Student Teacher AS STARTER PACK September 015 City and Islington Sixth Form College Mathematics Department www.candimaths.uk
CONTENTS INTRODUCTION 3 SUMMARY NOTES 4 WS CALCULUS 1 ~ Indices, powers and differentiation 6 WS CALCULUS ~ Applications of differentiation 8 WS CURVE SKETCHING 1 ~ Introduction 10 WS CURVE SKETCHING ~ Sketching quadratics 13 WS CURVE SKETCHING 3 ~ Equation of a straight line 15 WS QUADRATICS 1 ~ Factorising, quadratic simultaneous equations 18 WS QUADRATICS ~ Completing the square 0 WS QUADRATICS 3 ~ Quadratic formula and discriminant WS 9 TANGENTS AND NORMALS 5 WS SEQUENCES 1 ~ General sequences and summation notation 8 WS SEQUENCES ~ Arithmetic sequences 31 WS INTEGRATION 33 IMPORTANT INFORMATION 36
INTRODUCTION Over the next 7 weeks you will be studying new topics in mathematics. Each of these new topics builds upon GCSE work. At AS level you must ensure that you achieve a high standard of written mathematics, which is clear, logical and fluent. You will need to think deeply about the concepts and put in regular practice. Homework: You will be given homework each week to support your learning. Some homework will involve pre-learning that prepares you for the next lesson so it is very important that you complete it! You will be expected to mark the homework yourself and your teachers will check the working out and lay out of your work. Week Skills Test 1: This test is to check that you have not forgotten GCSE maths! If you do not do well on this test then you will be given extra work to make sure you are ready for the A- level mathematics. Week 6 Skills Test : This test is to make sure you are ready for the harder parts of the course. HW6: Practice Test in your C1 Homework Pack is an example of this test. You will need to work hard - the pass mark for this test is 70%. Week 1 Week Week 3 Week 4 Week 5 Week 6 Lesson 1 Lesson Lesson 3 Induction 1 Introduction to course SKILLS TEST 1 Curves Sketching 1 Differentiation 1 Differentiation Curve Sketching (Quadratics) Quadratics 1 Quadratics Quadratics 3 Curve Sketching 3 (Equation of a straight line) Tangents and Normals 1 Tangents and Normals Induction ICT skills Sequences 1 Sequences Induction 3 Study skills SKILLS TEST Integration Integration Week 7 Week 8 Transformations of graphs Transformations of graphs HALF TERM Transformations of graphs Extra resources, links and digital copies of the booklets can be found at our website: www.candimaths.uk 3
SUMMARY NOTES Number N Natural 1,, 3,.. [counting] Z Integers -, -1, 0, 1,, [counting ±] Q R Rational,,86,0 [fractions ±, all except Irrational] Real [All including irrationals numbers eg,,] Irrational numbers cannot be written as fractions. As decimals they are infinite and non-recurring. Equivalent fractions Indices (powers) 1 Algebra 4 Linear 311 8 4 Quadratic 6160 80,8 Complete the Square 6160 3 9160 3 5 3 5,8 Formula Fraction arithmetic Directed Numbers 345 BIDMAS ( ) 51 4 563 1 3 Factorising Completed Square Form 81 4 161 4 5 111535 693 693 933 5 191543 Simultaneous Equations 416 350 4 5 81 4 161 4 5 Elimination method Substitution method
Geometry Line 13 5 or 50 Circle 3 1 16 Graph Sketching 9 11 When When centre3,1 4, 15,38 51 83 gradient 83 51 5 4 4 5 Normal (perpendicular line) 5,8 Sequences 1,3 Arithmetic Sequence First term:, Common difference: Number of terms: th term: 1 Sum to terms: 1 5
