St Andrew s Academy Department of Mathematics. Course Textbook BOOK TWO

Transcription

1 St Andrew s Academ Department of Mathematics Course Tetbook BOOK TWO

2 Book Two Section Quadratic Functions. Completing the Square.... Equations of Quadratic Functions.. Sketching Graphs of Quadratic Functions 7. Features of Quadratic Functions.. 8. Solving Quadratic Equations from Graphs....6 Solving Quadratic Equations b Factorising.7 Solving Quadratic Equations using the Quadratic Formula..8 Identifing Intercepts and Turning Points.9 Problems Involving Quadratic Equations. 6. Quadratic Functions Eam Questions 7. The Discriminant... The Discriminant - Eam Questions Section Surds and Indices. Working with Surds... Problem Solving with Surds. 9. Indices... Indices - Eam Questions. Calculations Using Scientific Notation 6.6 Scientific Notation - Eam Questions. 8.7 Significant Figures 9 Section Vectors. Working with D Vectors.. Addition of Vectors Using Directed Line Segments. Subtraction of Vectors Using Directed Line Segments. Vector Journes in D. 7. Working with D Coordinates and Vectors. 9.6 Using Vector Components Section Algebraic Fractions. Simplifing Fractions 8. Calculations Involving Algebraic Fractions Algebraic Fractions - Eam Questions. 6 Section Triangle Formulae. Areas of Triangles. 6. Using the Sine Rule to Calculate Sides 68. Using the Sine Rule to Calculate Angles. 69. Using the Cosine Rule to Calculate Sides. 7. Using the Cosine Rule to Calculate Angles Choosing the Correct Formula Triangle Formulae - Eam Questions Problems Involving Bearings 8.9 Problems Involving Bearings - Eam Questions. 8 Section 6 Trigonometric Functions 6. Drawing Graphs of Trig Functions Equations of Trig Functions Working with Angles in Quadrants Eact Values The Period of a Function Solving Trig Equations Trig Identities 6.8 Trig Functions Eam Questions.

3 Section Quadratic Functions. Completing the Square. Write the following in the form and write down the minimum value of each one. (f) (g) (h). Write the following in the form and write down the minimum value of each one. (f) (g) (h) (i) (j) (k) (l). Write the following in the form and write down the maimum value of each one. (f)

4 . Equations of Quadratic Functions.Write down the equation of the graphs shown below, which have the form = k. (,) (,) (,) (f) (,)) (,7) (,6) (g) (h) (i) (,9) (,8) (,) (j) (k) (l) (,) (,) (,) (m) (n) (o) (,6) (,) (,)

5 .Write down the equation of the graphs shown below, which have the form = a b. (, ) (, ) (, ) (f) (, ) (, ) (,) (g) (h) (i) (,9) (, 9) (, ) (, ) (, ) (, ) (j) (k) (l) (, 8) (, 9) (, ) (, 6) (, ) (m) (, ) (, 9) (n) (, ) (, 6) (, )

6 . Write down the equation of the graphs shown below, which have the form = ( a) b. (, ) (, 6) (, ) (f) (, ) (, ) (, ) (g) (h) (i) (, ) (, ) (6, ) (j) (k) (l) (, ) (, ) (, ) 6

7 Rel (m) (n) (, 6) (, ). Sketching Graphs of Quadratic Functions. Sketch the graphs with the following equations = ( ) = ( ) = ( ) 7 = ( ) = ( ) (f) = ( ) (g) = ( ) 6 (h) = ( ) (i) = ( 8) (j) = ( ) (k) = ( ½ ) ¾ (l) = ( ) (m) = ( ) (n) = ( 6) (o) = ( 7) (p) = ( ) (q) = ( ) (r) = ( ) 7. Sketch the graphs with the following equations = ( )( ) = ( )( ) = ( )( 7) = ( 6)( 8) = ( )( ) (f) = ( 8)( ) (g) = ( )( ) (h) = ( )( ) (i) = ( )( 6) (j) = ( )( ) (k) = ( 9)( ) (l) = ( )( 8). Sketch the graphs with the following equations = ( )( ) = ( )(7 ) = ( )( ) = ( 8)( ) = ( )( 7) (f) = ( )(7 ) (g) = ( )( 9) (h) = ( )( ) (i) = ( 9)( 7) (j) = ( )( 6) (k) = ( )( ) (l) = ( )( 6) (m) = ( )( ) (n) = ( 7)( ) (o) = ( )( 6) 7

8 . Features of Quadratic Functions. For each of the graphs below, write down (i) the turning point (ii) its nature and (iii) the equation of the ais of smmetr a b c d e f g h 8

9 . For each of the equations below, write down (i) the turning point (ii) its nature and (iii) the equation of the ais of smmetr = ( ) = ( ) = ( ) 7 = ( ) = ( ) (f) = ( ) (g) = ( ) 6 (h) = ( ) (i) = ( 8) (j) = ( ) (k) = ( ½ ) ¾ (l) = ( ) (m) = ( ) (n) = ( 6) (o) = ( 7) (p) = ( ) (q) = ( ) (r) = ( ) 7 9

10 . Solving Quadratic Equations from Graphs. Use the sketches below to solve the quadratic equations. = 6 = = = 8 = (f) = 8

11 . (i) Cop and complete the tables below. (ii) Make a sketch of the graph. (iii) Write down the roots of the quadratic equation =. = ² = ² 6 6 = ² = 8 ². For each equation, draw a suitable sketch and find the roots. = 6 = = 8 = 6 9 = (f) = (g) 6 8 = (h) 8 = (i) 7 = (j) = (k) 6 = (l) = (m) = (n) = (o) =

12 .6 Solving Quadratic Equations b Factorising. Solve these quadratic equations, which are alread in factorised form. ( ) = ( 7) = ( ) = b(b ) = a(a ) = (f) m(m ) = (g) (a )(a ) = (h) ( )( ) = (i) (c )(c ) = (j) (w )(w ) = (k) (s )(s ) = (l) (z 7)(z 8) = (m) ( )( ) = (n) (t )(t ) = (o) ( )( 9) = (p) (a )(a ) = (q) (p 7)(p 7) = (r) (c )(c ) = (s) (d )(d ) = (t) ( )( ) = (u) (s )(s ) =. Solve these quadratic equations b factorising first. = c c = 8 = p p = z z = (f) n 7n = (g) t t = (h) = (i) 6b 8b = (j) 6 = (k) 6a 9a = (l) = (m) = (n) 9b b = (o) m m = (p) 6w w = (q) 9c c = (r) =. Solve these quadratic equations b factorising first. = b = = a 6 = z 9 = (f) k 6 = (g) 6 = (h) p = (i) m = (j) t 9 = (k) a 8 = (l) s = (m) a 8 = (n) c 8 = (o) 6 =

