Solutionbank C2 Edexcel Modular Mathematics for AS and A-Level
|
|
- Baldric Phelps
- 7 years ago
- Views:
Transcription
1 Heinemann Solutionbank: Core Maths C Page of Solutionbank C Revision Eercises Eercise A, Question Epand and simplify ( ) 5. ( ) 5 = + 5 ( ) + 0 ( ) + 0 ( ) + 5 ( ) + ( ) 5 = Compare ( + ) n with ( ) n. Replace n by 5 and by. Pearson Education Ltd 008
2 Heinemann Solutionbank: Core Maths C Page of Solutionbank C Revision Eercises Eercise A, Question In the diagram, ABC is an equilateral triangle with side 8 cm. PQ is an arc of a circle centre A, radius 6 cm. Find the perimeter of the shaded region in the diagram. Remember: The length of an arc of a circle is L = rθ. The area of a sector is A = r θ. Draw a diagram. Remember: 60 = π radius Length of arc PQ = rθ π = 6 ( ) = π cm Perimeter of shaded region = π = + π = 8.8 cm Pearson Education Ltd 008
3 Heinemann Solutionbank: Core Maths C Page of Solutionbank C Revision Eercises Eercise A, Question The sum to infinity of a geometric series is 5. Given that the first term is 5, (a) find the common ratio, (b) find the third term. (a) a r 5 r = 5, a = 5 = 5 Remember: s = a r a = 5 so that 5 =. 5 r, where r <. Here s = 5 and r = r = (b) ar = 5 ( ) = 5 9 = 0 9 Remember: nth term = ar n. Here a = 5, r = and n =, so that ar n = 5 ( ) = 5 ( ) Pearson Education Ltd 008
4 Heinemann Solutionbank: Core Maths C Page of Solutionbank C Revision Eercises Eercise A, Question Sketch the graph of y = sin θ in the interval π θ<π. Remember: 80 =π radians Pearson Education Ltd 008
5 Heinemann Solutionbank: Core Maths C Page of Solutionbank C Revision Eercises Eercise A, Question 5 Find the first three terms, in descending powers of b, of the binomial epansion of ( a + b ) 6, giving each term in its simplest form. ( a + b ) 6 = ( a ) ( ) ( a ) 5 6 ( b ) + ( ) ( a ) ( b ) + = 6 a a 5 b + 5 a b + = 6a a 5 b + 60a b + Compare ( a + b ) n with ( a + b ) n. Replace n by 6, a by a and b by b. Pearson Education Ltd 008
6 Heinemann Solutionbank: Core Maths C Page of Solutionbank C Revision Eercises Eercise A, Question 6 AB is an arc of a circle centre O. Arc AB = 8 cm and OA = OB = cm. (a) Find, in radians, AOB. (b) Calculate the length of the chord AB, giving your answer to significant figures. Draw a diagram. Let AOB =θ. (a) AOB =θ θ = 8 Use l = rθ. Here l = 8 and r =. so θ = (b) = AB + ( ) ( ) cos ( ) = 6.66 AB = 7.85 cm ( s.f. ) Use the cosine formula c = a + b ab cos C. Here c = AB, a =, b = and C =θ= change your to radians.. Remember to Pearson Education Ltd 008
7 Heinemann Solutionbank: Core Maths C Page of Solutionbank C Revision Eercises Eercise A, Question 7 A geometric series has first term and common ratio r. The sum of the first three terms of the series is 7. (a) Show that r + r = 0. (b) Find the two possible values of r. Given that r is positive, (c) find the sum to infinity of the series. (a), r, r, Use ar n to write down epressions for the first terms. Here a = and n =,,. + r + r = 7 r + r = 0 (as required) (b) r + r = 0 ( r ) ( r + ) = 0 r =, r = Factorize r + r. ac =. ( ) + ( + 6 ) = +, so r r + 6r = r ( r ) + ( r ) = ( r ) ( r + ). (c) r = a r = = 8 a Use S =. Here a = and r =, so that a r = r = = 8. Pearson Education Ltd 008
8 Heinemann Solutionbank: Core Maths C Page of Solutionbank C Revision Eercises Eercise A, Question 8 (a) Write down the number of cycles of the graph y = sin n in the interval (b) Hence write down the period of the graph y = sin n. (a) n Consider the graphs of y = sin, y = sin, y = sin y = sin has cycle in the interval y = sin has cycles in the interval y = sin has cycles in the interval etc. So y = sin n has n cycles in the internal (b) 60 n π ( or ) n Period = length of cycle. If there are n cycles in the interval 0 60, 60 the length of each cycle will be. n Pearson Education Ltd 008
9 Heinemann Solutionbank: Core Maths C Page of Solutionbank C Revision Eercises Eercise A, Question 9 (a) Find the first four terms, in ascending powers of, of the binomial epansion of ( + p ) 7, where p is a non-zero constant. Given that, in this epansion, the coefficients of and are equal, (b) find the value of p, (c) find the coefficient of. (a) ( + p ) 7 n ( n ) = + n + + n ( n ) ( n )!! + 7 ( 6 ) = + 7 ( p ) + ( p ) + 7 ( 6 ) ( 5 )!! ( p ) + = + 7p + p + 5p + Compare ( + ) n with ( + p ) n. Replace n by 7 and by p. (b) 7p = p p 0, so 7 = p The coefficients of and are equal, so 7p = p. p = (c) 5p = 5 ( ) = 5 7 The coefficient of is 5p. Here p =, so that 5p = 5 ( ). Pearson Education Ltd 008
10 Heinemann Solutionbank: Core Maths C Page of Solutionbank C Revision Eercises Eercise A, Question 0 A sector of a circle of radius 8 cm contains an angle of θ radians. Given that the perimeter of the sector is 0 cm, find the area of the sector. Draw a diagram. Perimeter of sector = 0cm, so arc length = cm. 8θ = θ = 8 Area of sector = ( 8 ) θ = ( 8 ) ( ) = 56 cm 8 Find the value of θ. Use L = rθ. Here L = and r = 8 so that 8θ =. Use A = r θ. Here r = 8 and θ =, so that A = ( 8 ) ( ). 8 8 Pearson Education Ltd 008
