Questions — Edexcel (10514 questions)

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Edexcel M3 2014 June Q3
11 marks Challenging +1.2
One end \(A\) of a light elastic string \(AB\), of modulus of elasticity \(mg\) and natural length \(a\), is fixed to a point on a rough plane inclined at an angle \(\theta\) to the horizontal. The other end \(B\) of the string is attached to a particle of mass \(m\) which is held at rest on the plane. The string \(AB\) lies along a line of greatest slope of the plane, with \(B\) lower than \(A\) and \(AB = a\). The coefficient of friction between the particle and the plane is \(\mu\), where \(\mu < \tan \theta\). The particle is released from rest.
  1. Show that when the particle comes to rest it has moved a distance \(2a(\sin \theta - \mu \cos \theta)\) down the plane. [6]
  2. Given that there is no further motion, show that \(\mu \geqslant \frac{1}{3} \tan \theta\). [5]
Edexcel M3 2014 June Q4
16 marks Challenging +1.2
\includegraphics{figure_2} A smooth sphere of radius \(a\) is fixed with a point \(A\) of its surface in contact with a fixed vertical wall. A particle is placed on the highest point of the sphere and is projected towards the wall and perpendicular to the wall with horizontal speed \(\sqrt{\frac{2ag}{5}}\), as shown in Figure 2. The particle leaves the surface of the sphere with speed \(V\).
  1. Show that \(V = \sqrt{\frac{4ag}{5}}\) [7]
The particle strikes the wall at the point \(X\).
  1. Find the distance \(AX\). [9]
Edexcel M3 2014 June Q5
13 marks Standard +0.8
\includegraphics{figure_3} A uniform solid right circular cylinder has height \(h\) and radius \(r\). The centre of one plane face is \(O\) and the centre of the other plane face is \(Y\). A cylindrical hole is made by removing a solid cylinder of radius \(\frac{1}{4}r\) and height \(\frac{1}{4}h\) from the end with centre \(O\). The axis of the cylinder removed is parallel to \(OY\) and meets the end with centre \(O\) at \(X\), where \(OX = \frac{1}{4}r\). One plane face of the cylinder removed coincides with the plane face through \(O\) of the original cylinder. The resulting solid \(S\) is shown in Figure 3.
  1. Show that the centre of mass of \(S\) is at a distance \(\frac{85h}{168}\) from the plane face containing \(O\). [7]
The solid \(S\) is freely suspended from \(O\). In equilibrium the line \(OY\) is inclined at an angle arctan(17) to the horizontal.
  1. Find \(r\) in terms of \(h\). [6]
Edexcel M3 2014 June Q6
14 marks Standard +0.8
A light elastic string, of natural length \(l\) and modulus of elasticity \(4mg\), has one end attached to a fixed point \(A\). The other end is attached to a particle \(P\) of mass \(m\). The particle hangs freely at rest in equilibrium at the point \(E\). The distance of \(E\) below \(A\) is \((l + e)\).
  1. Find \(e\) in terms of \(l\). [2]
At time \(t = 0\), the particle is projected vertically downwards from \(E\) with speed \(\sqrt{gl}\).