13 . Solve these quadratic equations b factorising first. = 6 = a 8a 7 = m m 6 = c 6c 8 = (f) z 7z = (g) ² = (h) b 8b 6 = (i) 7 = (j) w w 7 = (k) 8 7 ² = (l) k k = (m) 8 ² = (n) 6 m m² = (o) t 7t = (p) a a = (q) c c = (r) p p² =. Solve these quadratic equations b factorising first. 7 = p p = t t = k 7k = 8 = (f) 6 7a a² = (g) w w² = (h) d d = (i) 6 = (j) m m 8 = (k) 7 c c² = (l) 6² = (m) = (n) q q = 6 (o) t(t ) = (p) m m = (q) 6v = v (r) 7s = 7s.7 Solving Quadratic Equations using the Quadratic Formula. Solve these equations using the quadratic formula. 7 = a a = c 8c = p p 9 = = (f) d d 6 = (g) 7 = (h) a a = (i) p 7p 6 = (j) b 7b = (k) 6 7 = (l) 6 = (m) = (n) a a = (o) p p = (p) c 7c = (q) 6 = (r) w w 8 =

14 . Solve these equations using the quadratic formula, giving our answers correct to decimal places. = b 9b = p p = c c = 7 = (f) a 8a = (g) z z = (h) q q = (i) w 6w = (j) d d 8 = (k) = (l) m 7m = (m) 8 = (n) k k 6 = (o) c c 9 =. Solve these equations using the quadratic formula, giving our answers correct to decimal places. 8 = b 9b = p p = 6c c = 7 = (f) a 9a = (g) 8z 7z = (h) q q = (i) w 6w = (j) d d = (k) 7 = (l) 8m m = (m) 8 = (n) k 6k = (o) c c = (p) 8 9t t = (q) a 7a = (r) z z 9 =. Solve these equations using the quadratic formula, giving our answers correct to significant figures. = c c = m 8m = 7 7 = p 6p = (f) a 6a = (g) b b = (h) z 9z = (i) q 7q = (j) = (k) c 8c 8 = (l) w w = (m) k k = (n) d d = (o) s 8s 7 = (p) a a 9 = (q) = (r) c c = (s) 8 8 = (t) b b = 9 (u) p 9p = (v) 7m = 6m (w) = 8 () c = 9 c

15 .8 Identifing Intercepts and Turning Points. Sketch the graphs of the following quadratic functions marking all relevant points. Then. for each function answer the following questions (i) (ii) (iii) (iv) State the roots (or zeros) of the function; write down the equation of the ais of smmetr; state the coordinates and nature of the turning point; give the coordinates of the -intercept point; f ( ) = g ( ) = 8 h ( ) = f ( ) = 6 g( ) = (f) h( ) = 8 (g) f ( ) = 8 (h) g( ) = 7 6 (i) h ( ) = (j) f ( ) = (k) g ( ) = 7 6 (l) h( ) = (m) f ( ) = (n) g( ) = 6 (o) h ( ) = 9. Find the coordinates of the points marked with letters in the diagrams below. = 8 = 66 = 8 L A o B E o F I J o D C G H K (f) = = D = 8 7 M o N R o S o A B P Q U C T

16 .9 Problems Involving Quadratic Equations. The diagram opposite shows a right-angled triangle with sides measuring, 7 and centimetres. Form an equation and solve it to find. Hence calculate the perimeter of the triangle. 7. The sides of a right angled triangle are n, n and n 6 n 6 centimetres long. n n Find n b solving an equation, and hence calculate the area of this triangle.. Repeat question. for a right angled triangle with short sides measuring n and n and the hpotenuse measuring n millimetres.. A rectangular sheet of glass has an area of cm and a perimeter of 6 cm. Taking the length of the glass sheet as, write down an epression for the width of the sheet in terms of. Form an equation in for the area of the sheet and solve it to find. Hence state the dimensions of the glass sheet.. A farmer has 6 metres of clear plastic fencing. He uses all the fencing to create a rectangular holding pen. Taking as the length of the pen, write down an epression for the width of the pen in terms of. Given that the area of the pen is square metres, form an equation and solve it to find, the length of the pen. 6

17 . Quadratic Functions - EXAM QUESTIONS. For the quadratic function = ( ¾ ) ½, write down: the turning point its nature the equation of the ais of smmetr. A quadratic function is defined b the formula f () = ( ). Write down the turning point of the graph of the function.. Solve quadratic function = using an appropriate formula. Give our answers correct to decimal place.. The graph shown is a function of the form = ( b)( c). Establish the equation of the function. State the coordinates and nature of the turning point. 6 Write the equation of the function in the form = p( q) r.. Solve the quadratic equation 7 = using an appropriate formula. Give our answers correct to decimal place. 6. For the quadratic function = ( ½) ¾, write down the turning point and its nature. 7

18 7. Solve the quadratic equation = giving the roots correct to significant figures. 8. The equation of the parabola is of the form = ( p) q. (, 6) Write down the equation of the parabola. 9. Solve the quadratic equation 8 = Give our answers correct to decimal place.. The graph shows a quadratic function of the form = ( a) b. State: The coordinates of the turning point and its nature. The equation of the ais of smmetr. The equation of the function. (, ). The diagram shows the graph of = 7 Find the - coordinates of the points where the graph crosses the - ais giving our answers correct to decimal place. = 7 o 8

19 . For the quadratic function = (¾ ) 6, write down the turning point and its nature. the equation of the ais of smmetr. Solve the equation 8 9 = giving the roots correct to significant figures.. The graph shown has the equation of the form = ( b)( c). Find the equation of the parabola. Find the coordinates of where the graph cuts the -ais. State the coordinates and nature of the turning point. State the equation of the ais of smmetr of the parabola.. A quadratic graph has equation = ( ) 7. What are the coordinates and nature of the turning point of the graph? Which of the following is the equation of its ais of smmetr? A = B = ` C = 7 D = 7 6. Solve the quadratic equation 9 = Give our answers correct to decimal place. 9

