DEPARTMENT OF MATHEMATICS

Transcription

1 DEPARTMENT OF MATHEMATICS AS level Mathematics Core mathematics 2 - C Name: Page

2 C2 workbook contents Algebra Differentiation Integration Coordinate Geometry Logarithms Geometric series Series Expansion Trigonometry Page

3 C2WB: Algebra & Functions Algebraic division - Notes BAT efficiently perform algebraic long division with answers exact answers and remainders

4 C2WB: Algebra & Functions WB1 Divide x 3 + 2x 2 17x + 6 by (x 3) WB2 Divide 6x x 2 7x + 15 by (x + 5)

5 C2WB: Algebra & Functions WB3 Divide x 3 3x 2 by (x 2) WB4 Divide 3x 3 3x 2 4x + 4 by (x 1)

6 C2WB: Algebra & Functions WB5 Find the remainder when 2x 3 5x 2 16x + 10 is divided by (x 4)

7 C2WB: Algebra & Functions Factor Theorem - Notes BAT factorise cubic functions and other expressions BAT recall and use the Factor Theorem

8 C2WB: Algebra & Functions WB6 Given that (x + 1) is a factor of 4x 4 3x 2 + a, find the value of a WB7 Factorise the cubic polynomial f(x) = x 3 2x 2 x + 2 and hence sketch the graph of the function

9 C2WB: Algebra & Functions WB8 Express f(x) = x 3 + x 2 5x + 3 as the product of three linear factors. Hence: a) Sketch the graph of the function. b) Solve the equation x 3 + x 2 5x + 3 = 0 WB9 Factorise the cubic polynomial f(x) = x 3 + 3x 2 12x 14.

10 C2WB: Algebra & Functions WB10 Exam Q The polynomial p(x)is given by p(x) = x 3 19x 30 a) Use the factor theorem to show that x + 2 is a factor of p(x) b) Express p(x) as the product of three linear factors

11 C2WB: Algebra & Functions Factor and Remainder Theorem - Notes BAT recall and use the Remainder theorem (and Factor Theorem) to solve problems

12 C2WB: Algebra & Functions WB11 Find each of the remainders when the polynomial x 3 + 5x 2 17x 21 is divided by a) x + 1 b) x - 4 WB12 Use the Remainder Theorem to find the remainder when f(x) = 3x 3 x 2 8x 3 is divided by a) x 1 b) 3x + 2

13 C2WB: Algebra & Functions WB13 The function, x 3 + ax 2 2x 5, has a remainder of 7 when divided by (x 2). Find the value of a. WB14 The expression, x 3 + bx 2-3x - 1, is divisible exactly by (x 1). Find the value of b and the remainder when the expression is divided by (x + 3).

14 C2WB: Algebra & Functions WB15 When the polynomial f(x) = x 3 + ax 2 + bx + 2 is divided by x 1 the remainder is 4, and when it is divided by x + 2 the remainder is also 4. Find the values of the constants a and b. WB16 Exam Q f(x) = 3x 3 2x 2 12x + 8 a) Find the remainder when f(x) is divided x 1 b) Use the factor theorem to show x + 2 is a factor of f(x) c) Factorise f(x) completely

15 C2WB: Differentiation Stationary Points- Notes BAT find the gradient of a function at a point and understand increasing and decreasing parts of functions BAT find and determine the nature of stationary points

16 C2WB: Differentiation WB1 Show that the function f(x) = x x + 3 is an increasing function WB2 Find the range of values where f(x) = x 3 + 3x 2 9x is a decreasing function

17 C2WB: Differentiation WB3 Find the coordinates of the turning point on the curve y = x 4 32x, and state whether it is a minimum or maximum. WB4 Find the maximum possible value for y in the formula y = 6x x 2. State the range of the function.