  1. Prove that, while the string is taut, \(P\) moves with simple harmonic motion. [5]
  2. Find the amplitude of the simple harmonic motion. [3]
  3. Find the time at which the string first goes slack. [4]
Edexcel S1 2023 June Q1
8 marks Moderate -0.8
The histogram shows the distances, in km, that 274 people travel to work. \includegraphics{figure_1} Given that 60 of these people travel between 10km and 20km to work, estimate
  1. the number of people who travel between 22km and 45km to work, [3]
  2. the median distance travelled to work by these 274 people, [2]
  3. the mean distance travelled to work by these 274 people. [3]
Edexcel S1 2023 June Q2
13 marks Moderate -0.3
Two students, Olive and Shan, collect data on the weight, \(w\) grams, and the tail length, \(t\) cm, of 15 mice. Olive summarised the data as follows \(S_tt = 5.3173\) \quad \(\sum w^2 = 6089.12\) \quad \(\sum tw = 2304.53\) \quad \(\sum w = 297.8\) \quad \(\sum t = 114.8\)
  1. Calculate the value of \(S_{ww}\) and the value of \(S_{tw}\) [3]
  2. Calculate the value of the product moment correlation coefficient between \(w\) and \(t\) [2]
  3. Show that the equation of the regression line of \(w\) on \(t\) can be written as $$w = -16.7 + 4.77t$$ [3]
  4. Give an interpretation of the gradient of the regression line. [1]
  5. Explain why it would not be appropriate to use the regression line in part (c) to estimate the weight of a mouse with a tail length of 2cm. [2]
Shan decided to code the data using \(x = t - 6\) and \(y = \frac{w}{2} - 5\)
  1. Write down the value of the product moment correlation coefficient between \(x\) and \(y\) [1]
  2. Write down an equation of the regression line of \(y\) on \(x\) You do not need to simplify your equation. [1]
Edexcel S1 2023 June Q3
9 marks Moderate -0.8
Jim records the length, \(l\) mm, of 81 salmon. The data are coded using \(x = l - 600\) and the following summary statistics are obtained. $$n = 81 \quad \sum x = 3711 \quad \sum x^2 = 475181$$
  1. Find the mean length of these salmon. [3]
  2. Find the variance of the lengths of these salmon. [2]
The weight, \(w\) grams, of each of the 81 salmon is recorded to the nearest gram. The recorded results for the 81 salmon are summarised in the box plot below. \includegraphics{figure_2}
  1. Find the maximum number of salmon that have weights in the interval $$4600 < w \leqslant 7700$$ [1]
Raj says that the box plot is incorrect as Jim has not included outliers. For these data an outlier is defined as a value that is more than \(1.5 \times\) IQR above the upper quartile \quad or \quad \(1.5 \times\) IQR below the lower quartile
  1. Show that there are no outliers. [3]
Edexcel S1 2023 June Q4
9 marks Moderate -0.8
A bag contains a large number of coloured counters. Each counter is labelled A, B or C 30% of the counters are labelled A 45% of the counters are labelled B The rest of the counters are labelled C It is known that 2% of the counters labelled A are red 4% of the counters labelled B are red 6% of the counters labelled C are red One counter is selected at random from the bag.
  1. Complete the tree diagram on the opposite page to illustrate this information. [2]
  2. Calculate the probability that the counter is labelled A and is not red. [2]
  3. Calculate the probability that the counter is red. [2]
  4. Given that the counter is red, find the probability that it is labelled C [3]
\includegraphics{figure_3}
Edexcel S1 2023 June Q5
13 marks Standard +0.3
A discrete random variable \(Y\) has probability function $$\mathrm{P}(Y = y) = \begin{cases} k(3 - y) & y = 1, 2 \\ k(y^2 - 8) & y = 3, 4, 5 \\ k & y = 6 \\ 0 & \text{otherwise} \end{cases}$$ where \(k\) is a constant.
