WEDNESDAY, 18 MAY 9.00 AM AM. 1 Full credit will be given only where the solution contains appropriate working.

Transcription

1 X00/0 NATINAL QUALIFICATINS 0 WEDNESDAY, 8 MAY 9.00 AM 0.0 AM MATHEMATICS HIGHER Paper (Non-calculator) Read carefull Calculators ma NT be used in this paper. Section A Questions 0 (40 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 (0 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. LI X00/0 6/00 *X00/0*

2 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. Do not change an of these details. 4 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 our 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 = 4. What is the gradient at the point where =? A 8 B C 0 D 4 The correct answer is A 8. The answer A has been clearl marked in pencil with a horizontal line (see below). A B C D 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. A B C D [X00/0] Page two

3 FRMULAE LIST Circle: The equation + + g + f + c = 0 represents a circle centre ( g, f) and radius The equation ( a) + ( b) = r represents a circle centre (a, b) and radius r. g + f c. Scalar Product: a.b = a b cos θ, where θ is the angle between a and b or a.b = a b + a b + a b where a = a a a b and b =. b b Trigonometric formulae: sin (A ± B) = sin A cos B ± cos A sin B ± cos (A ± B) = cos A cos B sin A sin B sin A = sin A cos A cos A = cos A sin A = cos A = sin A Table of standard derivatives: f ( ) f ( ) sin a cos a acosa asina Table of standard integrals: f ( ) fd ( ) sin a a cosa + C cosa a sin a + C [Turn over [X00/0] Page three

4 SECTIN A ALL questions should be attempted.. Given that p =, q = and r =, epress p q in component form.. A line l has equation + = 6. What is the gradient of an line parallel to l? Page four [X00/0] A B C D A 9 5 B 5 C 9 5 D 5 r

5 . The diagram shows the graph of = f(). (, ) (, ) Which of the following shows the graph of = f( + )? A ( 4, ) (, ) B (, ) C (, ) 4 D (, ) ( 4, 4) [Turn over [X00/0] Page five

6 4. A tangent to the curve with equation = is drawn at the point (, 4). What is the gradient of this tangent? A B C 4 D 0 5. If is written in the form ( p) + q, what is the value of q? A 9 B C 7 D 6. The point P(, ) lies on the circle with centre C as shown. The gradient of CP is. What is the equation of the tangent at P? A + = ( ) C B = ( + ) C + = ( ) D = ( + ) P(, ) 7. A function f is defined on the set of real numbers b f() = + +. What is the remainder when f() is divided b ( )? A 0 B C D 4 [X00/0] Page si

7 8. A line makes an angle of 0 with the positive direction of the -ais as shown. 0 What is the gradient of the line? A B C D 9. The discriminant of a quadratic equation is. Here are two statements about this quadratic equation: () the roots are real; () the roots are rational. Which of the following is true? A B C D Neither statement is correct. nl statement () is correct. nl statement () is correct. Both statements are correct. [Turn over [X00/0] Page seven

8 0. Solve cos = for, where 0 < π. π 5π A and π π B and π 5π C and 6 6 π π D and 6 6. Find 4 + d, where > 0. 4 A + c B + c 8 4 C + c 8 D + c [X00/0] Page eight

9 . The diagram shows two right-angled triangles with sides and angles as given. 5 q p What is the value of sin( p + q )? A B C + 5 D Given that f() = 4 sin, find f (0). A 0 B C D 6 [Turn over [X00/0] Page nine

10 4. An equilateral triangle of side units is shown. The vectors p and q are as represented in the diagram. What is the value of p.q? A 9 p B C 9 9 q D 0 5. Given that the points S( 4, 5, ), T( 6, 4, 6) and U( 4, 0, 6) are collinear, calculate the ratio in which T divides SU. A : B : C : 5 D : Find d, where 0. A B + c 9 + c C + c D + c [X00/0] Page ten

11 7. The diagram shows the graph of a cubic. (, ) What is the equation of this cubic? A = ( + )( ) B = ( )( + ) C = ( + )( ) D = ( )( + ) 8. If f() = ( )( + 5), for what values of is the graph of = f() above the -ais? A 5 < < B < < 5 C < 5, > D <, > 5 [Turn over [X00/0] Page eleven

