NATIONAL QUALIFICATIONS

Transcription

1 Mathematics Higher Mini-Prelim Eamination 00/0 NATIONAL QUALIFIATIONS Assessing Unit + revision from Units & Time allowed - hour 0 minutes Read carefull. alculators ma be used in this paper.. Full credit will be given onl where the solution contains appropriate working.. Answers obtained from readings from scale drawings will not receive an credit. Pegass 00

2 FORMULAE LIST ircle: 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 A B cos A B sina cosa sin Acos B cos Asin B cos Acos B 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 a a cos a sin a Pegass 00

3 SETION A In this section the correct answer to each question is given b one of the alternatives A, B, or D. Indicate the correct answer b writing A, B, or D opposite the number of the question on our answer paper. Rough working ma be done on the paper provided. marks will be given for each correct answer.. If 4 f ( ) ( ) then () f equals A 4 B D 8. The maimum value of the function g( ) sin cos is A B 0 D. The radius of the circle with equation 4 4 is A B D 4 4. When sin cos is written in the form k sin where 0 60, tan is equal to A B D Pegass 00

4 . The value of log 4 is A B 4 4 D 4 6. Given that the vectors 4 0 and p are perpendicular, the value of p is A 0 B 8 4 D 6 7. Part of the graph of log is shown in each diagram below as a broken line. 0 Which diagram also shows, as a full line, part of the graph of log 0? A B o o D o o Pegass 00

5 8. a g is a unit vector. Which of the following could be the value of g? A B D [ END OF SETION A ] SETION B ALL questions should be attempted 9. Triangle AB has vertices A(, 0, ), B(, 4, ) and ( 4, 6, 4 ) respectivel. A, B and D are collinear such that AB. BD A(, 0, ) B(, 4, ) D ( 4, 6, 4 ) (a) Show that the coordinates of D are (, 0, ). (b) Hence show clearl that angle AD is a right angle. 4 (c) Prove that angle AB is obtuse. Pegass 00

6 0. A function is defined as f ( ) 6cos sin. (a) B using the fact that cos (cos ) show clearl that this function can be epressed in the form f ( ) cos sin. (b) Epress cos sin in the form k cos( ) where 0 60 and k 0. (c) Hence solve the equation f ( ) 0 for A function f is given b ( ) f. (a) Find f () (b) Find algebraicall the values of for which f ( ).. Given that ( ) and ( ) are both factors of a b, find a and b. 4. Solve log ( 4) log ( ) [ END OF SETION B ] [ END OF QUESTION PAPER ] Pegass 00

7 Higher Mini Prelim - Unit 00/0 (Answers + Marking Scheme) Section A - Answers D A B 4 D 6 B 7 B 8 D Section B - Marking Scheme Give mark for each marks each (6 marks) Illustration(s) for awarding each mark 9(a) ans: (, 0, ) ( marks) valid method evidence of using stepping out/section formula answer (, 0, ) (b) ans: proof (4 marks) 0 finds DA DA = 0 7 finds D D = 6 finds DA.D DA.D = =0 4 conclusion 4 since DA.D = 0; AD is right angled (c) ans: proof ( marks) 4 finds BA and B BA = 4 B = finds BA.B BA.B = = 4 conclusion scalar product < 0 so obtuse angle Pegass 00

8 Give mark for each Illustration(s) for awarding each mark 0(a) ans: proof ( marks) applies given info to new function cos (cos ) knows to substitute in function 6[ (cos )] sin simplifies to required form (cos )] sin ;cos sin (b) ans: cos( 0) o + ( marks) finds k k 9 finds tan α tan finds α 0 (c) ans: 40 o (4 marks) equates to 0 cos( 0) o + = 0 simplifies cos( 0) o = finds values = 40 o ; 60 o 4 discards 4 40 o (a) ans: ( marks) differentiates bracket use of chain rule combines and simplifies (b) ans: = ( marks) equates derivative to = rearranges and squares : 4 solves to answer : : ans: a = 4; b = (4 marks) uses snthetic division to find one equation b a = 7 uses snthetic division to find other eq. b + a = 9 knows to use sstem of equations evidence 4 solves for a and b 4 a = 4; b = Pegass 00

9 Give mark for each Illustration(s) for awarding each mark (a) ans: = ( marks) applies difference of logs 4 log eponentiates 4 7 difference of two squares 4 4 simplifies 4 solves to answer 7 Sect. B (4 marks) Total: 0 marks Pegass 00