Back to start Quit Higher Maths www.maths4scotland.co.uk Click to start Strategies.

Back to start Quit Higher Maths www.maths4scotland.co.uk Click to start Strategies

Back to start Quit Maths4Scotland Higher Select topic – from which the questions are taken. The Straight Line Functions Differentiation Sequences Integration Polynomials Quadratic Theory Circle Compound Angles Vectors Exponential & Log Function Wave Function

Back to start Quit Higher Maths Strategies www.maths4scotland.co.uk Click to start The Straight Line Maths4Scotland The Straight Line Higher

Back to start Quit The Straight Line The following questions are on Non-calculator questions will be indicated Click to continue You will need a pencil, paper, ruler and rubber. Maths4Scotland The Straight Line Higher

Back to start Quit Hint PreviousNext Find the equation of the line which passes through the point (-1, 3) and is perpendicular to the line with equation Find gradient of given line: Find gradient of perpendicular: Find equation: Maths4Scotland The Straight Line Higher

Back to start Quit Hint PreviousNext Find the equation of the straight line which is parallel to the line with equation and which passes through the point (2, –1). Find gradient of given line: Gradient of parallel line is same: Find equation: Maths4Scotland The Straight Line Higher

Back to start Quit Hint PreviousNext Table of exact values Find gradient of the line: Use table of exact values Use Find the size of the angle a° that the line joining the points A(0, -1) and B(3  3, 2) makes with the positive direction of the x-axis. Maths4Scotland The Straight Line Higher

Back to start Quit Hint PreviousNext A and B are the points (–3, –1) and (5, 5). Find the equation of a)the line AB. b)the perpendicular bisector of AB Find gradient of the AB: Find mid-point of AB Find equation of AB Gradient of AB (perp): Use gradient and mid-point to obtain perpendicular bisector AB Maths4Scotland The Straight Line Higher

Back to start Quit Hint PreviousNext Table of exact values The line AB makes an angle of radians with the y-axis, as shown in the diagram. Find the exact value of the gradient of AB. Find angle between AB and x-axis: Use table of exact values Use (x and y axes are perpendicular) Maths4Scotland The Straight Line Higher

Back to start Quit Hint PreviousNext A triangle ABC has vertices A(4, 3), B(6, 1) and C(–2, –3) as shown in the diagram. Find the equation of AM, the median from A. Find mid-point of BC: Find equation of median AM Find gradient of median AM Maths4Scotland The Straight Line Higher

Back to start Quit Hint PreviousNext P(–4, 5), Q(–2, –2) and R(4, 1) are the vertices of triangle PQR as shown in the diagram. Find the equation of PS, the altitude from P. Find gradient of QR: Find equation of altitude PS Find gradient of PS (perpendicular to QR) Maths4Scotland The Straight Line Higher

Back to start Quit Hint PreviousNext The lines and make angles of a  and b  with the positive direction of the x-axis, as shown in the diagram. a)Find the values of a and b b)Hence find the acute angle between the two given lines. Find supplement of b Find gradient of Find a° Find b° Angle between two lines Use angle sum triangle = 180° 72° Maths4Scotland The Straight Line Higher

Back to start Quit Hint PreviousNext Triangle ABC has vertices A(–1, 6), B(–3, –2) and C(5, 2) Find: a) the equation of the line p, the median from C of triangle ABC. b) the equation of the line q, the perpendicular bisector of BC. c) the co-ordinates of the point of intersection of the lines p and q. Find mid-point of AB Find equation of p Find gradient of p (-2, 2) Find mid-point of BC (1, 0) Find gradient of BC Find gradient of q Find equation of q Solve p and q simultaneously for intersection (0, 2) Maths4Scotland The Straight Line Higher

Back to start Quit Hint PreviousNext Triangle ABC has vertices A(2, 2), B(12, 2) and C(8, 6). a) Write down the equation of l 1, the perpendicular bisector of AB b) Find the equation of l 2, the perpendicular bisector of AC. c) Find the point of intersection of lines l 1 and l 2 d) Hence find the equation of the circle passing through A, B and C. Mid-point AB Find mid-point AC (5, 4) Find gradient of AC Equ. of perp. bisector AC Gradient AC perp. Point of intersection (7, 1) This is the centre of circle Find radius (intersection to A) Equation of circle: Perpendicular bisector AB Maths4Scotland The Straight Line Higher

Back to start Quit Hint PreviousNext A triangle ABC has vertices A(–4, 1), B(12,3) and C(7, –7). a) Find the equation of the median CM. b) Find the equation of the altitude AD. c) Find the co-ordinates of the point of intersection of CM and AD Mid-point AB Equation of median CM Gradient of perpendicular AD Gradient BC Equation of AD Gradient CM (median) Solve simultaneously for point of intersection (6, -4) Maths4Scotland The Straight Line Higher

Back to start Quit Hint PreviousNext A triangle ABC has vertices A(–3, –3), B(–1, 1) and C(7,–3). a) Show that the triangle ABC is right angled at B. b) The medians AD and BE intersect at M. i) Find the equations of AD and BE. ii) Hence find the co-ordinates of M. Gradient AB Product of gradients Gradient of median AD Mid-point BC Equation AD Gradient BC Solve simultaneously for M, point of intersection Hence AB is perpendicular to BC, so B = 90° Gradient of median BE Mid-point AC Equation AD Maths4Scotland The Straight Line Higher

Back to start Quit Previous You have completed all 12 questions in this presentation Back to start Maths4Scotland The Straight Line Higher

Back to start Quit Graphs & Functions Strategies Higher Maths Click to start

Back to start Quit Graphs & Functons The following questions are on Non-calculator questions will be indicated Click to continue You will need a pencil, paper, ruler and rubber. Maths4Scotland Graphs & Functions Higher

Back to start Quit Hint Previous Next The diagram shows the graph of a function f. f has a minimum turning point at (0, -3) and a point of inflexion at (-4, 2). a) sketch the graph of y = f(-x). b) On the same diagram, sketch the graph of y = 2f(-x) a) Reflect across the y axis b) Now scale by 2 in the y direction Maths4Scotland Graphs & Functions Higher

Back to start Quit Hint Previous Next The diagram shows a sketch of part of the graph of a trigonometric function whose equation is of the form Determine the values of a, b and c a is the amplitude: a = 4 b is the number of waves in 2  b = 2 c is where the wave is centred vertically c = 1 2a 1 in  2 in 2  1 Maths4Scotland Graphs & Functions Higher