  1. Show that \(k = \frac{1}{30}\) [2]
Find the exact value of
  1. P\((1 < Y \leqslant 4)\) [2]
  2. E\((Y)\) [2]
The random variable \(X = 15 - 2Y\)
  1. Calculate P\((Y \geqslant X)\) [3]
  2. Calculate Var\((X)\) [4]
Edexcel S1 2023 June Q6
9 marks Moderate -0.3
Three events \(A\), \(B\) and \(C\) are such that $$\mathrm{P}(A) = 0.1 \quad \mathrm{P}(B|A) = 0.3 \quad \mathrm{P}(A \cup B) = 0.25 \quad \mathrm{P}(C) = 0.5$$ Given that \(A\) and \(C\) are mutually exclusive
  1. find P\((A \cup C)\) [1]
  2. Show that P\((B) = 0.18\) [3]
Given also that \(B\) and \(C\) are independent,
  1. draw a Venn diagram to represent the events \(A\), \(B\) and \(C\) and the probabilities associated with each region. [5]
Edexcel S1 2023 June Q7
14 marks Standard +0.3
A machine squeezes apples to extract their juice. The volume of juice, \(J\) ml, extracted from 1 kg of apples is modelled by a normal distribution with mean \(\mu\) and standard deviation \(\sigma\) Given that \(\mu = 500\) and \(\sigma = 25\) use standardisation to
    1. show that P\((J > 510) = 0.3446\) [2]
    2. calculate the value of \(d\) such that P\((J > d) = 0.9192\) [3]
Zen randomly selects 5 bags each containing 1 kg of apples and records the volume of juice extracted from each bag of apples.
  1. Calculate the probability that each of the 5 bags of apples produce less than 510ml of juice. [2]
Following adjustments to the machine, the volume of juice, \(R\) ml, extracted from 1 kg of apples is such that \(\mu = 520\) and \(\sigma = k\) Given that P\((R < r) = 0.15\) and P\((R > 3r - 800) = 0.005\)
  1. find the value of \(r\) and the value of \(k\) [7]
Edexcel S1 2002 January Q1
4 marks Easy -1.8
  1. Explain briefly what you understand by
    1. a statistical experiment, [1]
    2. an event. [1]
  2. State one advantage and one disadvantage of a statistical model. [2]
Edexcel S1 2002 January Q2
7 marks Moderate -0.8
A meteorologist measured the number of hours of sunshine, to the nearest hour, each day for 100 days. The results are summarised in the table below.
Hours of sunshineDays
116
2-432
5-628
712
89
9-112
121
  1. On graph paper, draw a histogram to represent these data. [5]
  2. Calculate an estimate of the number of days that had between 6 and 9 hours of sunshine. [2]
Edexcel S1 2002 January Q3
7 marks Moderate -0.8
A discrete random variable \(X\) has the probability function shown in the table below.
\(x\)012
P(\(X = x\))\(\frac{1}{3}\)\(a\)\(\frac{2}{3} - a\)
  1. Given that E(\(X\)) = \(\frac{2}{3}\), find \(a\). [3]
  2. Find the exact value of Var (\(X\)). [3]
  3. Find the exact value of P(\(X \leq 15\)). [1]
Edexcel S1 2002 January Q4
10 marks Moderate -0.8
A contractor bids for two building projects. He estimates that the probability of winning the first project is 0.5, the probability of winning the second is 0.3 and the probability of winning both projects is 0.2.
  1. Find the probability that he does not win either project. [3]
  2. Find the probability that he wins exactly one project. [2]
  3. Given that he does not win the first project, find the probability that he wins the second. [2]
  4. By calculation, determine whether or not winning the first contract and winning the second contract are independent events. [3]
Edexcel S1 2002 January Q5
11 marks Standard +0.3
The duration of the pregnancy of a certain breed of cow is normally distributed with mean \(\mu\) days and standard deviation \(\sigma\) days. Only 2.5\% of all pregnancies are shorter than 235 days and 15\% are longer than 286 days.