WS CALCULUS 1 ~ Indices, powers and differentiation Keywords BIDMAS, powers, indices, differentiation, evaluate Exercise A Simplify the following. 1.. 35 3. 0 4 4. 5. 6. 7. 8. 9. 4 10. 3 Exercise B 1. Simplify the following, giving each term in the form, where and are constants. (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) 3 (l) 34. Use your answers to Q1 to differentiate each of the above expressions. Exercise C more challenging 1. Evaluate: 36 16. Solve to find : 7 Exam Questions 1. [C1 May 014 Q] 1 5 (a) Write down the value of 3. 5 (b) Simplify fully (3x 5 ). (1) (3) 6
. [C1 Jan 014 Q1] Simplify fully (a) ( x) 3. [C1 Jan 014 Q] (a) y = x + 1, x > 0 x Find d y dx, giving each term in its simplest form. (3) 4 (1) Answers EXA 1.. 15 3. 5 4. 5. 6. 7. 8. 8 9. 16 10. 1 EXB 1. (a) (b) 3 (c) (f) (g) 3 (h) (k) 3 6 (l) 1 83 (d) (e) 3 (i) 3 (j). (a) (f) (b) (c) (d) (g) (h) (i) 5 (e) (j) 3 (k) 1 18 (l) 48 EXC 1.. 9 Exam Questions 1. (a) (b). 4 3. 4 7
WS CALCULUS ~ Applications of differentiation Exercise A - Complete the table Example Equation of curve Gradient Function dy dx Gradient of the curve at these points x = x = 1 x = 0 4 4 3 41 4 0 0 A B C D E F G H I J K L M N O 5 39 10 36 7 73 194 13 1 1 4 31 3 8
Exercise B - Complete the table Equation of curve Gradient = Example 8 Solve for x 58 58 560 30,3 A 4 B C 8 9 4 Answers EXA EXB A 5 80 5 0 A B 4-0 B C 3 3 3 3 C 1 3,5 D 0 40-0 0 E 3 1-5 -3 F 1 5-1 1 G 6 14 164 8 0 H 9-36 -9 0 I 1983 3 8 19 J 4 3-4 0 K 34 9-7 -3 L -1 0 M 1 4 1 4 1 4 1 4 N -6 0 - O 4 3 8 3 4 3 0 9
WS CURVE SKETCHING 1 ~ Introduction Keywords: curve, axes, intersection, maximum, minimum, linear, quadratic, cubic, reciprocal Exercise A Match the following graphs to their equations A B C D E F G H I Equation Graph Equation Graph 1 1 1 10
Exercise B [There is more than one possible answer for some of these questions!] (a) (b) On separate diagrams sketch the following the graphs. Make sure you label the axis correctly and use a ruler where necessary. Try and write an equation for each graph (some are difficult!). 1. A linear graph that crosses the axes at,0 and 0,.. A quadratic graph that crosses the axes at 0,3 only. 3. A cubic graph that crosses the axes at 0,, 4,0, 1,0 and 3,0. 4. A linear graph that has a gradient of 3 and crosses the -axis at. 5. A reciprocal graph that has asymptotes at 3 and 1. 6. A negative quadratic graph that passes through 1,0 and the origin. 7. A cubic graph that crosses the origin and,0. 8. A reciprocal graph that passes through 0, Exercise C For each equation fill in the following table and use the results to sketch the curves. 0 Very big Very small 1. 31. 1 3. 5 4. 14 5. 6. 1 0 Exam Question 1. [C1 Jan 01 Q8] The curve C 1 has equation y = x (x + ). d y (a) Find. dx (b) Sketch C 1, showing the coordinates of the points where C 1 meets the x-axis. (c) Find the gradient of C 1 at each point where C 1 meets the x-axis. The curve C has equation y = (x k) (x k + ), where k is a constant and k >. (d) Sketch C, showing the coordinates of the points where C meets the x and y axes. () (3) () (3) 11