Back to start Quit Hint Previous Next Functions and are defined on suitable domains. a)Find an expression for h(x) where h(x) = f(g(x)). b)Write down any restrictions on the domain of h. a) b) Maths4Scotland Graphs & Functions Higher

Back to start Quit Hint Previous Next a) Express in the form b) On the same diagram sketch i)the graph of ii)the graph of c) Find the range of values of x for which is positive a) c) Solve: 10 - f(x) is positive for -1 < x < 5 b) (2, 1) (2, -1) (2, 9) 5 y=f(x) y= -f(x) y= 10 - f(x) -5 Maths4Scotland Graphs & Functions Higher

Back to start Quit Hint Previous Next The graph of a function f intersects the x-axis at (–a, 0) and (e, 0) as shown. There is a point of inflexion at (0, b) and a maximum turning point at (c, d). Sketch the graph of the derived function m is + m is - f(x) Maths4Scotland Graphs & Functions Higher

Back to start Quit Hint Previous Next Functions f and g are defined on suitable domains by and a)Find expressions for: i) ii) b)Solve a) b) Maths4Scotland Graphs & Functions Higher

Back to start Quit Hint Previous Next The diagram shows the graphs of two quadratic functions Both graphs have a minimum turning point at (3, 2). Sketch the graph of and on the same diagram sketch the graph of y=g(x) y=f(x) Maths4Scotland Graphs & Functions Higher

Back to start Quit Hint Previous Next Functions are defined on a suitable set of real numbers. a) Find expressions for i) ii) b) i)Show that ii)Find a similar expression for and hence solve the equation a) b) Now use exact values Repeat for ii) equation reduces to Maths4Scotland Graphs & Functions Higher

Back to start Quit Hint Previous Next A sketch of the graph of y = f(x) where is shown. The graph has a maximum at A and a minimum at B(3, 0) a) Find the co-ordinates of the turning point at A. b) Hence, sketch the graph of Indicate the co-ordinates of the turning points. There is no need to calculate the co-ordinates of the points of intersection with the axes. c) Write down the range of values of k for which g(x) = k has 3 real roots. a) Differentiate for SP, f(x) = 0 when x = 1t.p. at A is: b) Graph is moved 2 units to the left, and 4 units up t.p.’s are: c) For 3 real roots, line y = k has to cut graph at 3 points from the graph, k  4 Maths4Scotland Graphs & Functions Higher

Back to start Quit Hint Previous Next a) Find b) If find in its simplest form. a) b) Maths4Scotland Graphs & Functions Higher

Back to start Quit Hint Previous Next Part of the graph of is shown in the diagram. On separate diagrams sketch the graph of a)b) Indicate on each graph the images of O, A, B, C, and D. a) b) graph moves to the left 1 unit graph is reflected in the x axis graph is then scaled 2 units in the y direction Maths4Scotland Graphs & Functions Higher

Back to start Quit Hint Previous Next Functions f and g are defined on the set of real numbers by a) Find formulae for i) ii) b) The function h is defined by Show that and sketch the graph of h. c) Find the area enclosed between this graph and the x-axis. a) b) c) Graph cuts x axis at 0 and 1Now evaluate Area Maths4Scotland Graphs & Functions Higher

Back to start Quit Hint Previous Next The functions f and g are defined on a suitable domain by a) Find an expression for b) Factorise a) Difference of 2 squares Simplify b) Maths4Scotland Graphs & Functions Higher

Back to start Quit You have completed all 13 questions in this section Previous Maths4Scotland Graphs & Functions Higher

Back to start Quit Higher Maths Strategies www.maths4scotland.co.uk Click to start Sequences Maths4Scotland Sequences Higher

Back to start Quit Sequences The following questions are on Non-calculator questions will be indicated Click to continue You will need a pencil, paper, ruler and rubber. Maths4Scotland Sequences Higher

Back to start Quit Hint PreviousNext Put u 1 into recurrence relation Solve simultaneously: A recurrence relation is defined by where -1 < p < -1 and u 0 = 12 a)If u 1 = 15 and u 2 = 16 find the values of p and q b)Find the limit of this recurrence relation as n   Put u 2 into recurrence relation (2) – (1) Hence State limit condition -1 < p < 1, so a limit L exists Use formula Limit = 16½ Maths4Scotland Sequences Higher

Back to start Quit Hint PreviousNext Construct a recurrence relation State limit condition -1 < 0.8 < 1, so a limit L exists Use formula Limit = 2.5 metres A man decides to plant a number of fast-growing trees as a boundary between his property and the property of his neighbour. He has been warned however by the local garden centre, that during any year, the trees are expected to increase in height by 0.5 metres. In response to this warning, he decides to trim 20% off the height of the trees at the start of any year. (a)If he adopts the “20% pruning policy”, to what height will he expect the trees to grow in the long run. (b)His neighbour is concerned that the trees are growing at an alarming rate and wants assurance that the trees will grow no taller than 2 metres. What is the minimum percentage that the trees will need to be trimmed each year so as to meet this condition. u n = height at the start of year Use formula again Minimum prune = 25% m = 0.75 Maths4Scotland Sequences Higher

Back to start Quit Hint PreviousNext Construct a recurrence relation On the first day of March, a bank loans a man £2500 at a fixed rate of interest of 1.5% per month. This interest is added on the last day of each month and is calculated on the amount due on the first day of the month. He agrees to make repayments on the first day of each subsequent month. Each repayment is £300 except for the smaller final amount which will pay off the loan. a) The amount that he owes at the start of each month is taken to be the amount still owing just after the monthly repayment has been made. Let u n and u n+1 and represent the amounts that he owes at the starts of two successive months. Write down a recurrence relation involving u n and u n+1 b) Find the date and amount of the final payment. u 0 = 2500 Calculate each term in the recurrence relation 1 Mar u 0 = 2500.00 1 Apr u 1 = 2237.50 1 May u 2 = 1971.06 1 Jun u 3 = 1700.62 1 Jul u 4 = 1426.14 1 Aug u 5 = 1147.53 1 Sept u 6 = 864.74 1 Oct u 7 = 577.71 1 Nov u 8 = 286.38 1 Dec Final payment £290.68 Maths4Scotland Sequences Higher