11 Heinemann Solutionbank: Core Maths C Page of Solutionbank C Revision Eercises Eercise A, Question A pendulum is set swinging. Its first oscillation is through 0. Each succeeding oscillation is What is the total angle described by the pendulum before it stops? 9 0 of the one before it. 0, 0 ( ), 0 ( ), a r = = ( ) 0 = Write down the first term. Use ar n. Here a = 0, r = 9 0 and n =,,. Use S =. Here a = 0 and r = so that S = a r 9 0 Pearson Education Ltd 008
12 Heinemann Solutionbank: Core Maths C Page of Solutionbank C Revision Eercises Eercise A, Question Write down the eact value (a) sin0, (b) cos0, (c) tan ( 60 ). (a) Draw a diagram for each part. Remember sin 0 = (b) cos 0 = cos 0 = \ 0 is in the fourth quadrant. (c)
13 Heinemann Solutionbank: Core Maths C Page of tan ( 60 ) = tan 60 = \ = \ 60 is in the fourth quadrant. Pearson Education Ltd 008
14 Heinemann Solutionbank: Core Maths C Page of Solutionbank C Revision Eercises Eercise A, Question (a) Find the first three terms, in ascending powers of, of the binomial epansion of ( a ) 8, where a is a non-zero integer. The first three terms are, and b, where b is a constant. (b) Find the value of a and the value of b. (a) ( a ) n ( n ) 8 = + n + +! = + 8 ( a ) + ( a ) + 8 ( 8 )! = 8a + 8a + Compare ( + ) n with ( a ) n Replace n by 8 and by a. (b) 8a = a = b = 8a = 8 ( ) = 5 so a = and b = 5 Compare coefficients of, so that 8a =. Pearson Education Ltd 008
15 Heinemann Solutionbank: Core Maths C Page of Solutionbank C Revision Eercises Eercise A, Question In the diagram, A and B are points on the circumference of a circle centre O and radius 5 cm. AOB =θ radians. AB = 6 cm. (a) Find the value of θ. (b) Calculate the length of the minor arc AB. (a) cos θ = ( 5 ) ( 5 ) = 7 5 θ =.87 radians Use the cosine formula cos C = a = 5, b = 5 and c = 6. a + b c ab. Here c =θ, (b) arc AB = 5θ = 5.87 = 6. cm Use C = rθ. Here C = arc AB, r = 5 and θ =.87 radians. Pearson Education Ltd 008
16 Heinemann Solutionbank: Core Maths C Page of Solutionbank C Revision Eercises Eercise A, Question 5 The fifth and sith terms of a geometric series are.5 and 6.75 respectively. (a) Find the common ratio. (b) Find the first term. (c) Find the sum of the first 0 terms, giving your answer to decimal places. (a) ar =.5, ar 5 = 6.75 ar = ar.5 r = Find r. Divide ar 5 by ar ar 5 so that = ar5 ar a 6.75 = r and =.5..5 (b) a (.5 ) =.5 a =.5 (.5 ) = 8 9 (c) S 0 = 8 ( (.5 ) 0 ) 9.5 = ( d.p. ) a ( r n ) Use S n =. Here a =, r =.5 and n = 0. r 8 9 Pearson Education Ltd 008
17 Heinemann Solutionbank: Core Maths C Page of Solutionbank C Revision Eercises Eercise A, Question 6 Given that θ is an acute angle measured in degrees, epress in term of cosθ (a) cos ( 60 + θ ), (b) cos ( θ ), (c) cos ( 80 θ ) (a) Draw a diagram for each part. cos ( 60 + θ ) = cos θ 60 + θ is in the first quadrant. (b) cos ( θ ) = cos θ θ is in the fourth quadrant. (c)
18 Heinemann Solutionbank: Core Maths C Page of cos ( 80 θ ) = cos θ 80 θ is in the second quadrant. Pearson Education Ltd 008
19 Heinemann Solutionbank: Core Maths C Page of Solutionbank C Revision Eercises Eercise A, Question 7 (a) Epand ( ) 0 in ascending powers of up to and including the term in. (b) Use your answer to part (a) to evaluate ( 0.98 ) 0 correct to decimal places. (a) ( ) 0 n ( n ) = + n + + n ( n ) ( n )!! + 0 ( a ) = + 0 ( ) + ( ) + 0 ( 9 ) ( 8 ) 6 ( ) + = Compare ( + ) n with ( ) n. Replace n by 0 and by. (b) ( ( 0.0 ) ) 0 = 0 ( 0.0 ) + 80 ( 0.0 ) 960 ( 0.0 ) ~ 0.87 ( d.p. ) Find the value of = 0.0 = ( 0.0 ) Pearson Education Ltd 008
20 Heinemann Solutionbank: Core Maths C Page of Solutionbank C Revision Eercises Eercise A, Question 8 In the diagram, AB = 0 cm, AC = cm. CAB = 0.6 radians. BD is an arc of a circle centre A and radius 0 cm. (a) Calculate the length of the arc BD. (b) Calculate the shaded area in the diagram. (a) arc BD = = 6 cm (b) Shaded area = ( 0 ) ( ) sin ( 0.6 ) ( 0 ) ( 0.6 ) = 6.70 cm ( s.f. ) Use L = rθ. Here L = arc BD, r = 0 and θ = 0.6 radians. Use area of triangle = bc sin A and area of sector = r θ. Here b =, c = 0 and A = (θ= ) 0.6 ; r = 0 and θ = 0.6. Pearson Education Ltd 008
21 Heinemann Solutionbank: Core Maths C Page of Solutionbank C Revision Eercises Eercise A, Question 9 The value of a gold coin in 000 was 80. The value of the coin increases by 5% per annum. (a) Write down an epression for the value of the coin after n years. (b) Find the year in which the value of the coin eceeds , 80 (.05 ), 80 (.05 ), (a) Value after n years = 80 (.05 ) n (b) 80 (.05 ) n > (.05 ) = (.05 ) 5 = 7. The value of the coin will eceed 60 in 0. Write down the first terms. Use ar n. Here a = 80, r =.05 and n =,,. Substitute values of n. The value of the coin after years is 56.9, and after 5 years is 7.. So the value of the coin will eceed 60 in the 5th year. Pearson Education Ltd 008
22 Heinemann Solutionbank: Core Maths C Page of Solutionbank C Revision Eercises Eercise A, Question 0 Given that is an acute angle measured in radians, epress in terms of sin (a) sin ( π ), (b) sin (π+), (c) cos. π (a) Draw a diagram for each part. Remember: π Radians = 80 sin ( π ) = sin π is in the fourth quadrant. (b) sin (π+) = sin π + is in the third quadrant. (c) 80 =π radians, so 90 π = radians.
