MATHEMATICS Higher Grade - Paper I (Non~calculator)

Transcription

1 Prelim Eamination 005 / 006 (Assessing Units & ) MATHEMATICS Higher Grade - Paper I (Non~calculator) Time allowed - hour 0 minutes Read Carefully. Calculators may not be used in this paper.. Full credit will be given only where the solution contains appropriate working.. Answers obtained by readings from scale drawings will not receive any credit.. This eamination paper contains questions graded at all levels. Pegasys 005

2 FORMULAE LIST Circle: The equation + y + g + fy + c 0 represents a circle centre ( g, f ) and radius g + f c. The equation ( a) + ( y b) r represents a circle centre ( a, b ) and radius r. Trigonometric formulae: ( A ± B) ( A ± B) sin cos sina cosa sin Acos B ± cos Asin B cos Acos B m sin Asin B sin Acos A cos A sin A cos A sin A Pegasys 005

3 All questions should be attempted. Part of the line with equation + y 9 is shown in the diagram. B lies on this line y + y 9 and has coordinates (, k). B(, k) (a) Find the value of k. (b) Given that the line AB is perpendicular to the line + y 9, find the O equation of the line AB. (c) Hence write down the coordinates of A. A (d) Calculate the area of the shaded triangle.. (a) A function f has as its derivative f ( ) a a. Find a if the function has a stationary point at. (b) Hence find the rate of change of this function at and comment on your result.. A quadratic function, defined on a suitable domain, is given as f ( ) The diagram shows part of the graph of this quadratic function, y f (). The graph passes through the points P(,) and Q(,0) as shown.. y P(,) (a) Sketch the graph of y f ( ) + 6 marking clearly the image points of P and Q and stating their coordinates. (b) Given that g ( ) f ( ) + 6, write down a formula for g (). O Q(,0). Find a given that ( + ) d 0 a, where a < Find all the values of in the interval 0 π for which sin cos. 6 Pegasys 005

4 6. Two functions are defined on suitable domains as f ( ) + and g ( ) Given that the function h is such that h ( ) g( f ( )), epress h in the form h ( ) ( + a) + b, where a and b are integers, and hence write down the minimum value of h and the corresponding replacement for The diagram below shows two congruent circles which touch at a single point T. The circle, centre A, has as its equation + y 6 8y The line with equation y is the common tangent to the two circles through T. y + y 6 8y A y o T B (a) Show algebraically that T has coordinates (6,). 5 (b) Hence establish the the coordinates of B, the centre of the lower circle, and find the equation of this circle Find f ( ) when f ( ), epressing your answer with positive indices, and hence calculate the value of the gradient of the tangent to the curve y f () at What can you say about p if the equation, in, + p p has no real roots? 6 [ END OF QUESTION PAPER ] Pegasys 005

5 Higher Grade Prelim 005/006 Give mark for each Marking Scheme - Paper I Illustration(s) for awarding each mark. (a) ans: k mark sub. to answer (b) ans: y 7 (or equiv.) marks for gradient of line for gradient of AB for equation (c) ans: A(0,-7) answer (y intercept) (d) ans: Area 5 square units y intercept of top line length of base perpendicular length calculation to answer mark marks (a) + k 9 k (b) m m AB y ( ) (c) A(0,-7) (d) (0,) distance between y intercepts 0 y -ais to B units A bh 0 5 units. (a) ans: a marks knowing to solve deriv. to zero setting up synthetic division completing synth. div. solving equ. to zero and answer (b) ans: zero, another stat. point marks substituting for a and - calculation finds zero + conclusion ** pupils may complete part (a) by substitution. (a) at s.p. f ( ) 0 (stated or implied) -a -a 0 -a -a 0 6-a 6-a -a 6-8a 0 6-a 0, a (b).. f ( ) ( ) ()( ) ()( ) f ( ) 0 another stat. point. (a) ans: see sketch marks for reflecting for translation up 6 for annotating, coordinates (b) ans: g ( ) + 6 marks for strategy for answer (any equivalent form) (a) P'(,-6) (b) strategy g ( ) ( ) + 6 Q'(,6) Pegasys 005

6 Give mark for each. ans: a 6 5 marks integrating first term integrating nd term sub. in limits simplifying to quad. equation 5 solving and choosing answer Illustration(s) for awarding each mark.. + (8 + ) (a + a ) 0 a + a 0 5 ( a )( a + 6) 0 a or a 6 5. ans: π 7 π π π {, 6,, 6 } 6 marks knowing to solve to zero replacement factorising and solutions st angle from one solution 5 st angle from other solution 6 for remaining two angles sin + cos 0 sin cos + cos 0 cos (sin + ) 0 cos 0 or sin π 7π π π, 6 6. ans: h ( ) ( + ) +, h 6 marks for f into g for epansion and simplifying bracket term number term 5 for minimum value 6 for g ( f ( )) ( + ) + 6( + ) + h ( ) [ ( + ) ] min - Pegasys 005

