MARKSCHEME SPECIMEN MATHEMATICS

Size: px
Start display at page:

Download "MARKSCHEME SPECIMEN MATHEMATICS"

Transcription

1 SPEC/5/MATHL/HP/ENG/TZ/XX/M MARKSCHEME SPECIMEN MATHEMATICS Higher Level Paper pages

2 5 SPEC/5/MATHL/HP/ENG/TZ/XX/M SECTION A. (a) 8 sin [ mark] 8 8 tan [ marks] (c) cos 6 cos [ marks] Total [6 marks]. (a) sum 45 9, product 4 9 [ mark] 45 4 it follows that and ( ) solving, ( ) the other two roots are 4, [6 marks] Total [7 marks]

3 6 SPEC/5/MATHL/HP/ENG/TZ/XX/M. (a) P(no heads from n coins tossed).5 n () P(no head) [ marks] P( coins and no heads) P( no heads) P(no heads) [ marks] Total [6 marks] 4. (a) E( ) ( )d X ( ) ( ) f at the mode therefore the mode ( ) ( ) f [ marks] [ marks] Total [6 marks] 5. (a) f( ) cos( ) ( )sin( ) cos sin f( ) therefore f is even f( ) sinsin cos ( sin cos ) f( ) coscos sin ( sin ) so f () AG [ marks] [ marks] continued

4 7 SPEC/5/MATHL/HP/ENG/TZ/XX/M Question 5 continued (c) John s statement is incorrect because either; there is a stationary point at (, ) and since f is an even function and therefore symmetrical about the y-ais it must be a maimum or a minimum or; f ( ) is even and therefore has the same sign either side of (, ) R [ marks] Total [7 marks] 6. (a) BC BC 7 9 area of triangle ABC 9 therefore AD AD 7 [ marks] [4 marks] Total [7 marks] 7. (a) using row operations, to obtain equations in the same variables for eample yz yz the fact that one of the left hand sides is a multiple of the other left hand side indicates that the equations do not have a unique solution, or equivalent RAG [4 marks] (i) (ii) put z then y and or equivalent [4 marks] Total [8 marks]

5 8 SPEC/5/MATHL/HP/ENG/TZ/XX/M 8. taking cross products with a, a( abc) a using the algebraic properties of vectors and the fact that aa, abac abca AG taking cross products with b, b( abc ) babc abbc AG this completes the proof [6 marks] 9. (a) (i) e ( ) e e f (ii) by inspection the two roots are, e [ marks] A Note: Award for maimum, for minimum and for general shape. [ marks] (c) e from the graph: e for all ecept e R e putting, conclude that e AG [ mark] Total [7 marks]

6 9 SPEC/5/MATHL/HP/ENG/TZ/XX/M SECTION B. (a) in Cartesian form z i i z z i i i ( i) ( i) ( i) i in modulus-argument form z cis5 z cis5 z cis5 cis5 z equating the two epressions for z cos5 sin5 cos5 tan 75 sin5 [7 marks] [5 marks] Total [ marks]

7 SPEC/5/MATHL/HP/ENG/TZ/XX/M. (a) let sin d cosd I cos cos d cos d Note: Award for limits and for epression. 4 ( cos )d sin 4 [7 marks].5.5 I d arcsin.5.5 arcsin [5 marks] (c) t 4 () dt ( t ) I () t ( t ) ( t ) dt t d sec d,, [,] arctan 6 [7 marks] Total [9 marks]

8 SPEC/5/MATHL/HP/ENG/TZ/XX/M. (a) f( ) e sin e cos f( ) e sine cose cos e sin e cos e sin AG f ( ) e sin e cos (4) f ( ) e sin e cos e cos e sin 4e cos 4e sin ( ) [ marks] [4 marks] (c) Note: the conjecture is that ( n ) ( ) n f e sin n for n, this formula gives f( ) e sin which is correct ( k) k k let the result be true for n k, ie.. f ( ) e sin (k consider k k k k f ( ) e sin e cos ( k ) k k k k k k k k f ( ) e sin e cos e cos e sin k k e cos k ( k ) e sin therefore true for nk true for nk and since true for n the result is proved by induction. R Award the final R only if the two M marks have been awarded. [8 marks] Total [5 marks]

9 SPEC/5/MATHL/HP/ENG/TZ/XX/M. (a) f continuous lim f( ) lim f( ) 4ab 8, f( ) a b, f continuous lim f( ) lim f( ) 4ab solve simultaneously to obtain a and b 6 for, f( ) for, f( ) 6 since f( ) for all values in the domain of f, f is increasing R therefore one-to-one AG (c) y y y 6y 5 y 6y5 y 4 therefore f, ( ) 4, 4 Note: Award for the first line and for the second line. [6 marks] [ marks] [5 marks] Total [4 marks]

10 SPEC/5/MATHL/HP/ENG/TZ/XX/M MARKSCHEME SPECIMEN MATHEMATICS Higher Level Paper pages

11 5 SPEC/5/MATHL/HP/ENG/TZ/XX/M SECTION A. f() 8 4ab 4 4ab 4 f() ab 4 6 ab solving, a, b 4 [6 marks] a. we are given that ar 9 and 64 r dividing, r ( r) 64 64r 64r 9 r.75, a 6 9 [5 marks]. (a) Interval Frequency ].,.] 6 ].,.] 8 ].,.] 8 ].,.4] 4 ].4,.5] ].5,.6] 4 A [ marks].6,. (c) no because the normal distribution is symmetric and these data are not R [ marks] [ marks] Total [6 marks] 4. (a) mod ( z), arg ( z) 5 [ marks] z (cos5 isin5 ) ().8.965i [ marks] (c) we require to find a multiple of 5 that is also a multiple of 6, so by any method, n Note: Only award mark for part (c) if n is based on arg( z). [ marks] Total [6 marks]

12 6 SPEC/5/MATHL/HP/ENG/TZ/XX/M 5. (a) (i) displacement vdt ().7 (m) (ii) total distance v dt ().5 (m) t cos( u ) d u solving the equation () t.9 (s) [4 marks] [ marks] Total [6 marks] 6. vertical asymptote 44bc horizontal asymptote y b b and c a a [6 marks]

13 7 SPEC/5/MATHL/HP/ENG/TZ/XX/M 7. (a) let the interception occur at the point P, t hrs after : then, SP t and MP t using the sine rule, SP sin MP sin5 whence 8. [4 marks] using the sine rule again, MP sin5 MS sin ( ) sin5 t sin t the interception occurs at :49 [5 marks] Total [9 marks]