Back to start Quit Hint PreviousNext Equate the two limits Cross multiply Sequence 1 Since limit exists a  1, so Use formula for each sequence Limit = 25 Two sequences are generated by the recurrence relations and The two sequences approach the same limit as n  . Determine the value of a and evaluate the limit. Sequence 2 Simplify Solve Deduction Maths4Scotland Sequences Higher

Back to start Quit Hint PreviousNext Equate the two limits Cross multiply Sequence 1 Use formula for each sequence Sequence 2 Rearrange Two sequences are defined by the recurrence relations If both sequences have the same limit, express p in terms of q. Maths4Scotland Sequences Higher

Back to start Quit Hint PreviousNext Sequence 2 Requirement for a limit List terms of 1 st sequence Two sequences are defined by these recurrence relations a)Explain why only one of these sequences approaches a limit as n   b)Find algebraically the exact value of the limit. c)For the other sequence find i)the smallest value of n for which the n th term exceeds 1000, and ii)the value of that term. First sequence has no limit since 3 is not between –1 and 1 2 nd sequence has a limit since –1 < 0.3 < 1 u 0 = 1 u 1 = 2.6 u 2 = 7.4 u 3 = 21.8 u 4 = 65 u 5 = 194.6 u 6 = 583.4 u 7 = 1749.8 Smallest value of n is 8; value of 8 th term = 1749.8 Maths4Scotland Sequences Higher

Back to start Quit Previous You have completed all 6 questions in this presentation Maths4Scotland Sequences Higher

Back to start Quit Differentiation Higher Mathematics www.maths4scotland.co.uk Next Maths4Scotland Differentiation Higher

Back to start Quit Back Next Differentiate Straight line form Hint Maths4Scotland Differentiation Higher

Back to start Quit Back Next Differentiate Hint Maths4Scotland Differentiation Higher

Back to start Quit Back Next Differentiate Chain Rule Simplify Hint Maths4Scotland Differentiation Higher

Back to start Quit Back Next Differentiate multiply out Differentiate Hint Maths4Scotland Differentiation Higher

Back to start Quit Back Next Differentiate Straight line form Hint Maths4Scotland Differentiation Higher

Back to start Quit Back Next Differentiate Straight line form Multiply out Differentiate Hint Maths4Scotland Differentiation Higher

Back to start Quit Back Next Differentiate Hint Maths4Scotland Differentiation Higher

Back to start Quit Back Next Differentiate Multiply out Differentiate Hint Maths4Scotland Differentiation Higher

Back to start Quit Quadratic Theory Strategies Higher Maths Click to start Maths4Scotland Quadratic Theory Higher

Back to start Quit Quadratic Theory The following questions are on Non-calculator questions will be indicated Click to continue You will need a pencil, paper, ruler and rubber. Maths4Scotland Quadratic Theory Higher

Back to start Quit Hint Previous Next Show that the line with equation does not intersect the parabola with equation Put two equations equal Use discriminant Show discriminant < 0 No real roots Maths4Scotland Quadratic Theory Higher

Back to start Quit Hint Previous Next a)Write in the form b)Hence or otherwise sketch the graph of a) b) This is graph ofmoved 3 places to left and 2 units up. minimum t.p. at (-3, 2)y-intercept at (0, 11) Maths4Scotland Quadratic Theory Higher

Back to start Quit Hint Previous Next Show that the equation has real roots for all integer values of k Use discriminant Consider when this is greater than or equal to zero Sketch graph cuts x axis at Hence equation has real roots for all integer k Maths4Scotland Quadratic Theory Higher

Back to start Quit Hint Previous Next The diagram shows a sketch of a parabola passing through (–1, 0), (0, p) and (p, 0). a) Show that the equation of the parabola is b) For what value of p will the line be a tangent to this curve? a) Use point (0, p) to find k b) Simultaneous equations Discriminant = 0 for tangency Maths4Scotland Quadratic Theory Higher

Back to start Quit Given, express in the form Hint Previous Next Maths4Scotland Quadratic Theory Higher

Back to start Quit Hint Previous Next For what value of k does the equation have equal roots? Discriminant For equal roots discriminant = 0 Maths4Scotland Quadratic Theory Higher

Back to start Quit You have completed all 6 questions in this section Previous Maths4Scotland Quadratic Theory Higher

Back to start Quit Integration Higher Mathematics www.maths4scotland.co.uk Next Maths4Scotland Integration Higher

Back to start Quit Back Next Integrate Integrate term by term simplify Hint Maths4Scotland Integration Higher

Back to start Quit Back Next Find Hint Maths4Scotland Integration Higher

Back to start Quit Back Next Integrate Multiply out brackets Integrate term by term simplify Hint Maths4Scotland Integration Higher

Back to start Quit Back Next Integrate Standard Integral (from Chain Rule) Hint Maths4Scotland Integration Higher

Back to start Quit Back Next Find p, given Hint Maths4Scotland Integration Higher

Back to start Quit Back Next Evaluate Straight line form Hint Maths4Scotland Integration Higher

Back to start Quit Back Next Find Use standard Integral (from chain rule) Hint Maths4Scotland Integration Higher

Back to start Quit Back Next Find Integrate term by term Hint Maths4Scotland Integration Higher

Back to start Quit Back Next Integrate Straight line form Hint Maths4Scotland Integration Higher

Back to start Quit Back Next Integrate Split into separate fractions Hint Maths4Scotland Integration Higher

Back to start Quit Back Next Find Use standard Integral (from chain rule) Hint Maths4Scotland Integration Higher

Back to start Quit Back Next Given the acceleration a is: If it starts at rest, find an expression for the velocity v where Starts at rest, so v = 0, when t = 0 Hint Maths4Scotland Integration Higher

Back to start Quit Back Next A curve for which passes through the point Find y in terms of x. Use the point Hint Maths4Scotland Integration Higher

Back to start Quit Back Next Integrate Split into separate fractions Multiply out brackets Hint Maths4Scotland Integration Higher

Back to start Quit Back Next If passes through the point express y in terms of x. Use the point Hint Maths4Scotland Integration Higher

Back to start Quit Back Next The graph of passes through the point (1, 2). express y in terms of x. If simplify Use the point Evaluate c Hint Maths4Scotland Integration Higher

Back to start Quit Back Next A curve for which passes through the point (–1, 2). Express y in terms of x. Use the point Hint Maths4Scotland Integration Higher

