DEPARTMENT OF MATHEMATICS

Transcription

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

2 C1 workbook contents Indices and Surds Simultaneous equations Quadratics Inequalities Graphs Arithmetic series Coordinate Geometry Differentiation Integration Page

3 C1WB: Indices & Surds Indices and Surds Numeracy - Notes BAT apply the laws of indices to calculations BAT simplify and rationalise surds BAT solve equations using the rules for indices and surds a 0 = 1 a m = 1 a m n a 1 n = a a m a n = a m+n a m a n = a m n a m n = a m n a m n 1 n m = a

4 C1WB: Indices & Surds WB1 Solve a) x 1 2 = 9 b) 2x 1 2 = 32 c) 3x 1 2 = 48 d) x 3 4 = 27 e) 7x 1 2 = 1 7 f) x 2 3 = 4 9 WB2 Rearrange to single powers of 2, 3 or 5 a) 8 2 x b) 16 2x c) 8 x d) 4 3 x e) x f) 2 2 x g) h)

5 C1WB: Indices & Surds WB3 a) Solve 27 x = 9 b) Solve 4 3 x = 8 x c) Solve 7 x+1 3 = 49 2x d) Solve 5 x 2 =

6 C1WB: Indices & Surds WB4 a = 2 x b = 3 x Write in terms of a and b a) 2 x x 1 b) 4 x + 27 x+1 c) 2 x+2 d) 8 x e) 4 8 x f) 2 2x + 3 x+2 g) 27 x 9 x+1 h) 3 9 2x+1

7 C1WB: Indices & Surds Indices & Surds Algebra - Notes BAT manipulate expressions using rules of indices

8 C1WB: Indices & Surds WB5 Write as a single power a) x b) 1 x Rearrange d) 9x 1 2 e) 9x 1 2 f) 3x 1 2 c) 3 x WB6 Rearrange and simplify a) 6x x b) 4x2 +10x+6 2x

9 C1WB: Indices & Surds Calculations with Surds - Notes BAT simplify and rationalise surds BAT solve equations using the rules for indices and surds

10 C1WB: Indices & Surds WB7 Simplify a) 12 b) c) 1200 d) e) f)

11 C1WB: Indices & Surds WB8 Find length BC and express your answer as an exact simplified value B 6 A 4 C WB9 B is (11, 6) and C is ( 1.5, 10) Show that length BC is B A C

12 C1WB: Indices & Surds WB10 work out the area of the rectangle give an exact simplified answer WB11 a special case: making an integer (2 + 3)(2-3)

13 C1WB: Indices & Surds WB12 rationalise each surd a) 2 5 b) WB13 Simplify

14 C1WB: Simultaneous equations Simultaneous equations - Notes BAT know and use three methods for solving simultaneous equations BAT solve simultaneous equations from index problems

15 C1WB: Simultaneous equations WB1 Solve these simultaneous equations a) y = 2x + 8 And y = 7x 2 b) y = x 2 2x And y = 3x 6

16 C1WB: Simultaneous equations WB2 Solve these simultaneous equations a) y = x 2 + 3x 8 And y 2x = 4 b) y = 4 + x x 2 And y = 7 3x

17 C1WB: Simultaneous equations WB3 Solve the simultaneous equations below algebraically y = x 2 2x + 3 And y + 2x = 7

18 C1WB: Simultaneous equations WB4 Solve the simultaneous equations below algebraically x 2 4y = 9 And y 2x = 4 WB5 Solve the simultaneous equations below algebraically x 2 + 3y 27 = 0 And x + y = 9

19 C1WB: Simultaneous equations WB6 Solve the simultaneous equations below algebraically x 2 + y 2 = 26 And y = 3x + 2 WB7 Solve the simultaneous equations below algebraically x 2 + y 2 = 29 And 3x + y = 11

20 C1WB: Simultaneous equations WB8 Solve the simultaneous equations below algebraically a) 16x 8 y = 1 4 And b) 4 a 2 b And 4 x 2 y = 16 8 b 2 3 = 4( 2a a ) c) 3c 3 d = 27 And 9c 3 d = 1 3

