Core Maths C2. Revision Notes


 Cody Ferguson
 8 years ago
 Views:
Transcription
1 Core Maths C Revision Notes November 0
2 Core Maths C Algebra... Polnomials: +,,,.... Factorising... Long division... Remainder theorem... Factor theorem... 4 Choosing a suitable factor... 5 Cubic equations... 6 Trigonometr... 6 Radians... 6 Connection between radians and degrees... 6 Arc length, area of a sector and area of a segment... 6 Trig functions... 6 Basic results... 6 Eact values for 0 o, 45 o and 60 o... 7 Graphs of trig functions... 7 Graphs of = sin n, = sin( ), = sin( + n) etc Sine & Cosine rules and area of triangle... 8 Identities... 8 Trigonometric equations... 9 Coordinate Geometr... 0 Mid point... 0 Circle... 0 Centre at the origin... 0 General equation... 0 Equation of tangent... 4 Sequences and series... Geometric series... Finite geometric series... Infinite geometric series... Proof of the formula for the sum of a geometric series... Binomial series for positive integral inde... 4 Pascal s triangle... 4 Factorials... 4 n Binomial coefficients or n C r or... 4 r 5 Eponentials and logarithms... 5 Graphs of eponentials and logarithms... 5 Rules of logarithms... 5 Changing the base of a logarithm... 6 Equations of the form a = b... 7 C 4/04/0 SDB
3 6 Differentiation... 7 Increasing and decreasing functions... 7 Stationar points and local maima and minima (turning points) Using second derivative... 8 Using gradients before and after... 9 Maimum and minimum problems Integration... Definite integrals... Area under curve... Numerical integration: the trapezium rule... Inde... C 4/04/0 SDB
4 Algebra Polnomials: +,,,. A polnomial is an epression of the form a n n + a n n +... a + a + a 0 where all the powers of the variable,, are positive integers or 0. +,, of polnomials are eas, must be done b long division. Factorising General eamples of factorising: ab + 6ac = a(b + c ) = ( )( ) 6 = ( 6) 6 0 = ( + )( 5) Standard results: = ( )( + ), difference of two squares ( + ) = + +, ( ) = + Long division See eamples in book Remainder theorem If 67 is divided b 6 the quotient is 04 and the remainder is. This can be written as 67 = In the same wa, if a polnomial P() = a 0 + a + a +... a n n is divided b (c + d) to give a quotient, Q() with a remainder r, then r will be a constant (since the divisor is of degree one) and we can write P() = (c + d) Q() + r If we now choose the value of which makes (c + d) = 0 = d / c then we have d P( c ) = 0 Q() + r d P( c ) = r. C 4/04/0 SDB
5 Theorem: If we substitute = d / c in the polnomial we obtain the remainder that we would have after dividing the polnomial b (c + d). Eample: The remainder when P() = a  4 is divided b (  ) is 7. Find a. Solution: Choose the value of which makes (  ) = 0, i.e. = /. Then the remainder is P( ) = 4 ( )  ( ) + a ( ) a ( )  4 = a ( ) = 7 a = = 7 You ma be given a polnomial with two unknown letters, a and b. You will also be given two pieces of information to let ou form two simultaneous equations in a and b. Factor theorem Theorem: If, in the remainder theorem, r = 0 then (c + d) is a factor of P() d P( c ) = 0 (c + d) is a factor of P(). Eample: A quadratic equation has solutions (roots) = / and =. Find the quadratic equation in the form a + b + c = 0 Solution: The equation has roots = ½ and = it must have factors ( + ) and ( ) b the factor theorem the equation is ( + )( ) = 0 5 = 0. Eample: Show that ( ) is a factor of P() = and hence factorise the epression completel. Solution: Choose the value of which makes ( ) = 0, i.e. = remainder = P() = = = 0 ( ) is a factor b the factor theorem. We have a cubic and so we can see that the other factor must be a quadratic of the form (6 + a ) and we can write = ( ) (6 + a ) Multipling out we see that the 6 The term is and +6 terms are correct. 4 C 4/04/0 SDB
6 Multipling out the term comes from must come to + a which a = 7. We must now check the term which is 9. Multipling out with a = 7 the term is ( 7) + ( ) 6 = 7 = 9, which works! = ( )(6 7 ) which is now factorised completel. = ( )( )( + ) Choosing a suitable factor To choose a suitable factor we look at the coefficient of the highest power of and the constant (the term without an ). Eample: Factorise Solution: is the coefficient of and has factors of and. 6 is the constant and 6 has factors of,, and 6 so the possible linear factors of are ( ± ), ( ± ), ( ± ), ( ± 6) ( ± ), ( ± ), ( ± ), ( ± 6) But ( ± ) = ( ± ) and ( ± 6) = ( ± ), so the are not new factors. We now test the possible factors using the factor theorem until we find one that works. Test ( ), put = giving Test ( + ), put = giving ( ) + ( ) ( ) Test ( ), put = giving = = 0 and since the result is zero ( ) is a factor. We now divide in, as in the previous eample, to give = ( )( + 5 ) = ( )( )( + ). C 4/04/0 SDB 5