14 8 SPEC/5/MATHL/HP/ENG/TZ/XX/M 8. (a) OC AB OAcos6 BCcos6 AB AB AB AB AG [ marks] OC AB ( ba ) OD OC CD OC AO baa ba OE BC babba [7 marks] Total [9 marks] 9. let, y (m) denote respectively the distance of the bottom of the ladder from the wall and the distance of the top of the ladder from the ground then, y d dy y dt dt when 4, y 84 and d.5 dt dy substituting, dt dy.8 ms dt (speed of descent is.8 ms ) [7 marks]

15 9 SPEC/5/MATHL/HP/ENG/TZ/XX/M SECTION B. (a) (i) OA OB i7 j5k (ii) area 5 i7j5 k (4.) (iii) equation of plane is 7y5z k 7y5z [5 marks] (i) direction of line (i j k) ( ij k) i j k equation of line is r ( ij k) ( i j k ) (ii) at a point of intersection, 4 solving the nd and rd equations, 4, these values do not satisfy the st equation so the lines are skew R [7 marks] Total [ marks]

16 SPEC/5/MATHL/HP/ENG/TZ/XX/M I. (a) (i) S P R I I S P R R I I P R AG (ii) etending this, n n I I I Sn P R... n I R n I P I n n I R I =P I AG [7 marks] (i) putting S6, P5, I R(. ) R ($). (ii) putting n, P5, I, R. S 5..(. ) ($)65 which is the outstanding amount [6 marks] Total [ marks]

17 SPEC/5/MATHL/HP/ENG/TZ/XX/M. (a) we are given that ,.4 [4 marks] (i) let X denote the number of birds weighing more than.5 kg then X is B(,.5) E( X ).5 (ii).584 (iii) to find the most likely value of X, consider p.56, p.877, p.85, p.5 therefore, most likely value [5 marks] (c) (i) we solve P( Y ).885 using the GDC. (ii) let X, X denote the number of eggs laid by each bird P( X X ) P( X )P( X ) P( X )P( X ) P( X )P( X ) 9 9 e e (e ) e e.446 (iii) P( X, X ) P( X, X X X ) P( X X ).5 [8 marks] Total [7 marks]

18 SPEC/5/MATHL/HP/ENG/TZ/XX/M. (a) f( ) sin( ) ln( )cos( ) [ marks] A4 Note: Award for graphs, for intercepts. [4 marks] (c).,. (d) f (.75).899 so equation of normal is y.9578 (.75).899 y [ marks] [ marks] (e) A(, ) c d B(.548,.4 ) e f C(.44,.88 ) Note: Accept coordinates for B and C rounded to significant figures. area ABC ( ) ( ) cidj ei fj ( de cf ).554 [6 marks] Total [8 marks]

Math 1000 Final Exam Review Solutions. (x + 3)(x 2) = lim. = lim x 2 = 3 2 = 5. (x + 1) 1 x( x ) = lim. = lim. f f(1 + h) f(1) (1) = lim

Math 1000 Final Exam Review Solutions. (x + 3)(x 2) = lim. = lim x 2 = 3 2 = 5. (x + 1) 1 x( x ) = lim. = lim. f f(1 + h) f(1) (1) = lim Math Final Eam Review Solutions { + 3 if < Consider f() Find the following limits: (a) lim f() + + (b) lim f() + 3 3 (c) lim f() does not eist Find each of the following limits: + 6 (a) lim 3 + 3 (b) lim

More information

Answers for NSSH exam paper 2 type of questions, based on the syllabus part 2 (includes 16)

Answers for NSSH exam paper 2 type of questions, based on the syllabus part 2 (includes 16) Answers for NSSH eam paper type of questions, based on the syllabus part (includes 6) Section Integration dy 6 6. (a) Integrate with respect to : d y c ( )d or d The curve passes through P(,) so = 6/ +

More information

Section A. A 5 12 sin cm. AC cos100 7 M15/5/MATHL/HP2/ENG/TZ2/XX/M. 1. (a) (M1) A1 [2 marks] (b) (M1) therefore AC 13.

Section A. A 5 12 sin cm. AC cos100 7 M15/5/MATHL/HP2/ENG/TZ2/XX/M. 1. (a) (M1) A1 [2 marks] (b) (M1) therefore AC 13. 7 M5/5/MATHL/HP/ENG/TZ/XX/M Section A. (a) A 5 sin00 9.5 cm (M) A [ marks] (b) AC 5 5 cos00 (M) therefore AC 3.8 (cm) A [ marks] Total [4 marks]. (a) 098 330 4 43 (M)A [ marks] (b) 5 6 54 65 50 M A [ marks]

More information

Topic 6: Calculus Integration Markscheme 6.10 Area Under Curve Paper 2

Topic 6: Calculus Integration Markscheme 6.10 Area Under Curve Paper 2 Topic 6: Calculus Integration Markscheme 6. Area Under Curve Paper. (a). N Standard Level (b) (i). N (ii).59 N (c) q p f ( ) = 9.96 N split into two regions, make the area below the -ais positive RR N

More information

Mock Exam 3. 1 Hong Kong Educational Publishing Company. Section A. 1. Reference: HKDSE Math M Q1 (a) (1 + 2x) 2 (1 - x) n

Mock Exam 3. 1 Hong Kong Educational Publishing Company. Section A. 1. Reference: HKDSE Math M Q1 (a) (1 + 2x) 2 (1 - x) n Mock Eam Mock Eam Section A. Reference: HKDSE Math M 0 Q (a) ( + ) ( - ) n nn ( ) ( + + ) n + + Coefficient of - n - n -7 n (b) Coefficient of nn ( - ) - n + (- ) - () + (). Reference: HKDSE Math M PP

More information

e x for x 0. Find the coordinates of the point of inflexion and justify that it is a point of inflexion. (Total 7 marks)

e x for x 0. Find the coordinates of the point of inflexion and justify that it is a point of inflexion. (Total 7 marks) Chapter 0 Application of differential calculus 014 GDC required 1. Consider the curve with equation f () = e for 0. Find the coordinates of the point of infleion and justify that it is a point of infleion.

