Mathematics Paper 1 (Non-Calculator)

Transcription

1 H National Qualifications CFE Higher Mathematics - Specimen Paper F Duration hour and 0 minutes Mathematics Paper (Non-Calculator) Total marks 60 Attempt ALL questions. You ma NOT use a calculator. Full credit will be given onl to solutions which contain appropriate working. State the units for our answer where appropriate. Write our answers clearl in the answer booklet provided. In the answer booklet ou must clearl identif the question number ou are attempting. Use blue or black ink. Before leaving the eamination room ou must give our answer booklet to the Invigilator; if ou do not ou ma lose all the marks for this paper. Pegass 05 CFE Higher Specimen Paper F

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. Trigonometric formulae: ( A± B) ( A± B) sin cos sina cosa sin Acos B± cos Asin B cos Acos Bm sin Asin B sin Acos A cos A sin A cos A sin A Scalar Product: a. b a b cosθ, where θ isthe anglebetween a and b. or a.b a b + a b + a b where a a a a and b b b b Table of standard derivatives: f ( ) sin a cosa f ( ) acosa asin a Table of standard integrals: f ( ) f ( ) d sin a cos a cos a + a sin a + a C C Pegass 05 CFE Higher Specimen Paper F

3 Attempt ALL questions Total marks 60. (a) If k 8 4is eactl divisible b ( + ), find the value of k. (b) Hence, solve the equation k when k takes this value.. For what range of values of k does the quadratic equation 4k 8+ k 0 have real roots?. Given that a, b 5 and b a. 7, find the value of a.( b a+ ). 4. A recurrence relation is defined as u 0 75u. n+ n + Given that U, find the difference between the limit of the sequence and the 0 third term, U The diagram shows two right-angled triangles with lengths as shown. 4 Calculate the eact value of cos( + ) Given that log ( + 8) log, find the value of where > 0. 4 h. t 7. A function is defined on a suitable domain as ( t) ( t ) Find the rate of change of this function when t. Pegass 05 CFE Higher Specimen Paper F

4 d 8. A curve has as its derivative 4. d Given that the point (, 7) lies on this curve, epress in terms of (a) Two functions, defined on suitable domains, are given as f ( ) and g( ). Show that f ( g( a)) can be epressed in the form pa + qa+ r and write down the values of p, q and r. (b) Sketch the graph of f(g()). 0. Find the equation of the tangent to the curve at the point where. 5. Epress the function f ( ) 6+ in the form p ( q) + r.. (a) The point (5, k) lies on the graph of log. Find the value of k. 5 (b) The diagram show part of the graph of a. State the value of a. P(, 9) o Pegass 05 CFE Higher Specimen Paper F

5 . Part of the graph of the curve with equation The diagram is not drawn to scale. is shown below. Q P O (a) Establish the coordinates of the stationar point P. 4 (b) The horizontal line through P meets the curve again at Q. Find the coordinates of Q. (c) Hence calculate the shaded area shown in the diagram below. Q P O 6 [ END OF QUESTION PAPER ] Pegass 05 CFE Higher Specimen Paper F

6 H National Qualifications CFE Higher Mathematics - Specimen Paper F Duration hour and 0 minutes Mathematics Paper (Calculator) Total marks 70 Attempt ALL questions. You ma use a calculator. Full credit will be given onl to solutions which contain appropriate working. State the units for our answer where appropriate. Write our answers clearl in the answer booklet provided. In the answer booklet ou must clearl identif the question number ou are attempting. Use blue or black ink. Before leaving the eamination room ou must give our answer booklet to the Invigilator; if ou do not ou ma lose all the marks for this paper. Pegass 05 CFE Higher Specimen Paper F

7 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. Trigonometric formulae: ( A± B) ( A± B) sin cos sina cosa sin Acos B± cos Asin B cos Acos Bm sin Asin B sin Acos A cos A sin A cos A sin A Scalar Product: a. b a b cosθ, where θ isthe anglebetween a and b. or a.b a b + a b + a b where a a a a and b b b b Table of standard derivatives: f ( ) sin a cosa f ( ) acosa asin a Table of standard integrals: f ( ) f ( ) d sin a cos a cos a + a sin a + a C C Pegass 05 CFE Higher Specimen Paper F

8 Attempt ALL questions Total marks 70. In the diagram A, B and C are the points ( 5, ), (, ) and (4, ) respectivel. A( 5, ) B(, ) O C(4, ) (a) Find the equation of the line through C parallel to the line AB. (b) Find the equation of the line perpendicular to BC which passes through the point A. (c) Find the coordinates of T, the point of intersection of these two lines. 4. Solve the equation (cos + 4sin ) 5 in the interval 0 π. 5. A is the point (, 4, 4 ), B is the point (,, 5 ) and C is (,, c ). Given that A, B and C are collinear, determine the value of c and state the ratio in which B divides AC. A.. B. C 4 Pegass 05 CFE Higher Specimen Paper F

