Coombe Sixth Form Compulsory Summer Work

Transcription

1 The Bridge to A level Coombe Sixth Form Compulsory Summer Work This pack contains a programme of activities and resources to prepare you to start A-level in Maths in September. It is aimed to be completed throughout the remainder of the Summer term and over the Summer Holidays to ensure you are ready to start your A Level Maths course in September. You MUST have this pack completed before your first A Level maths lesson The resources include: 1. Links with activities on six websites where you can research the topics you will be exploring in your sixth form courses and get a flavour of mathematics beyond GCSE key pre-knowledge topics that will help you to be successful in your course. The topics covered are a mixture of GCSE topics and topics which extend GCSE but which will be very useful on your A level course. 3. Past exam questions that will test your key knowledge of these 10 topics, with hints if you need a quick reminder. 4. Answers for the above questions. 5. Suggested therapies to help you with those topics with which you are having difficulty. 1

2 Websites NRich Mathwire The History of Maths Wikipedia The History of Maths Youtube video Decision Maths Videos Statistics v Mechanics student discussion 10 key Topics Topic Your Mark 1 Solving quadratic equations /10 2 Changing the subject /10 3 Simultaneous equations /10 4 Surds /10 5 Indices /10 6 Properties of Lines /10 7 Sketching curves /10 8 Transformation of functions /10 9 Trigonometric ratios /10 10 Sine / Cosine Rule /10 Total /100 2

3 Hints for Diagnostic Questions. Use these hints to help you complete the questions in all ten sections. There are answers provided for you to check your work. The questions are given after this list If you need more help or guidance use the resources listed in the Therapy section to answer the questions as fully as possible. 1 Hints Solving quadratic equations Factorise or use the quadratic equation to solve for x Recognise a quadratic in x 2 (replace y with x 2 and solve for x) Complete the square and analyse to find minimum and the value of x at which this occurs. 2 Hints Changing the subject Multiply by (C + 4) to get rid of fraction, re-arrange and then factorise for C 3 Hints Simultaneous equations and 2 Solve by elimination or substitution. Choose answers as fractions in place of recurring decimals. Remember to find both x and y. Solve substituting the linear into the quadratic. Remember there will be two pairs of solutions. 3

4 ... 4 Hints Surds (ii) To rationalise a denominator of the form (x + y ) multiply top and bottom by (x y ) (i) Always look for square number factors 5 Hints Indices Remember x a is 1 x a x½ is x x 0 = 1 for all x 6 Hints Properties of Lines (i) Find gradients m1 and m2 for lines AB and BC and show m1m2 = 1. (ii) Use Pythagoras theorem with coordinates of AC: L 2 = (x2 x1) 2 + (y2 y1) 2 (i) Use (y y1) = m(x x1) or equivalent (ii) Use mid-point = (½[x1 + x2], ½[y1 + y2]) 7 Hints Sketching curves Cubic with roots a, b and c is f(x) = (x a)(x b)(x c) (with coefficient of x 3 = 1). Use this to find x-axis intercepts (when y = 0) and y-axis intercept (when x = 0). The sketch is a transformation of negative x 2 graph. Use a table of values with x from 0 to 5 4

5 ... 8 Hints Transformation of functions, 2 and 3 f(x) is translated a units to the right, or ( a ), is f(x a) 0 Also: f(x) + a f(x) a f(x + a) f(x - a) 9 Hints Trigonometric ratios (i) (ii) Use Pythagoras theorem in the right angled triangle. Use SOH CAH TOA with the ladder as the hypotenuse. Sketch a right angled triangle with adjacent and hypotenuse 1 and 3, use Pythagoras to calculate the opposite. When a question asks for Exact values, give your answer in surd form (NOT decimal form). Consider key points when x = 0, 90, 180, 270, 360 and maximum and minimum values. 10 Hints Sine / Cosine Rule (i) (ii) Use the cosine rule for two lengths and included angle Use the area of triangle ABC = ½ab sin(c) Use the cosine rule for two lengths and included angle to find CB then find total length of all three sides Calculate the total length of the course for this race. 5