  1. Show that \(\mu - 235 = 1.96\sigma\). [2]
  2. Obtain a second equation in \(\mu\) and \(\sigma\). [3]
  3. Find the value of \(\mu\) and the value of \(\sigma\). [4]
  4. Find the values between which the middle 68.3\% of pregnancies lie. [2]
Edexcel S1 2002 January Q6
17 marks Easy -1.2
Hospital records show the number of babies born in a year. The number of babies delivered by 15 male doctors is summarised by the stem and leaf diagram below. Babies \quad (4|5 means 45) \quad Totals 0 \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad (0) 1|9 \quad \quad \quad \quad \quad \quad \quad \quad \quad (1) 2|1 6 7 7 \quad \quad \quad \quad \quad \quad (4) 3|2 2 3 4 8 \quad \quad \quad \quad \quad (5) 4|5 \quad \quad \quad \quad \quad \quad \quad \quad \quad (1) 5|1 \quad \quad \quad \quad \quad \quad \quad \quad \quad (1) 6|0 \quad \quad \quad \quad \quad \quad \quad \quad \quad (1) 7 \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad (0) 8|6 7 \quad \quad \quad \quad \quad \quad \quad \quad (2)
  1. Find the median and inter-quartile range of these data. [3]
  2. Given that there are no outliers, draw a box plot on graph paper to represent these data. Start your scale at the origin. [4]
  3. Calculate the mean and standard deviation of these data. [5]
The records also contain the number of babies delivered by 10 female doctors. 34 \quad 30 \quad 20 \quad 15 \quad 6 32 \quad 26 \quad 19 \quad 11 \quad 4 The quartiles are 11, 19.5 and 30.
  1. Using the same scale as in part (b) and on the same graph paper draw a box plot for the data for the 10 female doctors. [3]
  2. Compare and contrast the box plots for the data for male and female doctors. [2]
Edexcel S1 2002 January Q7
19 marks Moderate -0.3
A number of people were asked to guess the calorific content of 10 foods. The mean \(s\) of the guesses for each food and the true calorific content \(t\) are given in the table below.
Food\(t\)\(s\)
Packet of biscuits170420
1 potato90160
1 apple80110
Crisp breads1070
Chocolate bar260360
1 slice white bread75135
1 slice brown bread60115
Portion of beef curry270350
Portion of rice pudding165390
Half a pint of milk160200
[You may assume that \(\Sigma t = 1340\), \(\Sigma s = 2310\), \(\Sigma ts = 396775\), \(\Sigma t^2 = 246050\), \(\Sigma s^2 = 694650\).]
  1. Draw a scatter diagram, indicating clearly which is the explanatory (independent) and which is the response (dependent) variable. [3]
  2. Calculate, to 3 significant figures, the product moment correlation coefficient for the above data. [7]
  3. State, with a reason, whether or not the value of the product moment correlation coefficient changes if all the guesses are 50 calories higher than the values in the table. [2]
The mean of the guesses for the portion of rice pudding and for the packet of biscuits are outside the linear relation of the other eight foods.
  1. Find the equation of the regression line of \(s\) on \(t\) excluding the values for rice pudding and biscuits. [3]
[You may now assume that \(S_{tt} = 72587\), \(S_{st} = 63671.875\), \(\bar{t} = 125.625\), \(\bar{s} = 187.5\).]
  1. Draw the regression line on your scatter diagram. [2]
  2. State, with a reason, what the effect would be on the regression line of including the values for a portion of rice pudding and a packet of biscuits. [2]
Edexcel S1 2010 January Q1
5 marks Easy -1.3
A jar contains 2 red, 1 blue and 1 green bead. Two beads are drawn at random from the jar without replacement.
  1. In the space below, draw a tree diagram to illustrate all the possible outcomes and associated probabilities. State your probabilities clearly. [3]
  2. Find the probability that a blue bead and a green bead are drawn from the jar. [2]
Edexcel S1 2010 January Q2
9 marks Easy -1.2
The 19 employees of a company take an aptitude test. The scores out of 40 are illustrated in the stem and leaf diagram below. \(2|6\) means a score of 26 \begin{align} 0 & | 7 & (1)
1 & | 88 & (2)
2 & | 4468 & (4)
3 & | 2333459 & (7)
4 & | 00000 & (5) \end{align} Find
  1. the median score, [1]
  2. the interquartile range. [3]
The company director decides that any employees whose scores are so low that they are outliers will undergo retraining. An outlier is an observation whose value is less than the lower quartile minus 1.0 times the interquartile range.
