*X100/12/02* X100/12/02. MATHEMATICS HIGHER Paper 1 (Non-calculator) MONDAY, 21 MAY 1.00 PM 2.30 PM NATIONAL QUALIFICATIONS 2012

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1 X00//0 NTIONL QULIFITIONS 0 MONY, MY.00 PM.0 PM MTHEMTIS HIGHER Paper (Non-calculator) Read carefully alculators may NOT be used in this paper. Section Questions 0 (40 marks) Instructions for completion of Section are given on Page two. For this section of the examination you must use an H pencil. Section (0 marks) Full credit will be given only where the solution contains appropriate working. nswers obtained by readings from scale drawings will not receive any credit. LI X00//0 /80 *X00//0*

2 Read carefully heck that the answer sheet provided is for Mathematics Higher (Section ). For this section of the examination you must use an H pencil and, where necessary, an eraser. heck that the answer sheet you have been given has your name, date of birth, SN (Scottish andidate Number) and entre Name printed on it. o not change any of these details. 4 If any of this information is wrong, tell the Invigilator immediately. If this information is correct, print your name and seat number in the boxes provided. The answer to each question is either,, or. ecide what your answer is, then, using your pencil, put a horizontal line in the space provided (see sample question below). 7 There is only one correct answer to each question. 8 Rough working should not be done on your answer sheet. 9 t the end of the exam, put the answer sheet for Section inside the front cover of your answer book. Sample Question curve has equation y = x 4x. What is the gradient at the point where x =? The correct answer is 8. The answer has been clearly marked in pencil with a horizontal line (see below). hanging an answer If you decide to change your answer, carefully erase your first answer and, using your pencil, fill in the answer you want. The answer below has been changed to. [X00//0] Page two

3 FORMULE 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 ( ± ) = sin cos ± cos sin cos ( ± ) = cos cos sin sin sin = sin cos ± cos = cos sin = cos = sin 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 + a sin ax + [Turn over [X00//0] Page three

4 SETION LL questions should be attempted.. sequence is defined by the recurrence relation u n + = u n + 4, with u 0 =. Find the value of u What is the gradient of the tangent to the curve with equation y = x x + at the point where x =? 4. If x x + 4 is written in the form (x p) + q, what is the value of q? What is the gradient of the line shown in the diagram? y 0 O x [X00//0] Page four

5 . The diagram shows a right-angled triangle with sides and angles as marked. What is the value of cosa? a If y = x dy + x, x > 0, determine. dx x + x + x x + x x x x 7. If u = and v = t are perpendicular, what is the value of t? t [Turn over [X00//0] Page five

6 4 8. The volume of a sphere is given by the formula V = pr. What is the rate of change of V with respect to r, at r =? p p p p 9. The diagram shows the curve with equation of the form y = cos(x + a) + b for 0 x p. y O p 7p p x What is the equation of this curve? π y= cos x π y= cos x + y= cos x+ y = cos x + π π + [X00//0] Page six

7 0. The diagram shows a square-based pyramid P,QRST. TS, TQ and TP represent f, g and h respectively. P R h Q S f T g Express RP in terms of f, g and h. f + g h f g + h f g h f + g + h. Find dx, x 0. x x + c x + c x + c x + c. Find the maximum value of sin x π and the value of x where this occurs in the interval 0 x p. max value x π π π π [X00//0] Page seven [Turn over

8 . parabola intersects the axes at x =, x = and y =, as shown in the diagram. y O x What is the equation of the parabola? y = (x )(x ) y = (x + )(x + ) y = (x )(x ) y = (x + )(x + ) 4. Find ( x ) dx where x >. ( ) + x ( ) + x ( ) + x ( ) + x c c c c. If u = k, where k>0 and u is a unit vector, determine the value of k [X00//0] Page eight