18 C2WB: Differentiation WB5 Find the stationary points on the curve: y = 2x 3 15x x + 6, and state whether they are minima, maxima or points of inflexion

19 C2WB: Differentiation Second derivative - Notes BAT Find the second derivative of functions and use it to determine the nature of stationary points

20 C2WB: Differentiation WB6 Find the coordinates of the stationary points on the curve y = 24x 2x 3, and determine their nature using the second derivative WB7 Find the coordinates of thestationary point on the curve y = x x, and determine its nature using the second derivative

21 C2WB: Differentiation WB8 Find the stationary points on the curve: y = 2x 3 15x x + 6, and state whether they are minima, maxima or points of inflexion Hence sketch the graph of : y = 2x 3 15x x + 6,

22 C2WB: Differentiation Optimisation problems - Notes BAT apply differentiation to solve optimization problems BAT manipulate algebra to form equations that can be differentiated

23 C2WB: Differentiation WB9 A large tank (cuboid) is to be made from 54m 2 of sheet metal. It has no top. a) Show that the Volume of the tank will be given by V = 18x 2 3 x3 b) Calculate the values of x that will give the largest volume possible, and what this Volume is.

24 C2WB: Differentiation WB10 A wire of length 2m is bent into the shape shown, made up of a Rectangle and a Semi-circle a) Find an expression for y in terms of x b) Show that the Area is A = x (8 4x πx) 8 c) Find the maximum possible Area

25 C2WB: Differentiation WB11 A cuboid has a rectangular cross section where the length of the rectangle is equal to twice its width, x cm. The volume of the cuboid is 192 cubic cm a) Show that the total length, L cm, of the twelve edges of the cuboid is given by L = 12x x 2 b) Use calculus to justify that the value of L that you have found is a minimum

26 C2WB: Differentiation WB12

27 C2WB: Differentiation WB13

28 C2: Integration Definite Integrals - Notes BAT evaluate definite integrals

29 C2: Integration WB1 Evaluate the following: x 3x 1 dx 1 WB2 Evaluate the following: ( x 1) 1 2 dx

30 C2: Integration WB3 Evaluate these definite integrals: 3 a) (x + 1)(x 3)dx c) 10x 6x 2 dx 2 b) 4x 2 (x + 1) dx d) (x + 2) 2 dx

31 C2: Integration Areas under curves - Notes BAT use integration to find the area between a curve and the x-axis BAT evaluate areas under the x-axis BAT use integration to find the area between a line and curve or between two curves

32 C2: Integration WB4 Find the area of the region R bounded by the curve with equation y = (4 - x)(x + 2), and the y and x axes WB5 Find the value of R, where R is the area between the values of x = 1 and x = 3, and under the following curve: y x x

33 C2: Integration WB6 Find the area of the finite region bounded by the curve y = x(x 3) and the x- axis

34 C2: Integration WB7 Find the area between the curve: y = x(x + 1)(x 1) and the x-axis

35 C2: Integration WB8 Draw a diagram showing the equation y = x, as well as the curve y = x(4 x). Find the Area bounded by the line and the curve.

36 C2: Integration WB9 The diagram shows a sketch of the curve with equation y = x(x 3), and the line with Equation 2x. Calculate the Area of region R. y = x(x 3) y y = 2x A R O C B x

37 C2: Integration Trapezium rule - Notes BAT use the trapezium rule to approximate areas under curves

38 C2: Integration WB10 Using 4 strips, estimate the area under the curve: y = 2x + 3 Between the lines x = 0 and x =2 Extension: use 8 strips instead of 4

39 C2: Integration WB11 Use the trapezim rule with a) 3 strips b) 6 strips 3 To approximate the value of: 1 + x dx 0

40 C2: Integration WB12 Use the trapezium rule with 5 ordinates, find an estimate for the area under the graph y = cos x between x = 0 and x = π 2 State whether this is an over or under estimate of the actual area

41 C2WB: Logarithms Introduction to circle geometry- Notes BAT Find midpoints and distances BAT Solve problems with circle geometry using midpoints and distances

42 C2WB: Logarithms WB1 The points (3, 4) and B (-1, 6) are the end points of a diameter a) Find the coordinates of the centre of the circle b) Find the radius of the circle WB2 PQ is the diameter of a circle, where P and Q are (-1,3) and (6,-3) respectively. Find the radius of the circle

43 C2WB: Logarithms WB3 The line AB is the diameter of the circle, where A and B are (-3,21) and (7,-3) respectively. The point C (14,4) lies on the circumference of the circle. Find the values of AB 2, AC 2 and BC 2 and hence show that angle ACB is 90 WB4 The line AB is the diameter of the circle with centre C, where A and B are (-1, 4) and (5, 2) respectively. The line l passes through C and is perpendicular to AB. Find the equation of l.

