Questions — OCR MEI (4301 questions)

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OCR MEI S1 2007 June Q3
3 The marks \(x\) scored by a sample of 56 students in an examination are summarised by $$n = 56 , \quad \Sigma x = 3026 , \quad \Sigma x ^ { 2 } = 178890 .$$
  1. Calculate the mean and standard deviation of the marks.
  2. The highest mark scored by any of the 56 students in the examination was 93 . Show that this result may be considered to be an outlier.
  3. The formula \(y = 1.2 x - 10\) is used to scale the marks. Find the mean and standard deviation of the scaled marks.
OCR MEI S1 2007 June Q4
4 A local council has introduced a recycling scheme for aluminium, paper and kitchen waste. 50 residents are asked which of these materials they recycle. The numbers of people who recycle each type of material are shown in the Venn diagram.
\includegraphics[max width=\textwidth, alt={}, center]{5e4f3310-b96e-43db-9b6d-61da3270db06-3_803_803_406_671} One of the residents is selected at random.
  1. Find the probability that this resident recycles
    (A) at least one of the materials,
    (B) exactly one of the materials.
  2. Given that the resident recycles aluminium, find the probability that this resident does not recycle paper. Two residents are selected at random.
  3. Find the probability that exactly one of them recycles kitchen waste.
OCR MEI S1 2007 June Q5
5 A GCSE geography student is investigating a claim that global warming is causing summers in Britain to have more rainfall. He collects rainfall data from a local weather station for 2001 and 2006. The vertical line chart shows the number of days per week on which some rainfall was recorded during the 22 weeks of summer 2001.
\includegraphics[max width=\textwidth, alt={}, center]{5e4f3310-b96e-43db-9b6d-61da3270db06-4_720_1557_443_296} Number of days per week with rain recorded in summer 2001
  1. Show that the median of the data is 4 , and find the interquartile range.
  2. For summer 2006 the median is 3 and the interquartile range is also 3. The student concludes that the data demonstrate that global warming is causing summer rainfall to decrease rather than increase. Is this a valid conclusion from the data? Give two brief reasons to justify your answer.
OCR MEI S1 2007 June Q6
6 In a phone-in competition run by a local radio station, listeners are given the names of 7 local personalities and are told that 4 of them are in the studio. Competitors phone in and guess which 4 are in the studio.
  1. Show that the probability that a randomly selected competitor guesses all 4 correctly is \(\frac { 1 } { 35 }\). Let \(X\) represent the number of correct guesses made by a randomly selected competitor. The probability distribution of \(X\) is shown in the table.
    \(r\)01234
    \(\mathrm { P } ( X = r )\)0\(\frac { 4 } { 35 }\)\(\frac { 18 } { 35 }\)\(\frac { 12 } { 35 }\)\(\frac { 1 } { 35 }\)
  2. Find the expectation and variance of \(X\).
OCR MEI S1 2007 June Q7
7 A screening test for a particular disease is applied to everyone in a large population. The test classifies people into three groups: 'positive', 'doubtful' and 'negative'. Of the population, \(3 \%\) is classified as positive, \(6 \%\) as doubtful and the rest negative. In fact, of the people who test positive, only \(95 \%\) have the disease. Of the people who test doubtful, \(10 \%\) have the disease. Of the people who test negative, \(1 \%\) actually have the disease. People who do not have the disease are described as 'clear'.
  1. Copy and complete the tree diagram to show this information.
    \includegraphics[max width=\textwidth, alt={}, center]{5e4f3310-b96e-43db-9b6d-61da3270db06-5_830_1157_845_536}
  2. Find the probability that a randomly selected person tests negative and is clear.
  3. Find the probability that a randomly selected person has the disease.
  4. Find the probability that a randomly selected person tests negative given that the person has the disease.
  5. Comment briefly on what your answer to part (iv) indicates about the effectiveness of the screening test. Once the test has been carried out, those people who test doubtful are given a detailed medical examination. If a person has the disease the examination will correctly identify this in \(98 \%\) of cases. If a person is clear, the examination will always correctly identify this.
  6. A person is selected at random. Find the probability that this person either tests negative originally or tests doubtful and is then cleared in the detailed medical examination.
OCR MEI S1 2007 June Q8
8 A multinational accountancy firm receives a large number of job applications from graduates each year. On average \(20 \%\) of applicants are successful. A researcher in the human resources department of the firm selects a random sample of 17 graduate applicants.
