Questions - SPS - A-Level Maths

State the smallest possible value of $k$.
Show that $$\mathrm { f } ^ { \prime } ( x ) = \frac { a x ^ { 2 } + b x + c } { ( x + 4 ) ^ { 2 } }$$ where $a$, $b$ and $c$ are integers to be found.
Hence show that f is an increasing function.
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SPS SPS SM Mechanics 2022 February Q10

10. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{eacf7695-44c4-4937-8e92-5b0df8ad5f70-24_707_716_255_826} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} Figure 4 shows a sketch of the graph with equation $$y = | 2 x - 3 k |$$ where $k$ is a positive constant.

Sketch the graph with equation $y = \mathrm { f } ( x )$ where $$\mathrm { f } ( x ) = k - | 2 x - 3 k |$$ stating
- the coordinates of the maximum point
- the coordinates of any points where the graph cuts the coordinate axes
- 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.
Find, in terms of $k$, the coordinates of the minimum point of the graph with equation $$y = 3 - 5 \mathrm { f } \left( \frac { 1 } { 2 } x \right)$$ [BLANK PAGE]
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SPS SPS SM Mechanics 2022 February Q11

11. The curve $C$ has parametric equations $$x = \sin 2 \theta \quad y = \operatorname { cosec } ^ { 3 } \theta \quad 0 < \theta < \frac { \pi } { 2 }$$

Find an expression for $\frac { \mathrm { d } y } { \mathrm {~d} x }$ in terms of $\theta$
Hence find the exact value of the gradient of the tangent to $C$ at the point where $y = 8$
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SPS SPS SM Mechanics 2022 February Q12

12. Answer all the questions.
Two cyclists, $A$ and $B$, are cycling along the same straight horizontal track.
The cyclists are modelled as particles and the motion of the cyclists is modelled as follows:

At time $t = 0$, cyclist $A$ passes through the point $O$ with speed $2 \mathrm {~ms} ^ { - 1 }$
Cyclist $A$ is moving in a straight line with constant acceleration $2 \mathrm {~m} \mathrm {~s} ^ { - 2 }$
At time $t = 2$ seconds, cyclist $B$ starts from rest at $O$
Cyclist $B$ moves with constant acceleration $6 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ along the same straight line and in the same direction as cyclist $A$
At time $t = T$ seconds, $B$ overtakes $A$ at the point $X$

Using the model,

sketch, on the same axes, for the interval from $t = 0$ to $t = T$ seconds,
- a velocity-time graph for the motion of $A$
- a velocity-time graph for the motion of $B$
- explain why the two graphs must cross before time $t = T$ seconds,
- find the time when $A$ and $B$ are moving at the same speed,
- find the distance $O X$
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SPS SPS SM Mechanics 2022 February Q13

13.

\includegraphics[max width=\textwidth, alt={}]{eacf7695-44c4-4937-8e92-5b0df8ad5f70-34_328_1520_132_251}

A golfer hits a ball from a point $A$ with a speed of $25 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle of $15 ^ { \circ }$ above the horizontal. While the ball is in the air, it is modelled as a particle moving under the influence of gravity. Take the acceleration due to gravity to be $10 \mathrm {~m} \mathrm {~s} ^ { - 2 }$. The ball first lands at a point $B$ which is 4 m below the level of $A$ (see diagram).

Determine the time taken for the ball to travel from $A$ to $B$.
Determine the horizontal distance of $B$ from $A$.
Determine the direction of motion of the ball 1.5 seconds after the golfer hits the ball.
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SPS SPS SM Mechanics 2022 February Q14

