Questions - SPS - A-Level Maths

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.
[0pt] [BLANK PAGE]
[0pt] [BLANK PAGE]
[0pt] [BLANK PAGE]
[0pt] [TURN OVER FOR QUESTION 7]

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 $n \geq 1$
[0pt] [BLANK PAGE]

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.
[0pt] [BLANK PAGE]

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 $q$.
[0pt] [BLANK PAGE]

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 $\mathrm { f } ( x ) = 5$.
[0pt] [BLANK PAGE]

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).
[0pt] [BLANK PAGE]

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 $\pi ^ { \mathrm { e } }$.
[0pt] [BLANK PAGE]
[0pt] [BLANK PAGE]
[0pt] [BLANK PAGE]
[0pt] [BLANK PAGE]
[0pt] [BLANK PAGE]

SPS SPS FM Statistics 2022 February Q1

The random variable $X$ represents the clutch size (the number of eggs laid) by female birds of a particular species. The probability distribution of $X$ is given in the table.

$r$	2	3	4	5	6	7
$\mathrm { P } ( X = r )$	0.03	0.07	0.27	0.49	0.13	0.01

Find each of the following.
- $\mathrm { E } ( X )$
- $\operatorname { Var } ( X )$
On average $65 \%$ of eggs laid result in a young bird successfully leaving the nest.
1. Find the mean number of young birds that successfully leave the nest.
2. Find the standard deviation of the number of young birds that successfully leave the nest.
  [0pt] [BLANK PAGE]

SPS SPS FM Statistics 2022 February Q2

2. A shopper estimates the cost, $\pounds X$ per item, of each of 12 items in a supermarket. The shopper's estimates are compared with the actual cost, $\pounds Y$ per item, of each item. The results are summarised as follows.
$n = 12$
$\sum x ^ { 2 } = 28127$
$\sum x = 399$
$\Sigma y ^ { 2 } = 116509.0212$
$\Sigma y = 623.88$
$\sum x y = 45006.01$ Test at the $1 \%$ significance level whether the shopper's estimates are positively correlated with the actual cost of the items.
[0pt] [BLANK PAGE]

SPS SPS FM Statistics 2022 February Q3

3. A football player is practising taking penalties. On each attempt the player has a $70 \%$ chance of scoring a goal. The random variable $X$ represents the number of attempts that it takes for the player to score a goal.

Determine $\mathrm { P } ( X = 4 )$.
Find each of the following.
- $\mathrm { E } ( X )$
- $\operatorname { Var } ( X )$
- Determine the probability that the player needs exactly 4 attempts to score 2 goals.
  [0pt] [BLANK PAGE]

SPS SPS FM Statistics 2022 February Q4

(a) Using the scatter diagram below, explain what is meant by least squares in the context of a regression line of $y$ on $x$.
\includegraphics[max width=\textwidth, alt={}, center]{5a60e87d-7a09-4ef5-96ca-8f33030c8747-08_481_889_276_219}
(b) A set of bivariate data $( t , u )$ is summarised as follows.

$$\begin{array} { l l l } n = 5 & \sum t = 35 & \sum u = 54
\sum t ^ { 2 } = 285 & \sum u ^ { 2 } = 758 & \sum t u = 460 \end{array}$$

Calculate the equation of the regression line of $u$ on $t$.
The variables $t$ and $u$ are now scaled using the following scaling. $$v = 2 t , w = u + 4$$ Find the equation of the regression line of $w$ on $v$, giving your equation in the form $$w = \mathrm { f } ( v ) .$$ [BLANK PAGE]

SPS SPS FM Statistics 2022 February Q5

5. Charlie carried out a survey on the main type of investment people have. The contingency table below shows the results of a survey of a random sample of people.

\cline { 3 - 5 } \multicolumn{2}{c\|}{}	Main type of investment
\cline { 3 - 5 } \multicolumn{2}{c\|}{}	Bonds	Cash	Stocks
\multirow{2}{*}{Age}	$25 - 44$	$a$	$b - e$	$e$
\cline { 2 - 5 }	$45 - 75$	$c$	$d - 59$	59

Find an expression, in terms of $a , b , c$ and $d$, for the difference between the observed and the expected value $( O - E )$ for the group whose main type of investment is Bonds and are aged $45 - 75$
Express your answer as a single fraction in its simplest form. Given that $\sum \frac { ( O - E ) ^ { 2 } } { E } = 9.62$ for this information,
test, at the $5 \%$ level of significance, whether or not there is evidence of an association between the age of a person and the main type of investment they have. You should state your hypotheses, critical value and conclusion clearly. You may assume that no cells need to be combined.
[0pt] [BLANK PAGE]

