence, or otherwise, find the series expansion of $\mathrm { f } ( x )$, in ascending powers of $x$, up to and including the term in $x ^ { 2 }$. Simplify each term.
State, with a reason, whether your series expansion in part (b) is valid for $x = \frac { 1 } { 2 }$.
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SPS SPS FM Pure 2024 January Q6

6. $$\mathbf { M } = \left( \begin{array} { r r } - 2 & 5
6 & k \end{array} \right)$$ where $k$ is a constant.
Given that $$\mathbf { M } ^ { 2 } + 11 \mathbf { M } = a \mathbf { I }$$ where $a$ is a constant and $\mathbf { I }$ is the $2 \times 2$ identity matrix,

1. determine the value of $a$
2. show that $k = - 9$
Determine the equations of the invariant lines of the transformation represented by $\mathbf { M }$.
State which, if any, of the lines identified in (b) consist of fixed points, giving a reason for your answer.
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SPS SPS FM Pure 2024 January Q7

The point $P$ represents a complex number $z$ on an Argand diagram such that

$$| z - 6 \mathrm { i } | = 2 | z - 3 |$$

Show that, as $z$ varies, the locus of $P$ is a circle, stating the radius and the coordinates of the centre of this circle.
(6) The point $Q$ represents a complex number $z$ on an Argand diagram such that $$\arg ( z - 6 ) = - \frac { 3 \pi } { 4 }$$
Sketch, on the same Argand diagram, the locus of $P$ and the locus of $Q$ as $z$ varies.
Find the complex number for which both $| z - 6 \mathrm { i } | = 2 | z - 3 |$ and $\arg ( z - 6 ) = - \frac { 3 \pi } { 4 }$
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SPS SPS FM 2024 February Q1

O is the origin of a coordinate system whose units are cm .

The points $A , B , C$ and $D$ have coordinates $( 1,0 ) , ( 1,4 ) , ( 6,9 )$ and $( 0,9 )$ respectively.
The arc $B C$ is part of the curve with equation $x ^ { 2 } + ( y - 10 ) ^ { 2 } = 37$.
The closed shape $O A B C D$ is formed, in turn, from the line segments $O A$ and $A B$, the arc $B C$ and the line segments $C D$ and $D O$ (see diagram).
A funnel can be modelled by rotating $O A B C D$ by $2 \pi$ radians about the $y$-axis.
\includegraphics[max width=\textwidth, alt={}, center]{4e1bb995-ce3d-4d16-a0a2-72383489ffe1-04_510_894_443_226} Find the volume of the funnel according to the model.
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SPS SPS FM 2024 February Q2

2. The diagram below shows the graphs of $y = | 3 x - 2 |$ and $y = | 2 x + 1 |$.

\includegraphics[max width=\textwidth, alt={}, center]{4e1bb995-ce3d-4d16-a0a2-72383489ffe1-06_318_511_187_904} Give the coordinates of the points of intersection of the graphs with the coordinate axes.
Solve the equation $| 2 x + 1 | = | 3 x - 2 |$.
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SPS SPS FM 2024 February Q3

3. Show that $\int _ { 4 } ^ { \infty } x ^ { - \frac { 3 } { 2 } } d x = 1$
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SPS SPS FM 2024 February Q4

4. Two lines, $l _ { 1 }$ and $l _ { 2 }$, have the following equations. $$\begin{aligned} & l _ { 1 } : \mathbf { r } = \left( \begin{array} { c } - 11
10
3 \end{array} \right) + \lambda \left( \begin{array} { c } 2
- 2
1 \end{array} \right)
& l _ { 2 } : \mathbf { r } = \left( \begin{array} { l }

SPS SPS FM 2024 February Q5

5
2
4 \end{array} \right) + \mu \left( \begin{array} { c } 3
1
- 2 \end{array} \right) \end{aligned}$$ $P$ is the point of intersection of $l _ { 1 }$ and $l _ { 2 }$.

