Practice Unit tests Use this booklet to help you prepare for all unit tests in Higher Maths.

Transcription

1 Practice Unit tests Use this booklet to help you prepare for all unit tests in Higher Maths. Your formal test will be of a similar standard. Read the description of each assessment standard carefully to make sure you know what you could be tested on. Work through each practice test to check your understanding. There are slight differences and it is important that you do your best to prepare for these. Reasoning problems are identifiable in the booklet with a #.1 or #.1. If you come across something you don t understand or can t remember then do something about it: Look in your course notes for a similar eample Look in your tetbook for notes and worked eamples Look on SCHOLAR for further eplanation and more practice questions Look on the Internet notes and worked eamples (HSN is a useful site ) Ask your teacher or anyone else that is able to help you with Maths 1

2 Contents: Applications Basic skills Page My assessment record Page 4 Practice Tests and Answers: 1.1 Applying algebraic skills to rectilinear shapes Page 5 1. Applying algebraic skills to circles Page 9 1. Applying algebraic skills to sequences Page Applying calculus skills to optimisation and area Page 15 Epressions and Functions Basic skills Page 19 My assessment record Page 0 Practice Tests and Answers: 1.1 Applying algebraic skills to logarithms and eponentials Page 1 1. Applying trigonometric skills to manipulating epressions Page 1. Applying algebraic and trigonometric skills to functions Page Applying geometric skills to vectors Page 4 Relationships and Calculus Basic skills Page 41 My assessment record Page 4 Practice Tests and Answers: 1.1 Applying algebraic skills to solve equations Page 4 1. Applying trigonometric skills to solve equations Page Applying calculus skills of differentiation Page Applying calculus skills of integration Page 49 BANCHORY ACADEMY REVISION GUIDES Page 51

3 Unit title: Applications Assessment Standards 1.1 Applying algebraic skills to rectilinear shapes Making assessment judgements The sub-skills in the Assessment Standard are: finding the equation of a line parallel to, and a line perpendicular to, a given line using m tan to calculate a gradient or angle 1. Applying algebraic skills to circles The sub-skills in the Assessment Standard are: determining and using the equation of a circle using properties of tangency in the solution of a problem 1. Applying algebraic skills to sequences 1.4 Applying calculus skills to optimisation and area The sub-skills in the Assessment Standard are: determining a recurrence relation from given information and using it to calculate a required term finding and interpreting the limit of a sequence, where it eists The sub-skills in the Assessment Standard are: determining the optimal solution for a given problem finding the area between a curve and the -ais finding the area between two curves or a straight line and a curve.1 Interpreting a situation where mathematics can be used and identifying a valid strategy. Eplaining a solution and, where appropriate, relating it to contet Assessment Standard.1 and.1 are transferable across Units and can be attached to any of the sub-skills listed above. #.1 is mostly about choosing an appropriate strategy. This skill is usually required to make a good start on a problem. #. is about how well you answer a question, often with a summary statement at the end of a question.

4 My assessment record: Mathematics: Applications (Higher) Keep this up to date as you go along so that you know if you have any areas that you need to do more work on Assessment standard First attempt Second attempt (if required) 1.1 Applying algebraic skills to rectilinear shapes finding the equation of a line parallel to, and a line perpendicular to, a given line using m tan to calculate a gradient or angle 1. Applying algebraic skills to circles determining and using the equation of a circle using properties of tangency in the solution of a problem 1. Applying algebraic skills to sequences determining a recurrence relation from given information and using it to calculate a required term finding and interpreting the limit of a sequence, where it eists 1.4 Applying calculus skills to optimisation and area determining the optimal solution for a given problem finding the area between a curve and the -ais finding the area between two curves or a straight line and a curve.1 Interpreting a situation where mathematics can be used and identifying a valid strategy Mark(s) Pass/Fail Mark(s) Pass/Fail. Eplaining a solution and, where appropriate, relating it to contet To pass each of the assessment standards 1.1 to 1.4 you need to get at least half of the marks overall or half of the marks for each sub-skill. Assessment standards.1 and. can be gained across the whole course (you have to show that you can apply each skill on separate occasions, i.e. twice for #.1 and twice for #.) so if you don t manage to get it in this unit you need to make sure that you do in either of the other units. 4

5 Unit title: Applications Assessment standard: 1.1 Applying algebraic skills to rectilinear shapes Sub-skills finding the equation of a line parallel to, and a line perpendicular to, a given line using m tan to calculate a gradient or angle Practice test 1 1 Find the equation of the line passing thorough (5, -10), parallel to the line with equation 4 y 8 0. (). ABCD is a rhombus. Diagonal BD has equation y. A has coordinates ( 4, 1). A (-4,1) Find the equation of the diagonal AC (#.1 and 1) A ski slope is categorised by its gradient as shown in the table. Dry slope category Steepness (s) of slope Teaching and general skiing 0 s 0 5 Etreme skiing s > 0 5 (a) What is the gradient of the line shown in the diagram? (1) (b) To which category does the ski slope represented by the line in part (a) belong? Give a reason for your answer. (#. ) 5