6 Diagnostic Questions You may need extra paper to do your working out which you should keep with this booklet. 1 Solving quadratic equations Solve x 2 + 6x + 8 = 0 Solve the equation y 2 7y + 12 = 0 Hence solve the equation x 4 7x = 0 (4) (i) Express x 2 6x + 2 in the form (x a) 2 b (ii) State the coordinates of the minimum value on the graph of y = x 2 6x + 2 (1) Total / 10 2 Changing the subject Make v the subject of the formula E = 1 2 mv2 Make r the subject of the formula V = 4 3 Π r3 Make C the subject of the formula P = C C+4 (4) Total / 10 6

7 3 Simultaneous equations Find the coordinates of the point of intersection of the lines y = 3x + 1 and x + 3y = 6 Find the coordinates of the point of intersection of the lines 5x + 2y = 20 and y = 5 x Solve the simultaneous equations x 2 + y 2 = 5 y = 3x + 1 (4) Total / 10 4 Surds (i) Simplify (3 + 2 )(3 2 ) (ii) Express in the form a + b 2 where a and b are rational (i) Simplify Express your answer in the form a b where a and b are integers and b is as small as possible. (ii) Express in the form p + q 3 where p and q are rational 7 Total / 10

8 5 Indices Simplify the following (i) a 0 (ii) a 6 a 2 (iii) (9a 6 b 2 ) 0.5 (1) (1) (i) Find the value of ( 1 25 ) 0.5 (ii) Simplify (2x 2 y 3 z) 5 4y 2 z Total / 10 6 Properties of Lines A (0, 2), B (7, 9) and C (6, 10) are three points. (i) (ii) Show that AB and BC are perpendicular Find the length of AC (i) Find, in the form y = mx + c, the equation of the line passing through A (3, 7) and B (5, 1). (ii) Show that the midpoint of AB lies on the line x + 2y = 10 Total / 10 8

9 7 Sketching curves In the cubic polynomial f(x), the coefficient of x 3 is 1. The roots of f(x) = 0 are 1, 2 and 5. Sketch the graph of y = f(x) Sketch the graph of y = 9 x 2 The graph below shows the graph of y = 1 x On the same axes plot the graph of y = x 2 5x + 5 for 0 x 5 (4) Total / 10 9

10 8 Transformation of functions The curve y = x 2 4 is translated by ( 2 0 ) Write down an equation for the translated curve. You need not simplify your answer. This diagram shows graphs A and B. (i) (ii) State the transformation which maps graph A onto graph B The equation of graph A is y = f(x). Which one of the following is the equation of graph B? y = f(x) + 2 y = f(x) 2 y = f(x + 2) y = f(x 2) y = 2f(x) y = f(x + 3) y = f(x 3) y = 3f(x) (i) Describe the transformation which maps the curve y = x 2 onto the curve y = (x + 4) 2 (ii) Sketch the graph of y = x 2 4 Total / 10 10

11 9 Trigonometric ratios Sidney places the foot of his ladder on horizontal ground and the top against a vertical wall. The ladder is 16 feet long. The foot of the ladder is 4 feet from the base of the wall. (i) (ii) Work out how high up the wall the ladder reaches. Give your answer to 3 significant figures. Work out the angle the base of the ladder makes with the ground. Give your answer to 3 significant figures Given that cos Ɵ = 1 3 and Ɵ is acute, find the exact value of tan Ɵ Sketch the graph of y = cos x for 0 x Total / 10

12 10 Sine / Cosine Rule For triangle ABC, calculate (i) (ii) the length of BC the area of triangle ABC The course for a yacht race is a triangle as shown in the diagram below. The yachts start at A, then travel to B, then to C and finally back to A. Calculate the total length of the course for this race. (4) Total / 10 12

13 Answers for Diagnostic Questions. Use these to mark your answers after using hints and therapy if necessary. Don t worry too much if you are unsure how to mark partially correct answers estimate if not obvious. 1 Solving quadratic equations x = 2 or 4 y = 3 or 4 x = ± 3 or ± 2 (i) (x 3) 2 7 (ii) (3, 7) (1) 2 Changing the subject (4) C = V = ± 2E m 3 r = 3V 4π 4 p 1 alternative: 4 1 p (4) 3 Simultaneous equations ( 3 10, ) ( 10 3, 5 3 ) (x, y) = ( 2 5, 11 ) or ( 1, 2) 5 13