SPS SPS FM Statistics 2022 February Q6

6. The 20 members of a club consist of 10 Town members and 10 Country members.

All 20 members are arranged randomly in a straight line. Determine the probability that the Town members and the Country members alternate.
Ten members of the club are chosen at random. Show that the probability that the number of Town members chosen is no more than $r$, where $0 \leqslant r \leqslant 10$, is given by
$\frac { 1 } { N } \sum _ { i = 0 } ^ { r } \left( { } ^ { 10 } C _ { i } \right) ^ { 2 }$
where $N$ is an integer to be determined.
[0pt] [BLANK PAGE]

SPS SPS FM Statistics 2022 February Q7

7. (a) A substance emits particles randomly at a constant average rate of 3.2 per minute. A second substance emits particles randomly, and independently of the first source, at a constant average rate of 2.7 per minute. Find the probability that the total number of particles emitted by the two sources in a ten-minute period is less than 70 .
(b) The random variable $X$ represents the number of particles emitted by a substance in a fixed time interval $t$ minutes. It may be assumed that particles are emitted randomly and independently of each other. In general, the rate at which particles are emitted is proportional to the mass of the substance, but each particle emitted reduces the mass of the substance. Explain why a Poisson distribution may not be a valid model for $X$ if the value of $t$ is very large.
(c) The random variable $Y$ has the distribution $\operatorname { Po } ( \lambda )$. It is given that $$\begin{aligned} & \mathrm { P } ( Y = r ) = \mathrm { P } ( Y = r + 1 )
& \mathrm { P } ( Y = r ) = 1.5 \times \mathrm { P } ( Y = r - 1 ) \end{aligned}$$ Determine the following, in either order.

The value of $r$
The value of $\lambda$
[0pt] [BLANK PAGE]
[0pt] [BLANK PAGE]
[0pt] [BLANK PAGE]
[0pt] [BLANK PAGE]

SPS SPS FM Statistics 2022 February Q1

At a seaside resort the number $X$ of ice-creams sold and the temperature $Y ^ { \circ } \mathrm { F }$ were recorded on 20 randomly chosen summer days. The data can be summarised as follows.

$$\sum x = 1506 \quad \sum x ^ { 2 } = 127542 \quad \sum y = 1431 \quad \sum y ^ { 2 } = 104451 \quad \sum x y = 111297$$

Calculate the equation of the least squares regression line of $y$ on $x$, giving your answer in the form $y = a + b x$.
Explain the significance for the regression line of the quantity $\sum \left[ y _ { i } - \left( a x _ { i } + b \right) \right] ^ { 2 }$.
It is decided to measure the temperature in degrees Centigrade instead of degrees Fahrenheit. If the same temperature is measured both as $f ^ { \circ }$ Fahrenheit and $c ^ { \circ }$ Centigrade, the relationship between $f$ and $c$ is $c = \frac { 5 } { 9 } ( f - 32 )$. Find the equation of the new regression line.
[0pt] [BLANK PAGE]

SPS SPS FM Statistics 2022 February Q2

2. When babies are born, their head circumferences are measured. A random sample of 50 newborn female babies is selected. The sample mean head circumference is 34.711 cm . The sample standard deviation head circumference is 1.530 cm .

Determine a $95 \%$ confidence interval for the population mean head circumference of newborn female babies.
Explain why you can calculate this interval even though the distribution of the population of head circumferences of newborn female babies is unknown.
[0pt] [BLANK PAGE]

SPS SPS FM Statistics 2022 February Q3

1 marks

3. In air traffic management, air traffic controllers send radio messages to pilots. On receiving a message, the pilot repeats it back to the controller to check that it has been understood correctly. At a particular site, on average $4 \%$ of messages sent by controllers are not repeated back correctly and so have been misunderstood. You should assume that instances of messages being misunderstood occur randomly and independently.

Find the probability that exactly 2 messages are misunderstood in a sequence of 50 messages.
Find the probability that in a sequence of messages, the 10th message is the first one which is misunderstood.
[0pt]
Find the probability that in a sequence of 20 messages, there are no misunderstood messages. [1]
Determine the expected number of messages required for 3 of them to be misunderstood.
Determine the probability that in a sequence of messages, the 3rd misunderstood message is the 60th message in the sequence.
[0pt] [BLANK PAGE]

SPS SPS FM Statistics 2022 February Q4

4. Members of a photographic group may enter a maximum of 5 photographs into a members only competition.
Past experience has shown that the number of photographs, $N$, entered by a member follows the probability distribution shown below.