23 Heinemann Solutionbank: Core Maths C Page of π cos ( ) = sin ( = ). a c Pearson Education Ltd 008
24 Heinemann Solutionbank: Core Maths C Page of Solutionbank C Revision Eercises Eercise A, Question Epand and simplify 6 ( ) 6 = ( ) 5 6 ( ) + ( ) ( ) 6 + ( ) ( ) 5 + ( ) 6 ) 6 + ( ) ( ) 6 + ( ) ( ) 5 = ( ) + 5 ( ) + 0 ( + 5 ( ) + 6 ( ) = Compare ( ) n with ( a + b ) n. Replace n by 6, a with and b with. ( ) = = ( ) = = ( ) = = Pearson Education Ltd 008
25 Heinemann Solutionbank: Core Maths C Page of Solutionbank C Revision Eercises Eercise A, Question A cylindrical log, length m, radius 0 cm, floats with its ais horizontal and with its highest point cm above the water level. Find the volume of the log in the water. Draw a diagram. Let sector angle = φ. 6 cos φ = ( = 0.8 ) 0 Area above water level = r ( φ ) r sin ( φ ) ( 0 ) sin ( φ ) = ( 0 ) ( φ ) = 65.0 cm Area below water level =π(0 ) 65.0 = 9. cm Volume below water level = = 88 cm ( = 0.8 cm ) Use area of segment = r θ r sin θ. Here r = 0 cm and θ = cos ( 0.8 ) Pearson Education Ltd 008
26 Heinemann Solutionbank: Core Maths C Page of Solutionbank C Revision Eercises Eercise A, Question (a) On the same aes, in the interval 0 60, sketch the graphs of y = tan ( 90 ) and y = sin. (b) Hence write down the number of solutions of the equation tan ( 90 ) = sin in the interval (a) y = tan ( 90 ) is a translation of y = tan by + 90 in the -direction. (b) solutions in the interval From the sketch, the graphs of y = tan ( 90 ) and y = sin meet at two points. So there are solutions in the internal Pearson Education Ltd 008
27 Heinemann Solutionbank: Core Maths C Page of Solutionbank C Revision Eercises Eercise A, Question A geometric series has first term and common ratio sum eceeding 00.. Find the greatest number of terms the series can have without its a =, r = S n = ( ( ) n ) a ( r n ) Use S n =. Here a = and r =. r = ( ( ) n ) n ) = ( ( ) Now, ( ( ) n ) < 00 ( ) n < 00 ( ) n < + 00 ( ) n < 9 ( ) 7 = 7.9 ( ) 8 = Substitute values of n. The largest value of n for which ( ) n < 9 / is 7. so n = 7 Pearson Education Ltd 008
28 Heinemann Solutionbank: Core Maths C Page of Solutionbank C Revision Eercises Eercise A, Question 5 Describe geometrically the transformation which maps the graph of (a) y = tan onto the graph of y = tan ( 5 ), (b) y = sin onto the graph of y = sin, (c) y = cos onto the graph of y = cos, (d) y = sin onto the graph of y = sin. (a) A translation of + 5 in the direction (b) A stretch of scale factor in the y direction (c) A stretch of scale factor in the direction (d) A translation of in the y direction Pearson Education Ltd 008
29 Heinemann Solutionbank: Core Maths C Page of Solutionbank C Revision Eercises Eercise A, Question 6 If is so small that terms of and higher can be ignored, and ( ) ( + ) 5 a + b + c, find the values of the constants a, b and c. ( + ) 5 = + n + n ( n )! + ( ) + = + 5 ( ) + = ( ) ( ) ( ) = ( ) ( + ) Compare ( + ) n with ( + ) n. Replace n by 5 and by. Epand ( ) ( ) ignoring terms in, so that ( ) and = ( ) = 0 0 simplify so that = so a =, b = 9, c = 70 Compare with a + b + c + so that a =, b = 9 and c = 70. Pearson Education Ltd 008
30 Heinemann Solutionbank: Core Maths C Page of Solutionbank C Revision Eercises Eercise A, Question 7 A chord of a circle, radius 0 cm, divides the circumference in the ratio :. Find the ratio of the areas of the segments into which the circle is divided by the chord. Draw a diagram. Let the minor are l have angle θ and the major are l have angle θ. l : l = : 0θ : 0θ = : θ :θ = : Use l = rθ so that l = 0θ and l = 0θ. so θ = π = Shaded area = ( 0 ) θ ( 0 ) sin θ π = 5π 50 Area of large segment =π(0 ) ( 5π 50 ) = 00π 5π + 50 = 75π + 50 Use area of segment = r θ r sin θ. Here r = 0 and θ =θ = Ratio of small segment to large segment is 5π 50 : 75π + 50 Divide throughout by 5. π : π + or : π + π π Pearson Education Ltd 008
31 Heinemann Solutionbank: Core Maths C Page of Solutionbank C Revision Eercises Eercise A, Question 8, and + 8 are the fourth, fifth and sith terms of geometric series. (a) Find the two possible values of and the corresponding values of the common ratio. Given that the sum to infinity of the series eists, (b) find the first term, (c) the sum to infinity of the series. ar = ar = ar 5 = + 8 ar 5 ar = + 8 so = ar ar 5 = r and = r so =. ar ar ar ar ar ( + 8 ) = = 0 ( + 9 ) ( ) = 0 =, = 9 ar r = = ar ar ar5 Clear the fractions. Multiply each side by so that + 8 ar = ( + 8 ) and = 9. When =, r = Find r. Substitute =, then = 9, into =, so that When = 9, r = r = and r = =. 9 ar ar (b) r = ar = a ( ) = a = a Remember S = for r <, so r =. r (c)
32 Heinemann Solutionbank: Core Maths C Page of a r = = 6 Pearson Education Ltd 008
