8.05e Stationary points: where partial derivatives are zero

3 The surface \(S\) has equation \(z = f ( x , y )\), where \(f ( x , y ) = 4 x ^ { 2 } y - 6 x y ^ { 2 } - \frac { 1 } { 12 } x ^ { 4 }\) for all real values of \(x\) and \(y\). You are given that \(S\) has a stationary point at the origin, \(O\), and a second stationary point at the point \(P ( a , b , c )\), where \(\mathrm { c } = \mathrm { f } ( \mathrm { a } , \mathrm { b } )\). \begin{enumerate}[label=(\alph*)] \item Determine the values of \(a , b\) and \(c\). \item Throughout this part, take the values of \(a\) and \(b\) to be those found in part (a).

1. Evaluate \(\mathrm { f } _ { x }\) at the points \(\mathrm { U } _ { 1 } ( \mathrm { a } - 0.1 , \mathrm {~b} , \mathrm { f } ( \mathrm { a } - 0.1 , \mathrm {~b} ) )\) and \(\mathrm { U } _ { 2 } ( \mathrm { a } + 0.1 , \mathrm {~b} , \mathrm { f } ( \mathrm { a } + 0.1 , \mathrm {~b} ) )\).
2. Evaluate \(\mathrm { f } _ { y }\) at the points \(\mathrm { V } _ { 1 } ( \mathrm { a } , \mathrm { b } - 0.1 , \mathrm { f } ( \mathrm { a } , \mathrm { b } - 0.1 ) )\) and \(\mathrm { V } _ { 2 } ( \mathrm { a } , \mathrm { b } + 0.1 , \mathrm { f } ( \mathrm { a } , \mathrm { b } + 0.1 ) )\).
3. Use the answers to parts (b)(i) and (b)(ii) to sketch the portions of the sections of \(S\), given by

5 A trading company deals in two goods. The formula used to estimate \(z\), the total weekly cost to the company of trading the two goods, in tens of thousands of pounds, is \(z = 0.9 x + \frac { 0.096 y } { x } - x ^ { 2 } y ^ { 2 }\),
where \(x\) and \(y\) are the masses, in thousands of tonnes, of the two goods. You are given that \(x > 0\) and \(y > 0\).

1. In the first week of trading, it was found that the values of \(x\) and \(y\) corresponded to the stationary value of \(z\). Determine the total cost to the company for this week.
2. For the second week, the company intends to make a small change in either \(x\) or \(y\) in order to reduce the total weekly cost. Determine whether the company should change \(x\) or \(y\). (You are not expected to say by how much the company should reduce its costs.)