6 Practice test 1 A straight line has the equation 6 y 0. Write down the equation of the line parallel to the given line, which passes through the point (, 5). (). ABCD is a rhombus. Y Diagonal BD has equation y and point A has coordinates ( 1, ). y = - Note that the diagram is not to scale. Find the equation of the diagonal AC. A(-1, ) B (#.1 and 1) X D C A ramp is categorised by its gradient as shown in the table. Category Steepness (s) of ramp Safe 0 s 0. Dangerous s > 0. Which category does the ramp in the diagram below belong to? Eplain your answer fully. y 160º O (#.) 6

7 Practice test 1 A straight line has the equation 4 y 0. Write down the equation of the line perpendicular to the given line, which passes through the point (-,1). () ABCD is a kite. Diagonal BD has equation y and point A has coordinates (,5). Note that the diagram is not to scale. y Find the equation of the diagonal AC. A(-,5) B y=+ D C () Calculate the size of the obtuse angle between the line y 4 and the -ais. ( + #.) 4 The ramp on a livestock trailer is categorised by its gradient as shown in the table. Livestock category Steepness (s) of ramp Pigs/horses 0 < s < 0.6 Sheep/cattle 0 < s < 0.5 Which animals would be able to use the ramp in the diagram below? Eplain your answer fully. y º 0 O (1 + #.) 7

8 Unit title: Applications Assessment standard: 1.1 Applying algebraic skills to rectilinear shapes Answers to Practice Tests Practice test 1 1 y = y + 1 = 0 gradient, m = Therefore the steepness of the slope is 0 68 which means the slope is for teaching and general skiing because Practice test 1 y = 6 + y 8 = 0 angle = 0, gradient, m = 0 64 Therefore the steepness of the ramp is 0 64 which dangerous because 0 64 > 0 Practice test 1 4y + 7 = 0 + y 1 = 0 acute angle = 76 0 Obtuse angle = = angle =, gradient, m = 0 44 Therefore the steepness of the ramp is 0 44 which means it is only for sheep/cattle because 0 44 < 0 5 but it is not less than 0 5 8

9 Unit title: Applications Assessment standard: 1. Applying algebraic skills to circles Sub-skills determining and using the equation of a circle using properties of tangency in the solution of a problem Practice test 1 1 The diagram shows two identical circles, C1 and C. C1 has centre the origin and radius 8 units. y C 1 C O The circle, C, passes through the origin. The -ais passes through the centre of C. Find the equation of circle C. (#.1 and 1) Determine if the line y 10 is a tangent to the circle y 8 4y 0 0 ( and #. ) 9

10 Practice test 1 The diagram shows two congruent circles. One circle has centre the origin and diameter 18 units. y O Find the equation of the other circle which passes through the origin and whose centre lies on the -ais. (#.1 and 1) Determine algebraically if the line y 10 is a tangent to the circle ( 4) ( y ) 40 ( + #.) Practice test 1 The diagram shows two congruent circles. One circle has centre the origin and diameter 16 units. y O Find the equation of the other circle which passes through the origin and whose centre lies on the y-ais. () Determine algebraically if the line y = + 9 is a tangent to the circle ( ) ( y ) 8 ( + #.) 10

11 Unit title: Applications Assessment standard: 1. Applying algebraic skills to circles Answers to Practice Tests Practice test 1 1 ( 8) + y = 64 Solving simultaneously gives: = 0 Testing for discriminant gives: b 4ac = 0 Since b 4ac = 0 there are roots are real and equal (or repeated) so therefore the line a tangent to the circle. Practice test 1 ( 9) + y = 81 Solving simultaneously gives: = 0 Testing for discriminant gives: b 4ac = 56 Since b 4ac > 0 there are distinct points of intersection so therefore the line is not a tangent to the circle. Practice test 1 + (y 8) = 64 Solving simultaneously gives: = 0 Testing for discriminant gives: ( + 5) = 0, = 5 or = 5 Since the roots are real and equal there is only one point of intersection so therefore the line is a tangent to the circle. 11

12 Unit title: Applications Assessment standard: 1. Applying algebraic skills to sequences Sub-skills determining a recurrence relation from given information and using it to calculate a required term finding and interpreting the limit of a sequence, where it eists Practice test 1 1 A sequence is defined by the recurrence relation un 1 mun c where m and c are constants. It is known that u 1, u 7, and u Find the recurrence relation described by the sequence and use it to find the value of u 5. (4) On a particular day at 09:00, a doctor injects a first dose of 400 mg of medicine into a patient s bloodstream. The doctor then continues to administer the medicine in this way at 09:00 each day. The doctor knows that at the end of the 4-hour period after an injection, the amount of medicine in the bloodstream will only be 11% of what it was at the start. (a) Set up a recurrence relation which shows the amount of medicine in the bloodstream immediately after an injection. (1) The patient will overdose if the amount of medicine in their bloodstream eceeds 500 mg. (b) In the long term, if a patient continues with this treatment, is there a danger they will overdose? Eplain your answer. ( + #.) Practice Test 1 A Regular Saver Account offers % interest per year. Interest on the account is paid at the end of each year. You open this account with your first deposit of 180 at the start of a particular year and deposit 700 into the account at the start of each subsequent year. un represents the amount of money in the account n years after the account is opened, then u n 1 au n b. State the values of a and b. Given that u 0 is the initial deposit, calculate the value of u. Two sequences are generated by the recurrence relations un1 au n + and v n 1 4avn + with u0 = 1 and v0 = -1. The two sequences approach the same limit as n. Determine the value of a and hence evaluate the limit. () (#.1 and ) 1