Answers EXA Equation Graph Equation Graph = E = D = B = 1 F = I = 1 C = A =(+1)( ) H = G EXB 1.. 3 3. = 1 6 (+4)( 1)( 3) 4. =3+ 5. = 1 3 +1 6. =(+3) 7. =( ) 8. = 1 + EXC 1.. 3. 4. 5. 6. Exam question: Exam question (a) 3 +4 (b) 0, 4 (c) crosses axes at: (,0),(,0),(0, +) 1
WS CURVE SKETCHING ~ Sketching quadratics Exercise A Keywords: quadratic, negative, positive, factorise, intersection, axes, differentiation, Factorise each quadratic and sketch each curve on a different set of axes, stating clearly the coordinates of the points where the curve intersects the axes. 1. = +3 4. = + 6 3. = +1 4. = 5 6 5. = 9 5 6. = 4 7. = 5 +3+ 8. = 4 9. = +4+5 Exercise B Using differentiation, find the minimum or maximum points of each curve in Exercise A and write them on your diagrams. Exercise C Write down a possible equation for each of these curves. 1.. 3. 4. 5. 6. 7. 8. 9. 13
Answers EXA/B 1. =(+4)( 1) Crosses the axes at: ( 4,0), (1,0) and (0, 4) Minimum point:, 4. = (+3)(+) Crosses the axes at: ( 3,0), (,0) and (0, 6) Maximum point:, 7. = (5+)( 1) Crosses the axes at: ( /5,0), (1,0) and (0,) Maximum point:,. =(+3)( ) Crosses the axes at: ( 3,0), (,0) and (0, 6) Minimum point:, 5. =(+1)( 5) Crosses the axes at: (,0),(5,0) and (0, 5) Minimum point:. 8. =(+)( ) Crosses the axes at: (,0), (,0) and (0, 4) Minimum point: (0, 4) 3. =( 1) Crosses the axes at: (1,0) and (0,1) Minimum point: (1,0) 6. =( 4) Crosses the axes at: (0,0), (4,0) Minimum point: (, 4) 9. Can t be factorised! Crosses the axes at: (0,5) Minimum point: (,1) EXC 1. =(+1)( 3). =(+) 3. = (+1)( ) 4. = (+1)( ) 5. =(+3) 6. = ( ) 7. =( 1)(+1) 8. = 9. =( 1)(+1)( 3) 14
WS CURVE SKETCHING 3 ~ Equation of a straight line Exercise A Work out the gradient of each of these lines Keywords: gradient, = = = 1.. 3. 4. 5. 6. Exercise B Find the gradient of the straight line between the following points: 1. (1,3) (14,7). ( 1,5) (,8) 3. (0,1) (5, 9) 4. (3, ) ( 1,4) 5. (,3) (0,7) 6. ( 3, 1) ( 1,11) 7. (0,1) (15,7) 8. ( 1, 5) (, 8) 9. (4, 3) (13, 8) 10. ( 0, 4) (0, 1) Exercise C Use the equation =( ) to find the equation of the following lines in the form =+ 1. Passing through (,3) with gradient 4. Passing through (1,5) with gradient 3. Passing through ( 1,0) with gradient 3 4. Passing through (, 7) with gradient 5. Passing through (1,3) and (14,7) 6. Passing through ( 1,5) and (,8) 7. Passing through ( 1,5) and (1,9) 8. Passing through (8,0) and (,5) 9. Passing through (,) and (5, 7) 10. Passing through (0,10) and (35,5) 15
Exercise D Find the equations of the following lines in the form ++=0 where,, and are integers: Question Working Out Equation of line 1. Gradient is 4 and intercept is -. Gradient is 1 and crosses axis at 5 3. Gradient is -6 and goes through ( 0, ) 4. Gradient is 3 and passes through ( 1, ) 5. Gradient is -1 and passes through ( 4, 3) 6. Gradient is 3 and passes through ( 6, ) 7. Line passes through 3,11 ( 4,8 ) and ( ) 8. Line passes through 1,14 (,5) and ( ) 9. Line passes through ( 3,0 ) and is perpendicular to = 3 10. Line passes through ( 1, 4 ) and is perpendicular to = + 16
Answers EXA 1.. -1 3. 1 4. - 5. 3 4 6. 5 EXB 1.. 1 3. - 4. 3 5. - 6. 6 7. 6 5 8. 3 9. 5 9 10. 5 EXC 1. =4 5. = +7 3. =3+3 4. = 8 5. = 19 6. =+6 7. =+7 8. = +4 9. = 3+8 10. = + EXD 1. 4 =0. +10=0 3. 6+ =0 4. 3 +5=0 5. + 1=0 6. 3 6=0 7. 3+ 0=0 8. 3 +11=0 9. + 3=0 10. =+3=0 17