33 Heinemann Solutionbank: Core Maths C Page of Solutionbank C Revision Eercises Eercise A, Question 9 (a) Sketch the graph of y =.5 cos ( 60 ) in the interval 0 <60 (b) Write down the coordinates of the points where your graph meets the coordinate aes. (a) (b) When = 0, y =.5 cos ( 60 ) = 0.75 so ( 0, 0.75 ) y =.5 cos ( 60 ) The graph of y =.5 cos ( 60 ) meets the y ais when = 0. Substitute = 0 into y =.5 cos ( 60 ) so that y =.5 cos ( 60 ). cos ( 60 ) = cos 60 = so y =.5 = 0.75 y = 0, The graph of y =.5 cos ( 60 ) meets the -ais when = when y = 0. cos ( 60 ) represents a translation of = 50 cos by + 60 in the -direction cos meets the -ais at 90 and 70, so y =.5 ( cos 60 ) meets the - and = ais at = 50 and = 0. = 0 Pearson Education Ltd 008
34 Heinemann Solutionbank: Core Maths C Page of Solutionbank C Revision Eercises Eercise A, Question 0 Without using a calculator, solve sin ( 0 \ ) = in the interval sin 60 = \ sin = quadrants. \. sin is negative in the rd and th sin ( 0 ) = \ sin ( 00 ) = \ so 0 = 0 sin 0 \ = but sin ( 0 \ ) = so = 60 0 = 0, i.e. = 60. Similarly and 0 = 00 = 0 sin 00 = but sin ( 0 ) = so 0 = 00, i.e. = 0. \ \ Pearson Education Ltd 008
Core Maths C2. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations...
More informationD.3. Angles and Degree Measure. Review of Trigonometric Functions
APPENDIX D Precalculus Review D7 SECTION D. Review of Trigonometric Functions Angles and Degree Measure Radian Measure The Trigonometric Functions Evaluating Trigonometric Functions Solving Trigonometric
More informationMATHEMATICS Unit Pure Core 2
General Certificate of Education January 2008 Advanced Subsidiary Examination MATHEMATICS Unit Pure Core 2 MPC2 Wednesday 9 January 2008 1.30 pm to 3.00 pm For this paper you must have: an 8-page answer
More informationTrigonometric Functions and Triangles
Trigonometric Functions and Triangles Dr. Philippe B. Laval Kennesaw STate University August 27, 2010 Abstract This handout defines the trigonometric function of angles and discusses the relationship between
More informationSTRAND: 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
More informationYou must have: Ruler graduated in centimetres and millimetres, protractor, compasses, pen, HB pencil, eraser, calculator. Tracing paper may be used.
Write your name here Surname Other names Pearson Edexcel Certificate Pearson Edexcel International GCSE Mathematics A Paper 4H Centre Number Monday 1 January 015 Afternoon Time: hours Candidate Number
More informationAx 2 Cy 2 Dx Ey F 0. Here we show that the general second-degree equation. Ax 2 Bxy Cy 2 Dx Ey F 0. y X sin Y cos P(X, Y) X
Rotation of Aes ROTATION OF AES Rotation of Aes For a discussion of conic sections, see Calculus, Fourth Edition, Section 11.6 Calculus, Earl Transcendentals, Fourth Edition, Section 1.6 In precalculus
More information2010 Solutions. a + b. a + b 1. (a + b)2 + (b a) 2. (b2 + a 2 ) 2 (a 2 b 2 ) 2
00 Problem If a and b are nonzero real numbers such that a b, compute the value of the expression ( ) ( b a + a a + b b b a + b a ) ( + ) a b b a + b a +. b a a b Answer: 8. Solution: Let s simplify the
More informationBiggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress
Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation
More informationCircles - Past Edexcel Exam Questions
ircles - Past Edecel Eam Questions 1. The points A and B have coordinates (5,-1) and (13,11) respectivel. (a) find the coordinates of the mid-point of AB. [2] Given that AB is a diameter of the circle,
More informationALGEBRA 2/TRIGONOMETRY
ALGEBRA /TRIGONOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA /TRIGONOMETRY Tuesday, January 8, 014 1:15 to 4:15 p.m., only Student Name: School Name: The possession
More informationNumber Sense and Operations
Number Sense and Operations representing as they: 6.N.1 6.N.2 6.N.3 6.N.4 6.N.5 6.N.6 6.N.7 6.N.8 6.N.9 6.N.10 6.N.11 6.N.12 6.N.13. 6.N.14 6.N.15 Demonstrate an understanding of positive integer exponents
More informationHigher Education Math Placement
Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication
More informationwww.mathsbox.org.uk ab = c a If the coefficients a,b and c are real then either α and β are real or α and β are complex conjugates
Further Pure Summary Notes. Roots of Quadratic Equations For a quadratic equation ax + bx + c = 0 with roots α and β Sum of the roots Product of roots a + b = b a ab = c a If the coefficients a,b and c
More informationMath 0306 Final Exam Review
Math 006 Final Exam Review Problem Section Answers Whole Numbers 1. According to the 1990 census, the population of Nebraska is 1,8,8, the population of Nevada is 1,01,8, the population of New Hampshire
More informationCore Maths C3. Revision Notes
Core Maths C Revision Notes October 0 Core Maths C Algebraic fractions... Cancelling common factors... Multipling and dividing fractions... Adding and subtracting fractions... Equations... 4 Functions...