13 Practice Test 1 A sequence is defined by the recurrence relation un 1 mun c where m and c are constants. It is known that u, u, u 1. 1 Find the recurrence relation described by the sequence and use it to find the value of u 7 (4) On a particular day at 07:00, a vet injects a first dose of 65 mg of medicine into a dog s bloodstream. The vet then continues to administer the medicine in this way at 07:00 each day. The vet knows that at the end of the 4-hour period after an injection, the amount of medicine in the bloodstream will only be 18% of what it was at the start. (a) Set up a recurrence relation which shows the amount of medicine in the bloodstream immediately after an injection. The dog will overdose if the amount of medicine in its bloodstream eceeds 85 mg. (b) In the long term, if the dog continues with this treatment, is there a danger it will overdose? Eplain your answer. ( +#.) 1

14 Unit title: Applications Assessment standard: 1. Applying algebraic skills to sequences Answers to Practice Tests Practice test 1 1 u mu c 7 m c u 1 mu c 7m c m 4, c 5 u u n 1 4un U 0.11U 400 n1 n L 0.11L or L L < 500 so looks like the patient would not be in danger of overdosing Practice test 1 a = 1 0, b = 700 u 88.6, u ,... 1 u.0 know to equate, e.g. a = 1 (or -0 ) 5 L = 1 = 5 1 ( 1 5 ) = 1 a 1 4a Practice test 1 u = mu1 + c, = m+c u = mu + c, -1 = m + c m =, c = -7 un+1 = un - 7 u7 = -61 un+1 = 0.18un L = 0.18 L + 65 or L L = or < 85 so looks like the dog would not be in danger of overdosing 14

15 Unit title: Applications Assessment standard: 1.4 Applying calculus skills to optimisation and area Sub-skills determining the optimal solution for a given problem finding the area between a curve and the -ais finding the area between two curves or a straight line and a curve Practice test 1 1 A bo with a square base and open top has a surface area of 768 cm. The volume of the bo can be represented by the formula: 1 V ( ) 75 9 Find the value of which maimises the volume of the bo. (5) The curve with equation y ( ) is shown in the diagram. Calculate the shaded area. The line with equation y meets the curve with equation y (#.1 + 4) when = 0 and = 4 as shown in the diagram. Calculate the shaded area. (5) 15

16 Practice test 1 The area of a rectangle can be represented by the formula A() = 7, where > 0. Find the value of which maimises the area of the rectangle. Justify your answer. The curve with equation y = ( ) is shown in the diagram. (5) Calculate the shaded area. (4) The diagram shows graphs with equations y = 10 and y = + 10 y = + y = 10 (a) Which of the following integrals represents the shaded area? 10 A ( 8) d B (8 ) d C ( 8) d D 10 (8 ) d (1) (b) Calculate the shaded area. () 16

17 Practice test 1 A bo with a square base and open top has a surface area of 108 cm. The volume of the bo can be represented by the formula: 1 V ( ) 7. 4 Find the value of which maimises the volume of the bo. (4 + #.) The curve with equation y = (6 ) is shown In the diagram. 6 Calculate the shaded area. 6 (4) The line with equation y 7 and the curve with equation y 4 7 are shown in the diagram. The line and curve meet at the points where = 0 and = 5. Calculate the shaded area. (5) 5 (5) 17

18 Unit title: Applications Assessment standard: 1.4 Applying calculus skills to optimisation and area Answers to Practice Tests Practice Test 1 1 V = 75 1 = 0 stated eplicitly = ±15 use nd derivative or nature table maimum at = 15 7 or 6 square units 4 4 or 10 square units Practice Test = 0 stated eplicitly = ± Uses nature table or nd derivative Ma area when = 8 5 square units or equivalent Area = 1 1 square units, or equivalent Practice Test 1 V '( ) 4 = 6 nature table or nd derivative Maimum when = 6 7 and V () = square units 15 5 or 0 square units