Find the position vector of $P$.
Find, correct to 1 decimal place, the acute angle between $/ _ { 1 }$ and $/ _ { 2 }$.
$Q$ is a point on $/ 1$ which is 12 metres away from $P \cdot R$ is the point on $/ 2$ such that $Q R$ is perpendicular to $/ 1$.
Determine the length $Q R$.
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5. (a) Express $\frac { 5 + 4 x - 3 x ^ { 2 } } { ( 1 - 2 x ) ( 2 + x ) ^ { 2 } }$ in three partial fractions.
Hence find the first three terms in the expansion of $\frac { 5 + 4 x - 3 x ^ { 2 } } { ( 1 - 2 x ) ( 2 + x ) ^ { 2 } }$ in ascending powers of $x$.
State the set of values for which the expansion in part (b) is valid.
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SPS SPS FM 2024 February Q6

6. The matrices $\mathbf { A }$ and $\mathbf { B }$ are given by $= \left( \begin{array} { l l } 1 & a
3 & 0 \end{array} \right)$ and $\mathbf { B } = \left( \begin{array} { l l } 4 & 2
3 & 3 \end{array} \right)$.

Find the value of a such that $\mathbf { A B } = \mathbf { B A }$.
Prove by counter example that matrix multiplication for $2 \times 2$ matrices is not commutative.
A triangle of area 4 square units is transformed by the matrix $\mathbf { B }$. Find the area of the image of the triangle following this transformation.
Find the equations of the invariant lines of the form $y = m x$ for the transformation represented by matrix $\mathbf { B }$.
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SPS SPS FM 2024 February Q10

10
3 \end{array} \right) + \lambda \left( \begin{array} { c } 2
- 2
1 \end{array} \right)
& l _ { 2 } : \mathbf { r } = \left( \begin{array} { l } 5
2
4 \end{array} \right) + \mu \left( \begin{array} { c } 3
1
- 2 \end{array} \right) \end{aligned}$$ $P$ is the point of intersection of $l _ { 1 }$ and $l _ { 2 }$.

Find the position vector of $P$.
Find, correct to 1 decimal place, the acute angle between $/ _ { 1 }$ and $/ _ { 2 }$.
$Q$ is a point on $/ 1$ which is 12 metres away from $P \cdot R$ is the point on $/ 2$ such that $Q R$ is perpendicular to $/ 1$.
Determine the length $Q R$.
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5. (a) Express $\frac { 5 + 4 x - 3 x ^ { 2 } } { ( 1 - 2 x ) ( 2 + x ) ^ { 2 } }$ in three partial fractions.
Hence find the first three terms in the expansion of $\frac { 5 + 4 x - 3 x ^ { 2 } } { ( 1 - 2 x ) ( 2 + x ) ^ { 2 } }$ in ascending powers of $x$.
State the set of values for which the expansion in part (b) is valid.
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6. The matrices $\mathbf { A }$ and $\mathbf { B }$ are given by $= \left( \begin{array} { l l } 1 & a
3 & 0 \end{array} \right)$ and $\mathbf { B } = \left( \begin{array} { l l } 4 & 2
3 & 3 \end{array} \right)$.
Find the value of a such that $\mathbf { A B } = \mathbf { B A }$.
Prove by counter example that matrix multiplication for $2 \times 2$ matrices is not commutative.
A triangle of area 4 square units is transformed by the matrix $\mathbf { B }$. Find the area of the image of the triangle following this transformation.
Find the equations of the invariant lines of the form $y = m x$ for the transformation represented by matrix $\mathbf { B }$.
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7. (a) In this question you must show detailed reasoning. Find the roots of the equation $2 z ^ { 2 } - 2 z + 5 = 0$.
The loci $C _ { 1 }$ and $C _ { 2 }$ are given by $| z | = | z - 2 i |$ and $| z - 2 | = \sqrt { 5 }$ respectively.
i. Sketch on a single Argand diagram the loci $C _ { 1 }$ and $C _ { 2 }$, showing any intercepts with the imaginary axis.
ii. Indicate, by shading on your Argand diagram, the region $\{ z : | z | \leqslant | z - 2 \mathrm { i } | \} \cap \{ z : | z - 2 | \leqslant \sqrt { 5 } \}$.
i. Show that both of the roots of the equation $2 z ^ { 2 } - 2 z + 5 = 0$ satisfy $| z - 2 | < \sqrt { 5 }$.
ii. State, with a reason, which root of the equation $2 z ^ { 2 } - 2 z + 5 = 0$ satisfies $| z | < | z - 2 i |$.
On the same Argand diagram as part (b), indicate the positions of the roots of the equation $2 z ^ { 2 } - 2 z + 5 = 0$.
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SPS SPS SM Statistics 2024 April Q1