4 Let \(S\) be the set \(\{ 16,36,57,76,96 \}\) and \(\times _ { H }\) the operation of multiplication modulo 100 .

1. Given that \(a\) and \(b\) are odd positive integers, show that \(( 10 a + 6 ) ( 10 b + 6 )\) can also be written in the form \(10 n + 6\) for some odd positive integer \(n\).
2. Construct the multiplication table for \(\left( S , \times _ { H } \right)\) and show that it is a group. [You may use the result that \(\times _ { H }\) is associative on \(S\).]
3. Write down all generators of \(\left( S , \times _ { H } \right)\). Let \(\mathrm { f } ( x , y ) = x ^ { 3 } + y ^ { 3 } - 2 x y + 1\). The surface \(S\) has equation \(z = \mathrm { f } ( x , y )\).
4. (a) Find \(\mathrm { f } _ { x }\) and \(\mathrm { f } _ { y }\).
   (b) Show that f has a stationary point at \(P ( 0,0,1 )\) and find the coordinates of the second stationary point, \(Q\), of \(z\).
5. (a) State the equation of the plane of symmetry of the surface \(S\).
   (b) By considering the intersections of \(S\) at \(P\) with the planes \(y = x\) and \(y = - x\), show that \(P\) is a saddle-point of \(S\). A customer takes out a loan of \(\pounds P\) from a bank at an annual interest rate of \(6.32 \%\). Interest is charged monthly at an equivalent monthly interest rate. This interest is added to the outstanding amount of the loan at the end of each month, and then the customer makes a fixed monthly payment of \(\pounds 960\) in order to reduce the outstanding amount of the loan. Let \(L _ { n }\) denote the outstanding amount of the loan at the end of month \(n\) after the fixed payment has been made, with \(L _ { 0 } = P\).
6. Explain how the outstanding amount of the loan from one month to the next is modelled by the recurrence relation $$L _ { n + 1 } = 1.00512 L _ { n } - 960 ,$$ with \(L _ { 0 } = P , n \geq 0\).
7. Solve, in terms of \(n\) and \(P\), the first order recurrence relation given in part (i).
8. Given that the loan is fully repaid after 10 years, evaluate \(P\) (to the nearest whole number).
9. Comment on the practicality of the model.
10. Let \(N = 10 a + b\) and \(M = a - 5 b\) where \(a\) and \(b\) are integers such that \(a \geq 1\) and \(0 \leq b \leq 9\). \(N\) is to be tested for divisibility by 17 .
    (a) Prove that \(17 \mid N\) if and only if \(17 \mid M\).
    (b) Demonstrate step-by-step how an algorithm based on these forms can be used to show that \(17 \mid 4097\).
11. (a) Show that, for \(n \geq 2\), any number of the form \(1001 _ { n }\) is composite.
    (b) Given that \(n\) is a positive even number, provide a counter-example to show that the statement "any number of the form \(10001 _ { n }\) is prime" is false. \section*{Copyright Information:} }{www.ocr.org.uk}) after the live examination series.
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    Unit Y535 Additional Pure Mathematics SAMPLE MARK SCHEME Duration: 1 hour 15 minutes MAXIMUM MARK 60 \section*{DRAFT} \section*{Text Instructions} \section*{1. Annotations and abbreviations} \section*{2. Subject-specific Marking Instructions for AS Level Further Mathematics A} Annotations should be used whenever appropriate during your marking. The A, M and B annotations must be used on your standardisation scripts for responses that are not awarded either 0 or full marks. It is vital that you annotate standardisation scripts fully to show how the marks have been awarded. For subsequent marking you must make it clear how you have arrived at the mark you have awarded. An element of professional judgement is required in the marking of any written paper. Remember that the mark scheme is designed to assist in marking incorrect solutions. Correct solutions leading to correct answers are awarded full marks but work must not be judged on the answer alone, and answers that are given in the question, especially, must be validly obtained; key steps in the working must always be looked at and anything unfamiliar must be investigated thoroughly. Correct but unfamiliar or unexpected methods are often signalled by a correct result following an apparently incorrect method. Such work must be carefully assessed. When a candidate adopts a method which does not correspond to the mark scheme, escalate the question to your Team Leader who will decide on a course of action with the Principal Examiner.
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    The following types of marks are available. \section*{M} A suitable method has been selected and applied in a manner which shows that the method is essentially understood. Method marks are not usually lost for numerical errors, algebraic slips or errors in units. However, it is not usually sufficient for a candidate just to indicate an intention of using some method or just to quote a formula; the formula or idea must be applied to the specific problem in hand, e.g. by substituting the relevant quantities into the formula. In some cases the nature of the errors allowed for the award of an M mark may be specified. \section*{A} Accuracy mark, awarded for a correct answer or intermediate step correctly obtained. Accuracy marks cannot be given unless the associated Method mark is earned (or implied). Therefore M0 A1 cannot ever be awarded. \section*{B} Mark for a correct result or statement independent of Method marks. \section*{E} Mark for explaining a result or establishing a given result. This usually requires more working or explanation than the establishment of an unknown result.
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    e The abbreviation FT implies that the A or B mark indicated is allowed for work correctly following on from previously incorrect results. Otherwise, A and B marks are given for correct work only - differences in notation are of course permitted. A (accuracy) marks are not given for answers obtained from incorrect working. When A or B marks are awarded for work at an intermediate stage of a solution, there may be various alternatives that are equally acceptable. In such cases, what is acceptable will be detailed in the mark scheme. If this is not the case please, escalate the question to your Team Leader who will decide on a course of action with the Principal Examiner.
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    f Unless units are specifically requested, there is no penalty for wrong or missing units as long as the answer is numerically correct and expressed either in SI or in the units of the question (e.g. lengths will be assumed to be in metres unless in a particular question all the lengths are in km , when this would be assumed to be the unspecified unit.) We are usually quite flexible about the accuracy to which the final answer is expressed; over-specification is usually only penalised where the scheme explicitly says so. When a value is given in the paper only accept an answer correct to at least as many significant figures as the given value. This rule should be applied to each case. When a value is not given in the paper accept any answer that agrees with the correct value to 2 s.f. Follow through should be used so that only one mark is lost for each distinct accuracy error, except for errors due to premature approximation which should be penalised only once in the examination. There is no penalty for using a wrong value for \(g\). E marks will be lost except when results agree to the accuracy required in the question. Rules for replaced work: if a candidate attempts a question more than once, and indicates which attempt he/she wishes to be marked, then examiners should do as the candidate requests; if there are two or more attempts at a question which have not been crossed out, examiners should mark what appears to be the last (complete) attempt and ignore the others. NB Follow these maths-specific instructions rather than those in the assessor handbook.
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    M = Modelling

A surface \(S\) is defined by \(z = 4x^2 + 4y^2 - 4x + 8y + 11\).

1. Show that the point P\((0.5, -1, 6)\) is the only stationary point on \(S\). [2]
2. 1. On the axes in the Printed Answer Booklet, draw a sketch of the contour of the surface corresponding to \(z = 42\). [2]
   2. By using the sketch in part (b)(i), deduce that P must be a minimum point on \(S\). [3]
3. In the section of \(S\) corresponding to \(y = c\), the minimum value of \(z\) occurs at the point where \(x = a\) and \(z = 22\). Find all possible values of \(a\) and \(c\). [4]

In this question you must show detailed reasoning. A surface \(S\) is defined by \(z = \mathrm{f}(x, y)\) where \(\mathrm{f}(x, y) = x^3 + x^2 y - 2y^2\).

1. On the coordinate axes in the Printed Answer Booklet, sketch the section \(z = \mathrm{f}(2, y)\) giving the coordinates of any turning points and any points of intersection with the axes. [4]
2. Find the stationary points on \(S\). [7]

The function \(w = f(x, y, z)\) is given by \(f(x, y, z) = x^2yz + 2xy^2z + 3xyz^2 - 24xyz\), for \(x, y, z \neq 0\).

1. 1. Find
   2. Hence find the values of \(a\), \(b\), \(c\) and \(d\) for which \(w\) has a stationary value when \(d = f(a, b, c)\). [5]
2. You are given that this stationary value is a local minimum of \(w\). Find values of \(x\), \(y\) and \(z\) which show that it is not a global minimum of \(w\). [2]

A surface \(S\) has equation \(z = f(x, y)\), where \(f(x, y) = 2x^2 - y^2 + 3xy + 17y\). It is given that \(S\) has a single stationary point, \(P\).

1. Determine the coordinates, and the nature, of \(P\). [8]
2. Find the equation of the tangent plane to \(S\) at the point \(Q(1, 2, 38)\). [2]