19 Unit title: Epressions and Functions Assessment Standards 1.1 Applying algebraic skills to logarithms and eponentials 1. Applying trigonometric skills to manipulating epressions 1. Applying algebraic and trigonometric skills to functions 1.4 Applying geometric skills to vectors.1 Interpreting a situation where mathematics can be used and identifying a valid strategy. Eplaining a solution and, where appropriate, relating it to contet Making assessment judgements The sub-skills in the Assessment Standard are: simplifying an epression, using the laws of logarithms and eponents solving logarithmic and eponential equations, using the laws of logarithms and eponents The sub-skills in the Assessment Standard are: applying the addition or double angle formulae applying trigonometric identities converting acos bsin to kcos( ) or ksin( ), α in 1 st quadrant k 0 The sub-skills in the Assessment Standard are: identifying and sketching related algebraic functions identifying and sketching related trigonometric functions determining composite and inverse functions including basic knowledge of domain and range The sub-skills in the Assessment Standards are: determining the resultant of vector pathways in three dimensions working with collinearity determining the coordinates of an internal division point of a line evaluating a scalar product given suitable information and determining the angle between two vectors Assessment Standard.1 and.1 are transferable across Units and can be attached to any of the sub-skills listed above. #.1 is mostly about choosing an appropriate strategy. This skill is usually required to make a good start on a problem. #. is about how well you answer a question, often with a summary statement at the end of a question. 19

20 My assessment record: Mathematics: Epressions and Functions (Higher) Keep this up to date as you go along so that you know if you have any areas that you need to do more work on Assessment standard First attempt Second attempt (if required) Mark(s) Pass/Fail Mark(s) Pass/Fail 1.1 Applying algebraic skills to logarithms and eponentials simplifying an epression, using the laws of logarithms and eponents solving logarithmic and eponential equations, using the laws of logarithms and eponents 1. Applying trigonometric skills to manipulating epressions applying the addition or double angle formulae applying trigonometric identities converting acos bsin to kcos( ) or ksin( ) in 1 st quadrant k 0, α 1. Applying algebraic and trigonometric skills to functions identifying and sketching related algebraic functions identifying and sketching related trigonometric functions determining composite and inverse functions including basic knowledge of domain and range 1.4 Applying geometric skills to vectors determining the resultant of vector pathways in three dimensions working with collinearity determining the coordinates of an internal division point of a line evaluating a scalar product given suitable information and determining the angle between two vectors.1 Interpreting a situation where mathematics can be used and identifying a valid strategy. Eplaining a solution and, where appropriate, relating it to contet 0

21 Unit title: Epressions and Functions Assessment standard: 1.1 Applying algebraic skills to logarithms and eponentials Sub-skills simplifying an epression, using the laws of logarithms and eponents solving logarithmic and eponential equations, using the laws of logarithms and eponents Practice test 1 1 (a) Simplify log 5 6a log 5 7b. (b) Epress log 7 4 b log b in the form k log b [] Solve. log 4 ( 1) Practice test [] 1 (a) Simplify log 4 p log 4 q. (b) Epress log a log a in the form k log a [] Eplain why = 0 99 is a solution of the following equation to significant figures: 51 e 0 [ + #.] Practice test 1 (a) Simplifylog54 log56 y. (b) 7 4 Epress log log in the form k log. a a a [1] [] Solve 5 log 1 log 1

22 Unit title: Epressions and Functions Assessment standard: 1.1 Applying algebraic skills to logarithms and eponentials Answers to Practice Tests Practice test 1 1 (a) log 5 6a 7b log 5 4ab 7 log b log 4 b (b) log b 1 4 stated eplicitly 65 Practice test 1 (a) log 4 log p log 4 p q 4 q log 5 (b) log a a 5log a log a e 51 ln 0 51 e ln0 5 1 ln 0 ln e 51 0 OR log e 0 ln Apply logs to both sides OR convert from eponential to logarithmic form Rearrange equation for Practice test 1 (a) log 5 4y (b) log a 5 log 1 log 5 log

23 Unit title: Epressions and Functions Assessment standard: 1. Applying trigonometric skills to manipulating epressions Sub-skills applying the addition or double angle formulae applying trigonometric identities converting acos bsin to kcos( ) or ksin( ), α in 1 st quadrant k 0 Practice test 1 1 Epress sin + cos in the form k sin( a) where k 0 and 0 a 60. Calculate the values of k and a. The diagram below shows two right-angled triangles with measurements as shown. Find the eact value of sin (-y). [4] [] Show that ( + cos ) ( cos ) = 4 sin + 5. [#.1, ]

24 Practice test 1 Epress 4cos 5sin in the form k cos a where k 0 and 0 a 60. (4) The diagram below shows two right-angled triangles. sin y. Find the eact value of 7 y 11 4 (4) Show that 5sin 5sin 5cos 1. ( +.1) Practice test 1 Epress 5sin + 4cos in the form ksin( + a) 0 where k > 0 and 0 o a o 60 o. The diagram below shows two right-angled triangles. Find the eact value of cos( y). [4] [4] Show that sin( + y) sin( y) = sin sin y. [#.1 + ] 4