14.
\includegraphics[max width=\textwidth, alt={}, center]{eacf7695-44c4-4937-8e92-5b0df8ad5f70-36_527_1123_118_269} One end of a light inextensible string is attached to a particle $A$ of mass 2 kg . The other end of the string is attached to a second particle $B$ of mass 3 kg . Particle $A$ is in contact with a smooth plane inclined at $30 ^ { \circ }$ to the horizontal and particle $B$ is in contact with a rough horizontal plane. A second light inextensible string is attached to $B$. The other end of this second string is attached to a third particle $C$ of mass 4 kg . Particle $C$ is in contact with a smooth plane $\Pi$ inclined at an angle of $60 ^ { \circ }$ to the horizontal. Both strings are taut and pass over small smooth pulleys that are at the tops of the inclined planes. The parts of the strings from $A$ to the pulley, and from $C$ to the pulley, are parallel to lines of greatest slope of the corresponding planes (see diagram). The coefficient of friction between $B$ and the horizontal plane is $\mu$. The system is released from rest and in the subsequent motion $C$ moves down $\Pi$ with acceleration $a \mathrm {~ms} ^ { - 2 }$.

By considering an equation involving $\mu , a$ and $g$ show that $a < \frac { 1 } { 9 } g ( 2 \sqrt { 3 } - 1 )$.
Given that $a = \frac { 1 } { 9 } g$, determine the magnitude of the contact force between $B$ and the horizontal plane. Give your answer correct to $\mathbf { 3 }$ significant figures.
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SPS SPS SM Statistics 2022 February Q1

1. Answer all the questions. $$f ( x ) = 3 x ^ { 3 } - 7 x ^ { 2 } + 7 x - 10$$

Use the factor theorem to show that ( $x - 2$ ) is a factor of $\mathrm { f } ( x )$
Find the values of the constants $a , b$ and $c$ such that $$\mathrm { f } ( x ) \equiv ( x - 2 ) \left( a x ^ { 2 } + b x + c \right)$$
Using your answer to part (b) show that the equation $\mathrm { f } ( x ) = 0$ has only one real root.
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SPS SPS SM Statistics 2022 February Q2

2. \section*{In this question you must show all stages of your working.} \section*{Solutions relying entirely on calculator technology are not acceptable.}

Express as an integral $$\lim _ { \delta x \rightarrow 0 } \sum _ { x = 4 } ^ { 12 } ( 1 + 2 x ) ^ { \frac { 1 } { 2 } } \delta x$$
Using your answer to part (a) show that $$\lim _ { \delta x \rightarrow 0 } \sum _ { x = 4 } ^ { 12 } ( 1 + 2 x ) ^ { \frac { 1 } { 2 } } \delta x = \frac { 98 } { 3 }$$ [BLANK PAGE]

SPS SPS SM Statistics 2022 February Q3

3. The curve $C$ has equation $$y = 5 x ^ { 4 } - 24 x ^ { 3 } + 42 x ^ { 2 } - 32 x + 11 \quad x \in \mathbb { R }$$

Find
1. $\frac { \mathrm { d } y } { \mathrm {~d} x }$
2. $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$
1. Verify that $C$ has a stationary point at $x = 1$
2. Show that this stationary point is a point of inflection, giving reasons for your answer.
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SPS SPS SM Statistics 2022 February Q4

4. The functions $f$ and $g$ are defined by $$\begin{array} { l l l } \mathrm { f } ( x ) = \frac { k x } { 2 x - 1 } & x \in \mathbb { R } & x \neq \frac { 1 } { 2 }
\mathrm {~g} ( x ) = 2 + 3 x - x ^ { 2 } & x \in \mathbb { R } & \end{array}$$ where $k$ is a non-zero constant.

