St Peter the Apostle High. Mathematics Dept.

Transcription

1 St Peter the postle High Mathematics Dept. Higher Prelim Revision Paper I - Non~calculator Time allowed - hour 0 minutes FORMULE LIST Circle: The equation g f c 0 represents a circle centre ( g, f ) and radius g f c. The equation ( a) ( b) r represents a circle centre ( a, b ) and radius r. Trigonometric formulae: sin cos sin cos sin cos cos sin cos cos sin sin sin cos cos cos sin sin

2 ll questions should be attempted. Two functions, defined on suitable domains, are f ( ) ( ) & g ( ). (a) Show that the composite function, h ( ) f g ( ), can be written in the form h( ) a b c, where a, b and c are constants, and state the value(s) of a, b and c. (b) Hence solve the equation h () 6, for, showing clearl that there is onl one solution.. Part of the line, L, with equation, is shown in the diagram. The line L is parallel to L and passes through the point (0,-). Point lies on the -ais. (a) (b) Establish the equation of line L and write down the coordinates of the point. Given that the line is perpendicular to O both lines, find, algebraicall, the L - coordinates of point. 5 L (c) Hence calculate the eact shortest distance between the lines L and L.. For what value of p, where p 0, does the equation ( p ) p p 0 have equal roots? 6. Given that sin and 6 cos, with angles and both being acute, show clearl that cos( ) The diagram shows part of the graph of f (). (,) O 5 Sketch the graph of f ( ) marking clearl the new positions of the highlighted points and stating their new coordinates.

3 6. function, f, is defined on a suitable domain as f (). (a) Differentiate f with respect to, epressing our answer with positive indices. (b) Hence find when f () circle, centre C(8, k), has the points P(,-) and Q on its circumference as shown. M(0,) is the mid-point of the chord PQ. Q C(8, k) (a) Find the coordinates of Q. (b) Given that radius CQ is horizontal, M(0, ) write down the value of k, the O P(, -) -coordinate of C. (c) Hence establish the equation of the circle. 8. sequence is defined b the recurrence relation U au 0, where a is a constant. n n (a) Given that U 0 0 and U 6, find a. (b) Find the value of S, if S U U. d 9. curve has as its derivative. d Given that the point (, 7) lies on this curve, epress in terms of. 0. function is given as f ( ) cos 8cos 6 for 0. (a) Epress the function in the form f ( ) a (cos b) c and write down the values of a, b and c. (b) Hence state the minimum value of this function and the corresponding replacement for. [ END OF QUESTION PPER ]

4 St Peter the postle High Mathematics Dept. Higher Prelim Revision Paper II Calculator (Contains Unit Topics Clearl marked) Time allowed - hour 0 minutes FORMULE LIST Circle: The equation g f c 0 represents a circle centre ( g, f ) and radius g f c. The equation ( a) ( b) r represents a circle centre ( a, b ) and radius r. Trigonometric formulae: sin cos sin cos sin cos cos sin cos cos sin sin sin cos cos cos sin sin

5 ll questions should be attempted. Triangle PQR has as its vertices P(-7,-), Q(,8) and R(9,-0) as shown. Q(,8) (a) Find the equation of side PR. (b) Find the equation of the altitude QS. O (c) Hence find the coordinates of S, P(-7,-) the point where the altitude QS meets side PR. S (d) Establish the equation of the circle which passes through the points Q, S and R. R(9,-0). recurrence relation is defined as u 0 75u. n n Given that U 0, find the difference between the limit of the sequence and the third term, U. 5. curve has as its equation ( 6) 8. Given that the line with equation 5 is a tangent to this curve, establish the coordinates of the point T, the point of contact between the curve and the line.. Part of the graph of the curve with equation is shown below. The curve passes through the point (-,0). - O Find, algebraicall, the coordinates of the points and. 7

6 5. The diagram below shows the graph of 6 cos sin for 0. The line with equation first intersects this curve at the point as shown. Unit Wave Function! O 6 cos Ө sin (a) Show that 6 cos sin cos( 5.66) (b) Find the -coordinate of the point, in radians, correct to decimal places. 6. The functions f and g, defined on suitable domains, are given as 5a f ( ) and g( ) a, where a is a constant. (a) Given that f ( a) g(), find the value of a, where a 0. (b) With a taking this value, find the rate of change of g. 7. small feeding trough is shown opposite. The end face has an ais of smmetr. 60cm Edge CD is perpendicular to the ais of smmetr. D C When the end face is rotated through 90 o and then halved along the ais of smmetr, shape C can be placed on a coordinate diagram as shown below. lies along the -ais with the curved edge C being part of the curve with equation 60. C (a) Establish the coordinates of and. 60 (b) (c) Hence calculate the area of shape C given that all the units are in centimetres. Given that the trough is a prism and measures 60cm from back to front, o calculate the volume of feed the trough can hold when full, giving our answer correct to the nearest litre.

