MATHEMATICS Higher Grade - Paper I (Non~calculator)

Transcription

1 Higher Mathematics - Practice Eamination G Please note the format of this practice eamination is the same as the current format. The paper timings are the same, however, there are some differences in the marks allocated. Calculators ma onl be used in Paper. MATHEMATICS Higher Grade - Paper I (Non~calculator) Time allowed - hour 0 minutes Read Carefull. Calculators ma not be used in this paper.. Full credit will be given onl where the solution contains appropriate working.. Answers obtained b readings from scale drawings will not receive an credit.. This eamination paper contains questions graded at all levels.

2 FORMULAE 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. Scalar Product: a. b a b cosθ, where θ is the angle between a and b or a a. b a b + ab + ab where a a and b a b b b 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 Table of standard derivatives: f ( ) sin a cosa f ( ) acosa asin a Table of standard integrals: f ( ) f ( ) d sin a cos a a cos a + a sin a + C C

3 All questions should be attempted. The diagram shows an arc AB of the circle with its centre at C. The coordinates of A and B are (, 7) and ( 8, ) respectivel. C (a) (c) Find the equation of the perpendicular bisector of the chord AB. B O Hence establish the coordinates of C, given that C is verticall above A. Write down the equation of the circle, centre C, passing A through the points A and B.. For what value(s) of p does the equation ( p + ) p + 0 have equal roots?. The diagram below shows part of the the graph of g( ). The function has stationar points at ( 0, ) and (,0) as shown. O - Sketch the graph of the related function g( ) +.. A function is defined as f ( θ ) cosθ + sin θ for π 0 < θ <. (a) Show that f ( θ ) sinθ. Hence calculate the rate of change of the function at θ. π

4 5. The daigram shows the graph of sin, for equation. π 0, and the line with P O π π Establish the coordinates of the point P. d 6. A curve has as its derivative 8. d Given that the point (, ) lies on this curve, epress in terms of. 7. A sequence of numbers is defined b the recurrence relation Un+ kun + c, where k and c are constants. (a) Given that U 70, U 65 and U 6 5, find algebraicall, the values of k and c. Hence find the limit of this sequence. (c) Epress the difference between the fifth term and the limit of this sequence as a percentage of the limit, correct to the nearest percent. 8. A function is given as f ( ) and is defined on the set of real numbers. (a) Show that the derivative of this function can be epressed in the form f ( ) a[ ( b) + c ] and write down the values of a, b and c. Eplain wh this function has no stationar points and is in fact increasing for all values of.

5 9. The functions f ( ) 9 and real numbers. h( ) + are defined on the set of (a) Evaluate h ( f ()). Find an epression, in its simplest form, for f ( h() ). (c) For what value(s) of does ( h() ) f f ()? 0. Three vertices of the quadrilateral PQRS are P(, -, - ) Q( 0,, ) and R( 7,, ). z R(7,, ) Q(0,, ) O S P(, -, -) (a) Given that QR PS, establish the coordinates of S. Hence show that angle PSR is a right angle.. Given that log 5 ) log, find the value of. ( [ END OF QUESTION PAPER ]

6 Higher Mathematics Practice Eam G Marking Scheme - Paper Give mark for each Illustration(s) for awarding each mark. (a) ans: marks (a) For mid-point of AB For gradient of AB For gradient of perpendicular For equation of bisector ans: C(,8) For knowing to use A For sub. in equation to answer (c) ans: ( ) + ( 8) 5 marks marks For finding radius For sub. in standard equ. to answer m AB m m -, m bis. + ( + ) C A () 8 C(,8) (c) M(-,-) CA is vertical radius 5 units C(,8), r 5 in equ. ( a) + ( b) r. ans: p, p marks 9 For discr. 0 (stated or implied) For selecting a, b and c Substituting and simplifing Factorising to answers For equal roots b ac 0 a p +, b p, c ( p) (( p + ).) 0 9 p 6 p 0 ( p + )( p ) 0 p or 9 9. ans: diagram marks For reflection in -ais For translating. up For annotating final sketch (-,) O. (a) ans: proof marks For differentiating first term For differentiating second term common factor (isolating double angle) for double angle + simplifing ans: marks answer 5. ans: P 5π, ) marks ( 8 For equating Angles for sin (choosing quad.) Answer 6. ans: marks For setting up integral For integrating For substituting Correct answer (a) sin θ sinθ cosθ... + (sinθ cosθ ) sinθ + (sin θ ) sin θ π π π f ( ) sin( ) sin 6 sin 5π quad ½ π or 6 6 5π. P( 5π, ) d 8 + c + c ( ) () + c c to answer