21 C1WB: Quadratics Quadratics - Notes BAT convert between completed square and normal form BAT rearrange and solve quadratics using completed square form

22 C1WB: Quadratics WB1 Rearrange into completed square form x x + 10 WB2 Rearrange into completed square form x 2 + 7x + 15

23 C1WB: Quadratics WB3 Rearrange into completed square form x x + 10 and solve WB4 Rearrange into completed square form x x + 4 and solve

24 C1WB: Quadratics WB5 Rearrange into completed square form 4x x + 12 WB6 Rearrange into completed square form 3x x + 10

25 C1WB: Quadratics WB7 Rearrange into completed square form 3x x + 10 and solve

26 C1WB: Quadratics Quadratics - Notes BAT manipulate quadratic expressions and solve using the quadratic formula BAT rearrange and solve disguised quadratics

27 C1WB: Quadratics WB8 Solve 3x 2 x 1 = 0 using the quadratic formula WB9 Solve 2x 2 7x + 4 = 0 using the quadratic formula

28 C1WB: Quadratics WB10 Solve 2x + 1 = 21 x WB11 Solve y 4 + 3y 2 28 = 0

29 C1WB: Quadratics WB12 Solve x + 9 x + 14 = 0

30 C1WB: Quadratics Quadratic graphs and discriminant - Notes BAT know how to use the discriminant to solve problems and understand properties of quadratics BAT Sketch quadratic graphs showing intersections and max/min point

31 C1WB: Quadratics WB12 f(x) = 2x x + c Given that f(x)=0 has equal roots, find the value of c and hence solve f(x)=0 WB13 f(x) = x 3 kx + 16, where k is a constant a) Find the set of values of k for which the equation f(x) = 0 has no real solutions b) Express f(x) in the form (x p) 2 + q c) find the minimum value of f(x) and the value of x for which this occurs

32 C1WB: Quadratics WB14 The equation 8x 2 4x (k + 3) = 0, where k is a constant has no real roots Find the set of possible values of k WB15 Sketch f(x) = x 2 10x + 28

33 C1WB: Quadratics WB16 Sketch f(x) = x x 16

34 C1WB: Inequalities Inequalities - Notes BAT Solve quadratic and linear inequalities BAT solve inequalities problems in context

35 C1WB: Inequalities WB1 Solve x 2 + 7x 18 > 0 WB2 Solve x 2 8x

36 C1WB: Inequalities WB3 i) Solve 5x 2 > 3x + 7 ii) Solve x 2 7x 18 < 0 iii) Solve to find when both inequalities hold true WB4 The specification for a new rectangular car park states that the length L is to be 18 m more than the breadth and the perimeter of the car park is to be greater than 68 m The area of the car park is to be less than or equal to 360 m 2 Form two inequalities and solve them to determine the set of possible values of L

37 C1WB: Inequalities

38 C1WB: Inequalities BAT - Notes

39 C1WB: Inequalities WB WB

40 C1WB: Inequalities BAT - Notes

41 C1WB: Inequalities WB WB

42 C1WB: Graphs Transformations - Notes BAT know and use the six types of transformations to graphs

43 C1WB: Graphs Transformations - Notes (i) Shifts f ( x A) is a shift in the x direction (x, y) (x + A, y) +A (x, y + A) +A (x, y) f ( x) A is a shift in the y direction (ii) Stretches f (Ax) is a stretch by scale ( A 1 x, y) (x, y) factor 1 in the x direction A A 1 (x, Ay) (x, y) A Af (x) is a stretch by scale factor A in the y direction iii) Reflections f ( x) is a reflection of (-x, y) (x, y) the graph in the y axis (x, y) f (x) is a reflection of the graph in the x axis (x, y)

44 C1WB: Graphs WB1 Draw a sketch graph of y = (x 3) WB2 Draw a sketch graph of f(x + 4) 2

45 C1WB: Graphs WB3 Draw a sketch graph of f(3x) WB4 Draw a sketch graph of 2f(x)

46 C1WB: Graphs WB5 Draw a sketch graph of f( x)

47 C1WB: Graphs WB6 Describe these transformations a) f(x) b) f(x) + 4 c) f(2x) d) 3f(x) + 1 Extension : if f(x) = 2x 3 Work out the equations of the transformed graphs