More information

y= sin3 x+sin6x x 1 1 cos(2x + 4 ) = cos x + 2 = C(x) (M2) Therefore, C(x) is periodic with period 2.

y= sin3 x+sin6x x 1 1 cos(2x + 4 ) = cos x + 2 = C(x) (M2) Therefore, C(x) is periodic with period 2. . (a).5 0.5 y sin x+sin6x 0.5.5 (A) (C) (b) Period (C) []. (a) y x 0 x O x Notes: Award for end points Award for a maximum of.5 Award for a local maximum of 0.5 Award for a minimum of 0.75 Award for the

More information

IB Practice - Calculus - Differentiation Applications (V2 Legacy)

IB Practice - Calculus - Differentiation Applications (V2 Legacy) IB Math High Level Year - Calc Practice: Differentiation Applications IB Practice - Calculus - Differentiation Applications (V Legacy). A particle moves along a straight line. When it is a distance s from

More information

(b) g(x) = 4 + 6(x 3) (x 3) 2 (= x x 2 ) M1A1 Note: Accept any alternative form that is correct. Award M1A0 for a substitution of (x + 3).

(b) g(x) = 4 + 6(x 3) (x 3) 2 (= x x 2 ) M1A1 Note: Accept any alternative form that is correct. Award M1A0 for a substitution of (x + 3). Paper. Answers. (a) METHOD f (x) q x f () q 6 q 6 f() p + 8 9 5 p METHOD f(x) (x ) + 5 x + 6x q 6, p (b) g(x) + 6(x ) (x ) ( + x x ) Note: Accept any alternative form that is correct. Award A for a substitution

More information

CALCULUS AB/BC SUMMER REVIEW PACKET (Answers)

CALCULUS AB/BC SUMMER REVIEW PACKET (Answers) Name CALCULUS AB/BC SUMMER REVIEW PACKET (Answers) I. Simplify. Identify the zeros, vertical asymptotes, horizontal asymptotes, holes and sketch each rational function. Show the work that leads to your

More information

1985 AP Calculus AB: Section I

1985 AP Calculus AB: Section I 985 AP Calculus AB: Section I 9 Minutes No Calculator Notes: () In this eamination, ln denotes the natural logarithm of (that is, logarithm to the base e). () Unless otherwise specified, the domain of

More information

1. Find the area enclosed by the curve y = arctan x, the x-axis and the line x = 3. (Total 6 marks)

1. Find the area enclosed by the curve y = arctan x, the x-axis and the line x = 3. (Total 6 marks) 1. Find the area enclosed by the curve y = arctan, the -ais and the line = 3. (Total 6 marks). Show that the points (0, 0) and ( π, π) on the curve e ( + y) = cos (y) have a common tangent. 3. Consider

More information

1. (a) B, D A1A1 N2 2. A1A1 N2 Note: Award A1 for. 2xe. e and A1 for 2x.

1. (a) B, D A1A1 N2 2. A1A1 N2 Note: Award A1 for. 2xe. e and A1 for 2x. 1. (a) B, D N (b) (i) f () = e N Note: Award for e and for. (ii) finding the derivative of, i.e. () evidence of choosing the product rule e.g. e e e 4 e f () = (4 ) e AG N0 5 (c) valid reasoning R1 e.g.

More information

Limits, Continuity, and Differentiability Solutions

Limits, Continuity, and Differentiability Solutions Limits, Continuity, and Differentiability Solutions We have intentionally included more material than can be covered in most Student Study Sessions to account for groups that are able to answer the questions

More information

IB Math SL 1: Trig Practice Problems: MarkScheme Circular Functions and Trig - Practice Problems (to 07) MarkScheme

IB Math SL 1: Trig Practice Problems: MarkScheme Circular Functions and Trig - Practice Problems (to 07) MarkScheme IB Math SL : Trig Practice Problems: MarkScheme Circular Functions and Trig - Practice Problems (to 07) MarkScheme. (a) Evidence of using the cosine rule p + r q eg cos P Qˆ R, q p + r pr cos P Qˆ R pr

More information

AP Calculus (BC) Summer Assignment (104 points)

AP Calculus (BC) Summer Assignment (104 points) AP Calculus (BC) Summer Assignment (0 points) This packet is a review of some Precalculus topics and some Calculus topics. It is to be done NEATLY and on a SEPARATE sheet of paper. Use your discretion

More information

H2 MATHS SET D PAPER 1

H2 MATHS SET D PAPER 1 H Maths Set D Paper H MATHS Exam papers with worked solutions SET D PAPER Compiled by THE MATHS CAFE P a g e b The curve y ax c x 3 points, and, H Maths Set D Paper has a stationary point at x 3. It also

More information

1969 AP Calculus BC: Section I

1969 AP Calculus BC: Section I 969 AP Calculus BC: Section I 9 Minutes No Calculator Note: In this eamination, ln denotes the natural logarithm of (that is, logarithm to the base e).. t The asymptotes of the graph of the parametric

More information

Section Inverse Trigonometry. In this section, we will define inverse since, cosine and tangent functions. x is NOT one-to-one.

Section Inverse Trigonometry. In this section, we will define inverse since, cosine and tangent functions. x is NOT one-to-one. Section 5.4 - Inverse Trigonometry In this section, we will define inverse since, cosine and tangent functions. RECALL Facts about inverse functions: A function f ) is one-to-one if no two different inputs

More information

Special Maths Exam Paper 2 November 2013 Solutions

Special Maths Exam Paper 2 November 2013 Solutions Special Maths Eam Paper 2 November 2013 Solutions Question One 1.1 sin θ = 4/5 > 0, 270 o < θ 360 o. If 4 and 5 are the lengths of sides of a right-angled triangle, with 5 the hpotenuse, then the third

More information

d (5 cos 2 x) = 10 cos x sin x x x d y = (cos x)(e d (x 2 + 1) 2 d (ln(3x 1)) = (3) (M1)(M1) (C2) Differentiation Practice Answers 1.

d (5 cos 2 x) = 10 cos x sin x x x d y = (cos x)(e d (x 2 + 1) 2 d (ln(3x 1)) = (3) (M1)(M1) (C2) Differentiation Practice Answers 1. . (a) y x ( x) Differentiation Practice Answers dy ( x) ( ) (A)(A) (C) Note: Award (A) for each element, to a maximum of [ marks]. y e sin x d y (cos x)(e sin x ) (A)(A) (C) Note: Award (A) for each element.