20 7. There is an arch built over the new Wemble football stadium in London. It can be represented on suitable aes b the parabola with equation = 9 ( ). A = 9 ( ) B Write down the coordinates of A, the maimum turning point of the parabola. What is the equation of the ais of smmetr of the parabola? The parabola cuts the - ais at the origin and the point B. Find the coordinates of B. 8. The graph shown is a function of the form = ( ). Write down the turning point of the graph and state its nature. What is the equation of the ais of smmetr of the graph? Write the co-ordinates of the points where the graph cuts the -ais. 9. A parabola has equation = 9 ( ) What are the coordinates and nature of its turning point? Write down the equation of the ais of smmetr of the parabola,

21 . The diagram shows part of the graph of the parabola which has a minimum turning point at (, 9). B A(, 9). Write down the equation of the parabola in the form = ( a) b The parabola cuts the - ais at the origin and the point B. What is the length of OB?. The parabola in the diagram has minimum turning point (, ) and crosses the - ais at points A and B. Write its equation in the form = ( a) b. State the equation of the ais of smmetr of the parabola. Find the coordinates of the point where the parabola cuts the - ais. A B B is the point (, ). What are the coordinates of A?. Show that the area of this L shaped room is given b the function A( ) =. m Given that the area is 6 m, calculate the value of. m m m

22 . The Discriminant. Find the discriminant for each of these quadratic equations = 6 9= 8 7= w w² = 7 = (f) 6= (g) 7 = (h) 7 9= (i) 6 = (j) 6 = (k) 8 9= (l) 7= (m) 7 = (n) = (o) = (p) = (q) 7 = (r) 8 6=. Use the discriminants from Q to state the nature of the roots of each of the quadratic equations.. Here are some graphs of quadratic functions. What can ou sa about the discriminant for each one? (f)

23 (g) (h) (i) (j) (k) (l). Find the value of a so that these quadratic equations have equal roots. a= a= a = a (a ) a= 8 a= (f) a 7 =. Find the value of k so that these quadratic equations have equal roots. k 8 = k 6 8= k = 6 k = k = (f) k 6=

24 . The Discriminant - EXAM QUESTIONS. The following words can be used to describe the roots of a quadratic. I Real II Equal III Distinct IV Non-real V Rational VI Irrational Which of the above words can be used to describe the roots of the equation =?. Find the value of the discriminant for the quadratic equation = Use the discriminant to state the nature of the roots in part.. For what values of p does the equation p= have equal roots?. The roots of a quadratic equation can be described as: I Real II Equal III Distinct IV Non-real V Rational VI Irrational Which of the above can be used to describe the roots of the equation =?. For the quadratic equation =, find the value of the discriminant. Use the words from question to describe the nature of the roots of the equation.

25 Section Surds and Indices. Working with Surds. Epress each of the following in its simplest form: 8 (f) 8 (g) 6 (h) 7 (i) (j) 7 (k) 96 (l) 8 (m) (n) 98 (o) 9 (p) 8 (q) 8 (r) 8 (s) (t) 6 (u) (v) (w) 6 () 7. Simplif: 8 8 (f) (g) 7 (h) 8 (i) 8 (j) (k) 6 (l). Epress each of the following in its simplest form: (f) (g) (h) (i) (j) (k) (l) 7. Epress each of the following in its simplest form: (f) (g) 8 (h) 9 (i) 8 (j) (k) 7 8 (l) 8 (m) 8 (n) 8 (o) 7 (p) 98 (q) 8 (r) (s) 8 (t) 7 (u) 8 8

26 . Simplif: a a 6 6 (f) c c (g) k k (h) 6 (i) 8 (j) 6 (k) (l) (m) 8 (n) (o) (p) (q) a b (r) (s) p q (t) k 6 (u) (v) (w) () 6 () (z) (f) 6 (g) 8 (h) 7. Simplif: (f) 8 (g) (h) 7 6 (i) 8 7 (j) 6 (k) 88 8 (l) 9 (m) 8 6 (n) (o) 98 7 (p) 6

27 8. Epand and simplif: ( ) ( ) ( ) ( ) ( 6) (f) ( 8 ) (g) ( 6 8) (h) ( ) (i) 6( 6 8) (j) 8( ) (k) ( 6) (l) ( ) (m) ( ) (n) ( 8 ) (o) ( 6) (p) ( ) 9. Epand and simplif where possible: ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) (f) ( )( ) (g) ( )( - ) (h) ( 8 )( 8 ) (i) ( )( ) (j) ( ) (k) ( ) (l) ( ) (m) ( 7 ) (n) ( ) (o) ( )( ) (p) ( 7 ) (q) ( 6 ) (r) ( )( ). Epress each of the following with a rational denominator and simplif where possible: 6 (f) (g) (h) (i) (j) (k) 6 (l) (m) (n) 7 7

28 . Epress each of the following with a rational denominator and simplif where possible: 6 8 (f) 7 (g) (h). Epress each of the following in its simplest form with a rational denominator. 8 8 (f) (g) (h) 8 6 (i) (j) (k) 7 (l) (m) 8 (n) (o) (p) (q) 6 8 (r) (s) (l). Epress each of the following with a rational denominator and simplif where possible: (f) 8 (g) (h) (i) (j) (k) (l) 8

29 . Rationalise the denominator, in each fraction, using the appropriate conjugate surd. (f) (g) (h) 6 (i) (j) (k) 7 (l) (m) 6 (n) (o) 6 (p) 9. Problem Solving with Surds. A right angled triangle has sides a, b and c as shown. For each case below calculate the length of the third side, epressing a c our answer as a surd in its simplest form. b Find a if b = 6 and c =. Find c if a = and b =. Find c if a = 8 and b = Find b if a = 8 and c = 6.. Given that = and =, simplif: ( )( ). Given that p = and q =, simplif: p q pq p q 9

30 . A rectangle has sides measuring ( ) cm and ( ) cm. Calculate the eact value of its area the length of a diagonal.. A curve has as its equation =. If the point P(, k) lies on this curve find the eact value of k. Find the eact length of OP where O is the origin. 6. In ABC, AB = AC = cm and BC = 8cm. Epress the length of the altitude from A to BC as a surd in its simplest form. [The line AM in the diagram] A cm cm 7. An equilateral triangle has each of its sides measuring a metres. Find the eact length of an altitude of the triangle in terms of a. Hence find the eact area of the triangle in terms of a. [Draw a diagram to help ou with this question] B M 8cm C 8. The eact area of a rectangle is ( 6 ) square centimetres. Given that the breadth of the rectangle is 6 cm, show that the length is equal to ( ) cm. 9. (a challenge) Given that tan 7 =, show that tan 7 =.