  1. Find the probability that at least 4 of the 17 applicants are successful.
  2. Find the expected number of successful applicants in the sample.
  3. Find the most likely number of successful applicants in the sample, justifying your answer. It is suggested that mathematics graduates are more likely to be successful than those from other fields. In order to test this suggestion, the researcher decides to select a new random sample of 17 mathematics graduate applicants. The researcher then carries out a hypothesis test at the \(5 \%\) significance level.
  4. (A) Write down suitable null and alternative hypotheses for the test.
    (B) Give a reason for your choice of the alternative hypothesis.
  5. Find the critical region for the test at the \(5 \%\) level, showing all of your calculations.
  6. Explain why the critical region found in part (v) would be unaltered if a \(10 \%\) significance level were used.
OCR MEI S1 2008 June Q1
1 In a survey, a sample of 44 fields is selected. Their areas ( \(x\) hectares) are summarised in the grouped frequency table.
Area \(( x )\)\(0 < x \leqslant 3\)\(3 < x \leqslant 5\)\(5 < x \leqslant 7\)\(7 < x \leqslant 10\)\(10 < x \leqslant 20\)
Frequency3813146
  1. Calculate an estimate of the sample mean and the sample standard deviation.
  2. Determine whether there could be any outliers at the upper end of the distribution.
OCR MEI S1 2008 June Q2
2 In the 2001 census, people living in Wales were asked whether or not they could speak Welsh. A resident of Wales is selected at random.
  • \(W\) is the event that this person speaks Welsh.
  • \(C\) is the event that this person is a child.
You are given that \(\mathrm { P } ( W ) = 0.20 , \mathrm { P } ( C ) = 0.17\) and \(\mathrm { P } ( W \cap C ) = 0.06\).
  1. Determine whether the events \(W\) and \(C\) are independent.
  2. Draw a Venn diagram, showing the events \(W\) and \(C\), and fill in the probability corresponding to each region of your diagram.
  3. Find \(\mathrm { P } ( W \mid C )\).
  4. Given that \(\mathrm { P } \left( W \mid C ^ { \prime } \right) = 0.169\), use this information and your answer to part (iii) to comment very briefly on how the ability to speak Welsh differs between children and adults.
OCR MEI S1 2008 June Q3
3 In a game of darts, a player throws three darts. Let \(X\) represent the number of darts which hit the bull's-eye. The probability distribution of \(X\) is shown in the table.
\(r\)0123
\(\mathrm { P } ( X = r )\)0.50.35\(p\)\(q\)
  1. (A) Show that \(p + q = 0.15\).
    (B) Given that the expectation of \(X\) is 0.67 , show that \(2 p + 3 q = 0.32\).
    (C) Find the values of \(p\) and \(q\).
  2. Find the variance of \(X\).
OCR MEI S1 2008 June Q4
4 A small business has 8 workers. On a given day, the probability that any particular worker is off sick is 0.05 , independently of the other workers.
  1. A day is selected at random. Find the probability that
    (A) no workers are off sick,
    (B) more than one worker is off sick.
  2. There are 250 working days in a year. Find the expected number of days in the year on which more than one worker is off sick.
OCR MEI S1 2008 June Q5
5 A psychology student is investigating memory. In an experiment, volunteers are given 30 seconds to try to memorise a number of items. The items are then removed and the volunteers have to try to name all of them. It has been found that the probability that a volunteer names all of the items is 0.35 . The student believes that this probability may be increased if the volunteers listen to the same piece of music while memorising the items and while trying to name them. The student selects 15 volunteers at random to do the experiment while listening to music. Of these volunteers, 8 name all of the items.
  1. Write down suitable hypotheses for a test to determine whether there is any evidence to support the student's belief, giving a reason for your choice of alternative hypothesis.
  2. Carry out the test at the \(5 \%\) significance level.
OCR MEI S1 2008 June Q6
6 In a large town, 79\% of the population were born in England, 20\% in the rest of the UK and the remaining 1\% overseas. Two people are selected at random. You may use the tree diagram below in answering this question.
\includegraphics[max width=\textwidth, alt={}, center]{be764df3-ff20-415d-9c5c-10edabf350de-4_946_1119_580_513}
  1. Find the probability that
    (A) both of these people were born in the rest of the UK,
    (B) at least one of these people was born in England,
    (C) neither of these people was born overseas.