WS QUADRATICS 1 ~ Factorising, quadratic simultaneous equations Key words Quadratic, Factorise, Simultaneous, Solve, Gradient function Exercise A Solve the following equation by factorisation. 1. +6+5=0. + 8=0 3. +11+5=0 4. +7+5=0 5. 9 5=0 6. +9 5=0 7. 3 +40+13=0 8. 3 8 11=0 9. 9+5=0 10. + 11=0 Exercise B 1. Solve the simultaneous equations: (a) =4+ (b) = +5 + 10 (c) =3 +30 10 = +9+16 =5 =1. Solve each pair of simultaneous equations using substitution: (a) y = x 6x + y = 7 (b) y = x + 1 xy 4x = 10 (c) y = x x + 4 y + 5 = 3 (d) x = y x + y = 10 (e) y = x 3 y = x + 5 x (f) x = 3 y xy y + 8 = 0 (g) x + y = 6 x = 16 3y (h) y x = 3 x + xy + y = 1 (i) y + x = 5 3y + 4xy = 0 Exercise C Find the minimum and maximum points of these curves by solving =0. 1. = + +4 7. = +1 7 3. = +11+3 4. = 7+350 5. 10=0 18
Answers EXA 1. = 1, 5. =, 4 3. =, 5 4. =, 1 5. =, 1 6. =, 5 7. =, 13 8. =, 3 9. =, 5 10. =1± 3 EXB 1. (a) = 3, 4 (b) =, 5 (c) =, 11 3 9. (a) ( 7, 49)(1, 1) (b) (, 1)(5, 6) (c), 5 )( 1, 6) EXC ( 4 16 ( 1, )(4 3 5, 5 4 (d) ( 3, 1)(1, 3) (e) ( 3, 6)( 1, 4) (f) ) (g) ( 1, 5)(, 4) (h) ( 4, 1)(1, 4) (i) ( 3, 4)(5, 0) 1. = 1, 4. =3,4 3. =,11 4. =,7 5. =, 1 19
WS QUADRATICS ~ Completing the square Keywords express, solve, completed square form, solution, roots Exercise A Multiply the brackets (revision try this as a mental arithmetic exercise) 1. ( x + 4)( x + 4). ( x + 4)( x 4) 3. 5. ( x + 7) 4. x 3) 6. ( + ( x 6) ( 3x 5) 7. Bill thinks that ( x + p + p) = x Is he correct? Exercise B Express in completed square form: y = ( x + p) + q 1. y = x + 6x + 1. y = x + 1x + 3 3. y = x 8x + 6 4. y = x 10x + 3 5. y = x + 16x + 70 6. y = x + 5x + 8 7. y = x 11x + 5 8. y = x + x + 1 *9. y = x + 8x + 18 *10. y = x + 8x + 10 Exercise C Solve by completing the square and leave your answers as fractions or surds 1. x + 14x + 46 = 0. x + 8x 13 = 0 3. x 10x + = 0 4. x + 10x + 18 = 0 5. 3x + 36x + 30 = 0 6. x 5x + 6 = 0 *7. x + 8x 18= 0 *8. x + 10x + 6 = 0 Exercise D 1. Sketch the graphs of the functions below. Show the position of the vertex. [Hint: express in completed form first] a) y = x + 6x + 13 b) y = x 10x + 3 c) y = x + 8x + 10 d) y = 5x 15x + 11. Check some of your answers for EXC by substitution e.g. is x = 5+ 3 a solution for x 10x + = 0? 3. Check that x = 4 + 5 is a solution for x + 8x + 11= 0 4. Look up completing the square on Wikipedia! 0
Answers EXA 1. + 8x + 16 5. 4 + 1x + 9 x. x 16 3. x + 14x + 49 4. x 1x + 36 x 6. 9x 30x + 5 7. Not generally! EXB 1. ( 3) 8 y = x +. = ( x + 6) 33 y 3. y = ( x 4) + 10 4. y = ( x 5) + 7 5. y = ( x + 8) + 6 6. 5 ) 7 11 1 3 y = ( x + + 4 7. y = ( x ) 4 8. y = ( x + 1 ) + 4 9. ( 7) 80 ( 4) EXC y = x + 10. y = x + 6 1. x = 7 ± 3. x = 4 ± 9 3. x = 5± 3 4. x = 5 ± 7 x = 6 ± 6. x =, 3 7. x = ± 13 8. x = 5 ± 31 5. 6 EXD 1. a) b) y y (0, 13) ( 3, 4) (0, 3) c) d) y x y( 5, ) x (4, 6) (0, 10) (0,11) x ( 3, 3. Hint: (5 + 3) 10(5 + 3) + = 0 Expand and see if the LHS equals zero 1 4 ) x 1