$n$	0	1	2	3	4	5
$\mathrm { P } ( N = n )$	$a$	0.2	0.05	0.25	$b$	$c$

Given that $\mathrm { E } ( 4 N + 2 ) = 14.8$ and $\mathrm { P } ( N = 5 \mid N > 2 ) = \frac { 1 } { 2 }$

show that $\operatorname { Var } ( N ) = 2.76$ The group decided to charge a 50 p entry fee for the first photograph entered and then 20 p for each extra photograph entered into the competition up to a maximum of $\pounds 1$ per person. Thus a member who enters 3 photographs pays 90 p and a member who enters 4 or 5 photographs just pays £l Assuming that the probability distribution for the number of photographs entered by a member is unchanged,
calculate the expected entry fee per member.
[0pt] [BLANK PAGE]

SPS SPS FM Statistics 2022 February Q5

3 marks

5. A practice examination paper is taken by 500 candidates, and the organiser wishes to know what continuous distribution could be used to model the actual time, $X$ minutes, taken by candidates to complete the paper. The organiser starts by carrying out a goodness-of-fit test for the distribution $\mathrm { N } \left( 100,15 ^ { 2 } \right)$ at the $5 \%$ significance level. The grouped data and the results of some of the calculations are shown in the following table.

Time	$0 \leqslant X < 80$	$80 \leqslant X < 90$	$90 \leqslant X < 100$	$100 \leqslant X < 110$	$x \geqslant 110$
Observed frequency $O$	36	95	137	129	103
Expected frequency $E$	45.606	80.641	123.754	123.754	126.246
$\frac { ( O - E ) ^ { 2 } } { E }$	2.023	2.557	1.418	0.222	4.280

State suitable hypotheses for the test.
Show how the figures 123.754 and 0.222 in the column for $100 \leqslant X < 110$ were obtained. [3]
Carry out the test.
[0pt] [BLANK PAGE]

SPS SPS FM Statistics 2022 February Q6

6. The continuous random variable $Y$ has a uniform distribution on [ 0,2 ].

It is given that $\mathrm { E } [ a \cos ( a Y ) ] = 0.3$, where $a$ is a constant between 0 and 1 , and $a Y$ is measured in radians. Determine the value of the constant $a$.
Determine the $60 ^ { \text {th } }$ percentile of $Y ^ { 2 }$.
[0pt] [BLANK PAGE]
[0pt] [BLANK PAGE]
[0pt] [BLANK PAGE]
[0pt] [BLANK PAGE]

SPS SPS SM Mechanics 2022 February Q1

1. Answer all the questions.
Find $$\int \left( x ^ { 4 } - 6 x ^ { 2 } + 7 \right) \mathrm { d } x$$ giving your answer in simplest form.
(3)
[0pt] [BLANK PAGE]

SPS SPS SM Mechanics 2022 February Q2

2. Given that $$\mathrm { f } ( x ) = x ^ { 2 } - 4 x + 5 \quad x \in \mathbb { R }$$

express $\mathrm { f } ( x )$ in the form $( x + a ) ^ { 2 } + b$ where $a$ and $b$ are integers to be found. The curve with equation $y = \mathrm { f } ( x )$
- meets the $y$-axis at the point $P$
- has a minimum turning point at the point $Q$
- Write down
  1. the coordinates of $P$
  2. the coordinates of $Q$
    [0pt] [BLANK PAGE]

SPS SPS SM Mechanics 2022 February Q3

3. The sequence $u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots$ is defined by $$u _ { n + 1 } = k - \frac { 24 } { u _ { n } } \quad u _ { 1 } = 2$$ where $k$ is an integer.
Given that $u _ { 1 } + 2 u _ { 2 } + u _ { 3 } = 0$

show that $$3 k ^ { 2 } - 58 k + 240 = 0$$
Find the value of $k$, giving a reason for your answer.
Find the value of $u _ { 3 }$
[0pt] [BLANK PAGE]

SPS SPS SM Mechanics 2022 February Q4

4. Relative to a fixed origin $O$,

the point $A$ has position vector $5 \mathbf { i } + 3 \mathbf { j } - 2 \mathbf { k }$
the point $B$ has position vector $7 \mathbf { i } + \mathbf { j } + 2 \mathbf { k }$
the point $C$ has position vector $4 \mathbf { i } + 8 \mathbf { j } - 3 \mathbf { k }$
1. Find $| \overrightarrow { A B } |$ giving your answer as a simplified surd.

Given that $A B C D$ is a parallelogram,

find the position vector of the point $D$. The point $E$ is positioned such that

$A C E$ is a straight line
$A C : C E = 2 : 1$
Find the coordinates of the point $E$.
[0pt] [BLANK PAGE]

\section*{Solutions relying entirely on calculator technology are not acceptable.} \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{eacf7695-44c4-4937-8e92-5b0df8ad5f70-12_855_1104_340_589} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a sketch of part of the curve $C$ with equation $$y = x ^ { 3 } - 10 x ^ { 2 } + 27 x - 23$$ The point $P ( 5 , - 13 )$ lies on $C$
The line $l$ is the tangent to $C$ at $P$

Use differentiation to find the equation of $l$, giving your answer in the form $y = m x + c$ where $m$ and $c$ are integers to be found.