Find in terms of $k$
1. $\mathrm { fg } ( 4 )$
2. the range of f
3. $\mathrm { f } ^ { - 1 }$ Given that $$\mathrm { f } ^ { - 1 } ( 2 ) = \frac { 11 } { 3 \mathrm {~g} ( 2 ) }$$
find the exact value of $k$
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SPS SPS SM Statistics 2022 February Q5

5. $$f ( x ) = \frac { 10 } { \sqrt { 4 - 3 x } }$$

Show that the first 4 terms in the binomial expansion of $\mathrm { f } ( x )$, in ascending powers of $x$, are $$A + B x + C x ^ { 2 } + \frac { 675 } { 1024 } x ^ { 3 }$$ where $A , B$ and $C$ are constants to be found. Give each constant in simplest form. Given that this expansion is valid for $| x | < k$
state the largest value of $k$. By substituting $x = \frac { 1 } { 3 }$ into $\mathrm { f } ( x )$ and into the answer for part (a),
find an approximation for $\sqrt { 3 }$ Give your answer in the form $\frac { a } { b }$ where $a$ and $b$ are integers to be found.
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SPS SPS SM Statistics 2022 February Q6

6. \section*{In this question you must show all stages of your working.} \section*{Solutions relying entirely on calculator technology are not acceptable.}

Express $3 \cos x + \sin x$ in the form $R \cos ( x - \alpha )$ where
- $\quad R$ and $\alpha$ are constants
- $R > 0$
- $0 < \alpha < \frac { \pi } { 2 }$
Give the exact value of $R$ and the value of $\alpha$ in radians to 3 decimal places. The temperature, $\theta ^ { \circ } \mathrm { C }$, inside a rabbit hole on a particular day is modelled by the equation $$\theta = 6.5 + 3 \cos \left( \frac { \pi t } { 13 } - 4 \right) + \sin \left( \frac { \pi t } { 13 } - 4 \right) \quad 0 \leqslant t < 24$$ where $t$ is the number of hours after midnight.
Using the equation of the model and your answer to part (a)
1. deduce the minimum value of $\theta$ during this day,
2. find the time of day when this minimum value occurs, giving your answer to the nearest minute.
Find the rate of temperature increase in the rabbit hole at midday.
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SPS SPS SM Statistics 2022 February Q7

7. The time, $T$ seconds, that a pendulum takes to complete one swing is modelled by the formula $$T = a l ^ { b }$$ where $l$ metres is the length of the pendulum and $a$ and $b$ are constants.

Show that this relationship can be written in the form $$\log _ { 10 } T = b \log _ { 10 } l + \log _ { 10 } a$$ \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{59120894-c480-492b-a304-106ddbadacf0-18_613_926_699_699} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A student carried out an experiment to find the values of the constants $a$ and $b$.
The student recorded the value of $T$ for different values of $l$.
Figure 3 shows the linear relationship between $\log _ { 10 } l$ and $\log _ { 10 } T$ for the student's data. The straight line passes through the points $( - 0.7,0 )$ and $( 0.21,0.45 )$ Using this information,
find a complete equation for the model in the form $$T = a l ^ { b }$$ giving the value of $a$ and the value of $b$, each to 3 significant figures.
With reference to the model, interpret the value of the constant $a$.
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SPS SPS SM Statistics 2022 February Q8

Use the substitution $u = 1 + \sin ^ { 2 } x$ to show that $$\int _ { 0 } ^ { \frac { \pi } { 6 } } \frac { 8 \tan x } { 1 + \sin ^ { 2 } x } \mathrm {~d} x = \int _ { p } ^ { q } \frac { 4 } { u ( 2 - u ) } \mathrm { d } u$$ where $p$ and $q$ are constants to be found.
Hence, using algebraic integration, show that $$\int _ { 0 } ^ { \frac { \pi } { 6 } } \frac { 8 \tan x } { 1 + \sin ^ { 2 } x } \mathrm {~d} x = \ln A$$ where $A$ is a rational number to be found.
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SPS SPS SM Statistics 2022 February Q9

9. In this question you must show all stages of your working.
Solutions relying on calculator technology are not acceptable.
The first 3 terms of an arithmetic sequence are $$\ln 3 \quad \ln \left( 3 ^ { k } - 1 \right) \quad \ln \left( 3 ^ { k } + 5 \right)$$ Find the exact value of the constant $k$.
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SPS SPS SM Statistics 2022 February Q10

10. (i) Use proof by exhaustion to show that for $n \in \mathbb { N } , n \leqslant 4$ $$( n + 1 ) ^ { 3 } > 3 ^ { n }$$ (ii) Given that $m ^ { 3 } + 5$ is odd, use proof by contradiction to show, using algebra, that $m$ is even.
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SPS SPS SM Statistics 2022 February Q11

Answer all the questions.