7 Cubic equations Factorise using the factor theorem then solve. N.B. The quadratic factor might not factorise in which case ou will need to use the formula for this part. Eample: Solve the equation + = 0. Solution: Possible factors are ( ± ) and ( ± ). Put = we have + = 0 ( ) is not a factor Putting = we have + = = 0 ( ) is a factor + = ( )( + ) = 0 = or + = 0 this will not factorise so we use the formula = or = ± ( ) 4 = 0.68 or. Trigonometr Radians A radian is the angle subtended at the centre of a circle b an arc of length equal to the radius. Connection between radians and degrees 80 o = π c Degrees Radians π / 6 π / 4 π / π / π / π / 4 5π / 6 π π / π Arc length, area of a sector and area of a segment Arc length s = rθ and area of sector A = ½r θ. Area of segment = area sector area of triangle = ½r θ ½r sinθ. Trig functions Basic results sin A tan A= ; cos A sin( A) = sin A; cos( A) = cos A; tan( A) = tan A; 6 C 4/04/0 SDB
8 Eact values for 0 o, 45 o and 60 o From the equilateral triangle of side we can read off sin 60 o = / sin 0 0 = ½ cos 60 o = ½ cos 0 o = / 0 o tan 60 o = tan 0 o = / 60 o and from the isosceles right angled triangle with sides,, we can read off sin 45 0 = / cos 45 o = / tan 45 o = 45 o Graphs of trig functions = sin = cos = tan Graphs of = sin n, = sin( ), = sin( + n) etc. You should know the shapes of these graphs = sin is like = sin but repeats itself times between 0 o and 60 o = sin( ) = sin and = tan( ) = tan are the graphs of = sin and = tan reflected in the ais. = cos( ) = cos is just the graph of = cos. = sin( + 0) is the graph of = sin translated through 0. 0 C 4/04/0 SDB 7
9 Sine & Cosine rules and area of triangle a b c = = : be careful the sine rule alwas gives ou two answers for each sin A sin B sinc angle so if possible do not use the sine rule to find the largest angle as it might be obtuse; or ou might want both answers if it is possible to draw two different triangles from the information given. a = b + c bc cosa etc. Unique answers here! Area of a triangle = ab sinc = bcsin A = acsin B. Identities sin A tan A = cos A Eample: Solve sin = 4 cos. Solution: First divide both sides b cos sin = 4 tan = 4 cos = 5. o, or. o. tan = 4 / sin A + cos A = Eample: Given that cos A = 5 / and that 70 o < A < 60 o, find sin A and tan A. Solution: We know that sin A + cos A = sin A = cos A sin A = ( 5 / ) sin A = ± /. But 70 o < A < 60 o sin A is negative sin A = /. sin A Also tan A = cos A tan A = 5 = / 5 =.4. 8 C 4/04/0 SDB
10 Eample: Solve sin + sin cos = Solution: Rewriting cos in terms of sin will make life easier so using sin + cos = cos = sin sin + sin cos = sin + sin ( sin ) = sin + sin = 0 ( sin )(sin + ) = 0 sin = / or = 4.8 o, 8.9 o, or 70 o. N.B. If asked to give answers in radians, ou are allowed to work in degrees as above and then convert to radians b multipling b π / 80 So answers in radians would be = 4.80 π / 80 = 0.70, or π / 80 =.4, or 70 π / 80 = π /. Trigonometric equations Eample: Solve sin ( π / 4 ) = 0.5 for 0 c π c, giving our answers in radians in terms of π. Solution: First we know that sin 60 o = 0.5, and 60 o = π / radians π / 4 = π / or π π / = π / = 7π / or π  /. Eample: Solve sin = 0.47 for 0 o 60 o, giving our answers to the nearest degree. Solution: First put X = and find all solutions of sin X = 0.47 for 0 o X 70 o X = 8., or = 5.9 or = 88., or = 5.9 i.e. X = 8., 5.9, 88., 5.9 = 4 o, 76 o, 94 o, 56 o to the nearest degree. There are several eamples in the book. C 4/04/0 SDB 9
11 Coordinate Geometr Mid point The mid point of the line joining P (a, b ) and Q (a, b ) is ( ( a a ), ( b b )) Circle + +. Centre at the origin Take an point, P, on a circle centre the origin and radius 5. Suppose that P has coordinates (, ) P (, ) Using Pthagoras Theorem we have + = 5 + = 5 r which is the equation of the circle. and in general the equation of a circle centre (0, 0) and radius r is + = r. General equation In the circle shown the centre is C, (a, b), and the radius is r. CQ = a and PQ = b and, using Pthagoras r P (, ) b CQ + PQ = r ( a) + ( b) = r, C (a, b) a Q which is the general equation of a circle. Eample: Find the centre and radius of the circle whose equation is = 0. Solution: First complete the square in both and to give = = 5 ( ) + ( + ) = 5 which is the equation of a circle with centre (, ) and radius 5. 0 C 4/04/0 SDB