7 8. circle has as its equation ( 9) ( ) 7. (, k) lies on the circle, where k 0. (a) Show that is the point (, 8). (b) Find the equation of the tangent to this circle at the point. (c) Show clearl that this tangent passes through the centre of the circle with equation The captain of a small pleasure boat wishes to take a group of passengers from one island to the net, a journe of 00 kilometres. The amount of fuel used is dependent upon the speed, v kilometres per hour, of the boat. (a) Given that the rate of fuel used is ( v ) gallons per hour, show clearl that the total fuel used, F, for this 00 kilometre journe is given b 00 F v gallons. v (b) Hence find the speed which keeps the amount of fuel used to a minimum and the amount of fuel needed, at this speed, for the voage. 6 [ END OF QUESTION PPER ]

8 Higher Prelim Revision Give mark for each Marking Scheme - Paper I Illustration(s) for awarding each mark. (a) ans: a, b, c marks for knowing g through f for correct substitution (algebra) epanding and simplifing for a, b and c (b) ans: marks solving to zero strateg - snthetic division finding root showing onl one solution (a) strateg f ( g( )) ( ) ( ) (or equivalent) f ( g( )) a, b, c (b) no solution (or equivalent). (a) ans: ; (,0) marks for gradient writing down equation of line establishing the coordinates of (b) ans: (,) for perpendicular gradient equation of line for strateg of a sstem for first coordinate 5 second coordinate 5 marks (c) ans: 7 units marks strateg + lengths to use in Pth. calculation to answer. ans: p 5 6 marks for discriminant statement for a, b and c substituting correctl simplifing 5 factorising and answer(s) 6 discarding (a) m 0 0 (b) m 0 ( ) or equiv. 5 Pupils ma attempt to step out using - gradient and then check if point satisfies equation - award marks on our discretion. (c) Pth + using and d 7 d 7 for equal roots b ac 0 a p, b p, c p ( p ) ( p ) p 0 (or equivalent) 00 p p 0 5 p (5 p ) 0 p 0 or 5 6 p 5 discard 0 and -5

9 Give mark for each. ans: Proof 6 marks strateg of drawing R.. triangles calculating missing sides b Pth. correct epansion substitution 5 calculation and simplifing surd 6 rationalising denom. to answer Illustration(s) for awarding each mark and cos( ) cos cos sin sin (or equivalent) 5. ans: sketch as opposite marks for units to the left for reflection in the -ais new positions of S.P.'s (-,0) and (0,-) and for new position of root (,0) - (0,-) 6. (a) ans: f ( ) marks for preparing to differentiate epanding brackets differentiating epress with positive indices (b) ans: marks for forming equation for to subject answer various was to solve use own discretion (a) f ( ) ( ) f ( ) f ( ) f ( ) (b) 5 mult. b then 0 8 then 8 square both sides 6 alternative realise that then 8

10 Give mark for each 7. (a) ans: Q(-,6) mark answer (b) ans: k 6 mark answer (c) ans: ( 8) ( 6) 00 marks for strateg. radius and centre finding radius answer (a) (b) (c) Illustration(s) for awarding each mark stepping out to answer answer strateg r can be found from horiz. line but some pupils will use points P and C. r ( 8) ( 6) (a) ans: a 0 6 marks forming equation solving (b) ans: S 6 6 marks finding U finding S (a) 6 a (0) 0 0a 6 a 6 (b) U 0 6(6) S ans: 6 marks knows to integrate integrates correctl (with c) substitutes and finds c writes down answer ( ) d c 7 ( ) c c (a) ans: f ( ) (cos ) a, b, c marks common factor completing the square multipling back through answer (b) ans: min of at marks knowing minimum of knowing to solve to zero establishing answer no marks off if given in degrees (a) f ( ) (cos cos ) 6... (cos ) 6... (cos ) 6 f ( ) (cos ) a, b, c (b) min cos 0 cos (80) (90) Total 60 marks