31 . Indices. Write each of the following in its simplest inde form (f) (g) (h) (i) (j) c c 9 (k) a a (l) (m) b b (n) p p 9 (o) d d (p) q q 9 (q) t t 7 (r) f f (s) k k (t) z z (u) (v) 9 (w) a a 6 () b b. Write each of the following in its simplest inde form (f) (g) 8 (h) (i) 7 (j) a 9 a (k) (l) b b (m) p p (n) c 7 c 7 (o) q 8 q (p) d d 9 (q) 8 a (r) a (s) m 7 s (t) 7 m s d (u) d (v) (w) t w () t w. Write each of the following in its simplest inde form. ( ) (8 ) ( ) ( ) ( ) (f) ( 7 ) (g) ( ) (h) ( ) (i) ( ) (j) ( 8 ) (k) (a ) 7 (l) (m ) (m) (b ) 6 (n) (p ) (o) (k ) (p) (z 6 )

32 . Write the following without brackets. (7a) () () (ab) (f) () 7 (g) (wz) (h) (st) (i) (pq ) (j) ( ) (k) (a b ) (l) (6a ) (m) ( ) (n) (c ) (o) (ab ) (p) (m k). Simplif these epressions. a a p 7 p b b 6 ( ) (f) (q ) q (g) (c ) 8c (h) 7z (z ) (i) k (k k ) (j) m (m m ) (k) ( ) (l) a (a a ) (m) 6 ( m (n) 6 m ) c c (o) 6 c 7 (q ) q (p) 7 6q (q) ( ) 9 (a b ) (r) (ab) 6 ( p ) (s) 6 p 8 p (ab ) (t) a b ab 6. Write down the value of () (f) ½ (g) a (h) k (i) (mn) (j) (ab ) (k) ( ) (l) (6 z ) 7. Rewrite the following with positive indices. 6 (f) 7 (g) a (h) (i) p 7 (j) (k) b (l) q (m) (n) w (o) a (p) 8 c (q) t (r)

33 8. Rewrite the following with negative indices (f) (g) (h) a (i) p (j) (k) 6 q (l) 8 c 9. Simplif the following epressions. m m 7 p 8 p a a ( ) (f) (c ) (g) (q ) (h) (w ) (i) b b (j) (k) k k (l) 8d d (m) ( ) (n) p (p p 8 ) (o) a (a a ) (p) ½ m (m m 6 v v ) (q) v 7 h h (r) h c 9c (s) 6c 6 6 (t) 8. Find the value of (f) (g) (h) 8 (i) (j) 6 (k) 6 (l) 6 (m) (n) 6 (o) 9 (p) 7 (q) 6 (r) (s) 6 (t) 8 (u) 8 (v) ( 8) (w) 6 () () ( ) (z) ( ) 8

34 . Simplif the following epressions, giving our answers with positive indices. ( ) 6 ( p ) 6 ( ( a ) 8 ( ) 9 q ) (f) ( k ) (g) (g ) (h) ( m ) (i) 9 ) (c (j) ( h ) (k) 6 ( z ) (l) ( b ) (m) (n) (o) d d (p) s s (q) (r) 6 (u) (v) (s) (t) (w) 8 () Write the following in surd form. a b (f) (g) c (h) a (i) (m) c (j) z (k) m (l) k 7 p (n) (o) w (p) d. Write the following in inde form. a z c (f) (g) p (h) m (i) a (j) z (k) (l) a (m) b (n) m (o) (p) c

35 . Simplif each of the following: ( ) ( ) ( ) ( ) ( ) (f) (g) ( ) (h) (i) (j) (k) (l) ( ). Indices - EXAM QUESTIONS. Simplif 7a b a b If a = and b =, find the value of the epression in part.. Given that =, find when = 8.. Simplif ( ) m. Simplif m Evaluate p 8p. Epress p in its simplest form. 6. Simplif, writing our answer with a positive inde: a a 6 7. Simplif the fraction, giving our answer in positive inde form: 9 a a 8. Simplif a. 9. Remove the brackets and simplif: p ( p ). Hence, or otherwise, find the value of p ( p ) when p =.

36 . Calculations Using Scientific Notation. Rewrite these sentences with the numbers written out in full (f) (g) The speed of light is 8 metres per second. The diameter of the earth is 68 kilometres. A Building Societ has. 9 in its funds. The radius of the orbit of an electron is 8 mm. A space probe reached a speed of 9 m.p.h. The earth weighs 6 6 tonnes. A film of oil is 8 7 mm thick.. Use our calculator to answer the following, giving our answers in Standard Form. ( ) ( 6 ) ( 7 ) ( 8 ) ( 8 ) ( ) (9 6 ) ( ) ( ) ( 9 ) (f) ( ) ( 7 ) (g) ( ) ( 8 ) (h) ( 9 8 ) ( 9 ) (i) (8 7 ) (7 ) (j) ( ) ( 8 7 ) (k) ( ) ( 8 ) (l) ( ) ( ) (m) ( 8 ) ( 7 ) (n) ( ) ( 8 ) (o) ( 8 ) (8 7 ) (p) (8 ) ( 7 ) (q) ( ) ( ) (r) (8 8 ) (6 ) (s) (9 67 ) ( ) (t) (6 86 ) (6 ) 6 8 (u) 7 (v) 8 (w) 6

37 . Answer each of the following questions leaving our answers in standard form. Light travels at 8 miles per second. How far will it travel in an hour? The radius of the earth is 6 6 metres. What is its circumference (in km)? If a heart beats 7 times a minute, how man times will it beat in a lifetime of 8 ears?[take all ears to have 6 das] grams of water contains drops. How man drops would there be in a tank containing tonne of water? In gram of carbon there are there in kg of pure carbon? 6 6 atoms. How man carbon atoms are. Answer each of the following questions leaving our answers in standard form (f) The weight of a droplet of water is 8 7 grams. Calculate the weight of droplets. A space probe can travel at a speed of 6 6 miles per da. What distance will it travel in a week? A biscuit factor produces teacakes ever da. How man teacakes were produced in the month of Februar 8? The speed of light is approimatel 99 million metres per second. How far can light travel in a minute? Last ear 68 6 copies of a DVD were sold on its first da of release. If the cost of one DVD was, how much mone was collected on that first da? In a realit TV show there were calls made to vote for the contestants. If each call cost p calculate how much the calls cost in total. Give our answer in pounds. (g) There are April? 8 6 seconds in one da. How man seconds are there in the month of (h) (i) Organisers of the London Marathon provide enough water to give each runner 7 litres during the race. If 77 runners take part, how man litres of water are provided? The echange rate in Turke is = 67 Turkish Lira. Stephen is going on an Adriatic cruise and changes 7 into Turkish Lira. How much will he get in Lira? 7

