SPS SPS SM Mechanics (SPS SM Mechanics) 2021 January

Question 1
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1. Which of the options below best describes the correlation shown in the diagram below?
\includegraphics[max width=\textwidth, alt={}, center]{d1809ec7-dccf-446b-8a1d-f04a4252ebec-05_719_1198_408_340} Tick ( \(\checkmark\) ) one box.
moderate positive □
strong positive
\includegraphics[max width=\textwidth, alt={}, center]{d1809ec7-dccf-446b-8a1d-f04a4252ebec-05_103_109_1407_881}
moderate negative
\includegraphics[max width=\textwidth, alt={}, center]{d1809ec7-dccf-446b-8a1d-f04a4252ebec-05_103_109_1535_881}
strong negative □
Question 2 1 marks
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2. Lenny is one of a team of people interviewing shoppers in a town centre.
He is asked to survey 50 women between the ages of 18 and 29
Identify the name of this type of sampling.
Circle your answer.
[0pt] [1 mark]
simple random
stratified
quota
systematic
Question 3
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3. The Venn diagram shows the probabilities associated with four events, \(A , B , C\) and \(D\)
\includegraphics[max width=\textwidth, alt={}, center]{d1809ec7-dccf-446b-8a1d-f04a4252ebec-06_524_897_351_625}
  1. Write down any pair of mutually exclusive events from \(A , B , C\) and \(D\) Given that \(\mathrm { P } ( B ) = 0.4\)
  2. find the value of \(p\) Given also that \(A\) and \(B\) are independent
  3. find the value of \(q\) Given further that \(\mathrm { P } \left( B ^ { \prime } \mid C \right) = 0.64\)
  4. find
    1. the value of \(r\)
    2. the value of \(s\)
Question 4
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4. Each member of a group of 27 people was timed when completing a puzzle.
The time taken, \(x\) minutes, for each member of the group was recorded.
These times are summarised in the following box and whisker plot.
\includegraphics[max width=\textwidth, alt={}, center]{d1809ec7-dccf-446b-8a1d-f04a4252ebec-08_357_1454_523_335}
  1. Find the range of the times.
  2. Find the interquartile range of the times. For these 27 people \(\sum x = 607.5\) and \(\sum x ^ { 2 } = 17623.25\)
  3. calculate the mean time taken to complete the puzzle,
  4. calculate the standard deviation of the times taken to complete the puzzle. Taruni defines an outlier as a value more than 3 standard deviations above the mean.
  5. State how many outliers Taruni would say there are in these data, giving a reason for your answer. Adam and Beth also completed the puzzle in \(a\) minutes and \(b\) minutes respectively, where \(a > b\).
    When their times are included with the data of the other 27 people
    • the median time increases
    • the mean time does not change
    • Suggest a possible value for \(a\) and a possible value for \(b\), explaining how your values satisfy the above conditions.
    • Without carrying out any further calculations, explain why the standard deviation of all 29 times will be lower than your answer to part (d).
Question 5
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5. Patrick is practising his skateboarding skills. On each day, he has 30 attempts at performing a difficult trick. Every time he attempts the trick, there is a probability of 0.2 that he will fall off his skateboard. Assume that the number of times he falls off on any given day may be modelled by a binomial distribution.
    1. Find the mean number of times he falls off in a day.
  2. (ii) Find the variance of the number of times he falls off in a day.
    1. Find the probability that, on a particular day, he falls off exactly 10 times.
  4. (ii) Find the probability that, on a particular day, he falls off 5 or more times.
  5. Patrick has 30 attempts to perform the trick on each of 5 consecutive days.
    1. Calculate the probability that he will fall off his skateboard at least 5 times on each of the 5 days.
  7. (ii) Explain why it may be unrealistic to use the same value of 0.2 for the probability of falling off for all 5 days.