7 Give mark for each 7. (a) ans: proof 5 marks know to solve a system combining equations simplifying to quad. for st coordinate 5 for nd coordinate (b) ans: B(9,-), ( 9) + ( y + ) 5 6 marks knowing T mid-pt between centres drawing out centre of top circle finding B knowing r the same 5 finding r 6 writing down equation of lower circle Illustration(s) for awarding each mark (a) set up a system (y) + y 6(y) 8y y 0y ( y )( y ) 0 y 5 () 6 (or equivalent) (b) strategy A(,9) A(,9) T(6,) B(9,-) stated or implied r r 5 r ( 9) + ( y + ) 5 8. ans: f ( ) +, 9 6 marks preparing to differentiate diff. st term diff. nd term writing with positive indices 5 substituting 6 answer 9. ans: 6 < p < 6 (or equivalent) 6 marks dealing with the fractions manipulation to quad. form discriminant statement for a, b and c 5 finding discriminant 6 solution from quad. inequat. f ( ) ( ) f ( ) + 5 f ( ) + ( ) 6 f ( ) strategy. p (or equiv.) + 9 p p for no real roots b ac < 0 a, b p, c 9 5 p 6 < < p < 6 b -ac -6 6 p Total 60 marks Pegasys 005

8 Prelim Eamination 005 / 006 (Assessing Units & ) MATHEMATICS Higher Grade - Paper II Time allowed - hour 0 minutes Read Carefully. Calculators may be used in this paper.. Full credit will be given only where the solution contains appropriate working.. Answers obtained by readings from scale drawings will not receive any credit.. This eamination paper contains questions graded at all levels. Pegasys 005

9 FORMULAE LIST Circle: The equation + y + g + fy + c 0 represents a circle centre ( g, f ) and radius g + f c. The equation ( a) + ( y b) r represents a circle centre ( a, b ) and radius r. Trigonometric formulae: ( A ± B) ( A ± B) sin cos sina cosa sin Acos B ± cos Asin B cos Acos B m sin Asin B sin Acos A cos A sin A cos A sin A Pegasys 005

10 All questions should be attempted. Triangle ABC has vertices A(-5,5), B(,-) and C(7,9). Q(,8) lies on AC and AM is a median of the triangle. y Q(,8) C(7,9) A(-5,5) R P(, k) M o B(,-) (a) Given that A, P and B are collinear, find the value of k. (b) Hence find the equation of PQ. (c) Find the coordinates of R, the point of intersection between the line PQ and the median AM. 5. A scientist studying a large colony of bats in a cave has noticed that the fall in the population over a number of years has followed the recurrence relation U n+ 0 75U n + 00, where n is the time in years and 00 is the average number of bats born each year during a concentrated breeding week. (a) He estimates the colony size at present to be 00 bats with the breeding week just over. Calculate the estimated bat population in years time immediately before that years breeding week. (b) The scientist knows that if in the long term the colony drops, at any time, below 700 individuals it is in serious trouble and will probably be unable to sustain itself. Is this colony in danger of etinction? Eplain your answer with words and appropriate working. Pegasys 005

11 . The diagram below shows a rectangle and an isosceles triangle. The letter p is a constant. All lengths are in centimetres. ( p) cm cm ( + ) cm ( + 6) cm (a) Taking A as the area of the rectangle, and A as the area of the triangle, show clearly that the difference between the two areas can be written in the form A A (8 p + ) 8p (b) Given that A A cm, establish the value of p, where p is >, for this equation to have only one solution for. 6 (c) Hence find when p takes this value.. The diagram shows part of the graph of the curve 8 with equation y +, 0. y y 8 + (a) (b) Find the coordinates of the stationary point A. 5 B Also shown is the line with equation y 7 8 which is a tangent to the curve at B. A y 7 8 Establish the coordinates of B. 6 o Pegasys 005

12 5. A function is defined on a suitable domain as f ( ) a, where a is a constant. (a) Find a formula for h given that h ( ) f ( f ( )). (b) Given now that h ( ) 8, find a. 6. The diagram, which is not drawn to scale, shows the cross-section of an iron bar. The units are in centimetres. When placed in the coordinate diagram the curved section of the rod has as its equation y 6 +. y 7 y 6 + o a b (a) Show algebraically that the values of a and b are and respectively. (b) Calculate the shaded area in square centimetres Triangle PQR is isosceles with PQ PR as shown. M is the midpoint of QR. QR 8units, PM units and RPM. P Q M 8 R (a) Show clearly that the eact value of sin is. Pegasys 005