38 .6 Scientific Notatiom - EXAM QUESTIONS. The distance between the earth and mars is on average 8 approimatel 6 miles. A spaceship has been designed to travel between the earth and mars at an average speed of miles per hour. How man das will the spaceship take to reach mars? Give our answer correct to the nearest da.. Uranium is a radioactive isotope which has a half-life of 9 ears. This means that onl half of the original mass will be radioactive after 9 ears. How long will it take for the radioactivit of a piece of Uranium to reduce to one eighth of its original level? Give our answer in scientific notation.. The population of Scotland in June was 6.6 people. The population of China in June was approimatel times larger than that of Scotland. Calculate, correct to three significant figures, the population of China in, epressing our answer in standard form.. The Blackbird is a two-seater high speed jet. In December 96 it broke a world speed record b travelling at metres per second. Calculate, correct to three significant figures, the distance travelled if the jet were to maintain this speed for one hour. Epress our answer in scientific notation. 8

39 .7 Significant Figures. Round to significant figure :. 78 (f) 9 (g) 9 (h) 8.6 (i) 766 (j) 98 (k) 8 (l) 9.7 (m) 9 (n) 6 (o) 98 (p) (q) (r) 9 (s) 86 (t) 66. Round to significant figures : (f) 6 6 (g) (h) 8 (i) 7 (j) 8 76 (k) 697 (l) 99 (m) 6 (n) 6 (o) 87 (p) 8 (q) 6 (r) 9 (s) 9 (t) 76. Round to significant figures : (f) 8 (g) 68 (h) 9 (i) 968 (j) 68 (k) 98 (l) 7 (m) 8 (n) 6 (o) 7 (p) (q) 679 (r) 8 (s) 7 (t) 7. Round 88 correct to sig. figs sig. figs sig. figs sig. fig. Round 866 correct to sig. figs sig. figs sig. figs sig. fig 9

40 6. Calculate and give our answer correct to significant figures (f) 8 9 (g) 9 (h) 9 (i) ( 8 ) (j) ( 8 ) (k) ( ) (l) 7 7. Calculate and give our answer correct to significant figures (f) (g) 7 ( ) (h) 99 9 (i) 77 ( 9) (j) (6 9 8) (k) 6 9 (l) 6 8. The speed of light is approimatel 8 times faster than the speed of sound in air. If the speed of sound in air is 7 metres per second, calculate the speed of light. Give our answer in scientific notation correct to significant figures.

41 . Working with D Vectors Section Vectors. Name the following vectors in was and write down the components: A R (g) u s B C S D H M (f) u v v E F w w (h) t X (i) (j) (k) a G R L W P C T b P Q c F Q. Draw representations of the following vectors on squared paper. (i) v = w = u = AB = CD = (f) EF = (g) r = (h) p = q = (j) XY = (k) PQ = (l) ST = 6. Calculate the magnitude of each of the vectors in questions and above leaving our answers as surds in their simplest form.. Find (f)

42 . Addition of Vectors Using Directed Line Segments. (i) Draw diagrams on squared paper to illustrate u v for each pair of vectors given. (ii) State the components of the resultant vector and calculate its magnitude leaving our answers as a surd in its simplest form u u v v u v v (f) u u u v v (g) u v (h) u v (i) u v. (i) Draw diagrams on squared to illustrate a b for each the following pairs of vectors. (ii) (g) State the components of the resultant vector and calculate its magnitude. 9 a = ; b = a = ; b = 7 6 a = ; b = a = ; b = 6 a = ; b = (f) a = ; b = 6 a = ; b = (h) a = ; b =

43 . The diagram shows vectors a, b and c. a b c (i) Draw diagrams on squared paper to represent: a b a c b c (a b) c a ( b c) (ii) For each resultant vector, state the components and calculate its magnitude correct to one decimal place.. (i) For the vectors in question draw representations of these vectors. a b c b a (f) c (g) a b (h) c a (ii) State the components of each of the vectors above and calculate the magnitude leaving answers as a surd in its simplest form.. The diagram shows vectors, and z. z (i) Draw diagrams to represent: z z ( ) z ( z) (ii) Calculate, correct to one decimal place: z z ( ) z ( z)

44 6. For the vectors in question, calculate: z (f) z (g) (h). Subtraction of Vectors Using Directed Line Segments. (i) Draw diagrams on squared paper to illustrate u v for each pair of vectors given. (ii) State the components of the resultant vector and calculate its magnitude leaving our answers as surds in their simplest form. u u v u v v v (f) u u u v v (g) u v (h) u v (i) u v

45 . (i) Draw diagrams on squared to illustrate a b for each the following pairs of vectors. (ii) (g) (i) State the components of the resultant vector and calculate its magnitude correct to one decimal place. 9 8 a = ; b = a = ; b = 7 7 a = ; b = a = ; b = 8 a = ; b = (f) a = ; b = 6 a = ; b = (h) a = ; b = 7 6 a = ; b = (j) a = ; b =. The diagram shows vectors a, b and c. a b c (i) Draw diagrams on squared paper to represent: a b a c b c (a b) c a (b c) (ii) Calculate, correct to two decimal places: a b a c b c (a b) c a (b c)

46 . The diagram shows vectors, and z. z (i) Draw diagrams to represent: z z ( ) z ( z) (ii) For each resultant vector, state the components and calculate its magnitude correct to one decimal place.. The diagram shows vectors, and z. z (i) Draw diagrams on squared paper to show: z z z (f) z (g) (h) z (careful!) (ii) State the components of each resultant vector above and calculate its magnitude correct to significant figures. 6

47 .a Vector Journes in D Part. Epress each of the following displacements in terms of vectors a and b. PQ QP PR RQ QR R a b Q P. In the diagram AB = DC. Epress each of the following displacements in terms of vectors v and w. D v C CD CA AB w CB BD A B. In the diagram M is the mid point of BC. Epress each of the following displacements in terms of vectors p and q. B CB BC BM AM A q M. EFGH is a parallelogram. M is the mid point of side HG. p C Epress each of the following displacements in terms of vectors a and b. FG GH GM H M G FM b E a F. In the diagram AB is parallel to PR. Q PA = cm and PQ = cm Find in terms of and/or the vectors represented b AQ QB A B 7 P R