44 C2WB: Logarithms WB5 The line PQ is the Chord of the circle centre (-3,5), where P and Q are (5,4) and (1,12) respectively. The line l is perpendicular to PQ and bisects it. Show that it passes through the centre of the circle. WB6 The lines AB and CD are chords of a circle. The line y = 3x 11 is the perpendicular bisector of AB. The line y = -x 1 is the perpendicular bisector of CD. Find the coordinates of the circle s centre.

45 C2WB: Logarithms Equations of circles - Notes BAT Find and use equations of circles BAT find the centre and radius of a circle from its equation

46 C2WB: Logarithms WB7 Find the coordinates of the centre, and the radius of, the circle with the following equation: WB8 The line AB is the diameter of a circle, where A and B are (4,7) and (-8,3) respectively. Find the equation of the circle.

47 C2WB: Logarithms

48 C2WB: Logarithms WB9 The line 4x 3y 40 = 0 touches the circle (x 2) 2 + (y 6) 2 = 100 at P = (10, 0) Show that the radius at P is perpendicular to this line

49 C2WB: Logarithms More equations and geometry problems - Notes BAT Find and use equations of circles BAT find the centre and radius of a circle from its equation

50 C2WB: Logarithms WB10 Find the centre and radius of the circle with equation x 2 + y 2 + 4x 12y + 27 = 0 WB11 Find where the circle with the equation (x 5) 2 + (y 4) 2 = 65 meets the x-axis

51 C2WB: Logarithms WB12 Find the coordinates where the line y = x + 5 meets the circle x 2 + (y 2) 2 = 29 WB13 Show that the line y = x 7 does not touch the circle (x + 2) 2 + y 2 = 33

52 C2WB: Logarithms Exponential Graphs and Logarithms - Notes BAT recognize key features of exponential graphs and logarithmic graphs

53 C2WB: Logarithms Introduction - Notes BAT write numbers and expressions as logarithms

54 C2WB: Logarithms WB1 Write each of these equations using logarithm notation. a) 5 2 = 25 b) 6 3 = 216 WB2 Write log864 = 2 in the form a b = c. Hence find log264

55 C2WB: Logarithms Laws of logarithms - Notes BAT know and apply the laws of logarithms to numbers and expressions

56 C2WB: Logarithms WB3 Write each of these as a single logarithm: 1) Log36 + log37 2) log215 log23 3) 2log53 + 3log52 4) log103 4log10 (½)

57 C2WB: Logarithms WB4 Write in terms of logax, logay and logaz 1) Log a (x 2 yz 3 ) 2) Log a ( x y 3) 3) Log a ( x y z 4) Log a ( x a 4)

58 C2WB: Logarithms WB5 Simplify a) 9 10Log a a b) log 10a 5 +log 10 a log 10 a WB6 The point P on the curve y = 3k x, where k is a constant, has its y-coordinate equal to 9k 2 Show that the x-coordinate of P may be written as 2 + log k 3

59 C2WB: Logarithms Solving equations using logarithms - Notes BAT Solve equations using logarithms BAT Solve disguised quadratic equations

60 C2WB: Logarithms WB7 Solve: 3 x = 20 WB8 Solve: 7 x+1 = 3 x+2

61 C2WB: Logarithms WB 9 Solve: 5 2x + 7(5 x ) 30 = 0 WB 10 Use logarithms to solve the equation 6 x = 3 x+2 Giving the value of x correct to 3 significant figures

62 C2WB: Logarithms Change of base - Notes BAT Use the change of base formula for logarithms

63 C2WB: Logarithms WB 11 Find the value of log811 to 3.s.f WB 12 Solve the equation: log5x + 6logx5 = 5

64 C2WB: Geometric Series Intro to Geometric sequences and series - Notes BAT you need to be able to spot patterns to work out the common ratio and rule for a Geometric Sequence BAT use Percentages in Geometric Sequences BAT apply logarithms to solve problems

65 C2WB: Geometric Series WB1 finding a and r The second term of a Geometric sequence is 4, and the 4 th term is 8. Find the values of the common ratio and the first term WB2 The numbers 3, x, and (x + 6) form the first three terms of a positive geometric sequence. Calculate the 15 th term of the sequence

66 C2WB: Geometric Series WB 3 Percentage change If A is to be invested in a savings fund at a rate of 4%. How much should be invested so the fund is worth 10,000 in 5 years? WB 4 Using logarithms What is the first term in the sequence 3, 6, 12, 24 to exceed 1 million?