More informationFriday, January 29, 2016 9:15 a.m. to 12:15 p.m., only
ALGEBRA /TRIGONOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA /TRIGONOMETRY Friday, January 9, 016 9:15 a.m. to 1:15 p.m., only Student Name: School Name: The possession
More informationACT Math Vocabulary. Altitude The height of a triangle that makes a 90-degree angle with the base of the triangle. Altitude
ACT Math Vocabular Acute When referring to an angle acute means less than 90 degrees. When referring to a triangle, acute means that all angles are less than 90 degrees. For eample: Altitude The height
More informationStraight Line. Paper 1 Section A. O xy
PSf Straight Line Paper 1 Section A Each correct answer in this section is worth two marks. 1. The line with equation = a + 4 is perpendicular to the line with equation 3 + + 1 = 0. What is the value of
More informationMATH 60 NOTEBOOK CERTIFICATIONS
MATH 60 NOTEBOOK CERTIFICATIONS Chapter #1: Integers and Real Numbers 1.1a 1.1b 1.2 1.3 1.4 1.8 Chapter #2: Algebraic Expressions, Linear Equations, and Applications 2.1a 2.1b 2.1c 2.2 2.3a 2.3b 2.4 2.5
More informationYou must have: Ruler graduated in centimetres and millimetres, protractor, compasses, pen, HB pencil, eraser, calculator. Tracing paper may be used.
Write your name here Surname Other names Edexcel IGCSE Centre Number Mathematics A Paper 3H Monday 6 June 2011 Afternoon Time: 2 hours Candidate Number Higher Tier Paper Reference 4MA0/3H You must have:
More informationAlgebra. 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
More informationGrade 7 & 8 Math Circles Circles, Circles, Circles March 19/20, 2013
Faculty of Mathematics Waterloo, Ontario N2L 3G Introduction Grade 7 & 8 Math Circles Circles, Circles, Circles March 9/20, 203 The circle is a very important shape. In fact of all shapes, the circle is
More informationAdditional Topics in Math
Chapter Additional Topics in Math In addition to the questions in Heart of Algebra, Problem Solving and Data Analysis, and Passport to Advanced Math, the SAT Math Test includes several questions that are
More information13. Write the decimal approximation of 9,000,001 9,000,000, rounded to three significant
æ If 3 + 4 = x, then x = 2 gold bar is a rectangular solid measuring 2 3 4 It is melted down, and three equal cubes are constructed from this gold What is the length of a side of each cube? 3 What is the
More informationPaper Reference. Edexcel GCSE Mathematics (Linear) 1380 Paper 4 (Calculator) Monday 5 March 2012 Afternoon Time: 1 hour 45 minutes
Centre No. Candidate No. Paper Reference 1 3 8 0 4 H Paper Reference(s) 1380/4H Edexcel GCSE Mathematics (Linear) 1380 Paper 4 (Calculator) Higher Tier Monday 5 March 2012 Afternoon Time: 1 hour 45 minutes
More information6.1 Basic Right Triangle Trigonometry
6.1 Basic Right Triangle Trigonometry MEASURING ANGLES IN RADIANS First, let s introduce the units you will be using to measure angles, radians. A radian is a unit of measurement defined as the angle at
More informationMath 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.
Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used
More information4. How many integers between 2004 and 4002 are perfect squares?
5 is 0% of what number? What is the value of + 3 4 + 99 00? (alternating signs) 3 A frog is at the bottom of a well 0 feet deep It climbs up 3 feet every day, but slides back feet each night If it started
More informationALGEBRA 2/TRIGONOMETRY
ALGEBRA /TRIGONOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION ALGEBRA /TRIGONOMETRY Thursday, January 9, 015 9:15 a.m to 1:15 p.m., only Student Name: School Name: The possession
More informationVersion 1.0. General Certificate of Education (A-level) January 2012. Mathematics MPC4. (Specification 6360) Pure Core 4. Final.