48 .b Vector Journes in D Part. Epress in terms of a and b. (i) PS (ii) ST T S If QR = PQ, show that RS can be epressed as a a R (a b) PQ = b P b Q. Epress in terms of vectors v and w. D BD BC AC If v = and w =, find the components w of the displacement AC. C A v B. Epress in terms of p and q. AB AF OF If p = and q = find the components of OF and hence its magnitude correct to decimal A q F p B place. O. Epress in terms of a and b:- (i) AB (ii) AC (iii) OC B If M is the mid-point of OC show that:- AM = = b C M O a A 8

49 . Working With D Coordinates and Vectors. For each diagram, write down the coordinates of the point A and the components of the vector OA. z z A A 7 z z A A 6 7 z (f) z A 8 A 9

50 (g) z (h) z A A (i) (j) z z A A 7 (k) (l) z z A A 8

51 (m) (n) (o) (p) (q) (r) 6 A z A z z 7 8 A 9 6 A z z A 6 z A

52 . Calculate the magnitude of each of the vectors in question correct to one decimal place.. State the coordinates of each verte of the cuboid shown in the diagram. z G F (,, 6) D E C B A. A cube of side 6 units is placed on coordinate aes as shown in the diagram. Write down the coordinates of each verte of the cube. z D C G F E B O A. This shape is made up from congruent trapezia and congruent isosceles triangles. From the information given in the diagram, write down the coordinates of each corner of the shape. z D E(6, 7, 8) O C A B

53 6. State the coordinates of each verte of the square based pramid shown in the diagram. P z PT = S R T O Q 7. A cuboid is placed on coordinate aes as shown. The dimensions of the cuboid are in the ratio OA : AB : BF = : : The point F has coordinates (, p, q) as shown. z G F (, p, q) D C E B O A Establish the values of p and q and write down the coordinates of all the vertices of the cuboid. 8. Write in component form v = i j k w = i 6j k u = 6i k a = j k b = 7i j (f) c = 6j

54 9. For each of these diagrams epress v in terms of i, j and k. 9 6 A z z A A z v v v z 7 8 A v

55 .6 Using Vector Components. For each pair of vectors: (i) Write down the components of u and v. (ii) (iii) (iv) (v) Find the components of the resultant vector u v Find the components of the resultant vector v u Find the components of the resultant vector v u Find the components of the resultant vector v u u u v u v v v (f) u u u v v (g) u v (h) u v (i) u v. u, v and w are vectors with components, and respectivel. Find the components of the following: u v u 6v w v u w u v (f) w u (g) u 6v w (h) u v w (i) u v w. Calculate the magnitude of each of these vectors giving answers to one decimal place:

56 6 = p = 7 v = r = t = 6 u (f) = q (g) = a (h) = b. u, v and w are vectors with components and 8, respectivel. (i) Find the components of the following: u v u 6v w v u w u v (f) w u (g) u 6v w (h) u v w (ii) Calculate the magnitude of each resultant vector above giving answers to decimal place.. (i) If p = i j k and q = i j k, epress the following in component form: p q p q q p p q p q (f) q p (g) p q (h) q p (ii) Calculate the magnitude of each resultant vector above giving answers to decimal place. 6. Calculate the magnitude of these vectors, leaving ou answer a surd in its in simplest form. = u = AB t = i j k t where point T has coordinates(,, ) v = k j 7i 7. Given that v = k i k, u = i aj k have the same magnitude, calculate the value of a if a >.

57 8. A skater is suspended b three wires with forces, 8 and acting on them. Calculate the resultant force and its magnitude correct to significant figures where necessar. 9. If u = and v =, solve each vector equation for. u = v u = v v = u. (i) If r = 6, 6 s = 6 and t =, epress these in component form: r s t s (r s) t r (s t) (ii) Find: r s t s (r s) t r (s t). Two forces are represented b the vectors F = i j k Find the magnitude of the resultant force F F. ~ ~ ~ and F = i k. ~ ~. Two vectors are defined as V = and i j 8 k ~ ~ ~ where a is a constant and all coefficients of Given that V = V, calculate the value of a. ~ ~ ~ V = 8 i j a k ~ ~ ~ i, j and k are greater than zero.. Vector a has components a = k. If a =, calculate the value(s) of k.. Calculate the length of vector a defined as a= i j k.. Vectors a and b are defined b a = i j k and b = i - j. Find the components of a b and calculate its magnitude. 7

58 . Simplifing Fractions Section Algebraic Fractions. Epress these fractions in their simplest form: a (f) 9b 6 (g) 8 (h) c (i) 6c a (j) 8a p (k) p (l) 6ab 6bc a (m) a (n) v t (o) 9vt (p) ab a b p q (q) pq 8 (r) 6 (s) mn 6mn (t) 8def e f (u) ab c a c k m (v) 8km efg (w) e fg () 6. Simplif b first finding the common factor: a 6b 6 a a ab 6 ab 6b (f) 9b (g) b b b (h) pq s (i) a ab ac (j) (k) st 6rs st (l) c ac bc (m) p 8p 8 6 p (n) 8c 6ac d ad (o) 8n n n (p) 6 8

59 9. Simplif the following b first factorising the numerator and/or denominator: b b 9 8 a a c c (f) 6 6 a a (g) p p (h) 9 9 q q (i) b a b a (j) (k) 6 8 m m (l) 8 8 d d (m) (n) p p p (o) a a (p) a a a (q) b b b (r) c c c (s) (t) 8 6 (u) p p p p (v) 6 c c c c (w) () 6 a a a a () b b b b

60 . Calculations Involving Algebraic Fractions. Epress each sum as a fraction in its simplest form: (i) (m) 8 6 (f) (g) (h) 9 6 (j) (k) (l) (n) (o) (p) Epress each difference as a fraction in its simplest form: (i) (m) (f) (g) (h) (j) (k) (l) (n) (o) (p) Epress each product as a fraction in its simplest form: (i) (m) (f) (g) (h) (j) (k) (l) (n) (o) (p)

61 . Epress as a single fraction: (i) (m) (f) (g) (h) (j) (k) (l) (n) (o) (p) 9. Epress each sum as a fraction in its simplest form: (i) (m) (q) a a b b p p 8 6 (f) (g) (h) 9 m m a a 8 (j) (k) (l) p p a b (n) (o) (p) m n p q c d 9 7 (r) (s) (t) a b a b m n p 6q (u) a a (v) (w) b b () 8 m m 6. Epress each difference as a fraction in its simplest form: (i) a a b b p p (f) (g) (h) 9 m m a a 8 (j) (k) (l) 8 p p a b 6