More information

FP1 Mark Schemes from old P4, P5, P6 and FP1, FP2, FP3 papers (back to June 2002)

FP1 Mark Schemes from old P4, P5, P6 and FP1, FP2, FP3 papers (back to June 2002) FP1 Mar Schemes from old P4, P5, P6 and FP1, FP, FP papers (bac to June 00) Please note that the following pages contain mar schemes for questions from past papers which were not written at an AS standard

More information

Paper2Practice [303 marks]

Paper2Practice [303 marks] PaperPractice [0 marks] Consider the expansion of (x + ) 10. 1a. Write down the number of terms in this expansion. [1 mark] 11 terms N1 [1 mark] 1b. Find the term containing x. evidence of binomial expansion

More information

Mathematics Extension 2

Mathematics Extension 2 0 HIGHER SCHOOL CERTIFICATE EXAMINATION Mathematics Etension General Instructions Reading time 5 minutes Working time hours Write using black or blue pen Black pen is preferred Board-approved calculators

More information

FP1 Mark Schemes from old P4, P5, P6 and FP1, FP2, FP3 papers (back to June 2002)

FP1 Mark Schemes from old P4, P5, P6 and FP1, FP2, FP3 papers (back to June 2002) FP Mar Schemes from old P4, P5, P6 and FP, FP, FP papers (bac to June 00) Please note that the following pages contain mar schemes for questions from past papers which were not written at an AS standard

More information

Markscheme May 2017 Mathematics Higher level Paper 1

Markscheme May 2017 Mathematics Higher level Paper 1 M17/5/MATHL/HP1/ENG/TZ/XX/M Markscheme May 017 Mathematics Higher level Paper 1 0 pages M17/5/MATHL/HP1/ENG/TZ/XX/M This markscheme is the property of the International Baccalaureate and must not be reproduced

More information

Calculus-Lab ) f(x) = 1 x. 3.8) f(x) = arcsin( x+1., prove the equality cosh 2 x sinh 2 x = 1. Calculus-Lab ) lim. 2.7) lim. 2.

Calculus-Lab ) f(x) = 1 x. 3.8) f(x) = arcsin( x+1., prove the equality cosh 2 x sinh 2 x = 1. Calculus-Lab ) lim. 2.7) lim. 2. ) Solve the following inequalities.) ++.) 4 > 3.3) Calculus-Lab { + > + 5 + < 3 +. ) Graph the functions f() = 3, g() = + +, h() = 3 cos( ), r() = 3 +. 3) Find the domain of the following functions 3.)

More information

Precalculus A - Final Exam Review Fall, 2014

Precalculus A - Final Exam Review Fall, 2014 Name: Precalculus A - Final Exam Review Fall, 2014 Period: Find the measures of two angles, one positive and one negative, that are coterminal with the given angle. 1) 85 2) -166 3) 3 Convert the radian

More information

Directions: Please read questions carefully. It is recommended that you do the Short Answer Section prior to doing the Multiple Choice.

Directions: Please read questions carefully. It is recommended that you do the Short Answer Section prior to doing the Multiple Choice. AP Calculus AB SUMMER ASSIGNMENT Multiple Choice Section Directions: Please read questions carefully It is recommended that you do the Short Answer Section prior to doing the Multiple Choice Show all work

More information

(a) Show that there is a root α of f (x) = 0 in the interval [1.2, 1.3]. (2)

(a) Show that there is a root α of f (x) = 0 in the interval [1.2, 1.3]. (2) . f() = 4 cosec 4 +, where is in radians. (a) Show that there is a root α of f () = 0 in the interval [.,.3]. Show that the equation f() = 0 can be written in the form = + sin 4 Use the iterative formula

More information

Topic 6 Part 4 [317 marks]

Topic 6 Part 4 [317 marks] Topic 6 Part [7 marks] a. ( + tan ) sec tan (+c) M [ marks] [ marks] Some correct answers but too many candidates had a poor approach and did not use the trig identity. b. sin sin (+c) cos M [ marks] Allow

More information

Advanced Higher Grade

Advanced Higher Grade Prelim Eamination / 5 (Assessing Units & ) MATHEMATICS Advanced Higher Grade Time allowed - hours Read Carefully. Full credit will be given only where the solution contains appropriate woring.. Calculators

More information

APPM 1360 Final Exam Spring 2016

APPM 1360 Final Exam Spring 2016 APPM 36 Final Eam Spring 6. 8 points) State whether each of the following quantities converge or diverge. Eplain your reasoning. a) The sequence a, a, a 3,... where a n ln8n) lnn + ) n!) b) ln d c) arctan

More information

MATHEMATICS 9740/01 Paper 1 14 September 2012

MATHEMATICS 9740/01 Paper 1 14 September 2012 NATIONAL JUNIOR COLLEGE PRELIMINARY EXAMINATIONS Higher MATHEMATICS 9740/0 Paper 4 September 0 hours Additional Materials: Answer Paper List of Formulae (MF5) Cover Sheet 085 5 hours READ THESE INSTRUCTIONS

More information

Calculus 1 - Lab ) f(x) = 1 x. 3.8) f(x) = arcsin( x+1., prove the equality cosh 2 x sinh 2 x = 1. Calculus 1 - Lab ) lim. 2.

Calculus 1 - Lab ) f(x) = 1 x. 3.8) f(x) = arcsin( x+1., prove the equality cosh 2 x sinh 2 x = 1. Calculus 1 - Lab ) lim. 2. ) Solve the following inequalities.) ++.) 4 >.) Calculus - Lab { + > + 5 + < +. ) Graph the functions f() =, g() = + +, h() = cos( ), r() = +. ) Find the domain of the following functions.) f() = +.) f()

More information

DUNMAN HIGH SCHOOL Preliminary Examination Year 6

DUNMAN HIGH SCHOOL Preliminary Examination Year 6 Name: Inde Number: Class: DUNMAN HIGH SCHOOL Preliminary Eamination Year 6 MATHEMATICS (Higher ) 970/0 Paper 7 September 05 Additional Materials: Answer Paper Graph paper List of Formulae (MF5) 3 hours

More information

MATH TOURNAMENT 2012 PROBLEMS SOLUTIONS

MATH TOURNAMENT 2012 PROBLEMS SOLUTIONS MATH TOURNAMENT 0 PROBLEMS SOLUTIONS. Consider the eperiment of throwing two 6 sided fair dice, where, the faces are numbered from to 6. What is the probability of the event that the sum of the values

More information

Lesson-3 TRIGONOMETRIC RATIOS AND IDENTITIES

Lesson-3 TRIGONOMETRIC RATIOS AND IDENTITIES Lesson- TRIGONOMETRIC RATIOS AND IDENTITIES Angle in trigonometry In trigonometry, the measure of an angle is the amount of rotation from B the direction of one ray of the angle to the other ray. Angle

More information

Algebra y funciones [219 marks]

Algebra y funciones [219 marks] Algebra y funciones [9 marks] Let f() = 3 ln and g() = ln5 3. a. Epress g() in the form f() + lna, where a Z +. attempt to apply rules of logarithms e.g. ln a b = b lna, lnab = lna + lnb correct application

More information

Trig Practice 08 and Specimen Papers

Trig Practice 08 and Specimen Papers IB Math High Level Year : Trig: Practice 08 and Spec Papers Trig Practice 08 and Specimen Papers. In triangle ABC, AB = 9 cm, AC = cm, and Bˆ is twice the size of Ĉ. Find the cosine of Ĉ.. In the diagram

More information

Work the following on notebook paper. No calculator. Find the derivative. Do not leave negative exponents or complex fractions in your answers.