Question 6
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6. The discrete random variable \(D\) has the following probability distribution
\(d\)1020304050
\(\mathrm { P } ( D = d )\)\(\frac { k } { 10 }\)\(\frac { k } { 20 }\)\(\frac { k } { 30 }\)\(\frac { k } { 40 }\)\(\frac { k } { 50 }\)
where \(k\) is a constant.
  1. Show that the value of \(k\) is \(\frac { 600 } { 137 }\) The random variables \(D _ { 1 }\) and \(D _ { 2 }\) are independent and each have the same distribution as \(D\).
  2. Find P \(\left( D _ { 1 } + D _ { 2 } = 80 \right)\) Give your answer to 3 significant figures. A single observation of \(D\) is made.
    The value obtained, \(d\), is the common difference of an arithmetic sequence.
    The first 4 terms of this arithmetic sequence are the angles, measured in degrees, of quadrilateral \(Q\)
  3. Find the exact probability that the smallest angle of \(Q\) is more than \(50 ^ { \circ }\)
Question 7 1 marks
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7. A health centre claims that the time a doctor spends with a patient can be modelled by a normal distribution with a mean of 10 minutes and a standard deviation of 4 minutes.
  1. Using this model, find the probability that the time spent with a randomly selected patient is more than 15 minutes. Some patients complain that the mean time the doctor spends with a patient is more than 10 minutes. The receptionist takes a random sample of 20 patients and finds that the mean time the doctor spends with a patient is 11.5 minutes.
  2. Stating your hypotheses clearly and using a \(5 \%\) significance level, test whether or not there is evidence to support the patients' complaint. The health centre also claims that the time a dentist spends with a patient during a routine appointment, \(T\) minutes, can be modelled by the normal distribution where \(T \sim \mathrm {~N} \left( 5,3.5 ^ { 2 } \right)\)
  3. Using this model,
    1. find the probability that a routine appointment with the dentist takes less than 2 minutes
    2. find \(\mathrm { P } ( T < 2 \mid T > 0 )\)
    3. hence explain why this normal distribution may not be a good model for \(T\). The dentist believes that she cannot complete a routine appointment in less than 2 minutes.
      She suggests that the health centre should use a refined model only including values of \(T > 2\)
  4. Find the median time for a routine appointment using this new model, giving your answer correct to one decimal place. Name: □ \section*{U8th A LEVEL Single Mathematics Assessment Mechanics \(7 ^ { \text {th } }\) January 2021 } Instructions
    • Answer all the questions
    • Write your answer to each question in the space provided under each question. The question number(s) must be clearly shown.
    • Use black or blue ink. Pencil may be used for graphs and diagrams only.
    • You should clearly write your name at the top of this page and on any additional sheets that you use.
    • You are permitted to use a scientific or graphical calculator in this paper.
    • Final answers should be given to a degree of accuracy appropriate to the context.
    Information
    • The total mark for this paper is 55 marks.
    • The marks for each question are shown in brackets ( ).
    • You are reminded of the need for clear presentation in your answers.