9 4. The power, E, emitting from a wave generator is given b the formula o o E cos t + sin t + 0, where t is the time elapsed, in seconds, from switch on. o (a) Epress E in the form E k sin ( t+ θ ) + 0, where k > 0 and 0 θ (b) Hence state the maimum value of E and the corresponding replacement for t. π 5. Evaluate ( cosθ sinθ ) d θ A radioactive substance decas according to the formula M t M o 0 t 8, where M o is the intitial mass of the substance, M t is the mass remaining after t ears. Calculate, to the nearest da, how long a sample would take to half its original mass The points P, Q and R have coordinates (4,, 7), (,, ) and (, 9, k) respectivel, where k is a positive whole number. (a) Establish the least value of k which will make angle PQR acute. 4 (b) Calculate the size of angle PQR when k takes this value. 4 Pegass 05 CFE Higher Specimen Paper F

10 8. A small electrical compan has won a contract to re-wire local authorit houses. The profit that the compan will make depends on the number of houses that are re-wired in one da. A function which represents the profit ( ) made in an one da is: P ( ) where is the number of houses re-wired in one da. (a) Calculate the number of houses which have to be re-wired in one da to maimise the profit, justifing that it is a maimum. 5 (b) Calculate the maimum profit made in one da. 9. Three wheels are positioned in such a wa that their centres are collinear. When placed on a set of rectangular aes the equations of the two larger circles are and ( 6) + ( 4) 00, as shown. C ( 6) + ( 4) 00 o C (a) Write down the coordinates of the two centres C and C. (b) Calculate the radii of the two larger circles and the distance between the two centres C and C. 4 (c) Hence establish the centre and radius of the small circle and write down its equation. Pegass 05 CFE Higher Specimen Paper F

11 0. The diagram shows a sketch of the graph of log. 5 (b,) log5 o (a,0) (a) Write down the values of a and b. (b) Make a cop of the graph and on the same diagram show the graph of log 5 (c) Find the coordinates of the point where the line with equation intersects with the graph of log 5.. Triangle PQR is right-angled at R. (a) If the eact value of tan Q is, show that the eact value of cos Q can be 5 written in the form h and state the value of h. () (b) Hence, or otherwise, show that the eact value of sin Q is 0 7 () [ END OF QUESTION PAPER ] Pegass 05 CFE Higher Specimen Paper F

12 CFE Higher Specimen Paper F Marking Scheme Paper Give mark for each Illustration(s) for awarding each mark (a) ans: k ( marks) knows to use snthetic division k 8 4 completes snthetic division 9 k+ 7 9k k 9 k 9 9k equates remainder to 0 and solves for k 9 9k 0; k (b) ans:, ⅔, 4 ( marks) Pegass 05 CFE Higher Specimen Paper F k 8 starts to factorise ( + )( 0 8) 0 completes factorising ( + )(+ )( 4) 0 solves, ⅔, 4 ans: k ( marks) knows condition from real roots b 4ac 0 finds epression for discriminant 64 6k 0 solves k ans: 6 ( marks) multipling out brackets a.a + a.b substiutes values and answer ans: ans: (5 marks) finding U U 0 75() + 6 U and U U 0 75(6) + 9 U 0 75(9) knowing how to find limit b L (or equivalent) a 4 finding limit 4 L calculating difference 5 diff (4 marks) know to epand cos( + ) coscos sinsin calculates hpotenuse for both triangles 5 and 9 substitutes eact values simplifies to answer

13 Give mark for each Illustration(s) for awarding each mark 6 ans: 8 (4 marks) bringing up power... log combining logs + 8 log 9 removing log + 8 (or equivalent) 9 4 answer ans: ( marks) preparing to differentiate h( t) (4t t+ 9) t 4t + 9t differentiating h ( t) 4 9t substituting to answer 9 h ( ) ans: 6 (4 marks) knows to integrate ( 4) d integrates correctl (with c) + c substitutes and finds c 7 ( ) + c c 6 4 writes down answer 4 6 9(a) ans: p, q 4, r ( marks) substitutes f ( g( a)) ( a) multiplies out and reorganises 4 4a + a a 4a+ lists values of p, q and r p, q 4, r (a) ans: graph drawn ( marks) correct shape of graph parabola with minimum TP correct TP turning point at (, ) [annotated] correct - intercept (0, ) [annotated] 0 ans: 9 6 (5 marks) knows to differentiate d d finds derivative substitutes in derivative () 9 4 finds point on the line 4 () -() 8 6 ; (, ) 5 substitutes in equation 5 9( ) Pegass 05 CFE Higher Specimen Paper F