  2. Find the probability that both of these people were born in the rest of the UK given that neither was born overseas.
  3. (A) Five people are selected at random. Find the probability that at least one of them was not born in England.
    (B) An interviewer selects \(n\) people at random. The interviewer wishes to ensure that the probability that at least one of them was not born in England is more than \(90 \%\). Find the least possible value of \(n\). You must show working to justify your answer.
OCR MEI S1 2008 June Q7
7 The histogram shows the age distribution of people living in Inner London in 2001.
\includegraphics[max width=\textwidth, alt={}, center]{be764df3-ff20-415d-9c5c-10edabf350de-5_814_1383_349_379} Data sourced from the 2001 Census, \href{http://www.statistics.gov.uk}{www.statistics.gov.uk}
  1. State the type of skewness shown by the distribution.
  2. Use the histogram to estimate the number of people aged under 25.
  3. The table below shows the cumulative frequency distribution.
    Age2030405065100
    Cumulative frequency (thousands)66012401810\(a\)24902770
    (A) Use the histogram to find the value of \(a\).
    (B) Use the table to calculate an estimate of the median age of these people. The ages of people living in Outer London in 2001 are summarised below.
    Age ( \(x\) years)\(0 \leqslant x < 20\)\(20 \leqslant x < 30\)\(30 \leqslant x < 40\)\(40 \leqslant x < 50\)\(50 \leqslant x < 65\)\(65 \leqslant x < 100\)
    Frequency (thousands)1120650770590680610
  4. Illustrate these data by means of a histogram.
  5. Make two brief comments on the differences between the age distributions of the populations of Inner London and Outer London.
  6. The data given in the table for Outer London are used to calculate the following estimates. Mean 38.5, median 35.7, midrange 50, standard deviation 23.7, interquartile range 34.4.
    The final group in the table assumes that the maximum age of any resident is 100 years. These estimates are to be recalculated, based on a maximum age of 105, rather than 100. For each of the five estimates, state whether it would increase, decrease or be unchanged.
OCR MEI FP3 2006 June Q1
1 Four points have coordinates \(\mathrm { A } ( - 2 , - 3,2 ) , \mathrm { B } ( - 3,1,5 ) , \mathrm { C } ( k , 5 , - 2 )\) and \(\mathrm { D } ( 0,9 , k )\).
  1. Find the vector product \(\overrightarrow { \mathrm { AB } } \times \overrightarrow { \mathrm { CD } }\).
  2. For the case when AB is parallel to CD ,
    (A) state the value of \(k\),
    (B) find the shortest distance between the parallel lines AB and CD ,
    (C) find, in the form \(a x + b y + c z + d = 0\), the equation of the plane containing AB and CD .
  3. When AB is not parallel to CD , find the shortest distance between the lines AB and CD , in terms of \(k\).
  4. Find the value of \(k\) for which the line AB intersects the line CD , and find the coordinates of the point of intersection in this case.
OCR MEI FP3 2006 June Q2
2 A surface has equation \(x ^ { 2 } - 4 x y + 3 y ^ { 2 } - 2 z ^ { 2 } - 63 = 0\).
  1. Find a normal vector at the point \(( x , y , z )\) on the surface.
  2. Find the equation of the tangent plane to the surface at the point \(\mathrm { Q } ( 17,4,1 )\).
  3. The point \(( 17 + h , 4 + p , 1 - h )\), where \(h\) and \(p\) are small, is on the surface and is close to Q . Find an approximate expression for \(p\) in terms of \(h\).