Version.0 General Certificate of Education (A-level) January 0 Mathematics MPC (Specification 660) Pure Core Final Mark Scheme Mark schemes are prepared by the Principal Eaminer and considered, together
More informationLesson 9.1 Solving Quadratic Equations
Lesson 9.1 Solving Quadratic Equations 1. Sketch the graph of a quadratic equation with a. One -intercept and all nonnegative y-values. b. The verte in the third quadrant and no -intercepts. c. The verte
More informationIB Maths SL Sequence and Series Practice Problems Mr. W Name
IB Maths SL Sequence and Series Practice Problems Mr. W Name Remember to show all necessary reasoning! Separate paper is probably best. 3b 3d is optional! 1. In an arithmetic sequence, u 1 = and u 3 =
More informationSAT Math Hard Practice Quiz. 5. How many integers between 10 and 500 begin and end in 3?
SAT Math Hard Practice Quiz Numbers and Operations 5. How many integers between 10 and 500 begin and end in 3? 1. A bag contains tomatoes that are either green or red. The ratio of green tomatoes to red
More informationPythagorean Theorem: 9. x 2 2
Geometry Chapter 8 - Right Triangles.7 Notes on Right s Given: any 3 sides of a Prove: the is acute, obtuse, or right (hint: use the converse of Pythagorean Theorem) If the (longest side) 2 > (side) 2
More informationMATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education)
MATH 095, College Prep Mathematics: Unit Coverage Pre-algebra topics (arithmetic skills) offered through BSE (Basic Skills Education) Accurately add, subtract, multiply, and divide whole numbers, integers,
More informationTUESDAY, 6 MAY 9.00 AM 9.45 AM. 2 Full credit will be given only where the solution contains appropriate working.
X00//0 NATIONAL QUALIFICATIONS 04 TUESDAY, 6 MAY 9.00 AM 9.45 AM MATHEMATICS INTERMEDIATE Units, and Paper (Non-calculator) Read carefully You may NOT use a calculator. Full credit will be given only where
More informationHomework 2 Solutions
Homework Solutions 1. (a) Find the area of a regular heagon inscribed in a circle of radius 1. Then, find the area of a regular heagon circumscribed about a circle of radius 1. Use these calculations to
More informationGraphing Trigonometric Skills
Name Period Date Show all work neatly on separate paper. (You may use both sides of your paper.) Problems should be labeled clearly. If I can t find a problem, I ll assume it s not there, so USE THE TEMPLATE
More informationa cos x + b sin x = R cos(x α)
a cos x + b sin x = R cos(x α) In this unit we explore how the sum of two trigonometric functions, e.g. cos x + 4 sin x, can be expressed as a single trigonometric function. Having the ability to do this
More informationGive an expression that generates all angles coterminal with the given angle. Let n represent any integer. 9) 179
Trigonometry Chapters 1 & 2 Test 1 Name Provide an appropriate response. 1) Find the supplement of an angle whose measure is 7. Find the measure of each angle in the problem. 2) Perform the calculation.
More informationClick here for answers.
CHALLENGE PROBLEMS: CHALLENGE PROBLEMS 1 CHAPTER A Click here for answers S Click here for solutions A 1 Find points P and Q on the parabola 1 so that the triangle ABC formed b the -ais and the tangent
More informationAdvanced Math Study Guide
Advanced Math Study Guide Topic Finding Triangle Area (Ls. 96) using A=½ bc sin A (uses Law of Sines, Law of Cosines) Law of Cosines, Law of Cosines (Ls. 81, Ls. 72) Finding Area & Perimeters of Regular
More informationDear Accelerated Pre-Calculus Student:
Dear Accelerated Pre-Calculus Student: I am very excited that you have decided to take this course in the upcoming school year! This is a fastpaced, college-preparatory mathematics course that will also
More informationAdvanced GMAT Math Questions
Advanced GMAT Math Questions Version Quantitative Fractions and Ratios 1. The current ratio of boys to girls at a certain school is to 5. If 1 additional boys were added to the school, the new ratio of
More informationColegio del mundo IB. Programa Diploma REPASO 2. 1. The mass m kg of a radio-active substance at time t hours is given by. m = 4e 0.2t.
REPASO. The mass m kg of a radio-active substance at time t hours is given b m = 4e 0.t. Write down the initial mass. The mass is reduced to.5 kg. How long does this take?. The function f is given b f()
More informationSection 7.1 Solving Right Triangles
Section 7.1 Solving Right Triangles Note that a calculator will be needed for most of the problems we will do in class. Test problems will involve angles for which no calculator is needed (e.g., 30, 45,
More informationCore Maths C1. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Indices... Rules of indices... Surds... 4 Simplifying surds... 4 Rationalising the denominator... 4 Quadratic functions... 4 Completing the
More informationChapter 8 Geometry We will discuss following concepts in this chapter.
Mat College Mathematics Updated on Nov 5, 009 Chapter 8 Geometry We will discuss following concepts in this chapter. Two Dimensional Geometry: Straight lines (parallel and perpendicular), Rays, Angles
More informationG. GRAPHING FUNCTIONS
G. GRAPHING FUNCTIONS To get a quick insight int o how the graph of a function looks, it is very helpful to know how certain simple operations on the graph are related to the way the function epression
More informationCSU Fresno Problem Solving Session. Geometry, 17 March 2012
CSU Fresno Problem Solving Session Problem Solving Sessions website: http://zimmer.csufresno.edu/ mnogin/mfd-prep.html Math Field Day date: Saturday, April 21, 2012 Math Field Day website: http://www.csufresno.edu/math/news
More informationHow To Solve The Pythagorean Triangle
Name Period CHAPTER 9 Right Triangles and Trigonometry Section 9.1 Similar right Triangles Objectives: Solve problems involving similar right triangles. Use a geometric mean to solve problems. Ex. 1 Use
More informationEvaluating trigonometric functions
MATH 1110 009-09-06 Evaluating trigonometric functions Remark. Throughout this document, remember the angle measurement convention, which states that if the measurement of an angle appears without units,
More informationWednesday 15 January 2014 Morning Time: 2 hours
Write your name here Surname Other names Pearson Edexcel Certificate Pearson Edexcel International GCSE Mathematics A Paper 4H Centre Number Wednesday 15 January 2014 Morning Time: 2 hours Candidate Number
More informationSection 6-3 Double-Angle and Half-Angle Identities
6-3 Double-Angle and Half-Angle Identities 47 Section 6-3 Double-Angle and Half-Angle Identities Double-Angle Identities Half-Angle Identities This section develops another important set of identities
More informationPaper Reference. Ruler graduated in centimetres and millimetres, protractor, compasses, pen, HB pencil, eraser. Tracing paper may be used.