7 Give mark for each Illustration(s) for awarding each mark 7. (a) ans: k, c 0 marks Setting up a sstem of equ. Finding k Finding c ans: 60 Knowing how to find limit Calculating limit (c) ans: % For calculating U5 For percentage calculation marks marks 8. (a) ans: a 9, b, c marks For differentiating For common factor For the square ( ) For c (no need to list a,b and c) ans: eplanation marks For statement on solving to zero Derivative alwas +ve, alwas increasing 9. (a) ans: mark answer ans: f ( h( )) + marks For substitition Simplifing to answer (c) ans:, marks For equating For solving to answers 0. (a) ans: S(,0,0) marks For finding displacement QR For establishing coordinates of S. ans: proof marks For knowing SR. SP 0, for R.A. (stated or implied) For both displacements For scalar product calculation to zero ans: For combining logs For removing logs For finding marks (a) U ku + c 65 70k + c ku + c k + c 5 k 5 k c 0 b L, or equivalent a L 0 / ( ) 60 (c) U (6 5) (a) % 60 f ( ) [ + ] 9 ( )... [ ] 9 [... + ] 9 [( ) ] + (no marks off if b ) ( ) + 0 has no solution (a) ( ) + alwas +ve, alwas incr. f ( h( )) ( + ) 9 U f ( ) 0, h(0) (or equiv.) f ( h( )) + (c) ( + )( + ) 0 (a) 7 0 QR r a s + 0, or equiv. 0 Total 5 marks For right-angle SR. SP 0 6 SR. SP , r angled 5 log 5 (or equivalent) 5 - or -

8 Higher Mathematics - Practice Eamination G Please note the format of this practice eamination is the same as the current format. The paper timings are the same, however, there are some differences in the marks allocated. Calculators ma onl be used in this paper. MATHEMATICS Higher Grade - Paper II Time allowed - hour 0 minutes Read Carefull. Calculators ma be used in this paper.. Full credit will be given onl where the solution contains appropriate working.. Answers obtained b readings from scale drawings will not receive an credit.. This eamination paper contains questions graded at all levels.

9 FORMULAE 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. Scalar Product: a. b a b cosθ, where θ is the angle between a and b or a a. b a b + ab + ab where a a and b a b b b 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 Table of standard derivatives: f ( ) sin a cosa f ( ) acosa asin a Table of standard integrals: f ( ) f ( ) d sin a cos a a cos a + a sin a + C C

10 All questions should be attempted. Triangle PQR has vertices P (, k ), Q (,0) and R (,) as shown. Q (,0) R(,) P (, k ) (a) Given that the gradient of side PQ is, find the equation of PQ. Hence find k, the -coordinate of verte P. (c) Find the equation of the median from P to QR. (d) Show that this median is at right-angles to side QR. What tpe of triangle is PQR?. Evaluate f () when f ( ). 5. Two functions are defined as f ( ) a b and where a is a constant. 6b h( ), (a) Given that f ( ) h(), show clearl that a. If b p 6, show that f ( ) p +. (c) Hence state the values of p for which f ( ) 0 has no real roots.

11 . A fishing boat's fish hold is in the shape of the prism shown opposite. The length of the hold is metres. The cross-section of the hold is represented in the coordinate diagram below. All the units are in metres, with the floor of the hold represented b the curve [ ] 8 + m 8 P Q [ ] 8 O a b (a) Find the values of a and b, the -coordinates of P and Q. Show clearl that the area between the line PQ and the curve [ 6 00 ] can be calculated b evaluating the integral: A 8 b 8 + (6 8) d a. (c) Calculate this area in square metres. (d) Hence calculate the volume of the hold, in cubic metres, b first establishing the total cross-sectional area of the hold. 5. Two vectors are defined as V i + j - 5k and V i + 6j. Calculate the angle between these two vectors to the nearest degree. 5

12 6. Certain radioisotopes are used as tracers, to track down diseased tissue within the bod, and then be absorbed, to act as a long-term radio-therap treatment. Their passage through the bod and mass is ascertained b means of a Geiger-Müller counter. During trials of a particular radioisotope the following information was obtained. the isotope loses % of its mass ever hour the maimum recommended mass in the bloodstream is 65mgs 00mgs is the smallest mass detectable b the Geiger-Müller counter (a) An intial dose of 50mgs of the isotope is injected into a patient. Would the mass remaining after hours still be detectable b the Geiger-Müller counter? Your answer must be accompanied b appropriate working. After the initial dose, top-up injections of 50mgs are given ever hours. Comment on the long-term suitabilt of this plan. Your answer must be accompanied b appropriate working. 7. The diagram shows a circle, centre C, with equation Two common tangents have been drawn from the point P to the points S and T ( 7,6) on the circle. P T( 7,6) S O C (a) Find the centre and radius of the circle. Hence find the equation of the tangent PT. (c) Given now that the tangent PS is parallel to the -ais, determine the coordinates of S and P.