WS QUADRATICS 3 ~ Quadratic formula and discriminant Key words quadratic formula, discriminant, real, distinct, inequality, roots, solutions Exercise A For a quadratics equation in the form: = ++ The quadratic formula: b ± b 4ac x = can also be written a = b 4ac b ± x = a Example x 8x + 11 = 0 =6.4,1.76 Use the quadratic formulae to find the roots of the following equations (where possible). 1. (a) x + 9x + 14 = 0 (b) x 5x 4 = 0 (c) x 16x + 63 = 0 (d) x + 10x + 5 = 0 (e) x 6x + 13 = 0 (f) x + 7x 10 = 0 (g) x 6x + 3 = 0 (h) 3x + 4x 5 = 0. Sketch the graphs for questions (b), (d), (e), (g) Compare them with those of the person sitting next to you. Exercise B The discriminant: = b 4ac Calculate the discriminant for each equation and state whether there are two real distinct roots, one real root or no real roots. 1. x + x 3 = 0. x + x + 3 = 0 3. x 6x + 9 = 0 4. 5x + 8x + 6 = 0 5. x + 4x 3 = 0 6. 4x 1x + 9 = 0 Exercise C Solve the following by first sketching the graph (it will help to factorise these) 1. + 6+ 5<0. + 8>0 3. +11+5<0 4. +15 17>0 5. +>0 6. +13 11<0
Exam questions 1. [C1 May 006 Q] Find the set of values of x for which x 7x 18 > 0. (4). [C1 Jan 005 Q3] Given that the equation kx + 1x + k = 0, where k is a positive constant, has equal roots, find the value of k. (4) 3. [C1 Jan 007 Q5] The equation x 3x (k + 1) = 0, where k is a constant, has no real roots. Find the set of possible values of k. (4) 4. [C1 May 007 Q7] The equation x + kx + (k + 3) = 0, where k is a constant, has different real roots. (a) Show that k 4k 1 > 0. () (b) Find the set of possible values of k. (4) 5. [C1 Solomon B Q10] Figure 1 Figure 1 shows the curve = 3+ 5 and the straight line =+1. The curve and the line intersect at the points P and Q. (a) Using algebra, show that P has coordinates (1, 3) and find the coordinates of Q. (4) (b) Find an equation for the tangent to the curve at P. (4) (c) Show that the tangent to the curve at Q has the equation =5 11 () (d) Find the coordinates of the point where the tangent to the curve at P intersects the tangent to the curve at Q. (3) Exercise D 1. Write an equation that has no real roots then sketch the graph to show the vertex.. Write out the proof of the quadratic formulae. 3
Answers EXA 1. (a) -, -7 (b) 8, -3 (c) 9, 7 (d) -5, -5 (e) no real roots (f) 1., -8. (g).37, 0.63 (h) 0.79, -.1. (b) (d) (e) (g) Ex B 1. two real distinct roots. no real roots 3. one real root 4. no real roots 5. two real distinct roots 6. one real root Ex C 1. 5<<1. < 4 > 3. 5<< 5. <<1 6. <1 > Exam questions 1. < or >9. =6 3. < 5. (a) (4,9) (b) = +4 (d), 4. < 1 > 4. (b) < or >6 4
WS 9 TANGENTS AND NORMALS Key words quadratic formula, discriminant, real, distinct, inequality, roots, solutions Exercise A (Finding equations for the tangent and normal) For each question follow these steps: (a) Calculate the y coordinate (if necessary). (b) Differentiate the function. (c) Calculate the gradient at the point given. (d) Write down the following (,)= = (e) Find the equation of the tangent using =( ). (f) Find the equation of the normal using =( ) using the perpendicular gradient. 1. = 7+1 (5,). = +4+5 =3 3. = 6 = 1 4. = +3 5 (,3) Exercise B (Sketching graphs) For each question: (a) Sketch the given curve. (b) Calculate the tangent at the given point. (c) Add this to your sketch (Remember sketches should include any points of axes intersection) 1. = at (3,9). = 11+5 at (, 9) 3. = when =3 4. = 3 10 when = 3 5. = when =4 Exercise C (Using completing the square) Through completing the square, find the vertex of the following quadratics. 1. = 6+11 then calculate (3). = +4 1 then calculate ( ), what do you notice? 3. = 3+ Can you think of another way you could find this minimum point? 5