12 Eample: Find the equation of the circle on the line joining A, (, 5), and B, (8, 7), as diameter. Solution: The centre is the mid point of AB is ( 8), (5 7) ) ( and the radius is ½ AB = ½ (8 ) + ( 7 5) = 6. 5 equation is ( 5.5) + ( + ) = = (5½, ) Equation of tangent Eample: Find the equation of the tangent to the circle = 64 which passes through the point of the circle ( 6, 4). Solution: First complete the square in and in to give ( + ) + ( ) = 69. Net find the gradient of the radius from the centre (, ) to the point ( 6, 4) which is / 5 gradient of the tangent at that point is 5 /, since the tangent is perpendicular to the radius and product of gradients of perpendicular lines is equation of the tangent is 4 = 5 ( + 6) 5 = 98. Eample: Find the intersection of the line = + 4 with the circle + = 5. Solution: Put = + 4 in + = 5 to give + ( + 4) = = = 0 (5 + )( + ) = 0 =. or = 0.4 or line intersects circle at (., 0.4) and (, ) If the two points of intersection are the same point then the line is a tangent. Note. You should know that the angle in a semicircle is a right angle and that the perpendicular from the centre to a chord bisects the chord (cuts it eactl in half). C 4/04/0 SDB
13 4 Sequences and series Geometric series Finite geometric series A geometric series is a series in which each term is a constant amount times the previous term: this constant amount is called the common ratio. The common ratio can be or, and positive or negative. Eamples:, 6, 8, 54, 6, 486,..... with common ratio, 40, 0, 0, 5, ½, ¼,..... with common ratio ½, ½,, 8,, 8, 5,.... with common ratio 4. Generall a geometric series can be written as S n = a + ar + ar + ar + ar up to n terms where a is the first term and r is the common ratio. The nth term is u n = ar n. The sum of the first n terms of the above geometric series is n r S n = a r a r n = r. Eample: Find the n th term and the sum of the first terms of the geometric series whose rd term is and whose 6 th term is 6. Solution: 6 = r 6 = r r = 8 r = Now = r = r = ( ) = ½ n th term, n = ar n = ½ ( ) n and the sum of the first terms is S = ( ) = S = 4 ½ Infinite geometric series When the common ratio is between and + the series converges to a limit. S n = a + ar + ar + ar + ar up to n terms n r and S n = a. r Since r <, r n 0 as n and so a S n S = r C 4/04/0 SDB
14 Eample: Show that the following geometric series converges to a limit and find its sum to infinit. S = ¾ + Solution: Firstl the common ratio is / 6 = ¾ which lies between  and + therefore the sum converges to a limit. a 6 The sum to infinit S = = r S = 64 4 Proof of the formula for the sum of a geometric series You must know this proof. S n = a + ar + ar + ar + ar n + ar n, multipl through b r r S n = ar + ar + ar + ar n + ar n + ar n subtract S n r S n = a ar n ( r) S n = a ar n = a( r n ) n n r r S n = a = a. r r Notice that if < r < + then r n 0 and S n S = a r. C 4/04/0 SDB
15 Binomial series for positive integral inde Pascal s triangle When using Pascal s triangle we think of the top row as row 0. row 0 row row row row 4 row 5 row To epand (a + b) 6 we first write out all the terms of degree 6 in order of decreasing powers of a to give a 6 + a 5 b + a 4 b + a b + a b 4 + ab 5 + b 6 and then fill in the coefficients using row 6 of the triangle to give a 6 + 6a 5 b + 5a 4 b + 0a b + 5a b 4 + 6ab 5 + b 6 = a 6 + 6a 5 b + 5a 4 b + 0a b + 5a b 4 + 6ab 5 + b 6 Factorials Factorial n, written as n! = n (n ) (n ).... So 5! = 5 4 = 0 Binomial coefficients or n n C r or r If we think of row 6 starting with the 0 th term we use the following notation 0 th term st term nd term rd term 4 th term 5 th term 6 th term C 0 6 C 6 C 6 C 6 C 4 6 C 5 6 C where the binomial coefficients n C r or n are defined b r 4 C 4/04/0 SDB
16 n n n! C r = = r ( n r)! r! This is particularl useful for calculating the numbers further down in Pascal s triangle e.g. The fourth term in row 5 is 5 5 5! 5! 5 4 C 4 = = = = 5 7 = 65 4 (5 4)!4!!4! 4 You ma find an n C r button on our calculator. Eample: Find the coefficient of in the epansion of ( ) 5. Solution: The term in is 5 C ( ) = 0 ( 8) = 80 so the coefficient of is Eponentials and logarithms Graphs of eponentials and logarithms = is an eponential function and its inverse is the logarithm function = log. Remember that the graph of an inverse function is the reflection of the original graph in = = = log Rules of logarithms log a = = a log a = log a + log a log a ( ) = log a log a log a n = n log a log a = 0 log a a = Note: log 0 is often written as lg C 4/04/0 SDB 5
17 Eample: Find log 8. Solution: Write log 8 = 8 = = 4 log 8 = 4. To solve log equations we can either use the rules of logarithms to end with log a = log a = or log a = = a Eample: Solve log a 40 log a = log a 5 Solution: log a 40 log a = log a 5 log a 40 log a 5 = log a log a (40 5) = log a log a 8 = log a = 8 =. Eample: Solve log 5 = + ½ log 5. Solution: log 5 = + ½ log 5 log 5 ½ log 5 =.5 log 5 = log 5 = = 5 = 5. Changing the base of a logarithm log a b = log log c c b a Eample: Find log 4 9. log Solution: log 9 = = =. 4 log A particular case logb b log a b = log a 4. 0 b = This gives a source of eam questions. log a b Eample: Solve log 4 6 log 4 = Solution: log 4 6 = log 4 (log 4 ) log 4 6 = 0 (log 4 )( log 4 + ) = 0 log 4 = or = 4 or 4 = 64 or / 6. 6 C 4/04/0 SDB