Work the following on notebook paper. No calculator. Find the derivative. Do not leave negative exponents or complex fractions in your answers. ALULUS B WORKSHEET ON 8. & REVIEW Find the derivative. Do not leave negative eponents or comple fractions in your answers. sec. f 8 7. f e. y ln tan. y cos tan. f 7. f cos. y 7 8. y log 7 Evaluate the

More information

Integration Techniques for the AB exam

Integration Techniques for the AB exam For the AB eam, students need to: determine antiderivatives of the basic functions calculate antiderivatives of functions using u-substitution use algebraic manipulation to rewrite the integrand prior

More information

TABLE OF CONTENTS 2 CHAPTER 1

TABLE OF CONTENTS 2 CHAPTER 1 TABLE OF CONTENTS CHAPTER 1 Quadratics CHAPTER Functions 3 CHAPTER 3 Coordinate Geometry 3 CHAPTER 4 Circular Measure 4 CHAPTER 5 Trigonometry 4 CHAPTER 6 Vectors 5 CHAPTER 7 Series 6 CHAPTER 8 Differentiation

More information

2016 Mathematics. Advanced Higher. Finalised Marking Instructions

2016 Mathematics. Advanced Higher. Finalised Marking Instructions National Qualifications 06 06 Mathematics Advanced Higher Finalised ing Instructions Scottish Qualifications Authority 06 The information in this publication may be reproduced to support SQA qualifications

More information

WYSE MATH STATE 2012 SOLUTIONS. 1. Ans E: Trapezoids need only have one pair of parallel sides. Parallelograms are, by definition, forced to have two.

WYSE MATH STATE 2012 SOLUTIONS. 1. Ans E: Trapezoids need only have one pair of parallel sides. Parallelograms are, by definition, forced to have two. WYSE MATH STATE 01 SOLUTIONS 1. Ans E: Trapezoids need only have one pair of parallel sides. Parallelograms are, by definition, forced to have two.. Ans A: All the cans can be arranged in 10 P 10 = 10!

More information

MATHEMATICS SPECIALIST. Calculator-free. ATAR course examination Marking Key

MATHEMATICS SPECIALIST. Calculator-free. ATAR course examination Marking Key MATHEMATICS SPECIALIST Calculator-free ATAR course examination 8 Marking Key Marking keys are an explicit statement about what the examining panel expect of candidates when they respond to particular examination

More information

N13/5/MATHL/HP2/ENG/TZ0/XX/M MARKSCHEME. November 2013 MATHEMATICS. Higher Level. Paper pages

N13/5/MATHL/HP2/ENG/TZ0/XX/M MARKSCHEME. November 2013 MATHEMATICS. Higher Level. Paper pages N/5/MATHL/HP/ENG/TZ0/XX/M MARKSCHEME November 0 MATHEMATICS Higher Level Paper 0 pages N/5/MATHL/HP/ENG/TZ0/XX/M This markscheme is confidential and for the exclusive use of examiners in this examination

More information

2 nd ORDER O.D.E.s SUBSTITUTIONS

2 nd ORDER O.D.E.s SUBSTITUTIONS nd ORDER O.D.E.s SUBSTITUTIONS Question 1 (***+) d y y 8y + 16y = d d d, y 0, Find the general solution of the above differential equation by using the transformation equation t = y. Give the answer in

More information

FP3 mark schemes from old P4, P5, P6 and FP1, FP2, FP3 papers (back to June 2002)

FP3 mark schemes from old P4, P5, P6 and FP1, FP2, FP3 papers (back to June 2002) FP mark schemes from old P, P5, P6 and FP, FP, FP papers (back to June ) Please note that the following pages contain mark schemes for questions from past papers. Where a question reference is marked with

More information

Correct substitution. cos = (A1) For substituting correctly sin 55.8 A1

Correct substitution. cos = (A1) For substituting correctly sin 55.8 A1 Circular Functions and Trig - Practice Problems (to 07) MarkScheme 1. (a) Evidence of using the cosine rule eg cos = cos Correct substitution eg cos = = 55.8 (0.973 radians) N2 (b) Area = sin For substituting

More information

Nama Pelajar : 347/ Additional Mathematics Paper September 00 Tingkatan 5 :. PERSIDANGAN KEBANGSAAN PENGETUA-PENGETUA SEKOLAH MENENGAH NEGERI KEDAH DARUL AMAN PEPERIKSAAN PERCUBAAN SPM 00 ADDITIONAL MATHEMATICS

More information

NATIONAL QUALIFICATIONS

NATIONAL QUALIFICATIONS Mathematics Higher Prelim Eamination 04/05 Paper Assessing Units & + Vectors NATIONAL QUALIFICATIONS Time allowed - hour 0 minutes Read carefully Calculators may NOT be used in this paper. Section A -

More information

Some commonly encountered sets and their notations

Some commonly encountered sets and their notations NATIONAL UNIVERSITY OF SINGAPORE DEPARTMENT OF MATHEMATICS (This notes are based on the book Introductory Mathematics by Ng Wee Seng ) LECTURE SETS & FUNCTIONS Some commonly encountered sets and their

More information

TMTA Calculus and Advanced Topics Test 2010

TMTA Calculus and Advanced Topics Test 2010 . Evaluate lim Does not eist - - 0 TMTA Calculus and Advanced Topics Test 00. Find the period of A 6D B B y Acos 4B 6D, where A 0, B 0, D 0. Solve the given equation for : ln = ln 4 4 ln { } {-} {0} {}

