SPS SPS SM (SPS SM) 2021 January

1. In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) A ball, initially at rest, is dropped from a height of 40 m above the ground.
Calculate the speed of the ball when it reaches the ground.

2. An object of mass 5 kg is moving in a straight line.
As a result of experiencing a forward force of \(F\) newtons and a resistant force of \(R\) newtons it accelerates at \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) Which one of the following equations is correct? $$F - R = 0 \quad F - R = 5 \quad F - R = 3 \quad F - R = 0.6$$

3. A vehicle, which begins at rest at point \(P\), is travelling in a straight line.
For the first 4 seconds the vehicle moves with a constant acceleration of \(0.75 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) For the next 5 seconds the vehicle moves with a constant acceleration of \(- 1.2 \mathrm {~ms} ^ { - 2 }\) The vehicle then immediately stops accelerating, and travels a further 33 m at constant speed.

1. Draw a velocity-time graph for this journey.
2. Find the distance of the car from \(P\) after 20 seconds.

4. In this question use \(\boldsymbol { g } = \mathbf { 9 . 8 1 } \mathbf { m ~ s } ^ { \mathbf { - 2 } }\) Two particles, of mass 1.8 kg and 1.2 kg , are connected by a light, inextensible string over a smooth peg. \includegraphics[max width=\textwidth, alt={}, center]{1a7b75e9-eab4-4264-ab15-5292c504fb4d-04_563_695_477_669}

1. Initially the particles are held at rest 1.5 m above horizontal ground and the string between them is taut. The particles are released from rest.
   Find the time taken for the 1.8 kg particle to reach the ground.
2. State one assumption you have made in answering part (a).

5. A cyclist, Laura, is travelling in a straight line on a horizontal road at a constant speed of \(25 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) A second cyclist, Jason, is riding closely and directly behind Laura. He is also moving with a constant speed of \(25 \mathrm {~km} \mathrm {~h} ^ { - 1 }\)

1. The driving force applied by Jason is likely to be less than the driving force applied by Laura. Explain why.
2. Jason has a problem and stops, but Laura continues at the same constant speed. Laura sees an accident 40 m ahead, so she stops pedalling and applies the brakes.
   She experiences a total resistance force of 40 N
   Laura and her cycle have a combined mass of 64 kg
   (b) (i) Determine whether Laura stops before reaching the accident. Fully justify your answer.
   [0pt] [4 marks]
   (b) (ii) State one assumption you have made that could affect your answer to part (b)(i).

6. A remote-controlled toy car is moving over a horizontal surface. It moves in a straight line through a point \(A\). The toy is initially at the point with displacement 3 metres from \(A\). Its velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), at time \(t\) seconds is defined by $$v = 0.06 \left( 2 + t - t ^ { 2 } \right)$$

1. Find an expression for the displacement, \(r\) metres, of the toy from \(A\) at time \(t\) seconds.
2. In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) At time \(t = 2\) seconds, the toy launches a ball which travels directly upwards with initial speed \(3.43 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Find the time taken for the ball to reach its highest point. Name: \section*{U8th AS LEVEL Single Mathematics Assessment
   Statistics
   18 \({ } ^ { \text {th } }\) January 2021 } Instructions
   Information
   \section*{Formulae} \section*{AS Level Mathematics A (H230)} \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 \({ } ^ { n } \mathrm { C } _ { r } = { } _ { n } \mathrm { C } _ { r } = \binom { n } { r } = \frac { n ! } { r ! ( n - r ) ! }\) \section*{Differentiation from first principles} $$\mathrm { f } ^ { \prime } ( x ) = \lim _ { h \rightarrow 0 } \frac { \mathrm { f } ( x + h ) - \mathrm { f } ( x ) } { h }$$ \section*{Standard deviation} $$\sqrt { \frac { \sum ( x - \bar { x } ) ^ { 2 } } { n } } = \sqrt { \frac { \sum x ^ { 2 } } { n } - \bar { x } ^ { 2 } } \text { or } \sqrt { \frac { \sum f ( x - \bar { x } ) ^ { 2 } } { \sum f } } = \sqrt { \frac { \sum f x ^ { 2 } } { \sum f } - \bar { x } ^ { 2 } }$$ \section*{The binomial distribution} If \(X \sim \mathrm {~B} ( n , p )\) then \(P ( X = x ) = \binom { n } { x } p ^ { x } ( 1 - p ) ^ { n - x }\), mean of \(X\) is \(n p\), variance of \(X\) is \(n p ( 1 - p )\) \section*{Kinematics} \(v = u + a t\) \(s = u t + \frac { 1 } { 2 } a t ^ { 2 }\) \(s = \frac { 1 } { 2 } ( u + v ) t\) \(v ^ { 2 } = u ^ { 2 } + 2 a s\) \(s = v t - \frac { 1 } { 2 } a t ^ { 2 }\) 1. The table below shows the probability distribution for a discrete random variable \(X\).