25 Unit title: Epressions and Functions Assessment standard: 1. Applying trigonometric skills to manipulating epressions Answers to Practice tests Practice test 1 1 k 1 and a = 56.º know to use identities cos cos L.H.S. = Practice test = 9 4cos sin = = 9 4 4sin = 4sin 5 1 k = 41 and a = know to use identities L.H.S. = 4 5sin = 4 5(1 cos 4 5 5cos 5cos RHS 1 ) Practice test 1 k = 41 and a = know to use identities L.H.S. = sin( + y) sin( y) = (sin cos y + sin y cos )(sin cos y sin y cos ) = sin cos y sin y cos = sin (1 sin y) sin y(1 sin ) = sin sin sin y sin y + sin y sin = sin sin y = R.H.S. 5

26 Unit title: Epressions and Functions Assessment standard: 1. Applying algebraic and trigonometric skills to functions Sub-skills identifying and sketching related algebraic functions identifying and sketching related trigonometric functions determining composite and inverse functions including basic knowledge of domain and range Practice test 1 1 Sketch the graph of a cos( ) for 0 and a 0. Show clearly the intercepts on the -ais and the coordinates of the turning points. [4] The diagram shows the graph of y f () with a maimum turning point at (, ) and a minimum turning point at (1, ). Sketch the graph of y f ( ) 1 The diagram below shows the graph of y asin( b) c. 5 4 y [] 1 Write down the values of a, b and c. 1 π/ π [] 6

27 4 The diagram shows the graph of y log ( a). b (-1,1) Determine the values of a and b. (-,0) [] 5 The functions f and g, defined on suitable domains contained within the set of real numbers, f ( ) 5, g ( ) 1. A third function h() is defined as h( ) g( f ( )). (a) Find an epression for h ( ). (b) Eplain why = 0 is not in the domain of h ( ). 6 A function is given by f ( ) 4 6. Find the inverse function f 1 ( ). [] [#.] [] Practice test 1 The diagram shows the graph of y f () with a maimum turning point at (, 4) and a minimum turning point at (0, 0). Sketch the graph of y 1 f ( ). [4] 7

28 The diagram shows the graph of y log b( a). Determine the values of a and b. [] Sketch the graph of y cos( ) for 0. 4 Show clearly the intercepts on the -ais and the coordinates of the turning points. 4 The diagram shows the graph of y = acos(b) for 0. [4] State the values of a and b. [] 5 The functions f and g are defined on suitable domains contained within the set of real numbers, 1 f and g. 16 A third function, h, is defined as h f g. (a) Find an epression for h ( ). (b) Find a suitable domain for h ( ). [] [#.1 + ] 6 A function is given by f ( ). Find the inverse function f 1 ( ). [] 8

29 Practice test 1 Sketch the graph of y asin( ) for0 and a 0, clearly showing the maimum and 6 minimum values and where it cuts the -ais. [] The diagram shows the graph of y f () with a maimum turning point at (, ) and a minimum turning point at (1, ). Sketch the graph of y f ( 1). [] The diagram below shows the graph of y acos( b) c. y -1 π /4 π / Write down the values of a, b and c. [] 9

30 4 The diagram shows the graph of y log ( a). y b 1 (7,1) (4,0) 7 Determine the values of a and b. [] 5 The functions f and g, defined on suitable domains, are given by f ( ), g( ). A third function h() is defined as h( ) g( f ( )). (a) Find an epression for h ( ). (b) Eplain why the largest domain for h ( ) is given by. [] [#.] 6 A function is given by f ( ) 4. Find the inverse function f 1 ( ). [] 0

31 Unit title: Epressions and Functions Assessment standard: 1. Applying algebraic and trigonometric skills to functions Answers to Practice Tests Practice test 1 5 y 1 -intercepts:,0 6 1 π/ π/ π π/ π 1 Ma = a Min = -a 4,a,a 11 and,0 6 4 y (-4,), (-1,-) and (-,-) clearly annotated a =, b =, c = 1 4 a = -, b = 5 (a) h ( ) 5 (b) Because on the set of real numbers, the square root of a negative number cannot be found. 1 6 f ( 6) 1 4 Practice test 1 Horizontal translation ( units to the right) Reflection in the -ais Vertical translation (1 unit up) Each image annotated clearly: (-, 0) to (0, 1) (-, 4) to (1, -) (0, 0) to (, 1) a =, b = 1

32 Amplitude correct: Ma value =, Min value = - 7 Correct turning points: Min. T.P., Ma T.P., Correct -intercepts, 0, Correct shape, i.e. cosine curve Note: y-intercept = a =, b = 5 (a) h 1 16 (b) , 6 domain = { R :, 6} 6 f 1 ) ( Practice test 1 ma a at ( π, a) and min a at (5π, a) -intercepts ( π, 0) and 6 (7π, 0) 6 correct shape, i.e. sine wave Correct horizontal translation (1 units to the right) Correct vertical translation ( units down) Key points annotated clearly: (-, ) to (-1, 1), (1, -) to (, -4) a =, b =, c = 1 4 a =, b =