62 6. (continued) (m) (q) (n) (o) (p) m n p q c d 7 (r) (s) (t) a b a b m n p 6q (u) a a (v) 7 (w) b b () 7 p p 7. Epress each product as a fraction in its simplest form (i) a b p q c c 6 (f) (g) (h) 6 a a p p m (j) (k) (l) m m b c 6 m 7 (m) (n) 9 a 7a (o) p p (p) t s s 6t pq 7ab c m (q) (r) (s) pq 6c a mn n z z ab a cd a (t) (u) (v) 9 c bc 7a cd st (w) () 8s t () pq 6a a p 6

63 8. Epress as a single fraction: (g) a a ab a p p c c 6 (f) 6 t t 9 (h) k m (i) bc c (j) (k) 6q 9q (l) z z p p (m) 8ab 9b (n) c ac m 8mn a (o) n 9 a 9. Simplif the following: 6 a 6 a d d 6 a a a b a b (f) u v u v (g) (h) (i) 7 (j) (k) (l) 6

64 . Algebraic Fractions - EXAM QUESTIONS. Write as a single fraction in its simplest form :,.. Simplif this fraction 9. Simplif full the fraction 6e e e. Simplif. Write as a single fraction in its simplest form: a a 6. Epress as a single fraction in its simplest form:. 6

65 . Areas of Triangles Section Triangle Formulae The area of a triangle : A = absin C 6

66 . Find the area of the following triangles : 6cm cm o cm 6 o cm 7cm cm cm o cm o 8cm (f) 7cm (g) o 8cm 9 o cm 8 o 8 7cm cm (h) (i) (j) 8cm 79 o 6 cm cm o cm o cm 8cm 67 o. Mr. Fields is planting a rose-bed in his garden. It is to be in the shape of an equilateral triangle of side m. LAWN rose-bed LAWN What area of lawn will he need to remove to plant his rose-bed?. Calculate the area of triangle ABC where AB = cm, AC = 7cm, ABC = o and BCA = 7 o. A cm B o 7cm 7 o C 66

67 6. For safet reasons the sides of a footbridge are to be covered with triangular panels. Each panel is an isosceles triangle as shown. 7m 7m 7 o Find the area of each panel. If there are 7 panels on each side of the bridge, find the total area of material required to cover the bridge. 7. Given that the area of this triangle is cm, calculate the size of the obtuse angle ABC. 8cm B cm A C 8. In triangle ABC, AB = m and AC = m. Angle BAC = o. A o m C m B Given that sin o =, calculate the area of triangle ABC. 9. The area of a triangular flag is 9 cm. Calculate the size of the obtuse angle ABC. B cm cm A C 67

68 . Using the Sine Rule to Calculate Sides. Use the sine rule to calculate the side marked in each triangle below. cm 67 o o cm o 8 o o o 7 cm o o cm 6 o m o (f) (g) (h) o mm 6 cm cm o o 78 o 6 o o 7 m. Use the sine rule to calculate the length of the side marked in each of the triangles below. o o cm cm 6cm 6 o 6 o o 8cm 8 o o 6cm 88 o o 9 (f) (g) o (h) 7cm 6 o 9 cm 8 o 8 o o o o (i) (j) (k) 9 o o cm o 9cm o 8 o 68 7cm 99 o 7 cm

69 . Using the Sine Rule to Calculate Angles. Use the sine rule to calculate the length of the angle marked o in each of the triangles below. 7 o 8cm cm 6cm o 66 o o 8cm cm cm cm o cm (g) 9 o o 87 o 8 o (h) 6 cm o 8 cm. Use the sine rule to calculate the size of the angle marked in each triangle below. o cm 6 6cm o o 6cm cm 6 o (f) o cm 9cm o cm o 8 cm cm 7 o 8 cm mm mm 8 o cm (f) o 7 mm mm cm careful! o 6 o 8 cm cm 69

70 . Using the Cosine Rule to Calculate Sides. Use the cosine rule to calculate the side marked in each triangle below. cm 7 cm 8 o 8 cm cm cm o 67 o 6 cm 7 m 6 mm o mm 6 o m m 7 o (f) (g) (h) 8 mm cm 6 cm m mm o o. Use the cosine rule to calculate the length of the side marked in each of the triangles below. cm o cm 6cm cm 6cm 6 o 8cm 8 o cm o 7cm (f) (g) (h) 88 o cm 9 cm 7cm 8 cm 9cm 8 o 6cm o cm (i) (j) (k) o o cm 9 o o 99 o 9cm 9 cm 7 cm o 7 8 o 7cm 7 cm 8cm

71 . Using the Cosine Rule to Calculate Angles. Use the nd form of the cosine rule to calculate the size of the angle marked in each triangle below. cm 8 cm cm cm 7 cm cm mm mm mm (f) 7 cm mm 7 mm cm 6 cm cm mm cm cm. Use the cosine rule to calculate the angle marked o in each of the triangles below. o 6cm 8cm o 8cm cm cm cm o cm 8cm 8cm 9cm o 8cm 6cm o (f) cm cm 7cm (g) 8 cm cm cm o 7 cm 9cm (h) 9cm o 8cm o cm cm 7

72 .6 Choosing the Correct Formula. Calculate the value of in each triangle below.. Calculate the area of the triangle with sides measuring cm, cm and cm.. Calculate the length of BD. Calculate the length of AD. Calculate the area of triangle ABC. From the framework opposite: Calculate the length of AC. Calculate the size of BAC. A cm o B 9 o cm C Write down the size of ACD. Calculate the length of AD. Calculate the area of the quadrilateral ABCD. 8 o D 7

73 . Two golfers are aiming for the green. The golfers are 6 m apart and the angles are as shown in the diagram. What distance will each golfer have to hit the ball in order to reach the pin? o 78 o GOLFER GOLFER 6 m 6. An aircraft is picked up b two radar stations, P and Q, km apart. How far is the aircraft from station P? STATION P 8 o 8 o STATION Q 7. A large crane is being used in the construction of a block of flats. The crossbeam is supported b two metal stas. B A B C m m A 6 o C The length of AB is m and the length of BC is m. BCA is 6 o. Calculate the size of BAC and the length of the crossbeam AC. 8. A hot air balloon B is fied to the ground at F and G b ropes m and m long. If FBG is 86 o, how far apart are F and G? m B 86 o m F G 7