18 Equations of the form a = b Eample: Solve 5 = Solution: Take logs of both sides log 0 5 = log 0 log 0 5 = log 0 log0.9 = = =.59. log Differentiation Increasing and decreasing functions is an increasing function if its gradient is positive, is an increasing function if its gradient is negative, Eample: d > 0; d d < 0 d For what values of is = + 7 an increasing function. Solution: = + 7 d = d For an increasing function we want values of for which > 0 Find solutions of = 0 ( + )( ) = 0 =  / or so graph of meets ais at  / and and is above ais for <  / or > So = + 7 is an increasing function for < / or >. C 4/04/0 SDB 7
19 Stationar points and local maima and minima (turning points). An point where the gradient is zero is called a stationar point. minimum maimum Turning points Stationar points of inflection Local maima and minima are called turning points. The gradient at a local maimum or minimum is 0. Therefore to find ma and min firstl differentiate and find the values of which give gradient, d, equal to zero: d secondl find second derivative d and substitute value of found above d second derivative positive minimum, and second derivative negative maimum: N.B. If d = 0, it does not help! In this case ou will need to find the gradient d just before and just after the value of. Be careful: ou might have a stationar point of inflection thirdl substitute to find the value of and give both coordinates in our answer. Using second derivative Eample: Find the local maima and minima of the curve with equation = Solution: = First find d d = At maima and minima the gradient = d d = = = 0 ( + 4) = 0 8 C 4/04/0 SDB
20 ( + 4)( ) = 0 = 4, 0 or. d Second find d = +4 6 When = 4, When = 0, When =, d = = 80, positive min at = 4 d d = 6, negative, ma at = 0 d d = = 0, positive, min at =. d Third find values: when = 4, 0 or = 5, 7 or 0 Maimum at (0, 7) and Minimum at ( 4, 5) and (, 0). d N.B. If = 0, it does not help! You can have an of ma, min or stationar d point of inflection. Using gradients before and after Eample: Find the stationar points of = Solution: = d = 4 + = 0 for stationar points d ( + ) = 0 ( ) = 0 = 0 or. d d = which is (positive) when = 0 minimum at (0, 7) and which is 0 when =, so we must look at gradients before and after. = 0.9. d d = stationar point of inflection at (, ) N.B. We could have ma, min or stationar point of inflection when the second derivative is zero, so we must look at gradients before and after. C 4/04/0 SDB 9
21 Maimum and minimum problems Eample: A manufacturer of cans for baked beans wishes to use as little metal as possible in the manufacture of these cans. The cans must have a volume of 500 cm : how should he design the cans? Solution: h cm We need to find the radius and height needed to make cans of volume 500 cm using the minimum possible amount of metal. Suppose that the radius is cm and that the height is h cm. The area of top and bottom together is π cm and the area of the curved surface is πh cm the total surface area A = π + πh cm. (I) cm We have a problem here: A is a function not onl of, but also of h. But we know that the volume is 500 cm and that the volume can also be written as π h cm π h = 500 h = 500 π and so (I) can be written A = π + π 500 π A = π da d = π = 4π 000 = 4π. For stationar values of A, the area, da d = 0 4π = π = 000 = = = π = 4.0 to S.F. h = 500 = 860. π We do not know whether this value gives a maimum or a minimum value of A or a stationar point of inflection d A 000 so we must find = 4 π = 4 π +. d Clearl this is positive when = 4.0 and thus this gives a minimum of A minimum area of metal is 49 cm when the radius is 4.0 cm and the height is 8.60 cm. 0 C 4/04/0 SDB
22 7 Integration Definite integrals When limits of integration are given. Eample: Find d Solution: d = [ 4 + ] no need for +C as it cancels out = [ 4 + ] [ 4 + ] put top limit in first = [] [ ] =. Area under curve The integral is the area between the curve and the ais, but areas above the ais are positive and areas below the ais are negative. Eample: Find the area between the ais, = 0, = and = 4. Solution: 0 4 d 8 = = [ 8] ] [0 0] 0 6 = which is negative since the area is below the ais required area is Eample: Find the area between the ais, =, = 4 and =. Solution: First sketch the curve to see which bits are above (positive) and which bits are below (negative). = = ( ) A meets ais at 0 and 4 A so graph is as shown. Area A, between and, is above ais: area A, between and 4, is below ais 5 so we must find these areas separatel. C 4/04/0 SDB
