Advanced Higher Grade

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1 Practice Eamination A (Assessing Units & ) MATHEMATICS Advanced Higher Grade Time allowed - hours 0 minutes Read Carefully. Full credit will be given only where the solution contains appropriate working.. Calculators may be used in this paper.. Answers obtained by readings from scale drawings will not receive any credit. 4. This eamination paper contains questions graded at all levels. Pegasys 005

2 All questions should be attempted. Differentiate with respect to, simplifying your answer as far as possible: (a) + y tan (4) (b) y ln(sec ) (). Use Gaussian Elimination to solve the system + y 4z + y + z y z 6 (5) d n n. Prove by induction ( ) n for all positive integers, n. (5) 4. Using the substitution t, evaluate the integral t + t (6) 5. Find the coefficient of 5 in the epansion of 7 +. () 6. (a) Find partial fractions for ( + 9)( + ). (4) (b) Hence evaluate the integral ( ) ( + 9) +. () Pegasys 005

3 7. Suppose that and y are differentiable functions of t and that d y t + dy, t + t. Find (t) given that () 4. (6) y 8. PQ is a chord of the loop of the curve y (8 ), > 0. PQ is parallel to the y-ais. Calculate the maimum possible length of PQ. P Q (6) 9. (a) Find two numbers and y whose sum is 4 and whose product is 8. (4) (b) Plot the solutions on an Argand diagram. () 0. Use integration by parts to show that cos ( )cos + ( 6) sin + C. (5). (a) Find an epression for the sum of n terms of the series in its simplest form. (4) (b) If 4 S n, find the value of n. () 8 Pegasys 005

4 . An investor has 000 with which to open an account and plans to add a further 000 each year. All funds in the account will earn compound interest at a rate of 0% p.a.. Let (t) be the amount of money in the account at time t years. (a) Write down a first order differential equation representing the rate of change of money in the account each year. () (b) Hence show that t 0 ln ( ) 00. (7) (c) How many years would it take to save ? (). A function f () is defined by + f ( ). (a) Write down the equation of the vertical asymptote of f (). () + (b) For the function g(), show that there is a non-vertical asymptote and find its equation. () (c) Find the coordinates of the stationary points of g() and determine their nature. (5) (d) By first considering the graph of g(), sketch the graph of f () showing all its main features. (4) 4. The semi-circle y a is rotated about the -ais to generate a sphere. (a) Find an epression for the volume of the sphere. (8) (b) Find the volume of the sphere with equation y 5. () END OF QUESTION PAPER Pegasys 005

5 Marking Scheme - AH Practice Paper A (a) Give one mark for each Illustrations for awarding each mark dy ans: 4 marks + know how to differentiate tan - chain rule factor manipulating algebra answer in simplest form ( ) + + ( ) ( ) + + ( ) (b) ans: dy tan marks know how to differentiate log chain rule factor answer in simplest form. ans: (, -, ) 5 marks write system as an augmented matri with in top left-hand corner (optional) first modified system second modified system using back-substitution to find z using back-substitution to find and y sec sec tan tan z y -, Pegasys 005

6 Give one mark for each Illustrations for awarding each mark. ans: proof by induction 5 marks d show true for n LHS ( ) ; RHS So true when n state inductive hypothesis d k k Assume ( ) k consider the case for n k + d k + Consider ( ) d k k k k k carry out manipulation ( ) + k + k state conclusion ( k + ) k So, if the formula is valid for n, it is valid for n+. Since it is valid for n, it is therefore true for all n. 4. ans: ln 7 marks rewrite integral in terms of correct limits tidy up integral integrate evaluate limits manipulate surds final answer 5. ans: 560 marks correct general term put power of equal to 5 and solve for r calculate coefficient and + + ln ( +)] ln ( + ) ln + + ln + and ln ln r 7 7 r 7 4 r r 4r 5; r r r ( ) ln Pegasys 005