33 5 (a) g(f()) = (b) Because on the set of real numbers, the square root of a negative number cannot be found. Therefore domain: 0 6 f 1 () = 4

34 Unit title: Epressions and Functions Assessment standard: 1.4 Applying geometric skills to vectors Sub-skills determining the resultant of vector pathways in three dimensions working with collinearity determining the coordinates of an internal division point of a line evaluating a scalar product given suitable information and determining the angle between two vectors Practice test 1 1 An engineer positioning marker flags needs to ensure that the following two conditions are met: The poles are in a straight line. The distance between flag and flag is three times the distance between flag 1 and flag. Relative to suitable aes, the top of each flag can be represented by the points A (5, 8, 1), B (7, 6, ), and C (1,0, 9) respectively. All three poles are vertical. A(5, 8, 1) B(7, 6, ) C(1,0, 9) Flag 1 Flag Flag Has the engineer satisfied the two conditions? You must justify your answer. #.1 #. [4] The points R, S and T lie in a straight line, as shown. S divides RT in the ratio :4. T(6, 16, 7) R( 1,, 0) S Find the coordinates of S. [] 4

35 TPQRS is a pyramid with rectangular base PQRS. T Q The vectors SP and ST are given by: 4 SP = ( 10) ; ST = ( 16) 6 1 P R Epress PT in component form. S [] 4 The diagram shows vectors PR and PQ. P, Q and R have coordinates P(4, 1, ), Q(6,, ) and R(8,, 0). R Find the size of the acute angle QPR. P Q [5] Practice test 1 An engineer positioning concrete posts needs to ensure that the following two conditions are met: The poles are in a straight line. The distance between post 1 and post is twice the distance between post and post Relative to a suitable aes, the posts can be represented by the points A(0,, 10), B(1, 1, 9) and C(,, 7) respectively. 1 (0,, 10) (1, 1, 9) (,, 7) Has the engineer satisfied the two conditions? You must justify your answer. [4 +#.1 + #.] 5

36 The points P, Q and R lie in a straight line, as shown. Q divides PR in the ratio :5. P(, 1, 18) Q R(10, 41, 6) Find the coordinates of Q. [] ABCDEF is a triangular prism as shown. E The vectors AB, AD and AF are given by: F AB = ( 8 ) ; AD = ( 4 ) ; AF = ( 4) 4 1 D C Epress FB in component form. A B [] 4 Points S, T and U have coordinates S(, 0, ), T(7, 1, -5) and U(4,, -). U T S Find the size of the acute angle STU. [5 + #.1] 6

37 Practice test 1 An engineer laying flags needs to check that: they are in a straight line; the distance between Flag and Flag is times the distance between Flag 1 and Flag. Relative to suitable aes, the top-left corner of each flag can be represented by the points A (1,, 0), B (4, 0, ), and C (1, 6, 8) respectively. All three flags point vertically upwards. A(1,,0) B(4,0,) C(1,-6,8) Flag 1 Flag Flag Has the engineer laid the flags correctly? You must justify your answer. [#.1, #., 4] The points R, S and T lie in a straight line, as shown. S divides RT in the ratio :5. Find the coordinates of S. T(15, 6, 4) R( 1,, 0) S [] TPQRS is a pyramid with rectangular base PQRS. The vectors SP, SR, ST are given by: T Q SP i 4jk SR 1i jk P R ST i 7j 7k Epress PT in component form. S [] 7

38 4 Points P, Q and R have coordinates P(5,, 1), Q(7, 4, ) and R(8, 1, 0). R P Find the size of the acute angle QPR. Q [5] 8

39 Unit title: Epressions and Functions Assessment standard: 1.4 Applying geometric skills to vectors Answers to Practice Tests Practice test AB and BC 6 AB therefore BC : AB 1: 6 BC=AB hence vectors are parallel and B is a common point so A, B and C are collinear. Yes, the engineer has placed the flags correctly because A, B and C lie on the same straight line and BC : AB :1 so the distance between flags and is times bigger that the distance between flags 1 and. S (, 8,) 6 PT 6i 6 j 18k or QPR = 6 7 (or 0 64 radians) Practice test 1 1 AB = ( ) and BC = ( 4) = AB therefore BC : AB 1: 1 BC = AB hence the vectors are parallel. B is a common point so the A, B and C are collinear. The posts have not been laid out correctly because although the posts lie in a straight line the distance between post 1 and post is one half of the distance between posts and. Q = (5, 16, 9) i + 1j 9k OR ( 1) 9 4 STU = 5 6 or 0 6 (radians) 9

40 Practice test 9 1 AB = ( ) and BC = ( 6) = AB therefore BC = AB or AB = 1 BC 6 BC = AB hence the vectors are parallel. B is a common point so the A, B and C are collinear. Yes, the engineer has placed the flags correctly because A, B and C lie on the same straight line and BC : AB :1 so the distance between flags and is times bigger that the distance between flags 1 and. S (5, -1, 9) 4 4i + j + 8k OR ( ) 8 4 QPR = 74 8 or 1 06 (radians) 40