23 A = d = = [4.5] [ ] = / 6. and 4 5 d = = [ ] [4.5] = 6 and so area A (areas are positive) = + 5 / 6 so total area = A + A = / + 5 / 6 = 5 / 6. Note that 4 4 = [ ] [ ] = 6 d which is A A = / 5 / 6 = /. 4 Numerical integration: the trapezium rule Man functions can not be anti differentiated and the trapezium rule is a wa of estimating the area under the curve. Divide the area under = f () into n strips, each of width h. = f () Join the top of each strip with a straight line to form a trapezium. n Then the area under the curve sum of the areas of the trapezia 0 h h h a b b f ) d h ( + ) + h ( + ) + h ( + ) h ( + ) a b ( 0 n f ) d h ( ) a b ( 0 n f ) d h ( + + ( )) a ( 0 n n area under curve ½ width of each strip ( ends + middles ). n n C 4/04/0 SDB
24 Inde Area of triangle, 8 Binomial series, 4 binomial coefficients, 4 n C r or n, 4 r Circle Centre at origin, 0 General equation, 0 Tangent equation, Cosine rule, 8 Cubic equations, 6 Differentiation, 7 Equations a = b, 7 Eponential, 5 Factor theorem, 4 Factorials, 4 Factorising eamples, Functions decreasing, 7 increasing, 7 Integrals area under curve, definite, Logarithm, 5 change of base, 6 rules of logs, 5 Mid point, 0 Pascal s triangle, 4 Polnomials, long division, Radians arc length, 6 area of sector, 6 area of segment, 6 connection between radians and degrees, 6 Remainder theorem, Series Geometric, finite, Geometric, infinite, Geometric, proof of sum, Sine rule, 8 Stationar points gradients before and after, 9 maima and minima, 8 maimum and minimum problems, 0 second derivative, 8 Trapezium rule, Trig equations, 9 Trig functions basic results, 6 graphs, 7 identities, 8 Turning points, 8 C 4/04/0 SDB
Core 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 informationHigher. Polynomials and Quadratics 64
hsn.uk.net Higher Mathematics UNIT OUTCOME 1 Polnomials and Quadratics Contents Polnomials and Quadratics 64 1 Quadratics 64 The Discriminant 66 3 Completing the Square 67 4 Sketching Parabolas 70 5 Determining
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 informationACT Math Vocabulary. Altitude The height of a triangle that makes a 90degree 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 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 informationy intercept Gradient Facts Lines that have the same gradient are PARALLEL
CORE Summar Notes Linear Graphs and Equations = m + c gradient = increase in increase in intercept Gradient Facts Lines that have the same gradient are PARALLEL If lines are PERPENDICULAR then m m = or
More informationSolutions to Exercises, Section 5.1
Instructor s Solutions Manual, Section 5.1 Exercise 1 Solutions to Exercises, Section 5.1 1. Find all numbers t such that ( 1 3,t) is a point on the unit circle. For ( 1 3,t)to be a point on the unit circle
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 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 informationPRECALCULUS GRADE 12
PRECALCULUS 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 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 informationSAMPLE. Polynomial functions
Objectives C H A P T E R 4 Polnomial functions To be able to use the technique of equating coefficients. To introduce the functions of the form f () = a( + h) n + k and to sketch graphs of this form through
More informationSouth Carolina College and CareerReady (SCCCR) PreCalculus
South Carolina College and CareerReady (SCCCR) PreCalculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know
More informationPolynomials Past Papers Unit 2 Outcome 1
PSf Polnomials Past Papers Unit 2 utcome 1 Multiple Choice Questions Each correct answer in this section is worth two marks. 1. Given p() = 2 + 6, which of the following are true? I. ( + 3) is a factor
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 informationPOLYNOMIAL FUNCTIONS
POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a
More informationPROPERTIES OF ELLIPTIC CURVES AND THEIR USE IN FACTORING LARGE NUMBERS
PROPERTIES OF ELLIPTIC CURVES AND THEIR USE IN FACTORING LARGE NUMBERS A ver important set of curves which has received considerabl attention in recent ears in connection with the factoring of large numbers
More informationPolynomials. Jackie Nicholas Jacquie Hargreaves Janet Hunter
Mathematics Learning Centre Polnomials Jackie Nicholas Jacquie Hargreaves Janet Hunter c 26 Universit of Sdne Mathematics Learning Centre, Universit of Sdne 1 1 Polnomials Man of the functions we will
More informationExponential and Logarithmic Functions
Chapter 6 Eponential and Logarithmic Functions Section summaries Section 6.1 Composite Functions Some functions are constructed in several steps, where each of the individual steps is a function. For eample,
More informationINVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4. Example 1
Chapter 1 INVESTIGATIONS AND FUNCTIONS 1.1.1 1.1.4 This opening section introduces the students to man of the big ideas of Algebra 2, as well as different was of thinking and various problem solving strategies.