11 Higher Prelim Revision Marking Scheme - Paper Give mark for each Illustration(s) for awarding each mark. (a) ans: (or equiv.) marks for gradient for equation of line (b) ans: marks knowing gradients mult. to - for gradient equation of altitude (c) ans: S(-,-) marks knowing to solve as a sstem sstem strateg. (subst. or elimin.) first coordinate second coordinate (d) ans: ( 6) ( ) 90 marks realising strateg of R.. QR = diam. finding centre calculating value of r equation of circle (a) m ( 9) (or equivalent) (b) if perpen. m m stated or implied m 8 ( ) (or equiv.) (c) attempts to substitute or eliminate 5 0 (d) strateg C 9 8, ( 0) ) C(6, ) ( r 9 90 ( 6) ( ) 90. ans: marks for finding U for U and U knowing how to find limit finding limit 5 calculating difference U 0 75() 6 U 0 75(6) 9 U 0 75(9) 5 b L (or equivalent) a L diff ans: T(7,9) marks knows to solve a sstem and sub. for simplifies first coordinate second coordinate 5 ( 6) ( 7)( 7) 0 7 (7) 5 9

12 Give mark for each. ans: (,0), (,-) 7 marks to find set up snth. division use -.. or other find coordinate of and hence for know to diff. and solve to 0 5 differentiate correctl 6 find coordinate of 7 find coordinate of Illustration(s) for awarding each mark set up snth. division for root ( )( ) 0, (,0) know S.P. d d 0 5 d d 0 6 ( ) 0 ( discard ) 7 () (, ) 5. (a) ans: 5.66, k marks for strateg and epansion finding alpha finding k Unit Wave Function! (a) 6 cos sin k cos( )... k cos cos k sin sin tan k ( 6) ( ) 9 (b) ans: 0 6 marks solving to finding value in radians knows to subtract to answer (b) cos( 5 66) (a) ans: a marks for writing down f(a) for g() for equating and factorising answer (b) ans: 5 marks for g when a for differentiation to answer a (a) f ( a) 5a g( ) a a 5a a a a 0 ( a )( a ) 0 a (other root discard, implied) (b) g ( ) 5 ( ) (or equiv.) g () 5

13 7. (a) ans: (0,0), (,0) marks for solving to zero factorising and roots stating finding (b) ans: Give mark for each 85 cm marks for setting up integral for integration substitution correct calculation to answer Illustration(s) for awarding each mark (a) (60 ) 0 (0 )(6 ) 0 0 or 6 (0,0) half wa between roots. ( 0 ( 6)) (,0) (for.. pupils ma diff. find -coordinate of S.P.) (b) = 0 ( 5 d 5 0 ) ) (0 ) ( 6 ) ( ) 85 ( (or equiv.) (c) ans: 0 litres knows to double area finds volume answers to nearest litre marks (c) face cm V cm V 0 litres (to nearest litre) 8. (a) ans: k 8 marks for subst. point in equation for simplifing to quadratic factorise and solve for k discarding (or implied) and answer (a) ( 9) ( k ) 7 k k 80 0 ( k 0)( k 8) 0, k 0 or 8 k 8 (b) ans: 8 marks for coordinates of centre gradient of radius gradient of tangent equation of tangent (c) ans: proof marks drawing out centre show point satisfies equation of tangent (b) (c) C(9,-) m r 9 m 8 T 8 ( ) (or equivalent) C(-,) () ( ) proved

14 9. (a) ans: proof marks for epression for time of journe simplifing to answer D 00 (a) T S v F T rate of fuel used v v v v (b) ans: v 0 km/h, 7 5 gallons 6 marks knows to diff and solve to zero differentiates correctl strateg for solving equation solves equation 5 uses Nature Table to confirm Minimum 6 substitutes 0 into F and calc. fuel used (b) at min F (v) 0 (stated or implied) 00 F ( v) 0 05v v v 0 ( v ) v 00 0 v 05v \ _ / 6 F (0 ) Total 70 marks