41 Unit title: Relationships and Calculus Assessment Standards 1.1 Applying algebraic skills to solve equations 1. Applying trigonometric skills to solve equations 1. Applying calculus skills of differentiation 1.4 Applying calculus skills of integration.1 Interpreting a situation where mathematics can be used and identifying a valid strategy. Eplaining a solution and, where appropriate, relating it to contet Making assessment judgements The sub-skills in the Assessment Standard are: factorising a cubic polynomial epression with unitary coefficient solving cubic polynomial equations with unitary coefficient given the nature of the roots of an equation, use the discriminant to find an unknown The sub-skills in the Assessment Standard are: solve trigonometric equations in degrees and radian measure, involving trigonometric formulae, in a given interval The sub-skills in the Assessment Standard are: differentiating an algebraic function which is, or can be simplified to, an epression in powers of differentiating k sin, k cos determining the equation of a tangent to a curve at a given point by differentiation The sub-skills in the Assessment Standard are: integrating an algebraic function which is, or can be, simplified to an epression of powers of integrating functions of the form f ( ) ( q) n, n 1 integrating functions of the form f ( ) pcos and f ( ) psin calculating definite integrals of polynomial functions with integer limits Assessment Standard.1 and.1 are transferable across Units and can be attached to any of the sub-skills listed above. #.1 is mostly about choosing an appropriate strategy. This skill is usually required to make a good start on a problem. #. is about how well you answer a question, often with a summary statement at the end of a question. 41

42 My assessment record: Mathematics: Epressions and Functions (Higher) Keep this up to date as you go along so that you know if you have any areas that you need to do more work on Assessment standard First attempt Second attempt (if required) 1.1 Applying algebraic skills to solve equations factorising a cubic polynomial epression with unitary coefficient solving cubic polynomial equations with unitary coefficient given the nature of the roots of an equation, use the discriminant to find an unknown 1. Applying trigonometric skills to solve equations solve trigonometric equations in degrees and radian measure, involving trigonometric formulae, in a given interval 1. Applying calculus skills of differentiation differentiating an algebraic function which is, or can be simplified to, an epression in powers of differentiating k sin, k cos determining the equation of a tangent to a curve at a given point by differentiation 1.4 Applying calculus skills of integration integrating an algebraic function which is, or can be, simplified to an epression of powers of integrating functions of the form f ( ) ( q) n, n 1 integrating functions of the form f ( ) pcos, f ( ) psin calculating definite integrals of polynomial functions with integer limits.1 Interpreting a situation where mathematics can be used and identifying a valid strategy Mark(s) Pass/Fail Mark(s) Pass/Fail. Eplaining a solution and, where appropriate, relating it to contet 4

43 Unit title: Relationships and Calculus Assessment standard: 1.1 Applying algebraic skills to solve equations Sub-skills factorising a cubic polynomial epression with unitary coefficient solving cubic polynomial equations with unitary coefficient given the nature of the roots of an equation, use the discriminant to find an unknown Practice test 1 1 A function f is defined by 5 6 (a) (i) Show that 1 is a factor of (ii) Hence factorise f fully. (b) Solve 0 f. f where is a real number. f. Solve the cubic equation f( ) 0 given the following: when f( ) is divided by, the remainder is zero when the graph of y f ( ) 4 f passes through the point, 0 is a factor of The graph of the function k 8 4 f does not touch or cross the -ais. What is the range of values for k? [4] [1] [# ] [# 1, 1] Practice test 1 Solve the cubic equation f( ) 0 given the following: when f( ) is divided by, the remainder is zero when the graph of y f ( ) is drawn, it passes through the point ( 4,0) ( 1) is a factor of f( ). Solve the equation = 0 [# ] [#.1+ 5] 4

44 Unit title: Relationships and Calculus Assessment standard: 1.1 Applying algebraic skills to solve equations Answers to Practice Tests Practice test 1 1 (a) (i) f(-1) = 0 so ( +1) is a factor Either synthetic division or substitution may be used to show that (ii) ( )( )( 1) (b), 1,, 4, k>4 Practice test 1 = -4, = -, = 1 = -1, =, = 44

45 Unit title: Relationships and Calculus Assessment standard: 1. Applying trigonometric skills to solve equations Sub-skills solve trigonometric equations in degrees and radian measure, involving trigonometric formulae, in a given interval Practice test 1 1 Solve cos 1, for Solve 4sin t cos t 0, for 0 t 180 Given that 5sin cos 4 cos 49, solve 5sin cos 6, for 0 < < 180. [] [4] [#.1+] Practice test 1 Solve cos, for Solve sin wcos w 0 for 0 t 180. Given sin 5cos 4 cos( 1.0), solve sin 5cos.5, for [] [4] [#.1+] 45

46 Unit title: Relationships and Calculus Assessment standard: 1. Applying trigonometric skills to solve equations Answers to Practice Tests Practice test 1 1 = 5 0 and t = 7 0, 90 0, = (other value out of range) Practice test 1 = 15 0 and w = , 90 0, = , (solution out of range) 46