Back to start Quit Back Next Evaluate Cannot use standard integral So multiply out Hint Maths4Scotland Integration Higher

Back to start Quit Back Next Evaluate Straight line form Hint Maths4Scotland Integration Higher

Back to start Quit Back Next Evaluate Use standard Integral (from chain rule) Hint Maths4Scotland Integration Higher

Back to start Quit Back Next The curve passes through the point Find f(x ) use the given point Hint Maths4Scotland Integration Higher

Back to start Quit Back Next Integrate Integrate term by term Hint Maths4Scotland Integration Higher

Back to start Quit Back Next Evaluate Hint Maths4Scotland Integration Higher

Back to start Quit Higher Maths Strategies www.maths4scotland.co.uk Click to start Compound Angles Maths4Scotland Compound Angles Higher

Back to start Quit Compound Angles The following questions are on Non-calculator questions will be indicated Click to continue You will need a pencil, paper, ruler and rubber. Maths4Scotland Compound Angles Higher

Back to start Quit This presentation is split into two parts Using Compound angle formula for Exact values Solving equations Choose by clicking on the appropriate button Maths4Scotland Compound Angles Higher

Back to start Quit Hint PreviousNext A is the point (8, 4). The line OA is inclined at an angle p radians to the x -axis a) Find the exact values of: i) sin (2 p ) ii) cos (2 p ) The line OB is inclined at an angle 2 p radians to the x -axis. b) Write down the exact value of the gradient of OB. Draw trianglePythagoras Write down values for cos p and sin p Expand sin (2p) Expand cos (2p) Use m = tan (2p) 8 4 p Maths4Scotland Compound Angles Higher

Back to start Quit Hint PreviousNext In triangle ABC show that the exact value of Use Pythagoras Write down values for sin a, cos a, sin b, cos b Expand sin (a + b) Substitute values Simplify Maths4Scotland Compound Angles Higher

Back to start Quit Hint PreviousNext Using triangle PQR, as shown, find the exact value of cos 2x Use Pythagoras Write down values for cos x and sin x Expand cos 2x Substitute values Simplify Maths4Scotland Compound Angles Higher

Back to start Quit Hint PreviousNext On the co-ordinate diagram shown, A is the point (6, 8) and B is the point (12, -5). Angle AOC = p and angle COB = q Find the exact value of sin ( p + q ). Use Pythagoras Write down values for sin p, cos p, sin q, cos q Expand sin (p + q) Substitute values Simplify 6 8 5 12 10 13 Mark up triangles Maths4Scotland Compound Angles Higher

Back to start Quit Hint PreviousNext Draw triangles Use Pythagoras Expand sin 2A A and B are acute angles such that and. Find the exact value of a) b) c) 4 3 A 12 5 B Hypotenuses are 5 and 13 respectively 5 13 Write down sin A, cos A, sin B, cos B Expand cos 2A Expand sin (2A + B) Substitute Maths4Scotland Compound Angles Higher

Back to start Quit Hint PreviousNext Draw triangle Use Pythagoras Expand sin (x + 30) If x° is an acute angle such that show that the exact value of 3 4 x Hypotenuse is 5 5 Write down sin x and cos x Substitute Simplify Table of exact values Maths4Scotland Compound Angles Higher

Back to start Quit Hint PreviousNext Use Pythagoras Expand cos (x + y) Write down sin x, cos x, sin y, cos y. Substitute Simplify The diagram shows two right angled triangles ABD and BCD with AB = 7 cm, BC = 4 cm and CD = 3 cm. Angle DBC = x ° and angle ABD is y °. Show that the exact value of 5 Maths4Scotland Compound Angles Higher

Back to start Quit Hint PreviousNext Draw triangle Use Pythagoras The framework of a child’s swing has dimensions as shown in the diagram. Find the exact value of sin x° Write down sin ½ x and cos ½ x Substitute Simplify Table of exact values 3 3 4 x Draw in perpendicular 2 h Use fact that sin x = sin ( ½ x + ½ x ) Expand sin ( ½ x + ½ x ) Maths4Scotland Compound Angles Higher

Back to start Quit Hint PreviousNext Given that find the exact value of Write down values for cos a and sin a Expand sin 2a Substitute values Simplify 3 a Draw triangle Use Pythagoras Maths4Scotland Compound Angles Higher

Back to start Quit Hint PreviousNext Find algebraically the exact value of Expand sin (  +120) Use table of exact values Combine and substitute Table of exact values Expand cos (  +150) Simplify Maths4Scotland Compound Angles Higher

Back to start Quit Hint PreviousNext If find the exact value of a)b) Write down values for cos  and sin  Expand sin 2  Draw triangle Use Pythagoras 5  4 3 Opposite side = 3 Expand sin 4  (4  = 2  + 2  ) Expand cos 2  Find sin 4  Maths4Scotland Compound Angles Higher

Back to start Quit Hint PreviousNext Draw triangles Use Pythagoras Expand sin (P + Q) For acute angles P and Q Show that the exact value of 12 13 P 5 3 Q Adjacent sides are 5 and 4 respectively 5 4 Write down sin P, cos P, sin Q, cos Q Substitute Simplify Maths4Scotland Compound Angles Higher

Back to start Quit Previous You have completed all 12 questions in this section Back to start Using Compound angle formula for Solving Equations Next Maths4Scotland Compound Angles Higher

Back to start Quit Solving Equations Using Compound angle formula for Continue Maths4Scotland Compound Angles Higher

Back to start Quit Hint PreviousNext Solve the equation for 0 ≤ x ≤  correct to 2 decimal places Replace cos 2x with Substitute Simplify Factorise Hence Discard Find acute x Determine quadrants AS CT Maths4Scotland Compound Angles Higher

Back to start Quit Hint PreviousNext Table of exact values Solve simultaneously Rearrange Find acute 2 x Determine quadrants AS CT The diagram shows the graph of a cosine function from 0 to . a) State the equation of the graph. b) The line with equation y = -  3 intersects this graph at points A and B. Find the co-ordinates of B. Equation Check range Deduce 2x Maths4Scotland Compound Angles Higher

Back to start Quit Functions f and g are defined on suitable domains by f(x) = sin (x) and g(x) = 2x a)Find expressions for: i) f(g(x)) ii) g(f(x)) b)Solve 2 f(g(x)) = g(f(x)) for 0  x  360° Hint PreviousNext Table of exact values 2 nd expression Form equation Rearrange Determine quadrants AS CT 1 st expression Common factor Replace sin 2x Hence Determine x Maths4Scotland Compound Angles Higher