12 9. Which of the following diagrams represents the graph with equation log =? A (, ) B (, ) C (, ) D (, ) [X00/0] Page twelve

13 0. n a suitable domain, D, a function g is defined b g() = sin. Which of the following gives the real values of in D and the corresponding values of g()? A 0 and g() B 0 and 0 g() C and g() D and 0 g() [END F SECTIN A] [Turn over for SECTIN B [X00/0] Page thirteen

14 SECTIN B Marks ALL questions should be attempted.. A quadrilateral has vertices A(, 8), B(7, ), C(8, 5) and D(, ) as shown in the diagram. B A E C D (a) Find the equation of diagonal BD. (b) The equation of diagonal AC is + =. Find the coordinates of E, the point of intersection of the diagonals. (c) (i) Find the equation of the perpendicular bisector of AB. (ii) Show that this line passes through E. 5 [X00/0] Page fourteen

15 Marks. A function f is defined on the set of real numbers b f() = ( )( + ). (a) Find where the graph of = f() cuts: (i) the -ais; (ii) the -ais. (b) Find the coordinates of the stationar points on the curve with equation = f() and determine their nature. 8 (c) n separate diagrams sketch the graphs of: (i) = f(); (ii) = f().. (a) Solve cos cos + = 0 for 0 < 60. (b) Hence solve cos 4 cos + = 0 for 0 < [END F SECTIN B] [END F QUESTIN PAPER] [X00/0] Page fifteen

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17 X00/0 NATINAL QUALIFICATINS 0 WEDNESDAY, 8 MAY 0.50 AM.00 NN MATHEMATICS HIGHER Paper 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. LI X00/0 6/00 *X00/0*

18 FRMULAE 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 cos θ, where θ is the angle between a and b or a.b = a b + a b + a b where a = Trigonometric formulae: sin (A ± B) = sin A cos B ± cos A sin B ± cos (A ± B) = cos A cos B sin A sin B sin A = sin A cos A cos A = cos A sin A = cos A = sin A Table of standard derivatives: f ( ) f ( ) sin a acosa cos a asina Table of standard integrals: f ( ) fd ( ) sin a cosa a cosa + C a sin a + C [X00/0] Page two

19 ALL questions should be attempted. Marks. D,ABC is a square based pramid as shown in the diagram below. z D(,, 6) C B M A is the origin, D is the point (,, 6) and A = 4 units. M is the mid-point of A. (a) State the coordinates of B. (b) Epress DB and DM in component form. (c) Find the size of angle BDM. 5. Functions f, g and h are defined on the set of real numbers b f() = g() = + h() = 4 5. (a) Find g( f()). (b) Show that g( f()) + h() = (c) (i) Show that ( ) is a factor of (ii) Factorise full. (d) Hence solve g( f()) + h() = 0. 5 [Turn over [X00/0] Page three

20 . (a) A sequence is defined b un+ = un with u 0 = 6. Write down the values of u and u. (b) A second sequence is given b 4, 5, 7,,.... It is generated b the recurrence relation v n + = pv n + q with v = 4. Find the values of p and q. Marks (c) Either the sequence in (a) or the sequence in (b) has a limit. (i) Calculate this limit. (ii) Wh does the other sequence not have a limit? 4. The diagram shows the curve with equation = and the line with equation = + 4. The curve and the line intersect at the points (, 0), (0, 4) and (, 0). = = + 4 Calculate the total shaded area. 0 [X00/0] Page four

21 5. Variables and are related b the equation = k n. log Marks The graph of log against log is a straight line through the points (0, 5) and (4, 7), as shown in the diagram. (0, 5) (4, 7) Find the values of k and n. log 5 6. (a) The epression sin 5cos can be written in the form R sin( + a) where R > 0 and 0 a < π. Calculate the values of R and a. 4 (b) Hence find the value of t, where 0 t, for which 0 t (cos + 5sin ) d = Circle C has equation ( + ) + ( ) =. A circle C with equation p = 0 is drawn inside C. The circles have no points of contact. What is the range of values of p? 9 [END F QUESTIN PAPER] [X00/0] Page five

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