WEDNESDAY, 20 MAY 9.00 AM AM

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1 X00// NATIONAL QUALIFIATIONS 05 WENESAY, 0 MAY 9.00 AM 0.0 AM MATHEMATIS HIGHER Paper (Non-calculator) Read carefully alculators may NOT be used in this paper. Section A Questions 0 (0 marks) Instructions for completion of Section A are given on Page two. Section (0 marks) Full credit will be given only where the solution contains appropriate working. Answers obtained by readings from scale drawings will not receive any credit. Write your answers clearly in the spaces in the answer booklet provided. Additional space for answers is provided at the end of the answer booklet. If you use this space you must clearly identify the question number you are attempting. Use blue or black ink. efore leaving the examination room you must give your answer booklet to the Invigilator; if you do not, you may lose all the marks for this paper. HTP *X00//*

2 Read the questions for Section A and record your answers on the grid on Page three of the answer booklet.. The answer to each question is either A,, or. ecide what your answer is, then fill in the appropriate bubble (see sample question below).. There is only one correct answer to each question. o NOT use gel pens. Sample Question A curve has equation y = x x. What is the gradient at the point where x =? A 8 0 The correct answer is 8. The answer bubble has been clearly filled in (see below). hanging an answer If you decide to change your answer, cancel your first answer by putting a cross through it (see below) and fill in the answer you want. The answer below has been changed to. If you then decide to change back to an answer you have already scored out, put a tick () to the right of the answer you want, as shown below: or [X00//] Page two

3 FORMULAE LIST ircle: The equation x + y + gx + fy + c = 0 represents a circle centre ( g, f) and radius The equation (x a) + (y b) = r represents a circle centre (a, b) and radius r. g + f c. Scalar Product: a.b = a b cos q, where q is the angle between a and b or a.b = a b + a b + a b where a = a a a b b and b =. b Trigonometric formulae: sin (A ± ) = sin A cos ± cos A sin cos (A ± ) = cos A cos sin A sin sin A = sin A cos A ± cos A = cos A sin A = cos A = sin A Table of standard derivatives: f(x) f (x) sin ax cos ax a cos ax a sin ax Table of standard integrals: f(x) sin ax cos ax f (x)dx a cos ax + c a sin ax + c [Turn over [X00//] Page three

4 SETION A ALL questions should be attempted.. Given f(x) = x 7 find the value of f'(). A The line with equation y = x + 5 is perpendicular to the line with equation y = kx. What is the value of k? A. If x + x x + is divided by (x ), what is the remainder? A 0 [X00//] Page four

5 . The diagram shows the graph with equation of the form y = a cos bx for 0 x p. y O p p x What is the equation of this graph? A y = cos x y = cos x y = cos x y = cos x 5. A sequence is defined by the recurrence relation u n + = 0 u n + 9, u 5 =. What is the value of u? A The points P, Q and R are collinear. P is the point (, 6, ), Q is the point (, 0, ) and QR = alculate the ratio in which Q divides PR.. 6 A : : : 5 5 : [X00//] Page five [Turn over

6 ( )( ) 7. What is x+ x dx? A x + c x + c x 6x+ c x + x x x c + 8. A straight line makes an angle of 60 with the x-axis as shown in the diagram. y 60 O x What is the gradient of this line? A [X00//] Page six

7 9. Find the minimum value of sinx + 5 and the value of x where this occurs in the interval 0 x < p. min value A x p p p p 0. Solve cosx + = 0 for x, where p x p. A 5p 6 7p 6 5p p. The curve y = f(x) is such that f'(x) = x. The curve passes through the point (, 9). What is the equation of the curve? A y = x x 5 y = x + y = x x + y = x x [Turn over [X00//] Page seven

8 . Given that the point R is (,, ), RS = find the coordinates of T. and RT = RS, A (,, ) (,, ) (9,, 7) (9,, 7). The diagram shows a curve with equation of the form y = ax + bx + c. y O x Here are two statements about a, b and c: () a > 0 () b ac > 0 Which of the following is true? A Neither statement is correct. Only statement () is correct. Only statement () is correct. oth statements are correct. [X00//] Page eight

9 . If cosx =, what is the value of cos x? 5 A The graph of a cubic function, y = f(x), is shown below. It passes through the points (, 0), (, 0), (, 0) and (0, ). y O x What is the equation of this curve? A y = (x )(x + )(x + ) y = (x )(x + )(x + ) y = (x + )(x )(x ) y = (x + )(x )(x ) [Turn over [X00//] Page nine

10 6. If e t = 6, find an expression for t. A t = log e e loge 6 t = t = 6 log log t = log e e 6 7. Vectors u and v have components 5 0 t and respectively. Here are two statements about u and v: () when t =, u is a unit vector 5 () when t =, u and v are parallel Which of the following is true? A Neither statement is correct. Only statement () is correct. Only statement () is correct. oth statements are correct. 8. The circle with equation x + y x 0y + k = 0 meets the coordinate axes at exactly three points. What is the value of k? A [X00//] Page ten

11 9. The diagram shows part of the graph of y = a cos bx. The shaded area is unit. y O p p x p 0 What is the value of ( cos ) A a bx dx? 0. The only stationary point on the graph of y = f(x) is the point (a, b). What are the coordinates of the only stationary point on the graph of y = f(x)? A ( a, b) ( a, b) ( a, b) ( a, b) [EN OF SETION A] [Turn over [X00//] Page eleven

12 SETION ALL questions should be attempted. Marks. (a) Show that (x ) is a factor of x 6x + 9x and hence factorise x 6x + 9x fully. (b) The diagram shows the graph with equation y = x 6x + x. y y = x 6x + x A(, ) O x (i) Find the equation of the tangent to the curve y = x 6x + x at the point A(, ). (ii) Hence find the coordinates of, the point of intersection of this tangent with the curve. [X00//] Page twelve