  1. Explain why there is only one employee who will undergo retraining. [2]
  2. On the graph paper on page 5, draw a box plot to illustrate the employees' scores. [3]
Edexcel S1 2010 January Q3
11 marks Moderate -0.8
The birth weights, in kg, of 1500 babies are summarised in the table below.
Weight (kg)Midpoint, \(x\)kgFrequency, \(f\)
\(0.0 - 1.0\)\(0.50\)\(1\)
\(1.0 - 2.0\)\(1.50\)\(6\)
\(2.0 - 2.5\)\(2.25\)\(60\)
\(2.5 - 3.0\)\(280\)
\(3.0 - 3.5\)\(3.25\)\(820\)
\(3.5 - 4.0\)\(3.75\)\(320\)
\(4.0 - 5.0\)\(4.50\)\(10\)
\(5.0 - 6.0\)\(3\)
[You may use \(\sum fx = 4841\) and \(\sum fx^2 = 15889.5\)]
  1. Write down the missing midpoints in the table above. [2]
  2. Calculate an estimate of the mean birth weight. [2]
  3. Calculate an estimate of the standard deviation of the birth weight. [3]
  4. Use interpolation to estimate the median birth weight. [2]
  5. Describe the skewness of the distribution. Give a reason for your answer. [2]
Edexcel S1 2010 January Q4
9 marks Moderate -0.3
There are 180 students at a college following a general course in computing. Students on this course can choose to take up to three extra options. 112 take systems support, 70 take developing software, 81 take networking, 35 take developing software and systems support, 28 take networking and developing software, 40 take systems support and networking, 4 take all three extra options.
  1. In the space below, draw a Venn diagram to represent this information. [5]
A student from the course is chosen at random. Find the probability that this student takes
  1. none of the three extra options, [1]
  2. networking only. [1]
Students who want to become technicians take systems support and networking. Given that a randomly chosen student wants to become a technician,
  1. find the probability that this student takes all three extra options. [2]
Edexcel S1 2010 January Q5
10 marks Moderate -0.8
The probability function of a discrete random variable \(X\) is given by $$p(x) = kx^2 \quad x = 1, 2, 3$$ where \(k\) is a positive constant.
  1. Show that \(k = \frac{1}{14}\) [2]
Find
  1. P\((X \geq 2)\) [2]
  2. E\((X)\) [2]
  3. Var\((1-X)\) [4]
Edexcel S1 2010 January Q6
18 marks Moderate -0.8
The blood pressures, \(p\) mmHg, and the ages, \(t\) years, of 7 hospital patients are shown in the table below.
PatientABCDEFG
\(t\)42744835562660
\(p\)981301208818280135
[\(\sum t = 341\), \(\sum p = 833\), \(\sum t^2 = 18181\), \(\sum p^2 = 106397\), \(\sum tp = 42948\)]
  1. Find \(S_{tt}\), \(S_{pp}\) and \(S_t\) for these data. [4]
  2. Calculate the product moment correlation coefficient for these data. [3]
  3. Interpret the correlation coefficient. [1]
  4. On the graph paper on page 17, draw the scatter diagram of blood pressure against age for these 7 patients. [2]
  5. Find the equation of the regression line of \(p\) on \(t\). [4]
  6. Plot your regression line on your scatter diagram. [2]
  7. Use your regression line to estimate the blood pressure of a 40 year old patient. [2]
Edexcel S1 2010 January Q7
13 marks Standard +0.3
The heights of a population of women are normally distributed with mean \(\mu\) cm and standard deviation \(\sigma\) cm. It is known that 30% of the women are taller than 172 cm and 5% are shorter than 154 cm.
  1. Sketch a diagram to show the distribution of heights represented by this information. [3]
  2. Show that \(\mu = 154 + 1.6449\sigma\). [3]
  3. Obtain a second equation and hence find the value of \(\mu\) and the value of \(\sigma\). [4]
A woman is chosen at random from the population.
  1. Find the probability that she is taller than 160 cm. [3]