48 C1WB: Graphs Graphs - Notes BAT explore cubic and reciprocal graphs BAT explore graph properties asymptotes and limits

49 C1WB: Graphs WB7 Sketch y = (x 3)(x + 2)(x 5)

50 C1WB: Graphs WB8 Sketch the graph of a) f(x) = x 3 b) f(x) = 4 1 x

51 C1WB: Arithmetic series Series - Notes BAT Explore arithmetic series and derive (new) formulas for the nth term and sum of terms BAT practice solving series problems of all types up to build and solve simultaneous equations

52 C1WB: Arithmetic series WB1 a) The third term of an arithmetic sequence is 11 and the seventh term is 23. Find the first term and the common difference b) An arithmetic series has first term 6 and common difference 2 ½. Find the least value of n for which the nth term exceeds 1000 c) Find the number of terms in the arithmetic series

53 C1WB: Arithmetic series WB2 The 5 th term of an arithmetic sequence is 24 and the 9 th term is 4 a) Find the first term and the common difference b) The last term of the sequence is -36. How many terms are in this sequence

54 C1WB: Arithmetic series WB3 The first term of an arithmetic sequence is 3, the fourth term is -9. What is the sum of the first 24 terms? WB4 The first term of an arithmetic sequence is 2, the sum of the first 10 terms is 335. Find the common difference

55 C1WB: Arithmetic series WB5 An arithmetic sequence for building each step of a spiral has first two terms 7.5 cm and 9 cm What will be (i) the length of the 40 th line of the spiral (ii) the total length of the spiral after 40 steps? WB6 Sim eqn An arithmetic sequence is used for modelling population growth of a Squirrel colony starting at three thousand in the year The 2 nd and 5 th numbers in the sequence are 14 and 23 showing the increase in population those years. Find: (i) the first increase in population (ii) the 16 th increase (iii) the population after 16 years?

56 C1WB: Arithmetic series WB7 Sim eqn The first three terms of an arithmetic sequence are (4x 5), 3x and (x + 13) respectively a) Find the value of x b) Find the 23 rd term WB8 sim eqn The sum of an arithmetic sequence to n terms is 450 The 2 nd and 4 th terms are 40 and 36. Find the possible values of n

57 C1WB: Arithmetic series Sigma notation - Notes BAT use sigma notation and solve series problems

58 C1WB: Arithmetic series WB9 8 Evaluate 5 (r 2 + 1) WB10 Evaluate (7r 3) 46 1

59 C1WB: Arithmetic series WB11 Evaluate 22 1 (3r + 5) WB12 Show that n 1 (3r + 4) = 3 n 1 r + 4n

60 C1WB: Arithmetic series Recurrence relations - Notes BAT solve problems involving recurrence relations

61 C1WB: Arithmetic series WB13 The sequence of positive numbers u 1, u 2, u 3, is given by u n+1 = (u n 6) 2, u 1 = 9 a) Find u 2, u 3 andu 4 b) Write down the value of u 20 where WB14 The nth term of a sequence is u n, the sequence is defined by u n+1 = pu n + q, where p & q are constants The first three terms of the sequence are Find u 1 = 2, u 2 = 5 and u 3 = 14 a) Show that q = 1 and find the value of p b) Find the value of u 4

62 C1WB: Linear Geometry Lines - Notes BAT explore gradients of parallel and perpendicular line BAT rearrange and find equations of lines

63 C1WB: Linear Geometry WB1 For each of these equations, i) rearrange it into the form y = mx + c ii) give the gradient iii) give the intercept on the y-axis a) 2x + y 10 = 0 b) 5x 2y + 6 = 0 WB2 Give the General equation of the perpendicular line to 2x + y 8 = 0 that goes through (4, 9)