More information

THE INVERSE TRIGONOMETRIC FUNCTIONS

THE INVERSE TRIGONOMETRIC FUNCTIONS THE INVERSE TRIGONOMETRIC FUNCTIONS Question 1 (**+) Solve the following trigonometric equation ( x ) π + 3arccos + 1 = 0. 1 x = Question (***) It is given that arcsin x = arccos y. Show, by a clear method,

More information

( ) = ( ) ( ) = ( ) = + = = = ( ) Therefore: , where t. Note: If we start with the condition BM = tab, we will have BM = ( x + 2, y + 3, z 5)

( ) = ( ) ( ) = ( ) = + = = = ( ) Therefore: , where t. Note: If we start with the condition BM = tab, we will have BM = ( x + 2, y + 3, z 5) Chapter Exercise a) AB OB OA ( xb xa, yb ya, zb za),,, 0, b) AB OB OA ( xb xa, yb ya, zb za) ( ), ( ),, 0, c) AB OB OA x x, y y, z z (, ( ), ) (,, ) ( ) B A B A B A ( ) d) AB OB OA ( xb xa, yb ya, zb za)

More information

Summer Review Packet. for students entering. IB Math SL

Summer Review Packet. for students entering. IB Math SL Summer Review Packet for students entering IB Math SL The problems in this packet are designed to help you review topics that are important to your success in IB Math SL. Please attempt the problems on

More information

Candidates sitting FP2 may also require those formulae listed under Further Pure Mathematics FP1 and Core Mathematics C1 C4. e π.

Candidates sitting FP2 may also require those formulae listed under Further Pure Mathematics FP1 and Core Mathematics C1 C4. e π. F Further IAL Pure PAPERS: Mathematics FP 04-6 AND SPECIMEN Candidates sitting FP may also require those formulae listed under Further Pure Mathematics FP and Core Mathematics C C4. Area of a sector A

More information

(a) During what time intervals on [0, 4] is the particle traveling to the left?

(a) During what time intervals on [0, 4] is the particle traveling to the left? Chapter 5. (AB/BC, calculator) A particle travels along the -ais for times 0 t 4. The velocity of the particle is given by 5 () sin. At time t = 0, the particle is units to the right of the origin. t /

More information

AP Calculus (BC) Summer Assignment (169 points)

AP Calculus (BC) Summer Assignment (169 points) AP Calculus (BC) Summer Assignment (69 points) This packet is a review of some Precalculus topics and some Calculus topics. It is to be done NEATLY and on a SEPARATE sheet of paper. Use your discretion

More information

1993 AP Calculus AB: Section I

1993 AP Calculus AB: Section I 99 AP Calculus AB: Section I 90 Minutes Scientific Calculator Notes: () The eact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among

More information

Markscheme November 2016 Mathematics Standard level Paper 1

Markscheme November 2016 Mathematics Standard level Paper 1 N6/5/MATME/SP/ENG/TZ0/XX/M Markscheme November 06 Mathematics Standard level Paper 6 pages N6/5/MATME/SP/ENG/TZ0/XX/M This markscheme is the property of the International Baccalaureate and must not be

More information

Advanced Higher Grade

Advanced Higher Grade Practice Eamination A (Assessing Units & ) MATHEMATICS Advanced Higher Grade Time allowed - hours 0 minutes Read Carefully. Full credit will be given only where the solution contains appropriate working..

More information

90 Chapter 5 Logarithmic, Exponential, and Other Transcendental Functions. Name Class. (a) (b) ln x (c) (a) (b) (c) 1 x. y e (a) 0 (b) y.

90 Chapter 5 Logarithmic, Exponential, and Other Transcendental Functions. Name Class. (a) (b) ln x (c) (a) (b) (c) 1 x. y e (a) 0 (b) y. 90 Chapter 5 Logarithmic, Eponential, and Other Transcendental Functions Test Form A Chapter 5 Name Class Date Section. Find the derivative: f ln. 6. Differentiate: y. ln y y y y. Find dy d if ey y. y

More information

1993 AP Calculus AB: Section I

1993 AP Calculus AB: Section I 99 AP Calculus AB: Section I 9 Minutes Scientific Calculator Notes: () The eact numerical value of the correct answer does not always appear among the choices given. When this happens, select from among

More information

TOTAL NAME DATE PERIOD AP CALCULUS AB UNIT 4 ADVANCED DIFFERENTIATION TECHNIQUES DATE TOPIC ASSIGNMENT /6 10/8 10/9 10/10 X X X X 10/11 10/12

TOTAL NAME DATE PERIOD AP CALCULUS AB UNIT 4 ADVANCED DIFFERENTIATION TECHNIQUES DATE TOPIC ASSIGNMENT /6 10/8 10/9 10/10 X X X X 10/11 10/12 NAME DATE PERIOD AP CALCULUS AB UNIT ADVANCED DIFFERENTIATION TECHNIQUES DATE TOPIC ASSIGNMENT 0 0 0/6 0/8 0/9 0/0 X X X X 0/ 0/ 0/5 0/6 QUIZ X X X 0/7 0/8 0/9 0/ 0/ 0/ 0/5 UNIT EXAM X X X TOTAL AP Calculus

More information

STEP II, ax y z = 3, 2ax y 3z = 7, 3ax y 5z = b, (i) In the case a = 0, show that the equations have a solution if and only if b = 11.

STEP II, ax y z = 3, 2ax y 3z = 7, 3ax y 5z = b, (i) In the case a = 0, show that the equations have a solution if and only if b = 11. STEP II, 2003 2 Section A: Pure Mathematics 1 Consider the equations ax y z = 3, 2ax y 3z = 7, 3ax y 5z = b, where a and b are given constants. (i) In the case a = 0, show that the equations have a solution

More information

Summer Review Packet for Students Entering AP Calculus BC. Complex Fractions

Summer Review Packet for Students Entering AP Calculus BC. Complex Fractions Summer Review Packet for Students Entering AP Calculus BC Comple Fractions When simplifying comple fractions, multiply by a fraction equal to 1 which has a numerator and denominator composed of the common

More information

Objective Mathematics

Objective Mathematics Chapter No - ( Area Bounded by Curves ). Normal at (, ) is given by : y y. f ( ) or f ( ). Area d ()() 7 Square units. Area (8)() 6 dy. ( ) d y c or f ( ) c f () c f ( ) As shown in figure, point P is

More information

HIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 4 UNIT (ADDITIONAL) Time allowed Three hours (Plus 5 minutes reading time)

HIGHER SCHOOL CERTIFICATE EXAMINATION MATHEMATICS 4 UNIT (ADDITIONAL) Time allowed Three hours (Plus 5 minutes reading time) N E W S O U T H W A L E S HIGHER SCHOOL CERTIFICATE EXAMINATION 996 MATHEMATICS 4 UNIT (ADDITIONAL) Time allowed Three hours (Plus 5 minutes reading time) DIRECTIONS TO CANDIDATES Attempt ALL questions.