13 . The function f(x) = + x x numbers. is defined on the domain x > 0, x R, the set of real Marks Find the maximum and minimum values of f(x) on the closed interval x. 6. Solve log (x + 7) = + log (x ), x >.. Find the range of values for k such that kx + x + 9k = 0 has real roots. 5. A stunt performed by two members of a motorcycle display team requires them to travel, at speed, at right angles to each other across the arena. y (t 5, 0) O x (0, t 0) The positions of the motorcyclists, relative to suitable axes, t seconds after the stunt begins, are (t 5, 0) and (0, t 0). (a) Show that, at any given moment, the distance,, between them is given by = 5t 0t+ 5. (b) etermine whether the distance between the motorcyclists is increasing or decreasing 5 seconds after the start of the stunt. [EN OF SETION ] [EN OF QUESTION PAPER] [X00//] Page thirteen

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17 X00// NATIONAL QUALIFIATIONS 05 WENESAY, 0 MAY 0.50 AM.00 NOON MATHEMATIS HIGHER Paper Read carefully alculators may be used in this paper. Full credit will be given only to solutions which contain appropriate working. Answers obtained by readings from scale drawings will not receive any credit. Write your answers clearly in the answer booklet provided. Additional space for answers is provided at the end of the answer booklet. If you use this space you must clearly identify the question number you are attempting. Use blue or black ink. efore leaving the examination room you must give your answer booklet to the Invigilator; if you do not, you may lose all the marks for this paper. HTP *X00//*

18 FORMULAE LIST ircle: The equation x + y + gx + fy + c = 0 represents a circle centre ( g, f) and radius The equation (x a) + (y b) = r represents a circle centre (a, b) and radius r. g + f c. Scalar Product: a.b = a b cos q, where q is the angle between a and b a b or a.b = a b + a b + a b where a = a b and b=. a b Trigonometric formulae: sin (A ± ) = sin A cos ± cos A sin cos (A ± ) = cos A cos sin A sin sin A = sin A cos A ± cos A = cos A sin A = cos A = sin A Table of standard derivatives: f(x) f (x) sin ax cos ax a cos ax a sin ax Table of standard integrals: f(x) sin ax cos ax f (x)dx a cos ax + c a sin ax + c [X00//] Page two

19 ALL questions should be attempted. Marks. The vertices of triangle A are A( 5, 7), (, 5) and (, ) as shown in the diagram. The broken line represents the altitude from. y A x (a) Show that the equation of the altitude from is x y =. (b) Find the equation of the median from. (c) Find the coordinates of the point of intersection of the altitude from and the median from.. Functions f and g are defined on suitable domains by f(x) = 0 + x and g(x) = ( + x) ( x) +. (a) Find an expression for f(g(x)). (b) Express f(g(x)) in the form p(x + q) + r. (c) Another function h is given by hx ( )= f g x ( ( )) What values of x cannot be in the domain of h?. [Turn over [X00//] Page three

20 . A version of the following problem first appeared in print in the 6th entury. Marks A frog and a toad fall to the bottom of a well that is 50 feet deep. Each day, the frog climbs feet and then rests overnight. uring the night, it slides down of its height above the floor of the well. The toad climbs feet each day before resting. Overnight, it slides down of its height above the floor of the well. Their progress can be modelled by the recurrence relations: f = n+ f + n f =, t = n+ t + n t =, where f n and t n are the heights reached by the frog and the toad at the end of the nth day after falling in. (a) alculate t, the height of the toad at the end of the second day. (b) etermine whether or not either of them will eventually escape from the well. 5 [X00//] Page four

21 . A wall plaque is to be made to commemorate the 50th anniversary of the publication of Alice s Adventures in Wonderland. The edges of the wall plaque can be modelled by parts of the graphs of four quadratic functions as shown in the sketch. y Marks y = f(x) y = g(x) y = h(x) y = k(x) O x ( ) ( ) ( ) ( ) f x = x x+ g x = x x+ 5 9 hx = x x+ 8 kx = x x 8 (a) Find the x-coordinate of the point of intersection of the graphs with equations y = f(x) and y = g(x). The graphs of the functions f(x) and h(x) intersect on the y-axis. The plaque has a vertical line of symmetry. (b) alculate the area of the wall plaque. 7 [Turn over [X00//] Page five

22 5. ircle has equation x + y + 6x + 0y + 9 = 0. The centre of circle is (9, ). ircles and touch externally. Marks (a) etermine the radius of. A third circle,, is drawn such that: both and touch internally the centres of, and are collinear. (b) etermine the equation of. [X00//] Page six

23 6. Vectors p, q and r are represented on the diagram as shown. Marks E is a parallelogram AE is an equilateral triangle p = Angle A = 90 r E q A p (a) Evaluate p.(q + r). (b) Express E in terms of p, q and r. (c) Given that AE.E = 9 9, find r. [Turn over for Questions seven and eight on Page eight [X00//] Page seven

24 ( ) 7. (a) Find cosx+ dx. (b) Show that cos x + = cos x sin x. ( ) (c) Hence, or otherwise, find sin x cos x dx. Marks 8. The blades of a wind turbine are turning at a steady rate. The height, h metres, of the tip of one of the blades above the ground at time, t seconds, is given by the formula h = 6 sin( 5t) 5 cos( 5t) Express 6 sin( 5t) 5 cos( 5t) in the form k sin( 5t a), where k > 0 and 0 < a < π, and hence find the two values of t for which the tip of this blade is at a height of 00 metres above the ground during the first turn. 8 [EN OF QUESTION PAPER] [X00//] Page eight