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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*

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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*X100/12/02* X100/12/02. MATHEMATICS HIGHER Paper 1 (Non-calculator) NATIONAL QUALIFICATIONS 2014 TUESDAY, 6 MAY 1.00 PM 2.30 PM X00//0 NTIONL QULIFITIONS 0 TUESY, 6 MY.00 PM.0 PM MTHEMTIS HIGHER Paper (Non-calculator) Read carefully alculators may NOT be used in this paper. Section Questions 0 (0 marks) Instructions for completion

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GRAPHING IN POLAR COORDINATES SYMMETRY GRAPHING IN POLAR COORDINATES SYMMETRY Recall from Algebra and Calculus I that the concept of symmetry was discussed using Cartesian equations. Also remember that there are three types of symmetry - y-axis,

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STRAND: ALGEBRA Unit 3 Solving Equations CMM Subject Support Strand: ALGEBRA Unit Solving Equations: Tet STRAND: ALGEBRA Unit Solving Equations TEXT Contents Section. Algebraic Fractions. Algebraic Fractions and Quadratic Equations. Algebraic

Algebra 2 Chapter 1 Vocabulary. identity - A statement that equates two equivalent expressions. Chapter 1 Vocabulary identity - A statement that equates two equivalent expressions. verbal model- A word equation that represents a real-life problem. algebraic expression - An expression with variables.

GEOMETRIC MENSURATION GEOMETRI MENSURTION Question 1 (**) 8 cm 6 cm θ 6 cm O The figure above shows a circular sector O, subtending an angle of θ radians at its centre O. The radius of the sector is 6 cm and the length of the

Contents. 2 Lines and Circles 3 2.1 Cartesian Coordinates... 3 2.2 Distance and Midpoint Formulas... 3 2.3 Lines... 3 2.4 Circles... Contents Lines and Circles 3.1 Cartesian Coordinates.......................... 3. Distance and Midpoint Formulas.................... 3.3 Lines.................................. 3.4 Circles..................................

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6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives 6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise

Section 2.7 One-to-One Functions and Their Inverses Section. One-to-One Functions and Their Inverses One-to-One Functions HORIZONTAL LINE TEST: A function is one-to-one if and only if no horizontal line intersects its graph more than once. EXAMPLES: 1.

Unit 2: Number, Algebra, Geometry 1 (Non-Calculator) Write your name here Surname Other names Edexcel GCSE Centre Number Mathematics B Unit 2: Number, Algebra, Geometry 1 (Non-Calculator) Friday 14 June 2013 Morning Time: 1 hour 15 minutes Candidate Number

Mathematics I, II and III (9465, 9470, and 9475) Mathematics I, II and III (9465, 9470, and 9475) General Introduction There are two syllabuses, one for Mathematics I and Mathematics II, the other for Mathematics III. The syllabus for Mathematics I and

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Mark Scheme. Mathematics 6360. General Certificate of Education. 2006 examination June series. MPC1 Pure Core 1 Version 1.0: 0706 abc General Certificate of Education Mathematics 660 MPC1 Pure Core 1 Mark Scheme 006 examination June series Mark schemes are prepared by the Principal Examiner and considered, together PRE-CALCULUS GRADE 12 [C] Communication Trigonometry General Outcome: Develop trigonometric reasoning. A1. Demonstrate an understanding of angles in standard position, expressed in degrees and radians.

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Algebra. Exponents. Absolute Value. Simplify each of the following as much as possible. 2x y x + y y. xxx 3. x x x xx x. 1. Evaluate 5 and 123 Algebra Eponents Simplify each of the following as much as possible. 1 4 9 4 y + y y. 1 5. 1 5 4. y + y 4 5 6 5. + 1 4 9 10 1 7 9 0 Absolute Value Evaluate 5 and 1. Eliminate the absolute value bars from 12.1: Tangent Lines Congruent Circles: circles that have the same radius length Diagram of Examples Center of Circle: Circle Name: Radius: Diameter: Chord: Secant: Tangent to A Circle: a line in the plane SOLUTIONS Problem. Find the critical points of the function f(x, y = 2x 3 3x 2 y 2x 2 3y 2 and determine their type i.e. local min/local max/saddle point. Are there any global min/max? Partial derivatives