9 . If y = cos 4 dy x, find. dx cos x sin x cos x cos x sin x sin x 7. Given that a = 4 and a. +, what is the value of a.b? ( a b)= The graph of y = f(x) shown has stationary points at (0, p) and (q, r). Here are two statements about f(x): () f(x) < 0 for s < x < t; () f (x) < 0 for x < q. Which of the following is true? y p y = f(x) Neither statement is correct. Only statement () is correct. Only statement () is correct. O s (q, r) t x oth statements are correct. [Turn over [X00//0] Page nine

10 9. Solve x x < 0. < x < x <, x > < x < x <, x > log 9a b 0. Simplify, where a > 0 and b > 0. log a b a log b a log b (9a a) [EN OF SETION ] [X00//0] Page ten

11 SETION LL questions should be attempted. Marks. (a) (i) Show that (x 4) is a factor of x x + x + 8. (ii) Factorise x x + x + 8 fully. (iii) Solve x x + x + 8 = 0. (b) The diagram shows the curve with equation y = x x + x + 8. y y = x x + x + 8 P O Q R x The curve crosses the x-axis at P, Q and R. etermine the shaded area. ( ). (a) The expression cosx sin xcan be written in the form kcos x+ a where k > 0 and 0 a < p. alculate the values of k and a. (b) Find the points of intersection of the graph of y = cosx sin x with the x and y axes, in the interval 0 x p. 4 [Turn over for Question on Page twelve [X00//0] Page eleven

12 . (a) Find the equation of l, the perpendicular bisector of the line joining P(, ) to Q(, 9). (b) Find the equation of l which is parallel to PQ and passes through R(, ). (c) Find the point of intersection of l and l. (d) Hence find the shortest distance between PQ and l. Marks 4 [EN OF SETION ] [EN OF QUESTION PPER] [X00//0] Page twelve

13 X00//0 NTIONL QULIFITIONS 0 MONY, MY.0 PM 4.00 PM MTHEMTIS HIGHER Paper Read arefully alculators may be used in this paper. Full credit will be given only where the solution contains appropriate working. nswers obtained by readings from scale drawings will not receive any credit. LI X00//0 /80 *X00//0*

14 FORMULE 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 ( ± ) = sin cos ± cos sin cos ( ± ) = cos cos sin sin sin = sin cos ± cos = cos sin = cos = sin 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 + a sin ax + [X00//0] Page two

15 LL questions should be attempted. Marks. Functions f and g are defined on the set of real numbers by (a) Find expressions for: (i) f(g(x)); f(x) = x + g(x) = x + 4. (ii) g(f(x)). (b) Show that f(g(x)) + g(f(x)) = 0 has no real roots.. (a) Relative to a suitable set of coordinate axes, iagram shows the line x y + = 0 intersecting the circle x + y x y 0 = 0 at the points P and Q. Q P iagram Find the coordinates of P and Q. (b) iagram shows the circle from (a) and a second congruent circle, which also passes through P and Q. Q etermine the equation of this second circle. P iagram [X00//0] Page three [Turn over

16 . function f is defined on the domain 0 x by f(x) = x x 4x +. etermine the maximum and minimum values of f. Marks 7 4. The diagram below shows the graph of a quartic y = h(x), with stationary points at x = 0 and x =. y O x y = h(x) On separate diagrams sketch the graphs of: (a) y = h'(x); (b) y = h'(x).. is the point (,, 0), is (,, ) and is (4, k, 0). (a) (i) Express and in component form. ^ (ii) Show that cos = ( k + k+ 4) (b) If angle = 0, find the possible values of k. 7 [X00//0] Page four

17 π. For 0 < x <, sequences can be generated using the recurrence relation Marks u = (sin x) u + cos x,withu = n+ n 0. (a) Why do these sequences have a limit? (b) The limit of one sequence generated by this recurrence relation is sin x. Find the value(s) of x The diagram shows the curves with equations y = 4 and y. x = x y y = 4 x T O y = x x The graphs intersect at the point T. log p (a) Show that the x coordinate of T can be written in the form log a q, a for all a >. (b) alculate the y coordinate of T. [EN OF QUESTION PPER] [X00//0] Page five

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