14 (4) 4 Surds (i) 7 (ii) (i) (ii) Indices Simplify the following (i) 1 (ii) a 8 (iii) 3 1 a 3 b 1 (alternatively 1 3a 3 b ) (1) (1) (i) ±5 (ii) 8x 10 y 13 z 4 6 Properties of Lines (i) Gradients 2 ½ = 1 (ii) AC = 10 (i) y = 4x + 19 (ii) Midpoint (4, 3) satisfies x + 2y = 10 Show that the midpoint of AB lies on the line 7 Sketching curves f(x) = (x + 1)(x 2)(x 5) y = 9 x

15 -3 3 The graph below shows the graph of y = 1 x On the same axes plot the graph of y = x 2 5x + 5 for 0 x 5 8 Transformation of functions y = x 2 4 becomes (x 2) 2 4 (i) ( 2 ) or a translation 2 to the right, 2 in the direction of the positive x axis. 0 (ii) y = f(x - 2) [Note : f(x) + 2 f(x) 2 f(x + 2) ] (i) Translation ( 4 0 ) or a translation 4 to the right (4) 15

16 Trigonometric ratios (i) 240 = 15.5 (3sf) (ii) x = cos -1 ( 4 ) x = 75.5 (3sf) 16 tan Ɵ = 8 y = cos x for 0 x Sine / Cosine Rule (i) (ii) BC = = 8.96 (2dp) Area of triangle ABC = square units Total length of the course = = 1034m. (4) 16

17 The Bridge to A level Therapy 17

18 Therapy for Topics All therapy references are referenced to the MathsWatch website The MathsWatch website Use the Centre ID and log in details that your school has provided you with. The MathsWatch clips in this document will take you directly to the pages you will need. For more advance topics, please use the provided web links to the website via youtube. 18

19 Therapy for Topic 1 Quadratic equations 1. Recognise the shape of quadratic graph transformations. 2. Find the minimum point of a quadratic ( 3. Solve quadratic equations by factorising. 4. Solving quadratic equations using the quadratic formula. 5. Solving quadratics by completing the square. MathsWatch clips (in order) 116, 140a, 140b, 161a, 161b, 161c Therapy for Topic 2 Algebra 1. Re-arranging formulae. 2. Re-arrange a formula where the subject appears more than once. MathsWatch clips (in order) 107, 164 Therapy for Topic 3 Simultaneous equations 1. Simultaneous linear equations. 2. Simultaneous non-linear equations. 3. Solving simultaneous equations graphically. MathsWatch clips (in order) 142a, 142b, 165, 115 Therapy for Topic 4 Surds 1. Simplify a surd. 2. Operate with surds. 3. Rationalise a surd. MathsWatch clips (in order) 157, 158 Therapy for Topic 5 Indices 1. Evaluate positive, negative and fractional indices MathsWatch clips (in order) 156 Therapy for Topic 6 Properties of lines 1. Find the equation of a straight line. 2. Understand and use y =mx + c. 3. Gradients of parallel and perpendicular lines. MathsWatch clips (in order) 114, 143, 166 Therapy for Topic 7 Sketching curves 19

20 1. Recognise the shapes of functions. 2. Graphs of cubic and reciprocal functions. 3. Graphs of exponential functions. 4. Sketching polynomials ( MathsWatch clips (in order) 146, 145, 170 Therapy for Topic 8 Transformation of functions 1.Recognise trigonometric graph transformations. Stretches ( ). Translations ( ). Sketches graphs after they are transformed ( ). ( ). 2.Interpret values from a transformed trigonometric graph ( ( ( MathsWatch clips (in order) 167, 168, 169 Therapy for Topic 9 Trigonometric ratios 1. Calculate an unknown angle or side using SOH CAH TOA 2. Apply Pythagoras Theorem and SOH CAH TOA in 3D contexts 3. Interpret and recognise values from a trigonometric graph. MathsWatch clips (in order) 147a, 147b, 147c, 174, 175, 169 Therapy for Topic 10 Sine / Cosine Rule 1. Apply the sine and cosine rules to calculate a length or an angle 2. Calculate the area of a non-right angles triangle and of a segment. MathsWatch clips (in order) 173, 176, 178 END 20