    • You should allow approximately 60 minutes for this section of the test
    \section*{Formulae} \section*{A Level Mathematics A (H240)} \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 \({ } ^ { n } \mathrm { C } _ { r } = { } _ { n } \mathrm { C } _ { r } = \binom { n } { r } = \frac { n ! } { r ! ( n - r ) ! }\)
    \(( 1 + x ) ^ { n } = 1 + n x + \frac { n ( n - 1 ) } { 2 ! } x ^ { 2 } + \ldots + \frac { n ( n - 1 ) \ldots ( n - r + 1 ) } { r ! } x ^ { r } + \ldots \quad ( | x | < 1 , n \in \mathbb { R } )\) \section*{Differentiation}
    \(\mathrm { f } ( x )\)\(\mathrm { f } ^ { \prime } ( x )\)
    \(\tan k x\)\(k \sec ^ { 2 } k x\)
    \(\sec x\)\(\sec x \tan x\)
    \(\cot x\)\(- \operatorname { cosec } ^ { 2 } x\)
    \(\operatorname { cosec } x\)\(- \operatorname { cosec } x \cot x\)
    Quotient rule \(y = \frac { u } { v } , \frac { \mathrm {~d} y } { \mathrm {~d} x } = \frac { v \frac { \mathrm {~d} u } { \mathrm {~d} x } - u \frac { \mathrm {~d} v } { \mathrm {~d} x } } { v ^ { 2 } }\) \section*{Differentiation from first principles} \(\mathrm { f } ^ { \prime } ( x ) = \lim _ { h \rightarrow 0 } \frac { \mathrm { f } ( x + h ) - \mathrm { f } ( x ) } { h }\) \section*{Integration} \(\int \frac { \mathrm { f } ^ { \prime } ( x ) } { \mathrm { f } ( x ) } \mathrm { d } x = \ln | \mathrm { f } ( x ) | + c\)
    \(\int \mathrm { f } ^ { \prime } ( x ) ( \mathrm { f } ( x ) ) ^ { n } \mathrm {~d} x = \frac { 1 } { n + 1 } ( \mathrm { f } ( x ) ) ^ { n + 1 } + c\)
    Integration by parts \(\int u \frac { \mathrm {~d} v } { \mathrm {~d} x } \mathrm {~d} x = u v - \int v \frac { \mathrm {~d} u } { \mathrm {~d} x } \mathrm {~d} x\) Small angle approximations
    \(\sin \theta \approx \theta , \cos \theta \approx 1 - \frac { 1 } { 2 } \theta ^ { 2 } , \tan \theta \approx \theta\) where \(\theta\) is measured in radians \section*{Trigonometric identities} \(\sin ( A \pm B ) = \sin A \cos B \pm \cos A \sin B\)
    \(\cos ( A \pm B ) = \cos A \cos B \mp \sin A \sin B\)
    \(\tan ( A \pm B ) = \frac { \tan A \pm \tan B } { 1 \mp \tan A \tan B } \quad \left( A \pm B \neq \left( k + \frac { 1 } { 2 } \right) \pi \right)\) \section*{Numerical methods} Trapezium rule: \(\int _ { a } ^ { b } y \mathrm {~d} x \approx \frac { 1 } { 2 } h \left\{ \left( y _ { 0 } + y _ { n } \right) + 2 \left( y _ { 1 } + y _ { 2 } + \ldots + y _ { n - 1 } \right) \right\}\), where \(h = \frac { b - a } { n }\)
    The Newton-Raphson iteration for solving \(\mathrm { f } ( x ) = 0 : x _ { n + 1 } = x _ { n } - \frac { \mathrm { f } \left( x _ { n } \right) } { \mathrm { f } ^ { \prime } \left( x _ { n } \right) }\) \section*{Probability} \(\mathrm { P } ( A \cup B ) = \mathrm { P } ( A ) + \mathrm { P } ( B ) - \mathrm { P } ( A \cap B )\)
    \(\mathrm { P } ( A \cap B ) = \mathrm { P } ( A ) \mathrm { P } ( B \mid A ) = \mathrm { P } ( B ) \mathrm { P } ( A \mid B )\) or \(\mathrm { P } ( A \mid B ) = \frac { \mathrm { P } ( A \cap B ) } { \mathrm { P } ( B ) }\) \section*{Standard deviation} \(\sqrt { \frac { \sum ( x - \bar { x } ) ^ { 2 } } { n } } = \sqrt { \frac { \sum x ^ { 2 } } { n } - \bar { x } ^ { 2 } }\) or \(\sqrt { \frac { \sum f ( x - \bar { x } ) ^ { 2 } } { \sum f } } = \sqrt { \frac { \sum f x ^ { 2 } } { \sum f } - \bar { x } ^ { 2 } }\) \section*{The binomial distribution} If \(X \sim \mathbf { B } ( n , p )\) then \(\mathbf { P } ( X = x ) = \binom { n } { x } p ^ { x } ( 1 - p ) ^ { n - x }\), Mean of \(X\) is \(n p\), Variance of \(X\) is \(n p ( 1 - p )\) \section*{Hypothesis test for the mean of a normal distribution} If \(X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\) then \(\bar { X } \sim \mathrm {~N} \left( \mu , \frac { \sigma ^ { 2 } } { n } \right)\) and \(\frac { \bar { X } - \mu } { \sigma / \sqrt { n } } \sim \mathrm {~N} ( 0,1 )\) \section*{Percentage points of the normal distribution} If \(Z\) has a normal distribution with mean 0 and variance 1 then, for each value of \(p\), the table gives the value of \(z\) such that \(\mathrm { P } ( Z \leqslant z ) = p\).