More information

PACKET Unit 4 Honors ICM Functions and Limits 1

PACKET Unit 4 Honors ICM Functions and Limits 1 PACKET Unit 4 Honors ICM Functions and Limits 1 Day 1 Homework For each of the rational functions find: a. domain b. -intercept(s) c. y-intercept Graph #8 and #10 with at least 5 EXACT points. 1. f 6.

More information

Chapter 7 Trigonometric Identities and Equations 7-1 Basic Trigonometric Identities Pages

Chapter 7 Trigonometric Identities and Equations 7-1 Basic Trigonometric Identities Pages Trigonometric Identities and Equations 7- Basic Trigonometric Identities Pages 47 430. Sample answer: 45 3. tan, cot, cot tan cos cot, cot csc 5. Rosalinda is correct; there may be other values for which

More information

AP Calculus BC : The Fundamental Theorem of Calculus

AP Calculus BC : The Fundamental Theorem of Calculus AP Calculus BC 415 5.3: The Fundamental Theorem of Calculus Tuesday, November 5, 008 Homework Answers 6. (a) approimately 0.5 (b) approimately 1 (c) approimately 1.75 38. 4 40. 5 50. 17 Introduction In

More information

McKinney High School AP Calculus Summer Packet

McKinney High School AP Calculus Summer Packet McKinne High School AP Calculus Summer Packet (for students entering AP Calculus AB or AP Calculus BC) Name:. This packet is to be handed in to our Calculus teacher the first week of school.. ALL work

More information

abc Mathematics Pure Core General Certificate of Education SPECIMEN UNITS AND MARK SCHEMES

abc Mathematics Pure Core General Certificate of Education SPECIMEN UNITS AND MARK SCHEMES abc General Certificate of Education Mathematics Pure Core SPECIMEN UNITS AND MARK SCHEMES ADVANCED SUBSIDIARY MATHEMATICS (56) ADVANCED SUBSIDIARY PURE MATHEMATICS (566) ADVANCED SUBSIDIARY FURTHER MATHEMATICS

More information

Questions. x 2 e x dx. Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the functions g(x) = x cost2 dt.

Questions. x 2 e x dx. Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the functions g(x) = x cost2 dt. Questions. Evaluate the Riemann sum for f() =,, with four subintervals, taking the sample points to be right endpoints. Eplain, with the aid of a diagram, what the Riemann sum represents.. If f() = ln,

More information

Markscheme May 2016 Mathematical studies Standard level Paper 1

Markscheme May 2016 Mathematical studies Standard level Paper 1 M16/5/MATSD/SP1/ENG/TZ/XX/M Markscheme May 016 Mathematical studies Standard level Paper 1 4 pages M16/5/MATSD/SP1/ENG/TZ/XX/M This markscheme is the property of the International Baccalaureate and must

More information

All Rights Reserved Wiley India Pvt. Ltd. 1

All Rights Reserved Wiley India Pvt. Ltd. 1 Question numbers to carry mark each. CBSE MATHEMATICS SECTION A. If R = {(, y) : + y = 8} is a relation of N, write the range of R. R = {(, y)! + y = 8} a relation of N. y = 8 y must be Integer So Can

More information

AP Calculus AB SUMMER ASSIGNMENT. Dear future Calculus AB student

AP Calculus AB SUMMER ASSIGNMENT. Dear future Calculus AB student AP Calculus AB SUMMER ASSIGNMENT Dear future Calculus AB student We are ecited to work with you net year in Calculus AB. In order to help you be prepared for this class, please complete the summer assignment.

More information

SOLUTIONS TO CONCEPTS CHAPTER 2

SOLUTIONS TO CONCEPTS CHAPTER 2 SOLUTIONS TO CONCPTS CHAPTR 1. As shown in the figure, The angle between A and B = 11 = 9 A = and B = 4m Resultant R = A B ABcos = 5 m Let be the angle between R and A 4 sin9 = tan 1 = tan 1 (4/) = 5 4cos9

More information

ERRATA MATHEMATICS FOR THE INTERNATIONAL STUDENT MATHEMATICS HL (CORE) (2nd edition)

ERRATA MATHEMATICS FOR THE INTERNATIONAL STUDENT MATHEMATICS HL (CORE) (2nd edition) ERRATA MATHEMATICS FOR THE INTERNATIONAL STUDENT MATHEMATICS HL (CORE) (nd edition) Second edition - 010 reprint page 95 TEXT last paragraph on the page should read: e is a special number in mathematics.

More information

M14/5/MATHL/HP1/ENG/TZ1/XX/M MARKSCHEME. May 2014 MATHEMATICS. Higher Level. Paper pages

M14/5/MATHL/HP1/ENG/TZ1/XX/M MARKSCHEME. May 2014 MATHEMATICS. Higher Level. Paper pages 4/5/MATHL/HP/ENG/TZ/XX/M MARKSCHEME May 04 MATHEMATICS Higher Level Paper 8 pages 4/5/MATHL/HP/ENG/TZ/XX/M This markscheme is confidential and for the eclusive use of eaminers in this eamination session.

More information

x f(x)

x f(x) CALCULATOR SECTION. For y y 8 find d point (, ) on the curve. A. D. dy at the 7 E. 6. Suppose silver is being etracted from a.t mine at a rate given by A'( t) e, A(t) is measured in tons of silver and

More information

C) 2 D) 4 E) 6. ? A) 0 B) 1 C) 1 D) The limit does not exist.