Exercise D (Working backwards to find coordinates) Find the coordinates at which the following functions have their given gradient. 1. = 3+. = 3 + 5 3. = +5 1 4. = + 7+ 5. = = 4 =13 =9 = = Exercise E (Tangents, normals and simultaneous equations) 1. = 1 a) Sketch the quadratic function. b) Find the equation of the normal to the quadratic when =1 c) Find the coordinates where the normal to the quadratic intersects the curve again. d) Add the normal line to your sketch, indicating the point of intersection.. = +3 4 a) Sketch the cubic function including all intersections with the coordinate axes. b) Add on to your sketch the tangent to the curve at the origin. c) Find d) Find the tangent to the curve at the origin. e) Find the other point that the tangent intersects the curve again. 3. =+ a) Show that the point (1,4) lies on the curve b) Find c) Show that the gradient of the tangent to the curve at is is d) Find the equation of the normal to the curve at. e) Find the point where the normal at intersects the curve again 6
Answers EXA tangent normal 1. =3 13 = +. = +14 +13=0 3. = +3 =+5 4. = 1 = +4 EXB 1. =6 9. = 3 3 3. +9 6=0 4. =35+81 5. 4+4=0 EXC 1. (3,). (, 5) 3., 1 EXD 1.,. (, 19) 3., and (,5) 4., and, 5., EXE 1. Normal: + 1=0 Point of intersection:,. Tangent: = 4 Point of intersection: ( 3,1) 3. Normal: +7=0 Point of intersection:6,
WS SEQUENCES 1 ~ General sequences and summation notation Keywords sequence, arithmetic, geometric, converge, diverge, oscillating, periodic, increasing, decreasing, recurrence relation Exercise A 1. Write down the first five terms in each sequence for =1,,3, a) u =n 1 b) u =3n+1 c) u =n d) u =( 3) e) u =0 5n f) u =n+ g) u =5+( 1) h) u = (). Write down the first five terms in each sequence: i) u = a) = +3, = b) =3, =4 c) =5, =3 d) = 1, = e) =, =5 f) = ( +1), =1 g) =, =5 h) = +, =1, =1 Exercise B exam style questions 1. A sequence of positive numbers is defined by a) Find and in surd form. b) Show that =4 = +3, =. A sequence,, is defined by =, =4 7 where is a constant a) Write down an expression for in terms of. b) Find in terms of k, simplifying your answer. Given that =13 c) Find the value of k. 3. A sequence is defined by =3 5, = a) Find and in terms of Given that =9 b) Find. 8
4. A sequence is defined by = +, = a) Find and Given that =6 b) Find the possible values of 5. A sequence is defined by = 3, =1, where is a constant a) Find an expression for in terms of a. b) Show that = 3 3 Given that =7 c) Find the possible values of. 6. A sequence,, is defined by =1, = +5 where is a constant. a) Write down an expression for and in terms of b) Given that =41, find the possible values of 7. The sequence of positive numbers,, is given by =( 3), =1 a) Find, and b) Write down the value of. 8. A sequence is given by = (+ ), =1 where is a constant and 0 a) Show that =1+3+ Given that =1, b) Find the value of. c) Write down the value of. Exercise C 1. UKMT Maths Challenge Question: A sequence,, is defined for positive integer values of by = + Where =0, =, and =1. What is the sum of the first 100 terms of the sequence?. Calculate the first 10 terms of the following sequence =1, =1 and = +. What is the name of this sequence and why is it so famous? 9