1. The masses of a random sample of 120 boulders in a certain area were recorded. The results are summarized in the histogram.
\includegraphics[max width=\textwidth, alt={}, center]{d59e9fea-31cb-4b6d-b1d6-f09f912b5b37-04_773_1765_402_148}

Calculate the number of boulders with masses between 60 and 65 kg .
1. Use midpoints to find estimates of the mean and standard deviation of the masses of the boulders in the sample.
2. Explain why your answers are only estimates.
Use your answers to part (b)(i) to determine an estimate of the number of outliers, if any, in the distribution.
Give one advantage of using a histogram rather than a pie chart in this context.
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SPS SPS SM Statistics 2024 April Q2

11 marks

A certain five-sided die is biased with faces numbered 0 to 4 . The score, Y , on each throw is a random variable with probability distribution given by:
$Y$ 0 1 2 3 4
$\mathrm { P } ( Y = y )$ $a$ $b$ $c$ 0.1 0.15
where $a$, $b$ and $c$ are constants. $$\begin{aligned} & \mathrm { P } ( Y = 1 ) = \mathrm { P } ( Y \geq 3 )
& \mathrm { P } ( Y = 0 ) = \mathrm { P } ( Y = 2 ) - 0.1 \end{aligned}$$ Find the values of $a , b$ and $c$.
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The same die is thrown 10 times. Find the probability that there are not more than 4 throws on which the score is 3 , stating the distribution used as well as any modelling assumptions made.
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A game uses the same biased die. The die is thrown once. If it shows 1, 3 or 4 then this number is the final score. If it shows 0 or 2 then the die is thrown again and the final score is the sum of the numbers shown on the two throws.
(a) Find the probability that the final score is 3 .
(b) Given that the die is thrown twice, find the probability that the final score is 3 .
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SPS SPS SM Statistics 2024 April Q3

3. The table shows the increases, between 2001 and 2011, in the percentages of employees travelling to work by various methods, in the Local Authorities (LAs) in the North East region of the UK.

Geography code	Local authority	Work mainly at or from home	Underground, metro, light rail or tram	Bus, minibus or coach	Driving a car or van	Passenger in a car or van	On foot
E06000047	County Durham	0.74\%	0.05\%	-1.50\%	4.58\%	-2.99\%	-0.97\%
E06000005	Darlington	0.26\%	-0.01\%	-3.25\%	3.06\%	-1.28\%	0.99\%
E08000020	Gateshead	-0.01\%	-0.01\%	-2.28\%	4.62\%	-2.35\%	-0.18\%
E06000001	Hartlepool	0.03\%	-0.04\%	-1.62\%	4.80\%	-2.38\%	-0.26\%
E06000002	Middlesbrough	-0.34\%	-0.01\%	-2.32\%	2.19\%	-1.33\%	0.67\%
E08000021	Newcastle upon Tyne	0.10\%	-0.23\%	-0.67\%	-0.48\%	-1.51\%	1.75\%
E08000022	North Tyneside	0.05\%	0.54\%	-1.18\%	3.30\%	-2.21\%	-0.60\%
E06000048	Northumberland	1.39\%	-0.08\%	-0.95\%	3.50\%	-2.37\%	-1.44\%
E06000003	Redcar and Cleveland	-0.02\%	-0.01\%	-2.09\%	4.20\%	-2.06\%	-0.49\%
E08000023	South Tyneside	-0.36\%	2.03\%	-3.05\%	4.50\%	-2.41\%	-0.51\%
E06000004	Stockton-on-Tees	0.14\%	0.03\%	-2.02\%	3.52\%	-2.01\%	-0.15\%
E08000024	Sunderland	0.17\%	1.48\%	-3.11\%	4.89\%	-2.21\%	-0.52\%

\section*{Increase in percentage of employees travelling to work by various methods} The first two digits of the Geography code give the type of each of the LAs:
06: Unitary authority
07: Non-metropolitan district
08: Metropolitan borough