7 6(a) Give one mark for each Illustrations for awarding each mark 6 ans: + 4 marks know how to find partial fractions know how to find A, B and C finds A finds B and C 6(b) ans: 7 units 5 marks knows to epress integral in partial fractions and integrates terms correctly evaluates limits final answer 7. ans: (t) t + 6 marks knows formula for d dy finds ( ) d y in parametric form substitutes information into formula finds in simplest form integrates to find finds constant of integration A + B C A + B + C A 0 B 6 and C ( )( ) ( + 9) and tan + ln + 0 ( ( tan + ln tan ) + ln) 7 units d dy d y ( ) t + + t t + ( t) t + c () 4 ; c 8. ans: 8 units 6 marks knows to find ma. and min. turning points knows to use implicit differentiation differentiates correctly finds -coordinate of relevant turning point finds corresponding y-coordinates finds ma. distance dy ( 4 ) y -, 0 or and chooses from diagram y -4 or 4 8 Pegasys 005

8 Give one mark for each Illustrations for awarding each mark 9(a) ans: + i, i 4 marks set up system of equations use substitution to obtain quadratic use quadratic formula to solve quadratic correct answer 9(b) ans: Diagram marks Argand diagram correctly labelled both points plotted and labelled + y 4; y ± 6 4( )( 8) + i or i Im + i Re i 0. ans: Proof 5 marks first application of integration by parts second application of integration by parts knowing to use integ. by parts again third application of integration by parts answer in required form sin and sin sin [ cos + 6cos ] sin + cos 6cos sin + cos 6 sin 6cos + C ( ) cos + ( 6) sin + C (a) ans: n 4 marks correct ratio using correct formula substituting correctly into formula answer in simplest form r n a ( r ) Sn r n ( ) ( ) n n Pegasys 005

9 Give one mark for each Illustrations for awarding each mark (b) ans: n 5 marks use formula correctly manipulate formula answer 4 8 n n 4 4 n 4 n 5 (using logs or trial and error) (a) ans: marks amount of money going into account each year 0% (b) ans: t ln 7 marks 00 know to use method of separating variables separates variables correctly integrates LHS correctly integrates RHS correctly (incl. constant of integration) correct initial conditions finds correct value of C finds required solution and and 0 ln ( ) t + C 000 at t 0 C 0ln t 0ln 00 (c) ans: years marks substitute in value for answer t 0ln ln 6 years years (a) ans: mark states equation of vertical asymptote (b) ans: y marks knows to divide restating function correctly stating equation of asymptote + and ( ) y + Pegasys 005

10 Give one mark for each Illustrations for awarding each mark (c) ans: Ma at (0, -), Min at (, ) 5 marks dy knows to find dy knows to put 0 finds -coordinates finds y-coordinates determines nature of each by second derivative or nature table dy ( ) 0 ( ) 0 or (0, -), (, ) d y ; Ma at ( 0, ),Min at (, ) ( ) (d) ans: sketch 4 marks sketch showing all relevant points correctly shows how curve approaches asymptotes knows to reflect all parts of graph from below the -ais to above the -ais reflects correctly See sketch at end of marking scheme 4(a) ans: 4 a π 8 marks draws sketch showing semi-circle above -ais Roots of semi-circle at a and a knows how to find volume of revolution limits of integration as a and a applies formula correctly integrates correctly evaluates limits correct answer 4(b) ans: 5 6 units marks knows to put a 5 finds volume and and V V a a π a a π y ( a ) π [ a ] a a a ( ) [ ( ) ] a π a a π a ( a) 4 a π 4 π ( 5 ) 5 6 units -a a Total 00 Marks Pegasys 005

11 Sketch for question (d) y y (0, ) (, ) (0, -) This is g(), it is not technically part of the sketch of f(), however, no marks deducted if g() is shown Pegasys 005

G H. Extended Unit Tests B L L. Higher Still Advanced Higher Mathematics. (more demanding tests covering all levels) Contents. 3 Extended Unit Tests

G H. Extended Unit Tests B L L. Higher Still Advanced Higher Mathematics. (more demanding tests covering all levels) Contents. 3 Extended Unit Tests M A T H E M A T I C S H I G H E R Higher Still Advanced Higher Mathematics S T I L L Etended Unit Tests B (more demanding tests covering all levels) Contents 3 Etended Unit Tests Detailed marking schemes