Back to start Quit Functions are defined on a suitable set of real numbers a)Find expressions for i) f(h(x)) ii) g(h(x)) b)i) Show that ii) Find a similar expression for g(h(x)) iii) Hence solve the equation Hint PreviousNext Table of exact values 2 nd expression Simplify 1 st expr. Similarly for 2 nd expr. Determine quadrants AS CT 1 st expression Use exact values Form Eqn. Simplifies to Rearrange: acute x Maths4Scotland Compound Angles Higher

Back to start Quit a)Solve the equation sin 2x - cos x = 0 in the interval 0  x  180° b)The diagram shows parts of two trigonometric graphs, y = sin 2x and y = cos x. Use your solutions in (a) to write down the co-ordinates of the point P. Hint PreviousNext Table of exact values Determine quadrants for sin x AS CT Common factor Replace sin 2x Hence Determine x Solutions for where graphs cross By inspection (P) Find y value Coords, P Maths4Scotland Compound Angles Higher

Back to start Quit Hint PreviousNext Solve the equation for 0 ≤ x ≤ 360° Replace cos 2x with Substitute Simplify Factorise Hence Find acute x Determine quadrants AS CT Table of exact values AS CT Solutions are: x= 60°, 132°, 228° and 300° Maths4Scotland Compound Angles Higher

Back to start Quit Hint PreviousNext Solve the equation for 0 ≤ x ≤ 2  Rearrange Find acute x Determine quadrants AS CT Table of exact values Solutions are: Note range Maths4Scotland Compound Angles Higher

Back to start Quit Hint PreviousNext a) Write the equation cos 2  + 8 cos  + 9 = 0 in terms of cos  and show that for cos  it has equal roots. b) Show that there are no real roots for  Rearrange Divide by 2 Deduction Factorise Replace cos 2  with Equal roots for cos  Try to solve: Hence there are no real solutions for  No solution Maths4Scotland Compound Angles Higher

Back to start Quit Solve algebraically, the equation sin 2x + sin x = 0, 0  x  360 Hint PreviousNext Table of exact values Determine quadrants for cos x AS CT Common factor Replace sin 2x Hence Determine x x = 0°, 120°, 240°, 360° Maths4Scotland Compound Angles Higher

Back to start Quit Find the exact solutions of 4sin 2 x = 1, 0  x  2  Hint PreviousNext Table of exact values Determine quadrants for s in x AS CT Take square roots Rearrange Find acute x + and – from the square root requires all 4 quadrants Maths4Scotland Compound Angles Higher

Back to start Quit Hint PreviousNext Solve the equation for 0 ≤ x ≤ 360° Replace cos 2x with Substitute Simplify Factorise Hence Find acute x Determine quadrants AS CT Table of exact values Solutions are: x= 60°, 180° and 300° Maths4Scotland Compound Angles Higher

Back to start Quit Hint PreviousNext Solve algebraically, the equation for 0 ≤ x ≤ 360° Replace cos 2x with Substitute Simplify Factorise Hence Find acute x Determine quadrants Table of exact values AS CT Solutions are: x= 60° and 300° Discard above Maths4Scotland Compound Angles Higher

Back to start Quit Previous You have completed all 12 questions in this presentation Back to start Maths4Scotland Compound Angles Higher

Back to start Quit Higher Maths Strategies www.maths4scotland.co.uk Click to start The Circle Maths4Scotland The Circle Higher

Back to start Quit The Circle The following questions are on Non-calculator questions will be indicated Click to continue You will need a pencil, paper, ruler and rubber. Maths4ScotlandThe Circle Higher

Back to start Quit Hint PreviousNext Find the equation of the circle with centre (–3, 4) and passing through the origin. Find radius (distance formula): You know the centre: Write down equation: Maths4ScotlandThe Circle Higher

Back to start Quit Hint PreviousNext Explain why the equation does not represent a circle. Consider the 2 conditions Calculate g and f: 1. Coefficients of x 2 and y 2 must be the same. 2. Radius must be > 0 Evaluate Deduction: Equation does not represent a circle Maths4ScotlandThe Circle Higher

Back to start Quit Hint PreviousNext Calculate mid-point for centre: Calculate radius CQ: Write down equation; Find the equation of the circle which has P(–2, –1) and Q(4, 5) as the end points of a diameter. Make a sketch P(-2, -1) Q(4, 5) C Maths4ScotlandThe Circle Higher

Back to start Quit Hint PreviousNext Calculate centre of circle: Calculate gradient of OP (radius to tangent) Gradient of tangent: Find the equation of the tangent at the point (3, 4) on the circle Equation of tangent: Make a sketch O(-1, 2) P(3, 4) Maths4ScotlandThe Circle Higher

Back to start Quit Hint PreviousNext Find centre of circle: Calculate gradient of radius to tangent Gradient of tangent: The point P(2, 3) lies on the circle Find the equation of the tangent at P. Equation of tangent: Make a sketch O(-1, 1) P(2, 3) Maths4ScotlandThe Circle Higher

Back to start Quit Hint PreviousNext A is centre of small circle O, A and B are the centres of the three circles shown in the diagram. The two outer circles are congruent, each touches the smallest circle. Circle centre A has equation The three centres lie on a parabola whose axis of symmetry is shown the by broken line through A. a) i) State coordinates of A and find length of line OA. ii) Hence find the equation of the circle with centre B. b) The equation of the parabola can be written in the form Find OA (Distance formula) Find radius of circle A from eqn. Use symmetry, find B Find radius of circle B Find p and q. Eqn. of B Points O, A, B lie on parabola – subst. A and B in turn Solve: Maths4ScotlandThe Circle Higher

Back to start Quit Hint PreviousNext Find centre of circle P: Gradient of radius of Q to tangent: Equation of tangent: Solve eqns. simultaneously Circle P has equation Circle Q has centre (–2, –1) and radius 2  2. a) i) Show that the radius of circle P is 4  2 ii) Hence show that circles P and Q touch. b) Find the equation of the tangent to circle Q at the point (–4, 1) c) The tangent in (b) intersects circle P in two points. Find the x co-ordinates of the points of intersection, expressing your answers in the form Find radius of circle :P: Find distance between centres Deduction: = sum of radii, so circles touch Gradient tangent at Q: Soln: Maths4ScotlandThe Circle Higher