In what type of LA are the largest increases in percentages of people travelling by underground, metro, light rail or tram?
Identify two main changes in the pattern of travel to work in the North East region between 2001 and 2011. Now assume the following.
- The data refer to residents in the given LAs who are in the age range 20 to 65 at the time of each census.
- The number of people in the age range 20 to 65 who move into or out of each given LA, or who die, between 2001 and 2011 is negligible.
- Estimate the percentage of the people in the age range 20 to 65 in 2011 whose data appears in both 2001 and 2011.
- In the light of your answer to part (c), suggest a reason for the changes in the pattern of travel to work in the North East region between 2001 and 2011.
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SPS SPS SM Statistics 2024 April Q4

4. An online shopping company takes orders through its website. On average $80 \%$ of orders from the website are delivered within 24 hours. The quality controller selects 10 orders at random to check when they are delivered.

Find the probability that
(A) exactly 8 of these orders are delivered within 24 hours,
(B) at least 8 of these orders are delivered within 24 hours. The company changes its delivery method. The quality controller suspects that the changes will mean that fewer than $80 \%$ of orders will be delivered within 24 hours. A random sample of 18 orders is checked and it is found that 12 of them arrive within 24 hours.
Write down suitable hypotheses and carry out a test at the $5 \%$ significance level to determine whether there is any evidence to support the quality controller's suspicion.
A statistician argues that it is possible that the new method could result in either better or worse delivery times. Therefore it would be better to carry out a 2 -tail test at the $5 \%$ significance level. State the alternative hypothesis for this test. Assuming that the sample size is still 18, find the critical region for this test, showing all of your calculations.
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SPS SPS SM Statistics 2024 April Q5

5. In this question you must show detailed reasoning.
A disease that affects trees shows no visible evidence for the first few years after the tree is infected. A test has been developed to determine whether a particular tree has the disease. A positive result to the test suggests that the tree has the disease. However, the test is not $100 \%$ reliable, and a researcher uses the following model.

If the tree has the disease, the probability of a positive result is 0.95 .
If the tree does not have the disease, the probability of a positive result is 0.1 .
1. It is known that in a certain county, $A , 35 \%$ of the trees have the disease. A tree in county $A$ is chosen at random and is tested.

Given that the result is positive, determine the probability that this tree has the disease. A forestry company wants to determine what proportion of trees in another county, $B$, have the disease. They choose a large random sample of trees in county $B$. Each tree in the sample is tested and it is found that the result is positive for $43 \%$ of these trees.

By carrying out a calculation, determine an estimate of the proportion of trees in county $B$ that have the disease.
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SPS SPS FM Pure 2024 June Q1

1. The matrix $\mathbf { M }$ is such that $\mathbf { M } \left( \begin{array} { r r r } 1 & 0 & k
2 & - 1 & 1 \end{array} \right) = \left( \begin{array} { l l l } 1 & - 2 & 0 \end{array} \right)$.
Find

the matrix $\mathbf { M }$,
the value of the constant $k$.
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SPS SPS FM Pure 2024 June Q2

Relative to a fixed origin $O$,
the point $A$ has position vector $\mathbf { i } + 7 \mathbf { j } - 2 \mathbf { k }$,
the point $B$ has position vector $4 \mathbf { i } + 3 \mathbf { j } + 3 \mathbf { k }$,
and the point $C$ has position vector $2 \mathbf { i } + 10 \mathbf { j } + 9 \mathbf { k }$.
Given that $A B C D$ is a parallelogram,
1. find the position vector of point $D$.
The vector $\overrightarrow { A X }$ has the same direction as $\overrightarrow { A B }$.
Given that $| \overrightarrow { A X } | = 10 \sqrt { 2 }$,
find the position vector of $X$.
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SPS SPS FM Pure 2024 June Q3

3. (a) Sketch on the Argand diagram below the locus of points satisfying the equation $| z - 2 | = 2$.
\includegraphics[max width=\textwidth, alt={}, center]{ace492d8-1dd0-401e-af74-505ca19d5e9c-08_1260_1303_260_468}
(b) Given that $| z - 2 | = 2$ and $\arg ( z - 2 ) = - \frac { \pi } { 3 }$, express $z$ in the form $a + b i$ where $a , b \in \mathbb { R }$.
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\(Y\)	0	1	2	3	4
\(\mathrm { P } ( Y = y )\)	\(a\)	\(b\)	\(c\)	0.1	0.15

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