67 C2WB: Geometric Series Sum of a geometric series - Notes BAT work out the sum of a Geometric Sequence BAT solve problems using the formula for Sn

68 C2WB: Geometric Series Proof for sum of a geometric series You must be able to give a complete proof with explanation. This is a standard exam question that comes up every few years

69 C2WB: Geometric Series WB 5 A geometric sequence has first term 20 and common ratio ¾ Find the sum of the first ten terms of the series, Giving your answer to 3 dp WB 6 The third term of a geometric series is 5625 and the sixth term is 1215 a) Show that the common ratio of the sequence is 3 5 b) Find the first term of the sequence c) Find the sum of the first 18 terms of the sequence

70 C2WB: Geometric Series WB 7 Using logs a) Find the sum of the following series: b) Find the sum of the following series:

71 C2WB: Geometric Series WB 8 Find the value of n at which the sum of the following sequence is greater than 2,000, WB 9 Find the value of the following: 10 1 (3 2r)

72 C2WB: Geometric Series Infinite sum of a geometric series BAT understand how a series can converge or diverge BAT work out the sum to infinity of a Geometric Sequence

73 C2WB: Geometric Series WB 10 Find the sum to infinity of the following sequence: WB 11 The sum to infinity of a convergent series is 20. The first term is 16. Find the third term of the series

74 C2WB: Geometric Series WB 14 algebra The Sum to infinity of a Sequence is 16, and the sum of the first 4 terms is 15. Find the possible values of r, and the first term if all terms are positive WB 13 Exam Q A geometric progression has first term 22 and common ratio 0.6 a) Find the sum to infinity b) Find the sum of the first 34 terms c) Use logarithms to find the smallest value of p such that the pth term is less than 0.3

75 C2WB: Series expansion Introduction to series expansion - Notes BAT explore how to find coefficients in a series expansion BAT write out in full a series expansion of a bracket

76 C2WB: Series expansion WB1 Expand (2x - 5) 4 use three colours to help keep your working clear WB2 Expand (4 2x) 4

77 C2WB: Series expansion WB3 Expand (x + 2y) 3 WB4 Expand (1 + x 3 )3

78 C2WB: Series expansion Finding coefficients - Notes BAT find coefficients in a series expansion BAT understand and use the ncr formula

79 C2WB: Series expansion WB5 The coefficient of x 2 in the expansion of (2 - cx) 3 is 294 Find the value of c. WB6 (i) Expand (2 x) 10 in ascending powers of x as far as the term in x 4 (ii) Find the coefficient of the x 4 term in the expansion of (3-2x) 10 (iii) Calculate the value of the constant a if the coefficient of the x 3 term in the expansion of (a + 2x) 4 is 160

80 C2WB: Series expansion Solving problems - Notes BAT solve algebra problems in the Binomial expansion BAT use the Binomial expansion to make an estimate

81 C2WB: Series expansion WB7 a) Find the first 3 terms in ascending powers of x, of the binomial expansion of (3 + bx) 7 where b is a non-zero constant. Give each term in its simplest form Given that, in this expansion, the coefficient of x 2 is twice the coefficient of x b) Find the value of b WB8 Find the first 4 terms in ascending powers of x, in the binomial expansion of (1 + x 2 )12 Then, use your expansion to estimate the value of (1.02) 12

82 C2WB: Series expansion WB9 Find the first 4 terms in ascending powers of x of (1 x 4 )10 Then, by using a suitable substitution, find an approximate value for WB

83 C2WB: Trigonometry Triangles review - Notes BAT review the sine, cosine and area rules BAT understand the ambiguous case with the sin rule