64 C1WB: Linear Geometry WB3 Give the General equation of the perpendicular line to x + 5y 6 = 0 that goes through ( 3 5, 7) WB4 Two points A(1,2) and B(-3,6) are joined to make the line AB. Find the equation of the perpendicular bisector of AB

65 C1WB: Linear Geometry Points, Lines, Gradients - Notes BAT find distances between points BAT explore equations of lines, know the general equation of a line BAT use a new formula to find equations of lines

66 C1WB: Linear Geometry WB5 Find the line that joins these points (-2, 8) and (3,-7)

67 C1WB: Linear Geometry WB6 Find the equations of the lines that join these points (-6, 1) (2, 5) (-3, -5)

68 C1WB: Linear Geometry WB7 Find the line that joins points (4, 9) and (8, 12) in the form ax + by + c = 0 WB8 Find the line that joins points (-2, 8) and (3,-7) in the form ax + by + c = 0

69 C1WB: Linear Geometry WB9 Find the general equation of each line through (3, 7) and is perpendicular to y = 2x + 8 WB10 Find the equation of the line that goes through (3, 7) and is perpendicular to y = 2x + 8

70 C1WB: Linear Geometry WB11 The line l1 has gradient -3 goes through (-2, 3) Line l2 is perpendicular to l1 and goes through (-2, 3) Find the equations of lines l1 and L2 WB12 A line has equation 6x + 3y = 4 passes through the point (5, 5) Find the equation of the line parallel to this which

71 C1WB: Linear Geometry WB13 Find, in the form y = mx + c the equation of the line through (3, 11) which is parallel to y = 3x + 13

72 C1WB: Linear Geometry More Problems - Notes solve linear geometry problems

73 C1WB: Linear Geometry WB14 Line l1 joins points A (3, 6) and B (6, 4) a) What is the equation of the perpendicular line through midpoint of AB? b) Show this line goes through (3, 11 /4) WB15 L1 has equation 2x + y - 6 = 0 and goes through points A(0, p) and B(q, 0) a) Find the values of p and q b) What is the equation of the perpendicular line from point C(4, 5) to line L1? c) What is the area of triangle OAB?

74 C1WB: Linear Geometry WB16 Line L1 goes through points A(-3, 2) and B(3, -1) a) Find distance AB b) Find the equation of L1 in the form ax + by + c = 0 Perpendicular Line L2 has equation 2x y + 3 = 0 and crosses L1 at point D. c) Find coordinates of point D Line L2 crosses the y-axis at point Q d) Find the area of triangle AQB

75 C1WB: Linear Geometry WB17 The points A(-6, -5), B(2, -3) and C(4, -28) are the vertices of triangle ABC. Point D is the midpoint of the line joining A to B a) Show that CD is perpendicular to AB b) Find the equation of the line passing through A and B in the form ax + by + c = 0, where a, b and c are integers WB18 The straight line L 1 ha equation 4y +x = 0 The straight line L 2 has equation y = 5x - 4 a) The lines L 1 and L 2 intersect a the point A. Calculate, as exact fractions the coordinates of A b) Find an equation of the line though A which is perpendicular to L1. Give your answer in the form ax + by = c

76 C1WB: Linear Geometry WB19 The points A and B have coordinates (5, -1) and (10, 4) AB is a chord of a circle with centre C a) Find the gradient of AB The midpoint of AB is point M b) Find an equation for the line through C and M Given that the x-coordinate of point C is 6, b) Find the y coordinate of C c) Show that the radius of the circle is 17 WB20 The points A(3, 7) B(22, 7) and C(p, q) form the vertices of a triangle. Point D(9, 2) is the midpoint of AC a) Fins the values of p and q The line L, which passes through D and is perpendicular to AC, intersects AB at E b) Find an equation for line L in the form ax + by + c = 0 c) Find the exact x-coordinate of E

77 C1WB: Differentiation The Gradient function - Notes BAT explore differentiation and the gradient function of curves BAT differentiate polynomials and find the gradient of curves