  4. Show that there is no point on the surface where the normal line is parallel to the \(z\)-axis.
  5. Find the two values of \(k\) for which \(5 x - 6 y + 2 z = k\) is a tangent plane to the surface.
OCR MEI FP3 2006 June Q3
3 The curve \(C\) has parametric equations \(x = 2 t ^ { 3 } - 6 t , y = 6 t ^ { 2 }\).
  1. Find the length of the arc of \(C\) for which \(0 \leqslant t \leqslant 1\).
  2. Find the area of the surface generated when the arc of \(C\) for which \(0 \leqslant t \leqslant 1\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
  3. Show that the equation of the normal to \(C\) at the point with parameter \(t\) is $$y = \frac { 1 } { 2 } \left( \frac { 1 } { t } - t \right) x + 2 t ^ { 2 } + t ^ { 4 } + 3$$
  4. Find the cartesian equation of the envelope of the normals to \(C\).
  5. The point \(\mathrm { P } ( 64 , a )\) is the centre of curvature corresponding to a point on \(C\). Find \(a\).
OCR MEI FP3 2006 June Q4
\(\mathbf { 4 }\) The group \(G\) consists of the 8 complex matrices \(\{ \mathbf { I } , \mathbf { J } , \mathbf { K } , \mathbf { L } , - \mathbf { I } , - \mathbf { J } , - \mathbf { K } , - \mathbf { L } \}\) under matrix multiplication, where $$\mathbf { I } = \left( \begin{array} { l l } 1 & 0
0 & 1 \end{array} \right) , \quad \mathbf { J } = \left( \begin{array} { r r } \mathrm { j } & 0
0 & - \mathrm { j } \end{array} \right) , \quad \mathbf { K } = \left( \begin{array} { r r } 0 & 1
- 1 & 0 \end{array} \right) , \quad \mathbf { L } = \left( \begin{array} { c c } 0 & \mathrm { j }
\mathrm { j } & 0 \end{array} \right)$$
  1. Copy and complete the following composition table for \(G\).
    IJKL-I-J-K\(- \mathbf { L }\)
    IIJKL-I-J-K-L
    JJ-IL-K-JI-LK
    KK-L-I
    LLK
    -I-I-J
    -J-JI
    -K-KL
    -L-L-K
    (Note that \(\mathbf { J K } = \mathbf { L }\) and \(\mathbf { K J } = - \mathbf { L }\).)
  2. State the inverse of each element of \(G\).
  3. Find the order of each element of \(G\).
  4. Explain why, if \(G\) has a subgroup of order 4, that subgroup must be cyclic.
  5. Find all the proper subgroups of \(G\).
  6. Show that \(G\) is not isomorphic to the group of symmetries of a square.
OCR MEI FP3 2006 June Q5
5 A local hockey league has three divisions. Each team in the league plays in a division for a year. In the following year a team might play in the same division again, or it might move up or down one division. This question is about the progress of one particular team in the league. In 2007 this team will be playing in either Division 1 or Division 2. Because of its present position, the probability that it will be playing in Division 1 is 0.6 , and the probability that it will be playing in Division 2 is 0.4 . The following transition probabilities apply to this team from 2007 onwards.
  • When the team is playing in Division 1, the probability that it will play in Division 2 in the following year is 0.2 .
  • When the team is playing in Division 2, the probability that it will play in Division 1 in the following year is 0.1 , and the probability that it will play in Division 3 in the following year is 0.3 .
  • When the team is playing in Division 3, the probability that it will play in Division 2 in the following year is 0.15 .
This process is modelled as a Markov chain with three states corresponding to the three divisions.
  1. Write down the transition matrix.
  2. Determine in which division the team is most likely to be playing in 2014.
  3. Find the equilibrium probabilities for each division for this team. In 2015 the rules of the league are changed. A team playing in Division 3 might now be dropped from the league in the following year. Once dropped, a team does not play in the league again.
    -The transition probabilities from Divisions 1 and 2 remain the same as before.
    • When the team is playing in Division 3, the probability that it will play in Division 2 in the following year is 0.15 , and the probability that it will be dropped from the league is 0.1 .
    The team plays in Division 2 in 2015.
    The new situation is modelled as a Markov chain with four states: 'Division1', 'Division 2', 'Division 3' and 'Out of league'.
  4. Write down the transition matrix which applies from 2015.
  5. Find the probability that the team is still playing in the league in 2020.
  6. Find the first year for which the probability that the team is out of the league is greater than 0.5 .
OCR MEI FP3 2008 June Q1
1 A tetrahedron ABCD has vertices \(\mathrm { A } ( - 3,5,2 ) , \mathrm { B } ( 3,13,7 ) , \mathrm { C } ( 7,0,3 )\) and \(\mathrm { D } ( 5,4,8 )\).
  1. Find the vector product \(\overrightarrow { \mathrm { AB } } \times \overrightarrow { \mathrm { AC } }\), and hence find the equation of the plane ABC .