84 C2WB: Trigonometry WB1 Ambiguous case for sin In triangle ABC, AB = 4cm, BC = 3cm and angle BAC = 44. Work out the possible values of ACB. WB2 Triangle ABC is such that AC is 4 cm, angle ABC is 57 and angle ACB is 22 a) Calculate the length of AB b) Find the area of triangle ABC

85 C2WB: Trigonometry WB3 Triangle ABC is such that AC is 20.6 cm, angle BAC is 71 and angle ABC is 56 a) show that the length of BC is 23.5 cm b) Calculate the area of triangle ABC to the nearest cm square WB4 A TV presenter is at point P. A TV camera can move along track EF, which is of length 3.5 m. When the camera is at E the presenter is 2 m away. When the camera is at F the TV presenter is 3.2 m away a) Calculate in degrees, the size of angle EFP b) Calculate the shortest possible distance between the camera and the TV presenter

86 C2WB: Trigonometry Radian Measure - Notes BAT convert fluently between degrees and radians BAT calculate arcs, sectors and segments

87 C2WB: Trigonometry WB5 Arc AB of a circle, with centre O and radius r, subtends an angle of θ radians at O. The Perimeter of sector AOB is 10 cm. Express r in terms of θ. WB6 The border of a garden pond consists of a straight edge AB of length 2.4m, and a curved part C, as shown in the diagram below. The curved part is an arc of a circle, centre O and radius 2m. Find the length of C.

88 C2WB: Trigonometry WB7 In the diagram, the area of the minor sector AOB is 28.9cm 2. Given that angle AOB is 0.8 rad, calculate the value of r. WB8 A plot of land is in the shape of a sector of a circle of radius 55m. The length of fencing that is needed to enclose the land is 176m. Calculate the area of the plot of land.

89 C2WB: Trigonometry WB9 In the diagram AB is the diameter of a circle of radius r cm, and angle BOC = θ radians. Given that the Area of triangle AOC is 3 times that of the shaded segment, show that 3θ 4sinθ = 0 WB10 Patio PQRS is in the shape of a sector of a circle with centre Q and radius 6 m. Given that the length of the straight line PR is 6 3 m, (a) find the exact size of angle PQR in radians. (b) Show that the area of the patio PQRS is 12 m 2. (c) Find the exact area of the triangle PQR. (d) Find, in m 2 to 1 decimal place, the area of the segment PRS. (e) Find, in m to 1 decimal place, the perimeter of the patio PQRS.

90 C2WB: Trigonometry Trig Graphs - Notes BAT know the key features of trig graphs using both degrees and radians BAT use exact values of trig functions to solve problems BAT recognise and sketch transformations to trig graphs

91 C2WB: Trigonometry Trig Graphs - Symmetries

92 C2WB: Trigonometry Trig Identities and Equations Notes BAT derive and use the trig identities BAT rearrange and solve trig equations

93 C2WB: Trigonometry WB11 Exact values Given that Cosθ is - 3 / 5 and θ is reflex, find the value of Sinθ and Tanθ WB12 Exact values Given that Sinθ is 2 /5 and θ is obtuse, find the value of Cosθ and Tanθ

94 C2WB: Trigonometry WB13 Solve sin θ = 0.5 in the interval 0 θ 360 WB14 Solve 5sin θ = 2 in the interval 0 θ 360

95 C2WB: Trigonometry WB15 Solve sin θ = 2 cos θ in the interval 0 θ 360 WB16 radians Solve cos θ = 0.5 in the interval 0 θ 2π

96 C2WB: Trigonometry WB17 Solve cos 2θ = 1 in the interval 0 θ 360 WB18 Solve sin(2θ 35) = 1 in the interval 180 θ 180

97 C2WB: Trigonometry WB19 Solve tan(20 θ) = 3 in the interval 180 θ 180 WB20 Solve sin 2 (θ 30) = 1 2 in the interval 0 θ 360

98 C2WB: Trigonometry WB21 Solve 2cos 2 θ cos θ 1 = 0 in the interval 0 θ 360 WB22 Solve sin 2 θ 3 sin θ + 2 = 0 in the interval 0 θ 360