13 (b) Hence, or otherwise, show that sin RPQ. 8. An old fashioned bell tent is in the shape of a cone. The tent has radius r, vertical height h and a slant height of feet as shown. h ft r (a) Write down an epression for r in terms of h. (b) Hence show clearly that a function, in terms of h, for the volume of this cone can be epressed as V ( h) 8π h π h [note: the formula for the volume of a cone is V π r h ] (c) Hence find the eact value of h, the height of the tent, which would maimise the volume of the tent. Justify your answer. 5 [ END OF QUESTION PAPER ] Pegasys 005

14 Higher Grade Prelim 005/006. (a) ans: k marks gradient of AB gradient of AP equating gradients finding k (b) ans: Give mark for each y marks gradient of PQ equation (c) ans: R(,) coordinates of M equation of median setting up a system finding first coordinate 5 finding nd coordinate 5 marks Marking Scheme - Paper II Illustration(s) for awarding each mark 5 (a) m m k AP 6 k 5 6 k AB (b) 8 m PQ y ( ) (c) M(9,) no mark given for gradient of AM m AM 7 y 7 ( 9) 7 y + 0 y 5 y. (a) ans: 0 bats (ignore rounding) marks first two lines of calculation lines and of calculations answer (b) ans: Colony is in danger. 600 prior to breeding week is less than 700 bats marks knows to calculate limit + knows formula calculates limit correctly knows to subtract 00 eplanation (a) Low High U 0 75(00) U 0 75(775) U 0 75(5 5) U 0 75(8 ) 0 0 (b) b L a 00 L low population prior to breeding week is less than 700 bats so colony in danger Pegasys 005

15 Give mark for each. (a) ans: proof marks area of rectangle area of triangle subtracting areas tidy up and common factor (b) ans: p 6 marks equating to zero discriminant statement a, b and c substitution 5 to quadratic form 6 answer (c) ans: marks substitution solving to answer. (a) ans: A(,6) 5 marks knowing and preparing to differentiate differentiating solving to zero coordinate 5 y coordinate (b) ans: B(,0) know to form a system combining equations manipulation to polynomial form sets up synthetic division 5 finds coordinate 6 for y coordinate 6 marks Illustration(s) for awarding each mark (a) A rec ( + )( p) + 8 p 8p A tri ( + 6) + 6 A A + 8 p 8p ( + 6) A A (8 p + ) 8p (b) (8 p + ) 8p 0 b ac 0 for equal roots a, b (8 p + ), c 8 p (8 p + ) ( 8 p ) p + 96 p ( p + 5)( p + ) 0 5 p or p (c) ( 8( ) + ) 8( ) 0 ( )( ) 0, (a) y + 8 dy 8 8 d y 6 (b) 8 y + y y 7() 8 y 0 Pegasys 005

16 Give mark for each 5. (a) ans: h( ) a + a a marks for dealing with composition for formula (any equivalent form) (b) ans: a marks for differentiating substituting and solving to 8 answer Illustration(s) for awarding each mark (a) f ( a)... h( ) ( a) a a (b) h ( ) a ( ) () a 8 a + a a 6. (a) ans: proof marks for solving to for answer (a) 6 + ( )( ) 0 or (b) ans: 7 cm 6 marks setting up integral integrating substituting limits area under curve 5 area of rectangle 6 subtraction to answer (b) A 6 + d [ ] (6) + ()) ( + ) ( 6 cm 5 A 7 rec 6 A 6 7 cm Pegasys 005

17 Give mark for each 7. (a) ans: proof marks length of MR length of PR value of sin required form (b) ans: proof marks knowing angle is equiv. to sin use replacement for value of cos and substitution required answer Illustration(s) for awarding each mark (a) MR or 8 PR + 6 PR 6 sin 6 sin 6 (b) sin RPQ sin sin cos (a) ans: r answer h mark (b) ans: proof substituting for r required form marks (c) ans: h 8 ft 5 marks know to differentiate solve derivative to zero differentiate solve for h 5 justification (a) r h (b) V π ( h ) h V ( h) 8π h π h (c) strategy to diff. (stated or implied) solve V ( h) 0 (stated or implied) V ( h) 8π π h 8π π h 0 h 8 h ma Total 70 marks Pegasys 005