  2. Find the shortest distance from \(D\) to the plane \(A B C\).
  3. Find the shortest distance between the lines AB and CD .
  4. Find the volume of the tetrahedron ABCD . The plane \(P\) with equation \(3 x - 2 z + 5 = 0\) contains the point B , and meets the lines AC and AD at E and F respectively.
  5. Find \(\lambda\) and \(\mu\) such that \(\overrightarrow { \mathrm { AE } } = \lambda \overrightarrow { \mathrm { AC } }\) and \(\overrightarrow { \mathrm { AF } } = \mu \overrightarrow { \mathrm { AD } }\). Deduce that E is between A and C , and that F is between A and D.
  6. Hence, or otherwise, show that \(P\) divides the tetrahedron ABCD into two parts having volumes in the ratio 4 to 17.
OCR MEI FP3 2008 June Q2
2 You are given \(\mathrm { g } ( x , y , z ) = 6 x z - ( x + 2 y + 3 z ) ^ { 2 }\).
  1. Find \(\frac { \partial \mathrm { g } } { \partial x } , \frac { \partial \mathrm {~g} } { \partial y }\) and \(\frac { \partial \mathrm { g } } { \partial z }\). A surface \(S\) has equation \(\mathrm { g } ( x , y , z ) = 125\).
  2. Find the equation of the normal line to \(S\) at the point \(\mathrm { P } ( 7 , - 7.5,3 )\).
  3. The point Q is on this normal line and is close to P . At \(\mathrm { Q } , \mathrm { g } ( x , y , z ) = 125 + h\), where \(h\) is small. Find the vector \(\mathbf { n }\) such that \(\overrightarrow { \mathrm { PQ } } = h \mathbf { n }\) approximately.
  4. Show that there is no point on \(S\) at which the normal line is parallel to the \(z\)-axis.
  5. Find the two points on \(S\) at which the tangent plane is parallel to \(x + 5 y = 0\).
OCR MEI FP3 2008 June Q3
3 The curve \(C\) has parametric equations \(x = 8 t ^ { 3 } , y = 9 t ^ { 2 } - 2 t ^ { 4 }\), for \(t \geqslant 0\).
  1. Show that \(\dot { x } ^ { 2 } + \dot { y } ^ { 2 } = \left( 18 t + 8 t ^ { 3 } \right) ^ { 2 }\). Find the length of the arc of \(C\) for which \(0 \leqslant t \leqslant 2\).
  2. Find the area of the surface generated when the arc of \(C\) for which \(0 \leqslant t \leqslant 2\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
  3. Show that the curvature at a general point on \(C\) is \(\frac { - 6 } { t \left( 4 t ^ { 2 } + 9 \right) ^ { 2 } }\).
  4. Find the coordinates of the centre of curvature corresponding to the point on \(C\) where \(t = 1\).
OCR MEI FP3 2008 June Q4
4 A binary operation * is defined on real numbers \(x\) and \(y\) by $$x * y = 2 x y + x + y$$ You may assume that the operation \(*\) is commutative and associative.
  1. Explain briefly the meanings of the terms 'commutative' and 'associative'.
  2. Show that \(x * y = 2 \left( x + \frac { 1 } { 2 } \right) \left( y + \frac { 1 } { 2 } \right) - \frac { 1 } { 2 }\). The set \(S\) consists of all real numbers greater than \(- \frac { 1 } { 2 }\).
  3. (A) Use the result in part (ii) to show that \(S\) is closed under the operation *.
    (B) Show that \(S\), with the operation \(*\), is a group.
  4. Show that \(S\) contains no element of order 2 . The group \(G = \{ 0,1,2,4,5,6 \}\) has binary operation ∘ defined by $$x \circ y \text { is the remainder when } x * y \text { is divided by } 7 \text {. }$$
  5. Show that \(4 \circ 6 = 2\). The composition table for \(G\) is as follows.
    \(\circ\)012456
    0012456
    1140625
    2205164
    4461502
    5526041
    6654210
  6. Find the order of each element of \(G\).
  7. List all the subgroups of \(G\).
OCR MEI FP3 2008 June Q5
5 Every day, a security firm transports a large sum of money from one bank to another. There are three possible routes \(A , B\) and \(C\). The route to be used is decided just before the journey begins, by a computer programmed as follows. On the first day, each of the three routes is equally likely to be used.