    \(p\)0.750.900.950.9750.990.9950.9975.0 .9990.9995
    \(z\)0.6741.2821.6451.9602.3262.5762.8073.0903.291
    \section*{Kinematics} Motion in a straight line
    \(v = u + a t\)
    \(s = u t + \frac { 1 } { 2 } a t ^ { 2 }\)
    \(s = \frac { 1 } { 2 } ( u + v ) t\)
    \(v ^ { 2 } = u ^ { 2 } + 2 a s\)
    \(s = v t - \frac { 1 } { 2 } a t ^ { 2 }\) Motion in two dimensions
    \(\mathbf { v } = \mathbf { u } + \mathbf { a } t\)
    \(\mathbf { s } = \mathbf { u } t + \frac { 1 } { 2 } \mathbf { a } t ^ { 2 }\)
    \(\mathbf { s } = \frac { 1 } { 2 } ( \mathbf { u } + \mathbf { v } ) \boldsymbol { t }\)
    \(\mathbf { s } = \mathbf { v } t - \frac { 1 } { 2 } \mathbf { a } t ^ { 2 }\)
    [0pt] [BLANK PAGE]
    1. A vehicle is driven at a constant speed of \(12 \mathrm {~ms} ^ { - 1 }\) along a straight horizontal road. Only one of the statements below is correct. Identify the correct statement.
    Tick ( \(\checkmark\) ) one box. The vehicle is accelerating □ The vehicle's driving force exceeds the total force resisting its motion □ The resultant force acting on the vehicle is zero □ The resultant force acting on the vehicle is dependent on its mass □
    2. A number of forces act on a particle such that the resultant force is \(\binom { 6 } { - 3 } \mathrm {~N}\)
    One of the forces acting on the particle is \(\binom { 8 } { - 5 } \mathrm {~N}\)
    Calculate the total of the other forces acting on the particle.
    Circle your answer.
    [0pt] [1 mark] $$\binom { 2 } { - 2 } \mathrm {~N} \quad \binom { 14 } { - 8 } \mathrm {~N} \quad \binom { - 2 } { 2 } \mathrm {~N} \quad \binom { - 14 } { 8 } \mathrm {~N}$$ 3. A rough plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\)
    A brick \(P\) of mass \(m\) is placed on the plane.
    The coefficient of friction between \(P\) and the plane is \(\mu\)
    Brick \(P\) is in equilibrium and on the point of sliding down the plane.
    Brick \(P\) is modelled as a particle.
    Using the model,
  5. find, in terms of \(m\) and \(g\), the magnitude of the normal reaction of the plane on brick \(P\)
  6. show that \(\mu = \frac { 3 } { 4 }\) For parts (c) and (d), you are not required to do any further calculations.
    Brick \(P\) is now removed from the plane and a much heavier brick \(Q\) is placed on the plane. The coefficient of friction between \(Q\) and the plane is also \(\frac { 3 } { 4 }\)
  7. Explain briefly why brick \(Q\) will remain at rest on the plane. Brick \(Q\) is now projected with speed \(0.5 \mathrm {~ms} ^ { - 1 }\) down a line of greatest slope of the plane.
    Brick \(Q\) is modelled as a particle.
    Using the model,
  8. describe the motion of brick \(Q\), giving a reason for your answer.