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 informationNew HigherProposed OrderCombined Approach. Block 1. Lines 1.1 App. Vectors 1.4 EF. Quadratics 1.1 RC. Polynomials 1.1 RC
New HigherProposed OrderCombined Approach Block 1 Lines 1.1 App Vectors 1.4 EF Quadratics 1.1 RC Polynomials 1.1 RC Differentiationbut not optimisation 1.3 RC Block 2 Functions and graphs 1.3 EF Logs
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 informationMEMORANDUM. All students taking the CLC Math Placement Exam PLACEMENT INTO CALCULUS AND ANALYTIC GEOMETRY I, MTH 145:
MEMORANDUM To: All students taking the CLC Math Placement Eam From: CLC Mathematics Department Subject: What to epect on the Placement Eam Date: April 0 Placement into MTH 45 Solutions This memo is an
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 informationPolynomial Degree and Finite Differences
CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson you will learn the terminology associated with polynomials use the finite differences method to determine the degree of a polynomial
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 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 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 informationTrigonometry Review Workshop 1
Trigonometr Review Workshop Definitions: Let P(,) be an point (not the origin) on the terminal side of an angle with measure θ and let r be the distance from the origin to P. Then the si trig functions
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 informationFunctions and their Graphs
Functions and their Graphs Functions All of the functions you will see in this course will be realvalued functions in a single variable. A function is realvalued if the input and output are real numbers
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 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 informationColegio del mundo IB. Programa Diploma REPASO 2. 1. The mass m kg of a radioactive substance at time t hours is given by. m = 4e 0.2t.
REPASO. The mass m kg of a radioactive 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 informationRotated Ellipses. And Their Intersections With Lines. Mark C. Hendricks, Ph.D. Copyright March 8, 2012
Rotated Ellipses And Their Intersections With Lines b Mark C. Hendricks, Ph.D. Copright March 8, 0 Abstract: This paper addresses the mathematical equations for ellipses rotated at an angle and how to
More information135 Final Review. Determine whether the graph is symmetric with respect to the xaxis, the yaxis, and/or the origin.
13 Final Review Find the distance d(p1, P2) between the points P1 and P2. 1) P1 = (, 6); P2 = (7, 2) 2 12 2 12 3 Determine whether the graph is smmetric with respect to the ais, the ais, and/or the
More informationThe Mathematics Diagnostic Test
The Mathematics iagnostic Test Mock Test and Further Information 010 In welcome week, students will be asked to sit a short test in order to determine the appropriate lecture course, tutorial group, whether
More information2312 test 2 Fall 2010 Form B
2312 test 2 Fall 2010 Form B 1. Write the slopeintercept form of the equation of the line through the given point perpendicular to the given lin point: ( 7, 8) line: 9x 45y = 9 2. Evaluate the function
More informationCurriculum Map by Block Geometry Mapping for Math Block Testing 20072008. August 20 to August 24 Review concepts from previous grades.
Curriculum Map by Geometry Mapping for Math Testing 20072008 Pre s 1 August 20 to August 24 Review concepts from previous grades. August 27 to September 28 (Assessment to be completed by September 28)
More informationExact Values of the Sine and Cosine Functions in Increments of 3 degrees
Exact Values of the Sine and Cosine Functions in Increments of 3 degrees The sine and cosine values for all angle measurements in multiples of 3 degrees can be determined exactly, represented in terms
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 informationIntroduction Assignment
PRECALCULUS 11 Introduction Assignment Welcome to PREC 11! This assignment will help you review some topics from a previous math course and introduce you to some of the topics that you ll be studying
More informationReview of Intermediate Algebra Content
Review of Intermediate Algebra Content Table of Contents Page Factoring GCF and Trinomials of the Form + b + c... Factoring Trinomials of the Form a + b + c... Factoring Perfect Square Trinomials... 6
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 informationGeometry Notes RIGHT TRIANGLE TRIGONOMETRY
Right Triangle Trigonometry Page 1 of 15 RIGHT TRIANGLE TRIGONOMETRY Objectives: After completing this section, you should be able to do the following: Calculate the lengths of sides and angles of a right
More informationMathematics Placement Packet Colorado College Department of Mathematics and Computer Science
Mathematics Placement Packet Colorado College Department of Mathematics and Computer Science Colorado College has two all college requirements (QR and SI) which can be satisfied in full, or part, b taking
More informationPrecalculus REVERSE CORRELATION. Content Expectations for. Precalculus. Michigan CONTENT EXPECTATIONS FOR PRECALCULUS CHAPTER/LESSON TITLES
Content Expectations for Precalculus Michigan Precalculus 2011 REVERSE CORRELATION CHAPTER/LESSON TITLES Chapter 0 Preparing for Precalculus 01 Sets There are no statemandated Precalculus 02 Operations
More informationMath, Trigonometry and Vectors. Geometry. Trig Definitions. sin(θ) = opp hyp. cos(θ) = adj hyp. tan(θ) = opp adj. Here's a familiar image.