If route \(A\) was used on the previous day, route \(A\), \(B\) or \(C\) will be used, with probabilities \(0.1,0.4,0.5\) respectively.
If route \(B\) was used on the previous day, route \(A , B\) or \(C\) will be used, with probabilities \(0.7,0.2,0.1\) respectively.
If route \(C\) was used on the previous day, route \(A , B\) or \(C\) will be used, with probabilities \(0.1,0.6,0.3\) respectively. The situation is modelled as a Markov chain with three states.
  1. Write down the transition matrix \(\mathbf { P }\).
  2. Find the probability that route \(B\) is used on the 7th day.
  3. Find the probability that the same route is used on the 7th and 8th days.
  4. Find the probability that the route used on the 10th day is the same as that used on the 7th day.
  5. Given that \(\mathbf { P } ^ { n } \rightarrow \mathbf { Q }\) as \(n \rightarrow \infty\), find the matrix \(\mathbf { Q }\) (give the elements to 4 decimal places). Interpret the probabilities which occur in the matrix \(\mathbf { Q }\). The computer program is now to be changed, so that the long-run probabilities for routes \(A , B\) and \(C\) will become \(0.4,0.2\) and 0.4 respectively. The transition probabilities after routes \(A\) and \(B\) remain the same as before.
  6. Find the new transition probabilities after route \(C\).
  7. A long time after the change of program, a day is chosen at random. Find the probability that the route used on that day is the same as on the previous day. \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (OCR) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. OCR is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }
OCR MEI FP3 2010 June Q1
1 Four points have coordinates $$\mathrm { A } ( 3,8,27 ) , \quad \mathrm { B } ( 5,9,25 ) , \quad \mathrm { C } ( 8,0,1 ) \quad \text { and } \quad \mathrm { D } ( 11 , p , p ) ,$$ where \(p\) is a constant.
  1. Find the perpendicular distance from C to the line AB .
  2. Find \(\overrightarrow { \mathrm { AB } } \times \overrightarrow { \mathrm { CD } }\) in terms of \(p\), and show that the shortest distance between the lines AB and CD is $$\frac { 21 | p - 5 | } { \sqrt { 17 p ^ { 2 } - 2 p + 26 } }$$
  3. Find, in terms of \(p\), the volume of the tetrahedron ABCD .
  4. State the value of \(p\) for which the lines AB and CD intersect, and find the coordinates of the point of intersection in this case. Option 2: Multi-variable calculus
OCR MEI FP3 2010 June Q2
2 In this question, \(L\) is the straight line with equation \(\mathbf { r } = \left( \begin{array} { r } 2
1
- 1 \end{array} \right) + \lambda \left( \begin{array} { r } - 2
2
1 \end{array} \right)\), and \(\mathrm { g } ( x , y , z ) = \left( x y + z ^ { 2 } \right) \mathrm { e } ^ { x - 2 y }\).
  1. Find \(\frac { \partial \mathrm { g } } { \partial x } , \frac { \partial \mathrm {~g} } { \partial y }\) and \(\frac { \partial \mathrm { g } } { \partial z }\).
  2. Show that the normal to the surface \(\mathrm { g } ( x , y , z ) = 3\) at the point \(( 2,1 , - 1 )\) is the line \(L\). On the line \(L\), there are two points at which \(\mathrm { g } ( x , y , z ) = 0\).
  3. Show that one of these points is \(\mathrm { P } ( 0,3,0 )\), and find the coordinates of the other point Q .
  4. Show that, if \(x = - 2 \mu , y = 3 + 2 \mu , z = \mu\), and \(\mu\) is small, then $$\mathrm { g } ( x , y , z ) \approx - 6 \mu \mathrm { e } ^ { - 6 }$$ You are given that \(h\) is a small number.
  5. There is a point on \(L\), close to P , at which \(\mathrm { g } ( x , y , z ) = h\). Show that this point is approximately $$\left( \frac { 1 } { 3 } \mathrm { e } ^ { 6 } h , 3 - \frac { 1 } { 3 } \mathrm { e } ^ { 6 } h , - \frac { 1 } { 6 } \mathrm { e } ^ { 6 } h \right)$$
  6. Find the approximate coordinates of the point on \(L\), close to Q , at which \(\mathrm { g } ( x , y , z ) = h\).