Centre No. Candidate No. Paper Reference 1 3 8 0 3 H Paper Reference(s) 1380/3H Edexcel GCSE Mathematics (Linear) 1380 Paper 3 (Non-Calculator) Higher Tier Monday 18 May 2009 Afternoon Time: 1 hour 45
More informationWeek 13 Trigonometric Form of Complex Numbers
Week Trigonometric Form of Complex Numbers Overview In this week of the course, which is the last week if you are not going to take calculus, we will look at how Trigonometry can sometimes help in working
More informationUNIT 1: ANALYTICAL METHODS FOR ENGINEERS
UNIT : ANALYTICAL METHODS FOR ENGINEERS Unit code: A/60/40 QCF Level: 4 Credit value: 5 OUTCOME 3 - CALCULUS TUTORIAL DIFFERENTIATION 3 Be able to analyse and model engineering situations and solve problems
More informationAlgebra and Geometry Review (61 topics, no due date)
Course Name: Math 112 Credit Exam LA Tech University Course Code: ALEKS Course: Trigonometry Instructor: Course Dates: Course Content: 159 topics Algebra and Geometry Review (61 topics, no due date) Properties
More informationSan Jose Math Circle April 25 - May 2, 2009 ANGLE BISECTORS
San Jose Math Circle April 25 - May 2, 2009 ANGLE BISECTORS Recall that the bisector of an angle is the ray that divides the angle into two congruent angles. The most important results about angle bisectors
More informationIllinois State Standards Alignments Grades Three through Eleven
Illinois State Standards Alignments Grades Three through Eleven Trademark of Renaissance Learning, Inc., and its subsidiaries, registered, common law, or pending registration in the United States and other
More informationArc Length and Areas of Sectors
Student Outcomes When students are provided with the angle measure of the arc and the length of the radius of the circle, they understand how to determine the length of an arc and the area of a sector.
More informationSAT Subject Math Level 2 Facts & Formulas
Numbers, Sequences, Factors Integers:..., -3, -2, -1, 0, 1, 2, 3,... Reals: integers plus fractions, decimals, and irrationals ( 2, 3, π, etc.) Order Of Operations: Arithmetic Sequences: PEMDAS (Parentheses
More informationPaper Reference. Ruler graduated in centimetres and millimetres, protractor, compasses, pen, HB pencil, eraser, calculator. Tracing paper may be used.
Centre No. Candidate No. Paper Reference 1 3 8 0 4 H Paper Reference(s) 1380/4H Edexcel GCSE Mathematics (Linear) 1380 Paper 4 (Calculator) Higher Tier Friday 11 June 2010 Morning Time: 1 hour 45 minutes
More informationBell Baxter High School 0
Bell Bater High School Mathematics Department Fourth Level Homework Booklet Remember: Complete each homework in your jotter showing ALL working clearly Bell Bater High School 0 Evaluating Epressions and
More informationREVIEW OF ANALYTIC GEOMETRY
REVIEW OF ANALYTIC GEOMETRY The points in a plane can be identified with ordered pairs of real numbers. We start b drawing two perpendicular coordinate lines that intersect at the origin O on each line.
More informationCAMI Education linked to CAPS: Mathematics
- 1 - TOPIC 1.1 Whole numbers _CAPS curriculum TERM 1 CONTENT Mental calculations Revise: Multiplication of whole numbers to at least 12 12 Ordering and comparing whole numbers Revise prime numbers to
More informationThe University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Student Name:
GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Wednesday, August 18, 2010 8:30 to 11:30 a.m., only Student Name: School Name: Print your name and the name of
More informationShape, Space and Measure
Name: Shape, Space and Measure Prep for Paper 2 Including Pythagoras Trigonometry: SOHCAHTOA Sine Rule Cosine Rule Area using 1-2 ab sin C Transforming Trig Graphs 3D Pythag-Trig Plans and Elevations Area
More informationTSI College Level Math Practice Test
TSI College Level Math Practice Test Tutorial Services Mission del Paso Campus. Factor the Following Polynomials 4 a. 6 8 b. c. 7 d. ab + a + b + 6 e. 9 f. 6 9. Perform the indicated operation a. ( +7y)
More information1 TRIGONOMETRY. 1.0 Introduction. 1.1 Sum and product formulae. Objectives
TRIGONOMETRY Chapter Trigonometry Objectives After studying this chapter you should be able to handle with confidence a wide range of trigonometric identities; be able to express linear combinations of
More informationThe University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY. Tuesday, August 13, 2013 8:30 to 11:30 a.m., only.
GEOMETRY The University of the State of New York REGENTS HIGH SCHOOL EXAMINATION GEOMETRY Tuesday, August 13, 2013 8:30 to 11:30 a.m., only Student Name: School Name: The possession or use of any communications
More informationMath Placement Test Practice Problems
Math Placement Test Practice Problems The following problems cover material that is used on the math placement test to place students into Math 1111 College Algebra, Math 1113 Precalculus, and Math 2211
More informationYou must have: Ruler graduated in centimetres and millimetres, protractor, compasses, pen, HB pencil, eraser, calculator. Tracing paper may be used.