The discrete random variable $X$ has the probability function $$\mathrm { P } ( X = x ) = \left\{ \begin{array} { c c } c ( 7 - 2 x ) & x = 0,1,2,3
k & x = 4
0 & \text { otherwise } \end{array} \right.$$ where $c$ and $k$ are constants.

Show that $16 c + k = 1$
Given that $\mathrm { P } ( X \geq 3 ) = \frac { 5 } { 8 }$ find the value of $c$ and the value of $k$.
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SPS SPS SM Statistics 2022 February Q12

12. The four pie charts illustrate the numbers of employees using different methods of travel in four Local Authorities in 2011.
\includegraphics[max width=\textwidth, alt={}, center]{59120894-c480-492b-a304-106ddbadacf0-30_1234_1160_276_255} \begin{table}[h]

\captionsetup{labelformat=empty} \caption{Key:}

	Public transport
$\therefore$	Private motorised transport
$\therefore$	Bicycle
	All other methods of travel

\end{table}

State, with reasons, which of the four Local Authorities is most likely to be a rural area with many hills.
Two of the Local Authorities represent urban areas.
1. State with a reason which two Local Authorities are likely to be urban.
2. One urban Local Authority introduced a Park-and-Ride service in 2006. Users of this service drive to the edge of the urban area and then use buses to take them into the centre of the area. A student claims that a comparison of the corresponding pie charts for 2001 (not shown) and 2011 would enable them to identify which Local Authority this was. State with a reason whether you agree with the student.
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SPS SPS SM Statistics 2022 February Q13

13. Sam is playing a computer game. When Sam earns a reward in the game, she randomly receives either a Silver reward or a Gold reward. Each time that Sam earns a reward, the probability of receiving a Gold reward is 0.4 One day Sam plays the computer game and earns 11 rewards.

Find the probability that she receives
1. exactly 2 Gold rewards,
2. at least 5 Gold rewards. In the next month Sam earns 300 rewards.
  She decides to use a Normal distribution to estimate the probability that she will receive at least 135 Gold rewards.
1. Find the mean and variance of this Normal distribution.
2. Estimate the probability that Sam will receive at least 135 Gold rewards.
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SPS SPS SM Statistics 2022 February Q14

14. Zac is planning to write a report on the music preferences of the students at his college. There is a large number of students at the college.

State one reason why Zac might wish to obtain information from a sample of students, rather than from all the students.
Amaya suggests that Zac should use a sample that is stratified by school year. Give one advantage of this method as compared with random sampling, in this context. Zac decides to take a random sample of 60 students from his college. He asks each student how many hours per week, on average, they spend listening to music during term. From his results he calculates the following statistics.
Mean
Standard
deviation
Median
Lower
quartile
Upper
quartile
21.0 4.20 20.5 18.0 22.9
Sundip tells Zac that, during term, she spends on average 30 hours per week listening to music. Discuss briefly whether this value should be considered an outlier.
Layla claims that, during term, each student spends on average 20 hours per week listening to music. Zac believes that the true figure is higher than 20 hours. He uses his results to carry out a hypothesis test at the $5 \%$ significance level. Assume that the time spent listening to music is normally distributed with standard deviation 4.20 hours. Carry out the test.
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SPS SPS FM Statistics 2022 January Q1

A local authority official wishes to conduct a survey of households in the borough. He decides to select a stratified sample of 2000 households using Council Tax property bands as the strata. At the time of the survey there are 79368 households in the borough. The table shows the numbers of households in the different tax bands.