Back to start Quit Hint PreviousNext For what range of values of k does the equation represent a circle ? Determine g, f and c: State condition Put in values Simplify Complete the square So equation is a circle for all values of k. Need to see the position of the parabola Minimum value is This is positive, so graph is: Expression is positive for all k : Maths4ScotlandThe Circle Higher

Back to start Quit Hint PreviousNext For what range of values of c does the equation represent a circle ? Determine g, f and c: State condition Put in values Simplify Re-arrange: Maths4ScotlandThe Circle Higher

Back to start Quit Maths4ScotlandThe Circle Higher Hint PreviousNext The circle shown has equation Find the equation of the tangent at the point (6, 2). Calculate centre of circle: Calculate gradient of radius (to tangent) Gradient of tangent: Equation of tangent:

Back to start Quit Hint PreviousNext When newspapers were printed by lithograph, the newsprint had to run over three rollers, illustrated in the diagram by 3 circles. The centres A, B and C of the three circles are collinear. The equations of the circumferences of the outer circles are Find the equation of the central circle. Find centre and radius of Circle A Find centre and radius of Circle C Find distance AB (distance formula) Find diameter of circle B Use proportion to find B Centre of BEquation of B (24, 12) (-12, -15) 27 36 25 20 B Maths4ScotlandThe Circle Higher

Back to start Quit Previous You have completed all 11 questions in this presentation Back to start Maths4ScotlandThe Circle Higher

Back to start Quit Vectors Strategies Higher Maths Click to start Maths4Scotland Vectors Higher

Back to start Quit Vectors The following questions are on Non-calculator questions will be indicated Click to continue You will need a pencil, paper, ruler and rubber. Maths4Scotland Vectors Higher

Back to start Quit The questions are in groups Angles between vectors (5) Points dividing lines in ratios Collinear points (8) General vector questions (15) Maths4Scotland Vectors Higher

Back to start Quit General Vector Questions Continue Maths4Scotland Vectors Higher

Back to start Quit Hint Previous Next Vectors u and v are defined by and Determine whether or not u and v are perpendicular to each other. Is Scalar product = 0 Hence vectors are perpendicular Maths4Scotland Vectors Higher

Back to start Quit Hint Previous Next For what value of t are the vectors and perpendicular ? Put Scalar product = 0 Perpendicular  u.v = 0 Maths4Scotland Vectors Higher

Back to start Quit Hint Previous Next VABCD is a pyramid with rectangular base ABCD. The vectors are given by Express in component form. Ttriangle rule  ACV Re-arrange Triangle rule  ABC also Maths4Scotland Vectors Higher

Back to start Quit Hint Previous Next The diagram shows two vectors a and b, with | a | = 3 and | b | = 2  2. These vectors are inclined at an angle of 45° to each other. a) Evaluate i)a.a ii)b.b iii)a.b b) Another vector p is defined by Evaluate p.p and hence write down | p |. i) ii) iii) b) Since p.p = p 2 Maths4Scotland Vectors Higher

Back to start Quit Hint Previous Next Vectors p, q and r are defined by a)Express in component form b)Calculate p.r c)Find |r| a) b) c) Maths4Scotland Vectors Higher

Back to start Quit Hint Previous Next The diagram shows a point P with co-ordinates (4, 2, 6) and two points S and T which lie on the x-axis. If P is 7 units from S and 7 units from T, find the co-ordinates of S and T. Use distance formula hence there are 2 points on the x axis that are 7 units from P i.e. S and T and Maths4Scotland Vectors Higher

Back to start Quit The position vectors of the points P and Q are p = –i +3j+4k and q = 7 i – j + 5 k respectively. a) Express in component form. b) Find the length of PQ. Hint Previous Next a) b) Maths4Scotland Vectors Higher

Back to start Quit Hint Previous Next PQR is an equilateral triangle of side 2 units. Evaluate a.(b + c) and hence identify two vectors which are perpendicular. Diagram P R Q 60° ab c NB for a.c vectors must point OUT of the vertex ( so angle is 120° ) Hence so, a is perpendicular to b + c Table of Exact Values Maths4Scotland Vectors Higher

Back to start Quit Hint Previous Next Calculate the length of the vector 2i – 3j +  3k Length Maths4Scotland Vectors Higher

Back to start Quit Hint Previous Next Find the value of k for which the vectors and are perpendicular Put Scalar product = 0 Maths4Scotland Vectors Higher

Back to start Quit Hint Previous Next A is the point (2, –1, 4), B is (7, 1, 3) and C is (–6, 4, 2). If ABCD is a parallelogram, find the co-ordinates of D. D is the displacement from A hence Maths4Scotland Vectors Higher

Back to start Quit Hint Previous Next If and write down the components of u + v and u – v Hence show that u + v and u – v are perpendicular. look at scalar product Hence vectors are perpendicular Maths4Scotland Vectors Higher

Back to start Quit Hint Previous Next The vectors a, b and c are defined as follows: a = 2i – k, b = i + 2j + k, c = –j + k a) Evaluate a.b + a.c b) From your answer to part (a), make a deduction about the vector b + c a) b) b + c is perpendicular to a Maths4Scotland Vectors Higher

Back to start Quit Hint Previous Next A is the point ( –3, 2, 4 ) and B is ( –1, 3, 2 ) Find: a) the components of b) the length of AB a) b) Maths4Scotland Vectors Higher

Back to start Quit Hint Previous Next In the square based pyramid, all the eight edges are of length 3 units. Evaluate p.(q + r) Triangular faces are all equilateral Table of Exact Values Maths4Scotland Vectors Higher

Back to start Quit You have completed all 15 questions in this section Previous Maths4Scotland Vectors Higher

Back to start Quit Points dividing lines in ratios Collinear Points Continue Maths4Scotland Vectors Higher

Back to start Quit Hint Previous Next A and B are the points (-1, -3, 2) and (2, -1, 1) respectively. B and C are the points of trisection of AD. That is, AB = BC = CD. Find the coordinates of D Maths4Scotland Vectors Higher

Back to start Quit Hint Previous Next The point Q divides the line joining P(–1, –1, 0) to R(5, 2 –3) in the ratio 2:1. Find the co-ordinates of Q. Diagram P Q R 2 1 Maths4Scotland Vectors Higher