Hence verify that $l$ meets $C$ again on the $y$-axis. The finite region $R$, shown shaded in Figure 2, is bounded by the curve $C$ and the line $l$.

Use algebraic integration to find the exact area of $R$.
[0pt] [BLANK PAGE]
[0pt] [BLANK PAGE]
[0pt] [BLANK PAGE]
[0pt] [TURN OVER FOR QUESTION 6]

SPS SPS SM Mechanics 2022 February Q6

6. A scientist is studying the growth of two different populations of bacteria. The number of bacteria, $N$, in the first population is modelled by the equation $$N = A \mathrm { e } ^ { k t } \quad t \geqslant 0$$ where $A$ and $k$ are positive constants and $t$ is the time in hours from the start of the study.
Given that

there were 1000 bacteria in this population at the start of the study
it took exactly 5 hours from the start of the study for this population to double
1. find a complete equation for the model.
2. Hence find the rate of increase in the number of bacteria in this population exactly 8 hours from the start of the study. Give your answer to 2 significant figures.

The number of bacteria, $M$, in the second population is modelled by the equation $$M = 500 \mathrm { e } ^ { 1.4 k t } \quad t \geqslant 0$$ where $k$ has the value found in part (a) and $t$ is the time in hours from the start of the study.
Given that $T$ hours after the start of the study, the number of bacteria in the two different populations was the same,

find the value of $T$.
[0pt] [BLANK PAGE] \section*{Solutions relying entirely on calculator technology are not acceptable.}

Show that $$\frac { 1 - \cos 2 \theta } { \sin ^ { 2 } 2 \theta } \equiv k \sec ^ { 2 } \theta \quad \theta \neq \frac { n \pi } { 2 } \quad n \in \mathbb { Z }$$ where $k$ is a constant to be found.

Hence solve, for $- \frac { \pi } { 2 } < x < \frac { \pi } { 2 }$ $$\frac { 1 - \cos 2 x } { \sin ^ { 2 } 2 x } = ( 1 + 2 \tan x ) ^ { 2 }$$ Give your answers to 3 significant figures where appropriate.
[0pt] [BLANK PAGE]

\cline { 3 - 5 } \multicolumn{2}{c\|}{}	Main type of investment
\cline { 3 - 5 } \multicolumn{2}{c\|}{}	Bonds	Cash	Stocks
\multirow{2}{*}{Age}	\(25 - 44\)	\(a\)	\(b - e\)	\(e\)
\cline { 2 - 5 }	\(45 - 75\)	\(c\)	\(d - 59\)	59

\(n\)	0	1	2	3	4	5
\(\mathrm { P } ( N = n )\)	\(a\)	0.2	0.05	0.25	\(b\)	\(c\)

\(r\)	2	3	4	5	6	7
\(\mathrm { P } ( X = r )\)	0.03	0.07	0.27	0.49	0.13	0.01

Time	\(0 \leqslant X < 80\)	\(80 \leqslant X < 90\)	\(90 \leqslant X < 100\)	\(100 \leqslant X < 110\)	\(x \geqslant 110\)
Observed frequency \(O\)	36	95	137	129	103
Expected frequency \(E\)	45.606	80.641	123.754	123.754	126.246
\(\frac { ( O - E ) ^ { 2 } } { E }\)	2.023	2.557	1.418	0.222	4.280

Questions — SPS (1106 questions)

Browse by module

Browse by board

SPS SPS FM Pure 2022 February Q5

SPS SPS FM Pure 2022 February Q7

SPS SPS FM Pure 2022 February Q8

SPS SPS FM Pure 2022 February Q9

SPS SPS FM Pure 2022 February Q10

SPS SPS FM Pure 2022 February Q11

SPS SPS FM Pure 2022 February Q12

SPS SPS FM Statistics 2022 February Q1

SPS SPS FM Statistics 2022 February Q2

SPS SPS FM Statistics 2022 February Q3

SPS SPS FM Statistics 2022 February Q4

SPS SPS FM Statistics 2022 February Q5

SPS SPS FM Statistics 2022 February Q6

SPS SPS FM Statistics 2022 February Q7

SPS SPS FM Statistics 2022 February Q1

SPS SPS FM Statistics 2022 February Q2

SPS SPS FM Statistics 2022 February Q3

SPS SPS FM Statistics 2022 February Q4

SPS SPS FM Statistics 2022 February Q5

SPS SPS FM Statistics 2022 February Q6

SPS SPS SM Mechanics 2022 February Q1

SPS SPS SM Mechanics 2022 February Q2

SPS SPS SM Mechanics 2022 February Q3

SPS SPS SM Mechanics 2022 February Q4

SPS SPS SM Mechanics 2022 February Q6