Math, Trigonometr and Vectors Geometr Trig Definitions Here's a familiar image. To make predictive models of the phsical world, we'll need to make visualizations, which we can then turn into analtical
More informationMathematics 31 Precalculus and Limits
Mathematics 31 Precalculus and Limits Overview After completing this section, students will be epected to have acquired reliability and fluency in the algebraic skills of factoring, operations with radicals
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 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 informationAnswer Key for the Review Packet for Exam #3
Answer Key for the Review Packet for Eam # Professor Danielle Benedetto Math MaMin Problems. Show that of all rectangles with a given area, the one with the smallest perimeter is a square. Diagram: y
More informationJUST THE MATHS UNIT NUMBER 1.8. ALGEBRA 8 (Polynomials) A.J.Hobson
JUST THE MATHS UNIT NUMBER 1.8 ALGEBRA 8 (Polynomials) by A.J.Hobson 1.8.1 The factor theorem 1.8.2 Application to quadratic and cubic expressions 1.8.3 Cubic equations 1.8.4 Long division of polynomials
More informationUnit 6: Polynomials. 1 Polynomial Functions and End Behavior. 2 Polynomials and Linear Factors. 3 Dividing Polynomials
Date Period Unit 6: Polynomials DAY TOPIC 1 Polynomial Functions and End Behavior Polynomials and Linear Factors 3 Dividing Polynomials 4 Synthetic Division and the Remainder Theorem 5 Solving Polynomial
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 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 informationopp (the cotangent function) cot θ = adj opp Using this definition, the six trigonometric functions are welldefined for all angles
Definition of Trigonometric Functions using Right Triangle: C hp A θ B Given an right triangle ABC, suppose angle θ is an angle inside ABC, label the leg osite θ the osite side, label the leg acent to
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 informationSPECIFICATION. Mathematics 6360 2014. General Certificate of Education
Version 1.0: 0913 General Certificate of Education Mathematics 6360 014 Material accompanying this Specification Specimen and Past Papers and Mark Schemes Reports on the Examination Teachers Guide SPECIFICATION
More information2009 Chicago Area AllStar Math Team Tryouts Solutions
1. 2009 Chicago Area AllStar 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 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 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 informationDISTANCE, CIRCLES, AND QUADRATIC EQUATIONS
a p p e n d i g DISTANCE, CIRCLES, AND QUADRATIC EQUATIONS DISTANCE BETWEEN TWO POINTS IN THE PLANE Suppose that we are interested in finding the distance d between two points P (, ) and P (, ) in the
More informationGEOMETRIC 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
More informationGeorgia Department of Education Kathy Cox, State Superintendent of Schools 7/19/2005 All Rights Reserved 1
Accelerated Mathematics 3 This is a course in precalculus and statistics, designed to prepare students to take AB or BC Advanced Placement Calculus. It includes rational, circular trigonometric, and inverse
More informationMathematics Placement Examination (MPE)
Practice Problems for Mathematics Placement Eamination (MPE) Revised August, 04 When you come to New Meico State University, you may be asked to take the Mathematics Placement Eamination (MPE) Your inital
More informationSolutions to old Exam 1 problems
Solutions to old Exam 1 problems Hi students! I am putting this old version of my review for the first midterm review, place and time to be announced. Check for updates on the web site as to which sections
More informationUnderstanding Basic Calculus
Understanding Basic Calculus S.K. Chung Dedicated to all the people who have helped me in my life. i Preface This book is a revised and expanded version of the lecture notes for Basic Calculus and other
More informationDefinitions, Postulates and Theorems
Definitions, s and s Name: Definitions Complementary Angles Two angles whose measures have a sum of 90 o Supplementary Angles Two angles whose measures have a sum of 180 o A statement that can be proven
More informationSECTION 2.2. Distance and Midpoint Formulas; Circles
SECTION. Objectives. Find the distance between two points.. Find the midpoint of a line segment.. Write the standard form of a circle s equation.. Give the center and radius of a circle whose equation
More informationGCE Mathematics. Mark Scheme for June 2014. Unit 4722: Core Mathematics 2. Advanced Subsidiary GCE. Oxford Cambridge and RSA Examinations
GCE Mathematics Unit 4722: Core Mathematics 2 Advanced Subsidiary GCE Mark Scheme for June 2014 Oxford Cambridge and RSA Examinations OCR (Oxford Cambridge and RSA) is a leading UK awarding body, providing
More informationExtra Credit Assignment Lesson plan. The following assignment is optional and can be completed to receive up to 5 points on a previously taken exam.
Extra Credit Assignment Lesson plan The following assignment is optional and can be completed to receive up to 5 points on a previously taken exam. The extra credit assignment is to create a typed up lesson
More information2.6. The Circle. Introduction. Prerequisites. Learning Outcomes
The Circle 2.6 Introduction A circle is one of the most familiar geometrical figures and has been around a long time! In this brief Section we discuss the basic coordinate geometr of a circle  in particular
More informationCOMPONENTS OF VECTORS
COMPONENTS OF VECTORS To describe motion in two dimensions we need a coordinate sstem with two perpendicular aes, and. In such a coordinate sstem, an vector A can be uniquel decomposed into a sum of two
More informationMath Placement Test Study Guide. 2. The test consists entirely of multiple choice questions, each with five choices.
Math Placement Test Study Guide General Characteristics of the Test 1. All items are to be completed by all students. The items are roughly ordered from elementary to advanced. The expectation is that
More informationZero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.