    4. A particle \(P\) moves with acceleration \(( 4 \mathbf { i } - 5 \mathbf { j } ) \mathrm { ms } ^ { - 2 }\)
    At time \(t = 0 , P\) is moving with velocity \(( - 2 \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\)
  9. Find the velocity of \(P\) at time \(t = 2\) seconds. At time \(t = 0 , P\) passes through the origin \(O\).
    At time \(t = T\) seconds, where \(T > 0\), the particle \(P\) passes through the point \(A\).
    The position vector of \(A\) is \(( \lambda \mathbf { i } - 4.5 \mathbf { j } ) \mathrm { m }\) relative to \(O\), where \(\lambda\) is a constant.
  10. Find the value of \(T\).
  11. Hence find the value of \(\lambda\)
    5.
    1. At time \(t\) seconds, where \(t \geqslant 0\), a particle \(P\) moves so that its acceleration \(\mathbf { a } \mathrm { ms } ^ { - 2 }\) is given by $$\mathbf { a } = ( 1 - 4 t ) \mathbf { i } + \left( 3 - t ^ { 2 } \right) \mathbf { j }$$ At the instant when \(t = 0\), the velocity of \(P\) is \(36 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
  12. Find the velocity of \(P\) when \(t = 4\)
  13. Find the value of \(t\) at the instant when \(P\) is moving in a direction perpendicular to \(\mathbf { i }\)
    (ii) At time \(t\) seconds, where \(t \geqslant 0\), a particle \(Q\) moves so that its position vector \(\mathbf { r }\) metres, relative to a fixed origin \(O\), is given by $$\mathbf { r } = \left( t ^ { 2 } - t \right) \mathbf { i } + 3 t \mathbf { j }$$ Find the value of \(t\) at the instant when the speed of \(Q\) is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
    6. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{d1809ec7-dccf-446b-8a1d-f04a4252ebec-28_529_993_374_529} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} A small ball is projected with speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(O\) at the top of a vertical cliff.
    The point \(O\) is 25 m vertically above the point \(N\) which is on horizontal ground.
    The ball is projected at an angle of \(45 ^ { \circ }\) above the horizontal.
    The ball hits the ground at a point \(A\), where \(A N = 100 \mathrm {~m}\), as shown in Figure 2 .
    The motion of the ball is modelled as that of a particle moving freely under gravity.
    Using this initial model,
  14. show that \(U = 28\)
  15. find the greatest height of the ball above the horizontal ground \(N A\). In a refinement to the model of the motion of the ball from \(O\) to \(A\), the effect of air resistance is included. This refined model is used to find a new value of \(U\).
  16. How would this new value of \(U\) compare with 28 , the value given in part (a)?
  17. State one further refinement to the model that would make the model more realistic.
    7. Block \(A\), of mass 0.2 kg , lies at rest on a rough plane.
    The plane is inclined at an angle \(\theta\) to the horizontal, such that \(\tan \theta = \frac { 7 } { 24 }\)
    A light inextensible string is attached to \(A\) and runs parallel to the line of greatest slope until it passes over a smooth fixed pulley at the top of the slope. The other end of this string is attached to particle \(B\), of mass 2 kg , which is held at rest so that the string is taut, as shown in the diagram below.
    \includegraphics[max width=\textwidth, alt={}, center]{d1809ec7-dccf-446b-8a1d-f04a4252ebec-32_424_1070_815_486}
  18. \(\quad B\) is released from rest so that it begins to move vertically downwards with an acceleration of \(\frac { 543 } { 625 } \mathrm {~g} \mathrm {~m} \mathrm {~s} ^ { - 2 }\) Show that the coefficient of friction between \(A\) and the surface of the inclined plane is 0.17
  19. In this question use \(g = 9.81 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) When \(A\) reaches a speed of \(0.5 \mathrm {~ms} ^ { - 1 }\) the string breaks.
    1. Find the distance travelled by \(A\) after the string breaks until first coming to rest.
  21. (ii) State an assumption that could affect the validity of your answer to part (b)(i).