Back to start Quit Hint Previous Next a)Roadmakers look along the tops of a set of T-rods to ensure that straight sections of road are being created. Relative to suitable axes the top left corners of the T-rods are the points A(–8, –10, –2), B(–2, –1, 1) and C(6, 11, 5). Determine whether or not the section of road ABC has been built in a straight line. b)A further T-rod is placed such that D has co-ordinates (1, –4, 4). Show that DB is perpendicular to AB. a) are scalar multiples, so are parallel. A is common. A, B, C are collinear b) Use scalar product Hence, DB is perpendicular to AB Maths4Scotland Vectors Higher

Back to start Quit Hint Previous Next VABCD is a pyramid with rectangular base ABCD. Relative to some appropriate axis, represents – 7i – 13j – 11k represents 6i + 6j – 6k represents 8i – 4j – 4k K divides BC in the ratio 1:3 Find in component form. Maths4Scotland Vectors Higher

Back to start Quit Hint Previous Next The line AB is divided into 3 equal parts by the points C and D, as shown. A and B have co-ordinates (3, –1, 2) and (9, 2, –4). a)Find the components of and b)Find the co-ordinates of C and D. a) b) C is a displacement of from A similarly Maths4Scotland Vectors Higher

Back to start Quit Hint Previous Next Relative to a suitable set of axes, the tops of three chimneys have co-ordinates given by A(1, 3, 2), B(2, –1, 4) and C(4, –9, 8). Show that A, B and C are collinear are scalar multiples, so are parallel. A is common. A, B, C are collinear Maths4Scotland Vectors Higher

Back to start Quit Hint Previous Next A is the point (2, –5, 6), B is (6, –3, 4) and C is (12, 0, 1). Show that A, B and C are collinear and determine the ratio in which B divides AC are scalar multiples, so are parallel. B is common. A, B, C are collinear A B C 2 3 B divides AB in ratio 2 : 3 Maths4Scotland Vectors Higher

Back to start Quit Hint Previous Next Relative to the top of a hill, three gliders have positions given by R(–1, –8, –2), S(2, –5, 4) and T(3, –4, 6). Prove that R, S and T are collinear are scalar multiples, so are parallel. R is common. R, S, T are collinear Maths4Scotland Vectors Higher

Back to start Quit Angle between two vectors Continue Maths4Scotland Vectors Higher

Back to start Quit Hint Previous Next The diagram shows vectors a and b. If |a| = 5, |b| = 4 and a.(a + b) = 36 Find the size of the acute angle between a and b. Maths4Scotland Vectors Higher

Back to start Quit Hint Quit Previous Next The diagram shows a square based pyramid of height 8 units. Square OABC has a side length of 6 units. The co-ordinates of A and D are (6, 0, 0) and (3, 3, 8). C lies on the y-axis. a)Write down the co-ordinates of B b)Determine the components of c)Calculate the size of angle ADB. a) B(6, 6, 0) b) c) Maths4Scotland Vectors Higher

Back to start Quit Hint Previous Next A box in the shape of a cuboid designed with circles of different sizes on each face. The diagram shows three of the circles, where the origin represents one of the corners of the cuboid. The centres of the circles are A(6, 0, 7), B(0, 5, 6) and C(4, 5, 0) Find the size of angle ABC Vectors to point away from vertex Maths4Scotland Vectors Higher

Back to start Quit Hint Previous Next A cuboid measuring 11cm by 5 cm by 7 cm is placed centrally on top of another cuboid measuring 17 cm by 9 cm by 8 cm. Co-ordinate axes are taken as shown. a) The point A has co-ordinates (0, 9, 8) and C has co-ordinates (17, 0, 8). Write down the co-ordinates of B b) Calculate the size of angle ABC. a) b) Maths4Scotland Vectors Higher

Back to start Quit Hint Previous Next A triangle ABC has vertices A(2, –1, 3), B(3, 6, 5) and C(6, 6, –2). a)Find and b)Calculate the size of angle BAC. c)Hence find the area of the triangle. a) b)  BAC = c) Area of  ABC = Maths4Scotland Vectors Higher

Back to start Quit Higher Maths Strategies www.maths4scotland.co.uk Click to start The Wave Function Maths4Scotland The Wave Function Higher

Back to start Quit The Wave Function The following questions are on Non-calculator questions will be indicated Click to continue You will need a pencil, paper, ruler and rubber. Maths4Scotland The Wave Function Higher

Back to start Quit Part of the graph of y = 2 sin x + 5 cos x is shown in the diagram. a)Express y = 2 sin x + 5 cos x in the form k sin (x + a) where k > 0 and 0  a  360 b) Find the coordinates of the minimum turning point P. Hint Expand ksin(x + a): PreviousNext Quit Equate coefficients: Square and add Dividing: Put together: Minimum when: P has coords. a is in 1 st quadrant (sin and cos are +) Maths4Scotland The Wave Function Higher

Back to start Quit Hint Expand ksin(x - a): PreviousNext Equate coefficients: Square and add Dividing: Put together: Sketch Graph a)Write sin x - cos x in the form k sin (x - a) stating the values of k and a where k > 0 and 0  a  2  b) Sketch the graph of sin x - cos x for 0  a  2  showing clearly the graph’s maximum and minimum values and where it cuts the x-axis and the y-axis. Table of exact values a is in 1 st quadrant (sin and cos are +)

Back to start Quit Hint Expand kcos(x + a): PreviousNext Equate coefficients: Square and add Dividing: Put together: Express in the form a is in 1 st quadrant (sin and cos are +) Maths4Scotland The Wave Function Higher

Back to start Quit Hint Express as Rcos(x - a): PreviousNext Equate coefficients: Square and add Dividing: Put together: Find the maximum value of and the value of x for which it occurs in the interval 0  x  2 . a is in 4 th quadrant (sin is - and cos is +) Max value:when Table of exact values Maths4Scotland The Wave Function Higher

Back to start Quit Hint Expand ksin(x - a): PreviousNext Equate coefficients: Square and add Dividing: Put together: a is in 1 st quadrant (sin and cos are both +) Express in the form Maths4Scotland The Wave Function Higher