13 8. Solve algebraicall the equation sin o o 7 cos, where 0 < The curve below has as its equation 6 + k +, where k is a constant. A (,8) is a stationar point. A (,8) 6 + k + B O (a) Using the -coordinate of A, to help ou, find the value of k. Hence find the coordinates of the other stationar point at B. [ END OF QUESTION PAPER ]

14 Higher Mathematics Practice Eam G Marking Scheme - Paper Give mark for each. (a) ans: + marks For using Q For answer Illustration(s) for awarding each mark (a) Q(,0), m 0 ( ) ans: k mark ( ) + k For subst. to answer (c) ans: marks For mid-point of QR For calculating gradient answer (d) ans: proof, isosceles marks For knowing m.m - For calculation to prove For isosceles (no eplanation required) (c) M QR (7,6) 6 ( ) m med 7 ( ) 6 ( 7) (d) If perp m QR m Pm (stated or implied) isosceles. ans: f ( ) 5 marks For preparing to differentiate Differentiating first term Differentiating second term Subst. in derivative 5 Calculating answer f ( ) ( f ( )... f ( )... f ( ) 5 5 f ( ) + ) 5. (a) ans: proof marks For sub in f and h For equating For solving to required answer ans: proof For sub. for a and b and adjusting to required answer (c) ans: < p < mark marks For discr. statement (or implied) For values of a, b and c For subst. and factorising For final statement (worded ans. o.k.) (a) 6b f () a b, h() 6b a b (a b) 6b a 6b 6b... a f ( ) ( p 6) f ( ) p + (c) for no real roots b ac < 0 a, b p, c ( p) (..) < 0 ( p )( p + ) < 0 p has to lie between - and

15 Give mark for each Illustration(s) for awarding each mark. (a) ans: a, b marks (a) ( ) For sub. for in order to solve manipulating equation to zero factorising and answers ans: proof marks Strateg of line minus curve Constant out + tid to required answer ( )( ) 0, A ( 6 00) d A A ( ( ) d 8) d (c) ans: 6 m For integrating (all terms) For substituting For simplifing each part For calculating answer (d) ans: 768 m For total surface area For volume marks marks (c) A (d) 8 A [ 8 8] 8 [ 8( ) 8()) (8( ) 8()) ] ( A [ 00 9] A [ ] 6. A 6 + ( ) 6 m V m 5. ans: 5 o 5 marks consruct appropriate vectors strateg of cos θ... calculate scalar product process denominator (magnitudes) 5 calculate angle (rounding onl a guide) 6. (a) ans: Yes, 0 08 > 00 marks For taking a 0 97 For calculation For consistent answer ans: Plan o.k., over the long-term between and 6 mgs marks For attempting lines of working For finding the limit For being aware of the lower limit as well as the upper limit Consistent comment on findings V, V cos θ... (formula ma onl appear when numbers are subst.) V V V V cosθ θ 5 (a) a 0 97 U (0 97) Yes, since U > 00 U U5 below 5 08, 56 9, 58 87, 60, L 6 (0 97) lower limit 6 50 Over the long-term the amount present would alwas be between and 6 which is ideal.. o 5 o

16 Give mark for each 7. (a) ans: C(,), r 5 marks For centre For radius (a) Illustration(s) for awarding each mark C(,) r + ( 5) 5 5 ans: + 5 (or equiv.) marks For gradient of radius For gradient of tangent For sub. using mtan & T(7,6) (c) ans: S(-,), P(-,) marks For c - 5 s -, then coord. of S For knowing to sub. - into equat. For coordinates of P 8. ans: 90 0 o 7 marks For re-arranging and realising For epansion & equating coefficients Tan ratio For α 5 For k 6 For solving to one-third 7 For angle m CT 6 7 m tan 6 ( 7) (c) 5 S(,) same as C -coordinate of P is same as S. Also on line ( ) 5 P(,) sin + 8 cos k cos cosα + k sin sinα k cos α 8, k sinα tan α 8 o α k 9 6 cos( 9 5) o 9. (a) ans: k 9 marks For knowing that at A the deriv. 0 For differentiating For solving deriv 0 and making Calculating k ans: B(,) Solving deriv. to zero when k 9 Finding -coordinate of B Sub. to find -coordinate of B marks (a) d A is a stat. point 0 at A d d + k d ( ) () + k 0 k 9 solve ( )( ) 0 for B 6( ) + 9() + Total 67 marks