C) 2 D) 4 E) 6. ? A) 0 B) 1 C) 1 D) The limit does not exist. . The asymptotes of the graph of the parametric equations = t, y = t t + are A) =, y = B) = only C) =, y = D) = only E) =, y =. What are the coordinates of the inflection point on the graph of y = ( +

More information

Mark Scheme (Results) Summer GCE Core Mathematics 3 (6665/01R)

Mark Scheme (Results) Summer GCE Core Mathematics 3 (6665/01R) Mark Scheme (Results) Summer GCE Core Mathematics (6665/R) Question Number Scheme Marks. (a) + ( + 4)( ) B Attempt as a single fraction (+ 5)( ) ( + ) ( + )( ) or + 5 ( + 4) M ( + 4)( ) ( + 4)( ), ( +

More information

Mat 270 Final Exam Review Sheet Fall 2012 (Final on December 13th, 7:10 PM - 9:00 PM in PSH 153)

Mat 270 Final Exam Review Sheet Fall 2012 (Final on December 13th, 7:10 PM - 9:00 PM in PSH 153) Mat 70 Final Eam Review Sheet Fall 0 (Final on December th, 7:0 PM - 9:00 PM in PSH 5). Find the slope of the secant line to the graph of y f ( ) between the points f ( b) f ( a) ( a, f ( a)), and ( b,

More information

IYGB. Special Paper U. Time: 3 hours 30 minutes. Created by T. Madas. Created by T. Madas

IYGB. Special Paper U. Time: 3 hours 30 minutes. Created by T. Madas. Created by T. Madas IYGB Special Paper U Time: 3 hours 30 minutes Candidates may NOT use any calculator Information for Candidates This practice paper follows the Advanced Level Mathematics Core Syllabus Booklets of Mathematical

More information

A-LEVEL Mathematics MPC3

A-LEVEL Mathematics MPC3 A-LEVEL Mathematics MPC UNIT: Pure Core Mark scheme 660 June 07 Version:.0 Final MARK SCHEME A LEVEL MATHEMATICS MPC JUNE 07 Mark schemes are prepared by the Lead Assessment Writer and considered, together

More information

TRIGONOMETRY OUTCOMES

TRIGONOMETRY OUTCOMES TRIGONOMETRY OUTCOMES C10. Solve problems involving limits of trigonometric functions. C11. Apply derivatives of trigonometric functions. C12. Solve problems involving inverse trigonometric functions.

More information

2013 Mathematics. Advanced Higher. Finalised Marking Instructions

2013 Mathematics. Advanced Higher. Finalised Marking Instructions 0 Mathematics Advanced Higher Finalised ing Instructions Scottish Qualifications Authority 0 The information in this publication may be reproduced to support SQA qualifications only on a noncommercial

More information

OC = $ 3cos. 1 (5.4) 2 θ = (= radians) (M1) θ = 1. Note: Award (M1) for identifying the largest angle.

OC = $ 3cos. 1 (5.4) 2 θ = (= radians) (M1) θ = 1. Note: Award (M1) for identifying the largest angle. 4 + 5 7 cos α 4 5 5 α 0.5. Note: Award for identifying the largest angle. Find other angles first β 44.4 γ 4.0 α 0. (C4) Note: Award (C) if not given to the correct accuracy.. (a) p (C) 4. (a) OA A is

More information

AP Calculus BC Summer Review

AP Calculus BC Summer Review AP Calculus BC 07-08 Summer Review Due September, 07 Name: All students entering AP Calculus BC are epected to be proficient in Pre-Calculus skills. To enhance your chances for success in this class, it

More information

January Further Pure Mathematics FP1 Mark Scheme

January Further Pure Mathematics FP1 Mark Scheme January 6667 Further Pure Mathematics FP Mar Q z + 8i + i (a) z i + i + i + 8i 8 + 5i z (b) ( ) + 5 z (or awrt 5.8) () ft () (c) 5 5 tan α or z arg π.... z + i (a) and attempt to multiply out for + i -

More information

1. The following problems are not related: (a) (15 pts, 5 pts ea.) Find the following limits or show that they do not exist: arcsin(x)

1. The following problems are not related: (a) (15 pts, 5 pts ea.) Find the following limits or show that they do not exist: arcsin(x) APPM 5 Final Eam (5 pts) Fall. The following problems are not related: (a) (5 pts, 5 pts ea.) Find the following limits or show that they do not eist: (i) lim e (ii) lim arcsin() (b) (5 pts) Find and classify

More information

3.5 Continuity of a Function One Sided Continuity Intermediate Value Theorem... 23

3.5 Continuity of a Function One Sided Continuity Intermediate Value Theorem... 23 Chapter 3 Limit and Continuity Contents 3. Definition of Limit 3 3.2 Basic Limit Theorems 8 3.3 One sided Limit 4 3.4 Infinite Limit, Limit at infinity and Asymptotes 5 3.4. Infinite Limit and Vertical

More information

FIRST SEMESTER DIPLOMA EXAMINATION IN ENGINEERING/ TECHNOLIGY- OCTOBER, TECHNICAL MATHEMATICS- I (Common Except DCP and CABM)

FIRST SEMESTER DIPLOMA EXAMINATION IN ENGINEERING/ TECHNOLIGY- OCTOBER, TECHNICAL MATHEMATICS- I (Common Except DCP and CABM) TED (10)-1002 (REVISION-2010) Reg. No.. Signature. FIRST SEMESTER DIPLOMA EXAMINATION IN ENGINEERING/ TECHNOLIGY- OCTOBER, 2010 TECHNICAL MATHEMATICS- I (Common Except DCP and CABM) (Maximum marks: 100)

More information

ADVANCED PROGRAMME MATHEMATICS: PAPER I MODULE 1: CALCULUS AND ALGEBRA

ADVANCED PROGRAMME MATHEMATICS: PAPER I MODULE 1: CALCULUS AND ALGEBRA GRADE 1 EXAMINATION NOVEMBER 017 ADVANCED PROGRAMME MATHEMATICS: PAPER I MODULE 1: CALCULUS AND ALGEBRA Time: hours 00 marks PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY 1. This question paper consists

More information

2014 Mathematics. Advanced Higher. Finalised Marking Instructions

2014 Mathematics. Advanced Higher. Finalised Marking Instructions 0 Mathematics Advanced Higher Finalised ing Instructions Scottish Qualifications Authority 0 The information in this publication may be reproduced to support SQA qualifications only on a noncommercial

More information

The stationary points will be the solutions of quadratic equation x

The stationary points will be the solutions of quadratic equation x Calculus 1 171 Review In Problems (1) (4) consider the function f ( ) ( ) e. 1. Find the critical (stationary) points; establish their character (relative minimum, relative maimum, or neither); find intervals

More information

SPECIALIST MATHEMATICS

SPECIALIST MATHEMATICS Victorian Certificate of Education 00 SUPERVISOR TO ATTACH PROCESSING LABEL HERE STUDENT NUMBER Letter Figures Words SPECIALIST MATHEMATICS Written eamination Monday November 00 Reading time:.00 pm to.5

More information