3. Research the Mandelbrot Set - this is all done using sequences. http://www.youtube.com/watch?v=g_gbwuyuoos 4. Calculate the first few terms of the following infinity series, where n is an integer. a) b) c) What do they value to they tend towards? Why? Answers 1.a) -1, 1, 3, 5, 7 b) 4,7,10,13,16 c) 1, 4, 9, 16, 5 d) -3, 9, -7, 81, -43 e) 15, 10, 5, 0, -5 f) 3, 8, 14, 4, 4 g) 4, 6, 4, 6, 4 h) 1, 3, 6, 10, 15 i) 1 /, 1 / 6, 1 / 1, 1 / 0, 1 / 30 a), 5, 8, 11, 14 b) 4, 1, 36, 108, 34 c) 3, 13, 63, 313, 1563 d), 3, 8, 63, 3968 e) 5, 1/5, 5, 1/5, 5 f) 1,, 6, 4, 1806 g) 5, -5, 5, -5, 5 h) 1, 1,, 3, 5 3a) 7, 10 4 4a) =4 7 b) =16 35 c) =3 5a) =3 5 =9 0 b) =9 6a) =+ = ++ b) =1, 7a) = 3 c) =5, 8a) =+5 = +5+5 b) =4, 9 9a) =4, =1, =4 b) =4 10b) = c) = 30
WS SEQUENCES ~ Arithmetic sequences Keywords sequence, arithmetic, geometric, converge, diverge, oscillating, periodic, increasing, decreasing, recurrence relation =+( 1) = (+)= [+( 1)] Exercise A Write an expression, in terms of and, for each of these statements: 1. 15 th term is 104. 7 th term is -5 3. 3 rd term is 78 4. 109 th term is 10 5. 11 th term is 0 6. Sum of the first 6 terms is 34 7 Sum of the first 40 terms is 109 8. Sum of the first 17 terms is 80 Exercise B Use either of the formulae at the top to calculate the missing quantity. Note you can use =+( 1) directly =+7 1. =3, =5 Write out,,, the calculate. =7, = Write out,,,,, then calculate Use and appropriate formulae to calculate the missing quantity. 3. =3 4. =? 5. =7 =5 =5 =3 =10 =1 =? =? =47 =31 6. =4 7. =? 8. =8 =7 =3 =? =1 =8 =10 =? =164 = 55 9. =7 10. +4=4 11. =15 =3 +9=38 =1 =? =? = 55 =? 31
Exercise C Exam Questions 1. [C1 Jan 01 Q9] A company offers two salary schemes for a 10-year period, Year 1 to Year 10 inclusive. Scheme 1: Salary in Year 1 is P. Salary increases by (T ) each year, forming an arithmetic sequence. Scheme : Salary in Year 1 is (P + 1800). Salary increases by T each year, forming an arithmetic sequence. (a) Show that the total earned under Salary Scheme 1 for the 10-year period is (10P + 90T ). For the 10-year period, the total earned is the same for both salary schemes. (b) Find the value of T. For this value of T, the salary in Year 10 under Salary Scheme is 9 850. (c) Find the value of P. () (4) (3). [C1 Jun 013 Q7] A company, which is making 00 mobile phones each week, plans to increase its production. The number of mobile phones produced is to be increased by 0 each week from 00 in week 1 to 0 in week, to 40 in week 3 and so on, until it is producing 600 in week N. (a) Find the value of N. The company then plans to continue to make 600 mobile phones each week. () (b) Find the total number of mobile phones that will be made in the first 5 weeks starting from and including week 1. (5) Answers Exercise A 1. a + 14d = 104. a + 6d = -5 3. a + d = 78 4.a + 108d = 10 5.a + 10d = 0 6. (1/)(6) (a + 5d) = 34 7. (1/)(40) (a + 39d) = 109 8. (1/)(17) (a + 16d) = 80 Exercise B 1) 3, 8, 13, 18 4 ) 7, 5, 3, 1, -1, -3 1 3) 48 4) -8 5) 9 6) 510 7) 10 8) -3 9) 8 10) 11, 3 11) 1, - Exercise C 1. (b) T = 400 (c) P = 4 450. (a) N = 1 (b) 7 000 3
WS INTEGRATION Keywords sequence, arithmetic, geometric, converge, diverge, oscillating, periodic, increasing, decreasing, recurrence relation = = 1 +1 +, 1 () () 33