14 Give mark for each Illustration(s) for awarding each mark ans: ( ) + 8 (marks) takes common factor ( ) + completes square in bracket [( ) ] + simplifies ( ) + ( ) + 8 (a) ans: k ( mark) substitutes and solves for k k log 5 5; k (b) ans: a ( mark) substitutes and solves for a 9 a ; a (a) ans: P(, 4) (4 marks) knows to make derivative equal to 0 d 0 d finds derivative d 6 0 d solves for ( ) 0; 0, 4 states coordinates of P 4 P(, 4) (b) ans: Q(, 4) ( marks) knows to equate functions 4 uses app method to factorise epression evidence leading to ( )( )( + ) solves and states coordinates of Q Q(, 4) (c) ans: 6 units (6 marks) 4 knows to use integration... uses correct integration d 4 integrates subs values 4 () ( ) () ( ) evaluates 5 ( 8 4) ( ) subtracts from to answer units 4 4 Total: 60 marks Pegass 05 CFE Higher Specimen Paper F

15 CFE Higher Specimen Paper F Marking Scheme Paper Give mark for each Illustration(s) for awarding each mark (a) ans: + 7 ( marks) knows to find gradient of AB m AB finds gradient m AB + 5 substitutes values in equation + ( 4) (b) ans: + 7 ( marks) finds gradient of BC + m BC 4+ 7 takes perpendicular gradient m PERP 7 substitutes values in equation 7( + 5) (c) ans: ( 8, ) (4 marks) knows to use simultaneous equations evidence finds value for 8 finds value for 4 states coordinates 4 ( 8, ) ans: π 5π ; 6 6 (5 marks) multiplies brackets and substitutes ( sin ) + 8sin 5 simplifies to quadratic in sin 4sin 8sin + 0 factorises ( sin )( sin ) 0 4 solves and discards 4 sin or sin (discard) 5 finds angles 5 π 5π ; 6 6 ans: c 7, AB : BC : (4 marks) considering two vectors AB, BC 4 c 5 aware of the relationship between vectors BC AB (or equiv.) finding c c 5 c 7 4 stating ratio 4 AB : BC : (or equi Pegass 05 CFE Higher Specimen Paper F

16 Give mark for each Illustration(s) for awarding each mark o 4(a) ans: E sin( t+ 0) + 0 (5 marks) correct epansion k sin cosα + k cos sinα equating coefficients kcosα ; ksin α finding k k ( ) + 4 finding tan θ tan θ 4 θ in degrees + answer 4 o o θ 0 E sin( t+ 0) + 0 (b) ans: o E t 60 ( marks) maimum value of E E + 0 ma corresponding replacement sin( t + 0) o sin90 o t 60 o 5 ans: (4 marks) integrating first term [ sin ] 0 integrating second term plus sign [... + cosθ ] 0 substituting π sinπ + cos ) ( sin 0 + cos0) θ ( ( 0+ ) ) (0+ 4 calculating answer 4 π π 6 ans: t 406 das (5 marks) realising to solve ep. to t 0 5 introducing logs log8 0 t log0 5 dealing correctl with power 0 t log8 log making t the subject 4 log0 5 t 0 log8 5 answer 5 t 406 das (... ears) 7(a) ans: k 4 (4 marks) strateg and selecting correct vectors if acute then QP. QR > 0 5 vectors in component form QP 5, QR 6 6 k scalar product inequation QP. QR 5() 5(6) + 6( k ) > 0 4 calculating answer 4 k > 0 k > k 4 6 Pegass 05 CFE Higher Specimen Paper F

17 Give mark for each Illustration(s) for awarding each mark 7(b) ans: o PQR 87 5 (4 marks) calculating scalar product QP. QR both magnitudes QP 86, QR 54 correct substitution cos P QR ˆ answer 4 PQR 87 5 o 8(a) ans: 0 (5 marks) knowing to differentiate P '( ) (or equiv) carring out differentiation 40 making derivative solving to answer justification 5 (b) ans: 70 ( marks) knows to sub. in P () 40(0) answer 70 9(a) ans: C ( 8, 6), C (6, 4) ( marks) first centre C ( 8, 6) second centre C (6, 4) (b) ans: r 0, r 0, d 6 (4 marks) finding r of C r ( 8) + ( 6) finding r of C r 00 0 method (dist. form, pth, etc.) d ( )... etc. 4 correct distance 4 d (c) ans: C (4, ), r ( marks) ( 4) + ( + ) 9 centre Centre must be mid-pt C (4, ) radius r ( 6 0) sub. into equation to answer ( 4) + ( + ) 9 or Pegass 05 CFE Higher Specimen Paper F

18 Give mark for each Illustration(s) for awarding each mark 0(a) ans: a ; b 5 ( mark) states values of a and b a and b 5 (b) ans: diagram ( mark) graph moved unit parallel to -ais (c) ans: ( 5, ) ( marks) 0 (,0) (,-) log5 log5 (5,) (5,0) (a) ans: equates functions log 5 rearranges log 5 or 5 answer ( ) 5 5 h ( marks) 9 uses rt.triangle to find hpotenuse states value of cos Q R 5 5 cos Q rationalises denominator and states value of h 5 5 cos Q 5 ; 9 h 9 (b) ans: proof ( marks) finds value of sin Q sin Q epands and substitutes for sin Q sin Q sin Q cosq 5.. rearranges to answer 0. 7 P 7 Q Total: 70 marks Pegass 05 CFE Higher Specimen Paper F