MATH 11011 FINDING REAL ZEROS KSU OF A POLYNOMIAL Definitions: Polynomial: is a function of the form P (x) = a n x n + a n 1 x n 1 + + a x + a 1 x + a 0. The numbers a n, a n 1,..., a 1, a 0 are called
More informationEvaluating trigonometric functions
MATH 1110 0090906 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 informationLESSON EIII.E EXPONENTS AND LOGARITHMS
LESSON EIII.E EXPONENTS AND LOGARITHMS LESSON EIII.E EXPONENTS AND LOGARITHMS OVERVIEW Here s what ou ll learn in this lesson: Eponential Functions a. Graphing eponential functions b. Applications of eponential
More informationImplicit Differentiation
Revision Notes 2 Calculus 1270 Fall 2007 INSTRUCTOR: Peter Roper OFFICE: LCB 313 [EMAIL: roper@math.utah.edu] Standard Disclaimer These notes are not a complete review of the course thus far, and some
More information15.1. Exact Differential Equations. Exact FirstOrder Equations. Exact Differential Equations Integrating Factors
SECTION 5. Eact FirstOrder Equations 09 SECTION 5. Eact FirstOrder Equations Eact Differential Equations Integrating Factors Eact Differential Equations In Section 5.6, ou studied applications of differential
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 informationPrentice Hall Mathematics: Algebra 2 2007 Correlated to: Utah Core Curriculum for Math, Intermediate Algebra (Secondary)
Core Standards of the Course Standard 1 Students will acquire number sense and perform operations with real and complex numbers. Objective 1.1 Compute fluently and make reasonable estimates. 1. Simplify
More informationD.2. The Cartesian Plane. The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles. D10 APPENDIX D Precalculus Review
D0 APPENDIX D Precalculus Review SECTION D. The Cartesian Plane The Cartesian Plane The Distance and Midpoint Formulas Equations of Circles The Cartesian Plane An ordered pair, of real numbers has as its
More informationNational 5 Mathematics Course Assessment Specification (C747 75)
National 5 Mathematics Course Assessment Specification (C747 75) Valid from August 013 First edition: April 01 Revised: June 013, version 1.1 This specification may be reproduced in whole or in part for
More informationStudents Currently in Algebra 2 Maine East Math Placement Exam Review Problems
Students Currently in Algebra Maine East Math Placement Eam Review Problems The actual placement eam has 100 questions 3 hours. The placement eam is free response students must solve questions and write
More informationWhen I was 3.1 POLYNOMIAL FUNCTIONS
146 Chapter 3 Polnomial and Rational Functions Section 3.1 begins with basic definitions and graphical concepts and gives an overview of ke properties of polnomial functions. In Sections 3.2 and 3.3 we
More information10.1. Solving Quadratic Equations. Investigation: Rocket Science CONDENSED
CONDENSED L E S S O N 10.1 Solving Quadratic Equations In this lesson you will look at quadratic functions that model projectile motion use tables and graphs to approimate solutions to quadratic equations
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 midpoint of AB. [2] Given that AB is a diameter of the circle,
More informationsin(θ) = opp hyp cos(θ) = adj hyp tan(θ) = opp adj
Math, Trigonometr and Vectors Geometr 33º What is the angle equal to? a) α = 7 b) α = 57 c) α = 33 d) α = 90 e) α cannot be determined α Trig Definitions Here's a familiar image. To make predictive models
More informationMATHS LEVEL DESCRIPTORS
MATHS LEVEL DESCRIPTORS Number Level 3 Understand the place value of numbers up to thousands. Order numbers up to 9999. Round numbers to the nearest 10 or 100. Understand the number line below zero, and
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 8page answer
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 informationMathematics programmes of study: key stage 4. National curriculum in England
Mathematics programmes of study: key stage 4 National curriculum in England July 2014 Contents Purpose of study 3 Aims 3 Information and communication technology (ICT) 4 Spoken language 4 Working mathematically
More informationEssential Mathematics for Computer Graphics fast
John Vince Essential Mathematics for Computer Graphics fast Springer Contents 1. MATHEMATICS 1 Is mathematics difficult? 3 Who should read this book? 4 Aims and objectives of this book 4 Assumptions made
More informationx(x + 5) x 2 25 (x + 5)(x 5) = x 6(x 4) x ( x 4) + 3
CORE 4 Summary Notes Rational Expressions Factorise all expressions where possible Cancel any factors common to the numerator and denominator x + 5x x(x + 5) x 5 (x + 5)(x 5) x x 5 To add or subtract 
More informationANALYTICAL METHODS FOR ENGINEERS
UNIT 1: Unit code: QCF Level: 4 Credit value: 15 ANALYTICAL METHODS FOR ENGINEERS A/601/1401 OUTCOME  TRIGONOMETRIC METHODS TUTORIAL 1 SINUSOIDAL FUNCTION Be able to analyse and model engineering situations
More informationEstimated Pre Calculus Pacing Timeline
Estimated Pre Calculus Pacing Timeline 20102011 School Year The timeframes listed on this calendar are estimates based on a fiftyminute class period. You may need to adjust some of them from time to
More informationSecondary Mathematics Syllabuses
Secondary Mathematics Syllabuses Copyright 006 Curriculum Planning and Development Division. This publication is not for sale. All rights reserved. No part of this publication may be reproduced without
More informationMathematics. GCSE subject content and assessment objectives
Mathematics GCSE subject content and assessment objectives June 2013 Contents Introduction 3 Subject content 4 Assessment objectives 11 Appendix: Mathematical formulae 12 2 Introduction GCSE subject criteria
More information