47 Unit title: Relationships and Calculus Assessment standard: 1. Applying calculus skills of differentiation Sub-skills Practice test 1 differentiating an algebraic function which is, or can be simplified to, an epression in powers of differentiating k sin, k cos determining the equation of a tangent to a curve at a given point by differentiation 7 5 > 0. 1 Find f, given that f, A bowler throws a cricket ball vertically upwards. The height (in metres) of the ball above the ground, t seconds after it is thrown, can be represented by the formula ht 16t 4t. -1 dh The velocity, v ms, of the ball at time t is given by v. dt [] Find the velocity of the cricket ball three seconds after it is thrown. Eplain what this means in the contet of the question. Differentiate the function f 6sin with respect to. 4 A curve has equation y 7 5. Find the equation of the tangent to the curve at the point where 1. Practice test 1 Find f, given that 6 f, 0. 4 Differentiate cos With respect to. [#. + ] [1] [4] [] [1] A particle moves in a horizontal line. The distance (in metres) of the particle after t seconds can be represented by the formula 4t 4t. d The velocity of the particle at time t is given by v. dt (a) Find the velocity of the particle after three seconds. (b) Eplain your answer in terms of the particle s movement. 4 A curve has equation y 5. Find the equation of the tangent to the curve at the point where. [] [#.] [4] 47

48 Unit title: Relationships and Calculus Assessment standard: 1. Applying calculus skills of differentiation Answers to Practice Tests Practice test f '( ) v = dh = 16 8t when t =, v = -8m/s dt e.g. The cricket ball is has turned and is now falling downwards with an instantaneous speed of 8m/s. d ( 6sin ) 6cos d 4 y = Practice test f '( ) 4 7 d ( cos ) sin d v = d = 8t 4 when t =, v = 0m/s dt e.g. The particle is instantaneously at rest. 4 y =

49 Unit title: Relationships and Calculus Assessment standard: 1.4 Applying calculus skills of integration Sub-skills integrating an algebraic function which is, or can be, simplified to an epression of powers of integrating functions of the form f ( ) ( q) n, n 1 integrating functions of the form f ( ) pcos and f ( ) psin calculating definite integrals of polynomial functions with integer limits Practice test Find 4, 0. d h 5 4 find, 5 Find 4cos d. h. [4] [] [1] 4 Find 8 16d. [] Practice test 1 Find 1 1 ( ) d, 0. 4 [4] h h ( ) ( ), find ( ),. [] Find sin d.. [1] 4 Find 1 ( ) d. [] 49

50 Unit title: Relationships and Calculus Assessment standard: 1.4 Applying calculus skills of integration Answers to Practice Tests Practice test c 1 h( ) ( 5) C 4 sin C 4 5 or 111 Practice test c 1 h( ) ( ) c cosθ + c

51 BANCHORY ACADEMY REVISION GUIDES Block 1 Test 1 Straight Line Equations and Graphs (APP 1.1) Page 5 Sequences (APP 1.) Page 1 You will find more basic questions as well as etended questions to practise from your Higher Maths tetbook Chapters 1 and 5. The Mied Question eercises at the end of each chapter are recommended revision for etension tests. Block Test Functions and Graphs of Functions (E&F 1.) Page 6 4 Trigonometry - Radian measure and solving equations (R&C 1.) Page 45 5 Differentiation (R&C 1. note chain rule to be covered later) Page 47 You will find more basic questions as well as etended questions to practise from your Higher Maths tetbook Chapters,, 4 and 6. Note that from Chapter 6 we will only have covered some of the eercises. The Mied Question eercises at the end of each chapter are recommended revision for etension tests. Block Test 6 Trigonometry - Addition formulae and Wave Function (E&F 1.) Page 7 Quadratic and Polynomial Functions (R&C 1.1) Page 4 Integration (R&C 1.4 note reverse of chain rule to be covered later) Page 49 8 Note: page 49 omit Practice test 1 question, Practice Test questions and 4 9 The Circle (App 1.) Page 9 You will find more basic questions as well as etended questions to practise from your Higher Maths tetbook Chapters 11, 16, 7, 8, 9 and 1. Note that from Chapter 9 we will only have covered some of the eercises. The Mied Question eercises at the end of each chapter are recommended revision for etension tests. Block 4 Test 10 Eponential and Logarithmic Functions (E&F 1.1) Page 1 Further Calculus - Chain Rule and applications (R&C 1., 1.4 and App 1.4) Page Note: page 49 only do Practice test 1 question, Practice Test questions and 4 Page 15 1 Vectors (E&F 1.4) Page 4 You will find more basic questions as well as etended questions to practise from your Higher Maths tetbook Chapters 15, 6, 9, 14 and 1. Note that from Chapter 6 and 9 some of the eercises were covered earlier in the Blocks and. The Mied Question eercises at the end of each chapter are recommended revision for etension tests. 51