78 C1WB: Differentiation WB1 Find the gradient function of a) y = 4x 7 b) y = 9x 3 c) y = 3 4 x 8 d) y = x e) y = 1 2 x2 WB2 Find the gradient function of a) a) y = 2x 2 + 5x 3 b) y = 2x 5 + 6x 8 c) y = 3x 2 7x + 6 d) y = 4x 3 3x 2 + 8x 10 e) y = 3 5 x x2

79 C1WB: Differentiation WB3 Determine the points on the curve y = x 3 + 5x + 4 Where the gradient is equal to 17 WB4 a) Find the coordinate of the point on the curve y = 4x 2 10x + 6 Where the gradient is -2 b) Sketch this graph and point on a diagram

80 C1WB: Differentiation WB5 Sketch the graph of y = x 2 4x 21 showing the minimum point and the places where the graph intersects the axes WB6 a) Given that y = 5x dy find 2 b) Given that y = 9x 3 8 x + 9x2 +4 x x dx in its simplest form find dy dx in its simplest form

81 C1WB: Differentiation Stationary points - Notes BAT determine the coordinates and nature of stationary points BAT find and use the second derivative

82 C1WB: Differentiation WB7 Find the coordinates of the points on each of these curves at which the Gradient is zero a) y = x 2 2x 3 b) y = 8x 2x 2 Now sketch the graphs of each curve

83 C1WB: Differentiation WB8 Find the coordinates of the points on each of these curves and determine their nature a) y = x 2 8x + 14 b) y = x 3 + 3x Now sketch the graphs of each curve

84 C1WB: Differentiation WB9 Differentiate 4x 2 8 x and hence find the x-coordinate of the curve y = 4x2 8 x WB10 Find the coordinates of the stationary point(s) on the curve y = 1 3 x3 2x 2 + 4x + 1 and determine their nature

85 C1WB: Differentiation WB WB

86 C1WB: Differentiation WB WB

87 C1WB: Differentiation BAT Tangents and normal - Notes

88 C1WB: Differentiation WB WB

89 C1WB: Integration - Notes BAT Integrate functions using the reverse process to differentiation BAT Integrate functions in context BAT find the value of + C

90 C1WB: Integration WB1 Find a) 3x 2 dx b) 7x x 2 c) (x3 + 3x + 2) dx d) x 6 + x 8 dx e) (x + 3)(x 2) dx

91 C1WB: Integration WB2 Find a) 2 3 dx x b) x dx c) 2x 1 dx x 3 d) 1 dx e) x3 3x dx x 2 x

92 C1WB: Integration WB3 The gradient of a curve at the point (x, y) on the curve is given by dy dx = 3x2 4x Given that the point (1, 2) lies on the curve, determine the equation of the curve WB4 A curve passes through the point (1, 5) and dy dx = 16x7 6x Find its equation

93 C1WB: Integration WB5 The gradient of a curve at the point (x, y) on the curve is given by dy dx = x2 (2x + 1) The curve passes through the point (1, 5) Find the equation of the curve WB6 a) Given that f (x) = 2 2 and f (1) = 0 x 3 b) Given further that f(1) = 8 find f(x) Find f (x)

94 C1WB: Integration WB7 The curve C has equation y = f(x) where f (x) = 2x 6 x + 8 Given that the point P (4, -14) lies on C a) Find f(x) and simplify your answer b) Find an equation of the normal to C at point P x 2 WB8 A curve has equation y = 12x 2 15x 2x 3 The curve crosses the x-axis at the origin, O, and the point A (2, 2) lies on the curve a) Find the gradient of the curve at point A b) Hence, find the equation of the normal to the curve at point A giving your answer in the form x + py + q = 0

95 C1WB: Integration WB9 The curve C with equation y = f(x) passes through the point (2, 9) Given that f (x) = 5x + 8 x 2 a) Find f(x) b) Verify that f( 2) = 12 c) Find an equation for the tangent at C at the point (-2, 12) giving your answer in the form ax + by + c = 0 WB10 dy = 9x+12x5 2 dx x a) Write dy dx in the form 9xp + 12x q b) Given that y = 90 when x = 1, find y in terms of x, simplifying the coefficient of each term Given that