Write your name here Surname Other names Pearson Edexcel International GCSE Mathematics A Paper 3HR Centre Number Tuesday 6 January 015 Afternoon Time: hours Candidate Number Higher Tier Paper Reference
More informationNorth Carolina Community College System Diagnostic and Placement Test Sample Questions
North Carolina Communit College Sstem Diagnostic and Placement Test Sample Questions 0 The College Board. College Board, ACCUPLACER, WritePlacer and the acorn logo are registered trademarks of the College
More informationINTERESTING PROOFS FOR THE CIRCUMFERENCE AND AREA OF A CIRCLE
INTERESTING PROOFS FOR THE CIRCUMFERENCE AND AREA OF A CIRCLE ABSTRACT:- Vignesh Palani University of Minnesota - Twin cities e-mail address - palan019@umn.edu In this brief work, the existing formulae
More informationSelf-Paced Study Guide in Trigonometry. March 31, 2011
Self-Paced Study Guide in Trigonometry March 1, 011 1 CONTENTS TRIGONOMETRY Contents 1 How to Use the Self-Paced Review Module Trigonometry Self-Paced Review Module 4.1 Right Triangles..........................
More informationPRE-CALCULUS GRADE 12
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.
More informationTrigonometric Functions: The Unit Circle
Trigonometric Functions: The Unit Circle This chapter deals with the subject of trigonometry, which likely had its origins in the study of distances and angles by the ancient Greeks. The word trigonometry
More informationCharlesworth School Year Group Maths Targets
Charlesworth School Year Group Maths Targets Year One Maths Target Sheet Key Statement KS1 Maths Targets (Expected) These skills must be secure to move beyond expected. I can compare, describe and solve
More informationUnit 6 Trigonometric Identities, Equations, and Applications
Accelerated Mathematics III Frameworks Student Edition Unit 6 Trigonometric Identities, Equations, and Applications nd Edition Unit 6: Page of 3 Table of Contents Introduction:... 3 Discovering the Pythagorean
More informationSOLVING TRIGONOMETRIC EQUATIONS
Mathematics Revision Guides Solving Trigonometric Equations Page 1 of 17 M.K. HOME TUITION Mathematics Revision Guides Level: AS / A Level AQA : C2 Edexcel: C2 OCR: C2 OCR MEI: C2 SOLVING TRIGONOMETRIC
More informationTriangle Trigonometry and Circles
Math Objectives Students will understand that trigonometric functions of an angle do not depend on the size of the triangle within which the angle is contained, but rather on the ratios of the sides of
More information6 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
More informationPolynomials and Factoring
7.6 Polynomials and Factoring Basic Terminology A term, or monomial, is defined to be a number, a variable, or a product of numbers and variables. A polynomial is a term or a finite sum or difference of
More informationThnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks
Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson
More information2009 Chicago Area All-Star Math Team Tryouts Solutions
1. 2009 Chicago Area All-Star Math Team Tryouts Solutions If a car sells for q 1000 and the salesman earns q% = q/100, he earns 10q 2. He earns an additional 100 per car, and he sells p cars, so his total
More informationCIRCLE COORDINATE GEOMETRY
CIRCLE COORDINATE GEOMETRY (EXAM QUESTIONS) Question 1 (**) A circle has equation x + y = 2x + 8 Determine the radius and the coordinates of the centre of the circle. r = 3, ( 1,0 ) Question 2 (**) A circle
More informationArithmetic Progression
Worksheet 3.6 Arithmetic and Geometric Progressions Section 1 Arithmetic Progression An arithmetic progression is a list of numbers where the difference between successive numbers is constant. The terms
More informationKey Topics What will ALL students learn? What will the most able students learn?
2013 2014 Scheme of Work Subject MATHS Year 9 Course/ Year Term 1 Key Topics What will ALL students learn? What will the most able students learn? Number Written methods of calculations Decimals Rounding
More informationNorth Carolina Community College System Diagnostic and Placement Test Sample Questions
North Carolina Community College System Diagnostic and Placement Test Sample Questions 01 The College Board. College Board, ACCUPLACER, WritePlacer and the acorn logo are registered trademarks of the College
More informationWEDNESDAY, 2 MAY 1.30 PM 2.25 PM. 3 Full credit will be given only where the solution contains appropriate working.
C 500/1/01 NATIONAL QUALIFICATIONS 01 WEDNESDAY, MAY 1.0 PM.5 PM MATHEMATICS STANDARD GRADE Credit Level Paper 1 (Non-calculator) 1 You may NOT use a calculator. Answer as many questions as you can. Full
More informationSection V.2: Magnitudes, Directions, and Components of Vectors
Section V.: Magnitudes, Directions, and Components of Vectors Vectors in the plane If we graph a vector in the coordinate plane instead of just a grid, there are a few things to note. Firstl, directions
More information16 Circles and Cylinders
16 Circles and Cylinders 16.1 Introduction to Circles In this section we consider the circle, looking at drawing circles and at the lines that split circles into different parts. A chord joins any two
More informationCircle Name: Radius: Diameter: Chord: Secant:
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
More informationIdentifying second degree equations
Chapter 7 Identifing second degree equations 7.1 The eigenvalue method In this section we appl eigenvalue methods to determine the geometrical nature of the second degree equation a 2 + 2h + b 2 + 2g +
More informationKey Stage 3 Mathematics. Level by Level. Pack E: Level 8
p Key Stage Mathematics Level by Level Pack E: Level 8 Stafford Burndred ISBN 89960 6 Published by Pearson Publishing Limited 997 Pearson Publishing 996 Revised February 997 A licence to copy the material
More information