Exercise A Write the following in a form that they can be integrated. 1. = ()= 4. = 5. = 3. f()= 6. f()= 7. =(3+) 8. =( 1) 9. = x(+1) 10. ()= x(x 3) 11. ()=(1+ Exercise B ) *1. ()= Now integrate the expressions in Exercise A (don t forget the constant of integration!). Exercise C (Finding the constant of integration) () 1. A function passes through (1,8) and has gradient function =5, find the equation of the curve.. A function passes through (1,9) and has gradient function =6 +5, find the equation of the curve. 3. A function passes through (,1) and has gradient function 4. Solve the differential =+ given the point (1,3). = Exercise D - Exam Questions 1. [C1 May 01 Q1] Find 6x + + 5 dx, giving each term in its simplest form. x, find the equation of the curve.. [C1 Jan 01 Q7] A curve with equation y = f(x) passes through the point (, 10). Given that find the value of f(1). 3. [C1 May 011 Q6] Given that 34 6x + 3x x 5 f (x) = 3x 3x + 5, can be written in the form 6x p + 3xq, (a) write down the value of p and the value of q. 5 + d y 6x 3 Given that = and that y = 90 when x = 4, dx xx (b) find y in terms of x, simplifying the coefficient of each term. (4) (5) () (5)
Answers EXA 1. =. ()= 3. ()=5 4. = 5. =4 6. ()= 7. =9 + 1+4 8. =8 1 +6 1 9. = + 10. ()= +3 11. ()=+4 1. ()= +3 EXB 1. +.! 3. 5 + 4. + 5.! 6. 7. 3 +6 +4+ 8. 4 +3 + 9. + + 10. + + 11. + + 1. + + EXC 1. =5+3. = +5+ 3. =3 4. = + + EXD 1. +5+. (1)= 3. (a) =,= (b) =4 + 6 35
IMPORTANT INFORMATION Maths and Computer Science Teachers: room email Bill Alexander 3 bill.alexander@candi.ac.uk Ceinwen Hilton 3 ceinwen.hilton@candi.ac.uk Elliot Henchy 18 elliot.henchy@candi.ac.uk Greg Jefferys 18 greg.jefferys@candi.ac.uk Flo Oakley 18 flo.oakley@candi.ac.uk Najm Anwar 14 najm.anwar@candi.ac.uk Dan Nelson 14 daniel.nelson@candi.ac.uk Mike Thiele 14 mike.thiele@candi.ac.uk Nadya de Villiers 14 nadya.devilliers@candi.ac.uk Website Please take some time to visit our website: www.candimaths.uk Homework Work outside lessons should take 4-5 hours. You will be set homework on all the main topics. Complete the set work thoughtfully; it is for your benefit. Remember to check and mark your answers, write any comments or questions to the teacher on your work and submit it on time. You should also review notes, revise for future tests and plan ahead as part of your homework. Support to help you succeed The department runs several support workshops at lunchtimes and after college where you can get extra help. This is also an opportunity for you to get to know other teachers and students. Expectations Students take increasing responsibility for their learning at the Sixth Form. Do join in the classes, volunteer answers and ask questions. Spend time at home organising your equipment, notes and learning. Learning demands, courage, determination and resourcefulness. Use other text books, YouTube, websites, work with other students and talk with teachers. Other Links www.examsolutions.net www.physicsandmathstutor.com www.numberphile.com www.nrichmaths.org www.geogebra.org www.mathscareers.org.uk/ www.supermathsworld.com Most popular site with past exam papers and video solutions. Also clear explanations of topics Exam revision site Short video clips of popular maths Problem solving challenges Geometry, graphs and animations Careers linked to mathematics Multiple choice practice with cartoons 36