H Mathematics Higher Paper Practice Paper E Time allowed hour minutes NATIONAL QUALIFICATIONS Read carefull Calculators ma NOT be used in this paper. Section A Questions ( marks) Instructions for completion of Section A are given on page two. For this section of the eamination ou must use an HB pencil. Section B ( marks).. Full credit will be given onl where the solution contains appropriate working.. Answers obtained b readings from scale drawings will not receive an credit. Mathematics Matters 8
Read Carefull Check that the answer sheet provided is for Mathematics Higher (Section A). For this section of the eamination ou must use an HB pencil and, where necessar, an eraser. Check that the answer sheet ou have been given has our name, date of birth, SCN (Scottish Candidate Number) and Centre Name printed on it. If an of this information is wrong, tell the invigilator immediatel. 5 If this information is correct, print our name and seat number in the boes provided. 6 The answer to each question is either A, B, C or D. Decide what our answer is, then, using our pencil, put a horizontal line in the space provided (see sample question below.) 7 There is onl one correct answer to each question. 8 Rough working should not be done on the answer sheet. 9 At the end of the eam, put the answer sheet for Section A inside the front cover of our answer book. Sample Question A curve has equation. What is the gradient at the point where? A 8 B C D The correct answer is A 8. The answer A has been clearl marked in pencil with a horizontal line (see below). Changing an answer If ou decide to change our answer, carefull erase our first answer and using our pencil, fill in the answer ou want. The answer below has been changed to D. Page
FORMULAE LIST Circle: The equation The equation g f c represents a circle centre ( g, f ) and radius ( a) ( b) r represents a circle centre (, ) a b and radius r. g f c. Scalar Product : a. b a b cos, where is the angle between a and b. or a b a. bab ab ab, where a a and b. b a b Trigonometric formulae: sin(ab) sinacosbcosasinb cos(a B) cos A cos B sin A sin B sin A sin Acos A cos A cos A sin A cos A sin A Table of standard derivatives : f ( ) f ( ) sin a acos a cos a asin a Table of standard integrals : f ( ) f ( ) d sin a cos a C a cosa sin a C a Page
SECTION A ALL questions should be attempted.. K and L have position vectors and respectivel. What is the magnitude of KL?. If f( ) 7, find f ( ).. Find d. A function f is defined on the set of real numbers b f( ) 5. Find an epression for f( f( )). 5. Evaluate. sin cos 6. A circle with centre (,) passes through the point (,). What is the equation of the circle? 7. f ( ) 5. What is the remainder when f( ) is divided b ( )? 8. The diagram shows the part of the graph of the cubic f( ). Sketch the graph of f( )? Page
9. The graph shown in the diagram has equation p sin( q). What are the values of p and q?. A sequence is generated b the recurrence relation u 7 u. n n If u 5, what is the value of u?. For what value of k does the equation k 6 have equal roots?. Find ( 7) d.. Given that and () 5, f ( ) 6 f find a formula for f( ) in terms of.. What are the coordinates of the centre of the circle with equation 685? 5. The diagram shows part of the graph of a cubic function. 6 What is the equation of this graph? Page 5
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6. The diagram shows part of the graph of the cubic f( ). There are stationar points at and. Sketch the graph of f( )? 7. If 8 is epressed in the form ( p) q, what is the value of q? 8. If log t log 5, find the value of t. 9. If p find the rate of change of p with respect to when.. Find the solutions for 8? End of Section A Page 7
SECTION B ALL questions should be attempted. Marks. A line joins the points P(, ) and Q(, 7). Find the equation of the perpendicular bisector of PQ. P Q. Show that the line with equation is a tangent to the circle with equation 5 and find the coordinates of the point of contact of the tangent and circle. 6. The diagram shows a right angled triangle with height units, base unit and an angle of p. (a) Find the eact values of: (i) cos p ; (ii) cos p. p (b) B writing p p p, find the eact value of cos p.. A function f is defined b f( ), where. Find the maimum and minimum values of f. 5 Page 8
5. (a) Epress cos sin in the form kcos( a), where k Marks and a 6. (b) Find: (i) the maimum value of sin cos ; (ii) a value of where this maimum value occurs in the interval 6. End of question paper Page 9