Back to start Quit Hint Max for sine occurs PreviousNext Max value of sine function: Max value of function: The diagram shows an incomplete graph of Find the coordinates of the maximum stationary point. Sine takes values between 1 and -1 3 Coordinates of max s.p. Maths4Scotland The Wave Function Higher

Back to start Quit Hint Expand kcos(x - a): PreviousNext Equate coefficients: Square and add Dividing: Put together: a is in 1 st quadrant (sin and cos are both + ) a) Express f (x) in the form b) Hence solve algebraically Solve equation. Cosine +, so 1 st & 4 th quadrants Maths4Scotland The Wave Function Higher

Back to start Quit Hint Use tan A = sin A / cos A PreviousNext Divide Sine and cosine are both + in original equations Solve the simultaneous equations where k > 0 and 0  x  360 Find acute angle Determine quadrant(s) Solution must be in 1 st quadrant State solution Maths4Scotland The Wave Function Higher

Back to start Quit Hint Use Rcos(x - a): PreviousNext Equate coefficients: Square and add Dividing: Put together: a is in 2 nd quadrant (sin + and cos - ) Solve equation. Cosine +, so 1 st & 4 th quadrants Solve the equation in the interval 0  x  360. Maths4Scotland The Wave Function Higher

Back to start Quit Previous You have completed all 9 questions in this presentation Back to start Maths4Scotland The Wave Function Higher

Back to start Quit Higher Maths Logarithms & Exponential www.maths4scotland.co.uk Next Maths4Scotland Logarithms & Exponential Higher

Back to start Quit Back Next Reminder All the questions on this topic will depend upon you knowing and being able to use, some very basic rules and facts. Click to show When you see this button click for more information Maths4Scotland Logarithms & Exponential Higher

Back to start Quit Back Next Three Rules of logs Maths4Scotland Logarithms & Exponential Higher

Back to start Quit Back Next Two special logarithms Maths4Scotland Logarithms & Exponential Higher

Back to start Quit Back Next Relationship between log and exponential Maths4Scotland Logarithms & Exponential Higher

Back to start Quit Back Next Graph of the exponential function Maths4Scotland Logarithms & Exponential Higher

Back to start Quit Back Next Graph of the logarithmic function Maths4Scotland Logarithms & Exponential Higher

Back to start Quit Back Next Related functions of Move graph left a units Move graph right a units Reflect in x axis Reflect in y axis Move graph up a units Move graph down a units Click to show Maths4Scotland Logarithms & Exponential Higher

Back to start Quit Back Next Calculator keys lnln = l og = Maths4Scotland Logarithms & Exponential Higher

Back to start Quit Back Next Calculator keys lnln = 2.5= = 0.916… l og = 7.6= = 0.8808… Click to show Maths4Scotland Logarithms & Exponential Higher

Back to start Quit Back Next Solving exponential equations Show Take log e both sides Use log ab = log a + log b Use log a x = x log a Use log a a = 1 Maths4Scotland Logarithms & Exponential Higher

Back to start Quit Back Next Solving exponential equations Take log e both sides Use log ab = log a + log b Use log a x = x log a Use log a a = 1 Show Maths4Scotland Logarithms & Exponential Higher

Back to start Quit Back Next Solving logarithmic equations Change to exponential form Show Maths4Scotland Logarithms & Exponential Higher

Back to start Quit Back Next Simplify expressing your answer in the form where A, B and C are whole numbers. Show Maths4Scotland Logarithms & Exponential Higher

Back to start Quit Back Next Simplify Show Maths4Scotland Logarithms & Exponential Higher

Back to start Quit Back Next Find x if Show Maths4Scotland Logarithms & Exponential Higher

Back to start Quit Back Next Givenfind algebraically the value of x. Show Maths4Scotland Logarithms & Exponential Higher

Back to start Quit Back Next Find the x co-ordinate of the point where the graph of the curve with equation intersects the x-axis. When y = 0 Exponential form Re-arrange Show Maths4Scotland Logarithms & Exponential Higher

Back to start Quit Back Next The graph illustrates the law If the straight line passes through A(0.5, 0) and B(0, 1). Find the values of k and n. Gradient y-intercept Show Maths4Scotland Logarithms & Exponential Higher

Back to start Quit Back Next Before a forest fire was brought under control, the spread of fire was described by a law of the form where is the area covered by the fire when it was first detected and A is the area covered by the fire t hours later. If it takes one and a half hours for the area of the forest fire to double, find the value of the constant k. Show Maths4Scotland Logarithms & Exponential Higher

Back to start Quit Back Next The results of an experiment give rise to the graph shown. a)Write down the equation of the line in terms of P and Q. It is given that and stating the values of a and b. b) Show that p and q satisfy a relationship of the form Gradient y-intercept Show Maths4Scotland Logarithms & Exponential Higher

Back to start Quit Back Next The diagram shows part of the graph of. Determine the values of a and b. Use (7, 1) Use (3, 0) Hence, from (2) and from (1) Show Maths4Scotland Logarithms & Exponential Higher

Back to start Quit Back Next The diagram shows a sketch of part of the graph of a)State the values of a and b. b)Sketch the graph of Graph moves 1 unit to the left and 3 units down Show Maths4Scotland Logarithms & Exponential Higher

Back to start Quit Back Next a) i) Sketch the graph of ii) On the same diagram, sketch the graph of b)Prove that the graphs intersect at a point where the x-coordinate is Show Maths4Scotland Logarithms & Exponential Higher

Back to start Quit Back Next Part of the graph of is shown in the diagram. This graph crosses the x-axis at the point A and the straight line at the point B. Find algebraically the x co-ordinates of A and B. Show Maths4Scotland Logarithms & Exponential Higher

Back to start Quit Back Next The diagram is a sketch of part of the graph of a)If (1, t) and (u, 1) lie on this curve, write down the values of t and u. b)Make a copy of this diagram and on it sketch the graph of c)Find the co-ordinates of the point of intersection of with the line a)b) c) Show Maths4Scotland Logarithms & Exponential Higher

Back to start Quit Back Next The diagram shows part of the graph with equation and the straight line with equation These graphs intersect at P. Solve algebraically the equation and hence write down, correct to 3 decimal places, the co-ordinates of P. Show Maths4Scotland Logarithms & Exponential Higher

Back to start Quit Maths4Scotland Higher Return 30°45°60° sin cos tan1 Table of exact values

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Back to start Quit Higher Maths www.maths4scotland.co.uk Click to start Strategies.

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