   \(\boldsymbol { x }\)	0	1	2	3	4 or more
   \(\mathbf { P } ( \boldsymbol { X } = \boldsymbol { x } )\)	0.35	0.25	\(k\)	0.14	0.1

   Find the value of \(k\).
   2. Given that \(\sum x = 364 , \sum x ^ { 2 } = 19412 , n = 10\), find \(\sigma\), the standard deviation of \(X\).
   3. Nicola, a darts player, is practising hitting the bullseye. She knows from previous experience that she has a probability of 0.3 of hitting the bullseye with each dart. Nicola throws eight practice darts.
   1. Using a binomial distribution, calculate the probability that she will hit the bullseye three or more times.
   2. Nicola throws eight practice darts on three different occasions. Calculate the probability that she will hit the bullseye three or more times on all three occasions.
   3. State two assumptions that are necessary for the distribution you have used in part (a) to be valid.
      4. Kevin is the Principal of a college.
      He wishes to investigate types of transport used by students to travel to college.
      There are 3200 students in the college and Kevin decides to survey 60 of them.
      Describe how he could obtain a simple random sample of size 60 from the 3200 students.
      5. Jennie is a piano teacher who teaches nine pupils.
      She records how many hours per week they practice the piano along with their most recent practical exam score.

      Student	Practice (hours per week)	Practical exam score (out of 100)
      Donovan	50	64
      Vazquez	6	71
      Higgins	3	55
      Begum	2.5	47
      Collins	1	80
      Coldbridge	4	61
      Nedbalek	4.5	65
      Carter	8	83
      White	11	92

      She plots a scatter diagram of this data, as shown below. \includegraphics[max width=\textwidth, alt={}, center]{1a7b75e9-eab4-4264-ab15-5292c504fb4d-09_880_1550_1361_246}
   4. Identify two possible outliers by name, giving a possible explanation for the position on the scatter diagram of each outlier.
      [0pt] [4 marks]
   5. Jennie discards the two outliers.
   6. 1. Describe the correlation shown by the scatter diagram for the remaining points. \section*{6.} A factory buys \(10 \%\) of its components from supplier \(A , 30 \%\) from supplier \(B\) and the rest from supplier \(C\). It is known that \(6 \%\) of the components it buys are faulty. Of the components bought from supplier \(A , 9 \%\) are faulty and of the components bought from supplier \(B , 3 \%\) are faulty.
   7. Find the percentage of components bought from supplier \(C\) that are faulty. A component is selected at random.
   8. Explain why the event "the component was bought from supplier \(B\) " is not statistically independent from the event "the component is faulty".

7. A biased spinner can only land on one of the numbers \(1,2,3\) or 4 . The random variable \(X\) represents the number that the spinner lands on after a single spin and \(\mathrm { P } ( X = r ) = \mathrm { P } ( X = r + 2 )\) for \(r = 1,2\) Given that \(\mathrm { P } ( X = 2 ) = 0.35\)

1. find the complete probability distribution of \(X\). Ambroh spins the spinner 60 times.
2. Find the probability that more than half of the spins land on the number 4 Give your answer to 3 significant figures.