74 9. 7 cm 7 cm o A set of compasses is shown where the angle between the arms is set at o Calculate the diameter of the circle which could be drawn with the arms in this position.. During a golf match, Ian discovers that he has forgotten his sand wedge, so to avoid the bunker he plas a shot from T to F and then from F to G. T BUNKER G His opponent Fred decides to pla directl from T to G. m o 9m How far will Fred need to hit his shot to land at G? F. N A N km 6 o 6 km The diagram shows the path of an aircraft from Glasgow to Aberdeen, a distance of km and then from Aberdeen to Edinburgh, a distance of 6 km. G E Calculate the distance from Glasgow to Edinburgh.. N B km The diagram shows the path of an aircraft from A to B to C. N 7 km Write down the size of ABC. A 8 o o C Calculate the distance AC. 7

75 .7 Triangle Formulae - EXAM QUESTIONS. The sketch below shows a plot of land purchased to build a house on. m 7 o m o m At present the land is valued 8 per square metre. Calculate the value of the plot shown to the nearest.. The distance from the centre of a regular octagon to one of its vertees is cm. Calculate the area of the octagon. cm. Two securit cameras are positioned on a beam in a warehouse metres apart. One camera has an angle of depression of 7 and the other camera has an angle of depression of 6. Calculate the height, h metres, of the beam above the ground. 6 m h m 7 7

76 . Triangle PQR has sides with lengths, in centimetres, as shown. P 8 R Q Show clearl that cos PQR = 7.. A flagpole is attached to a wall and is supported b a wire PQ as shown in the diagram. The wire is metres long and makes an angle of o with the vertical wall. cm o P Q Given that the point P is metres above R in the diagram, calculate the length of the flagpole. R 6. A triangular sail designed for a racing acht is shown below. Two of its edges measure 6 metres and metres. 6m m Given that the sail has a perimeter of metres, calculate the area of the sail. 76

77 7. A sketch of Lee's garden is shown below. B 8 m C 6 7m m A D Calculate the size of angle ABC. Hence, or otherwise, calculate the area of the garden. 8. The diagram below shows a steel plate ABCD. B 6cm C AB = 6cm, AD= 8cm A 8cm D o E o DAB = 9 Calculate the length of BD correct to decimal place. Find the size of angle BDC correct to the nearest degree. Hence calculate the length of BC given that DC= cm. 9. In triangle PQR, PR = cm QR = cm. The perimeter of the triangle is cm. cm Q Find the size of angle PQR. R cm P 77

78 . In the diagram shown SR = cm, angle SQR = 7 o, angle QPS = o and angle PQS = 68 o. Q P o 68 o 7 o S cm R Calculate the length of PS.. Calculate the size of angle BAC in this triangle. 8m B A m o C. In the diagram ABCD represents a steel framework with BCD being a triangular steel plate. Angle ADB is a right angle. AB= 6 cm, BAC=7, BDC= and DCB= 8 8 o C D o Find the length of DB. Calculate the area of triangle BCD. A 7 o 6cm B 78

79 . A triangular sail has measurements as shown in the diagram. All lengths are in metres. Calculate the size of the largest angle in the triangle. 8m 6m Calculate the area of the sail in square metres. m. A building compan has to fence off a triangular piece of waste ground. The plan of the ground is shown below. All lengths are in metres. 6m 6 o m If the fence costs 8. per metre to erect, how much will the compan have to pa in total to fence off this piece of ground? {Fencing is priced in whole metres onl}. Calculate the value of cos ABC in this triangle. B 7cm cm A cm C Without actuall calculating the size of the angle a pupil was able to sa that angle ABC was obtuse. B referring to our answer in, eplain wh the pupil was able to do this. 79

80 .8 Problems Involving Bearings. Cop the bearing diagram opposite fill in as man angles as ou can. N A o N Now answer the following questions.. What is the bearing of (i) B from A (ii) A from B o N o B (iii) (iv) (v) C from B A from C C from A C (vi) B from C N. Repeat question. for this bearing diagram. A N 7 o 8 o B o N C. A ship sails from harbour H on a bearing of 8 o for km until it reaches point P. It then sails on a bearing of o for 6km until it reaches point Q. Calculate the distance between point Q and the harbour. On what bearing must the ship sail to return directl to the harbour from Q? H N 8 o km N P o N 6km Q 8

81 . A and B represent two forest look-out towers. A is km and on a bearing of o from B. B A forest fire is sighted at F, on a bearing of 7 o from A and o from B. A F A fire-fighting helicopter leaves A for F. What distance does this helicopter have to travel to reach the fire?. A surveor is walking due west when he comes to a marsh. To avoid the marsh he turns at P and walks for 6 metres on a bearing of o and then for 8 9 o. He then calculates the distance PR, the direct distance across the marsh. What answer should he get? metres on a bearing of 6. Two ships leave Liverpool at the same time. One of them travels north-west at an average speed of km/h while the other travels at an average speed of km/h on a bearing of 8 o. How far apart are these ships after hours? 7. A ship leaves a port on a bearing of 7º and sails 6km. The ship then changes course and sails a further 6km on a bearing of o where it anchors. When it anchors it is 9km from the port. Calculate the bearing of the ship from the port at this point. N N 6km 6km PORT 9km 8

82 8. A ship's captain is plotting a course for the net voage. He knows that he has to sail from Port D to port E on a bearing of 67 o for a distance of 8km and from there to Port F on a bearing of o. His course is shown in the diagram below. N N 8km E o 67 o D F Make a cop of the diagram and calculate the size of angle DEF. New instructions come through which inform the captain that he has to sail directl from Port D to Port F, a distance of 7km. Calculate the bearing on which the ship should sail in order to carr out these instructions. Give the bearing to the nearest degree. 9. A ship is at position A. Lighthouse L is on a bearing of o from the ship. L N o A N 6 km B The ship then travels 6 kilometres on a bearing of o to position B. From position B the captain now observes the lighthouse on a bearing of o. Calculate the distance between the ship and the lighthouse when the ship is at position B. 8

83 . Two students, All and Cameron are plaing football and at one point the are in the positions shown in the diagram. All (A) is m due west of Cameron (C). The are both facing North. B A C The ball (B) is on a bearing of 6 o from A and on a bearing of o from C. B N N A m C Make a cop of the above diagram and mark the sizes of the angles in the triangle. Calculate how far Cameron is awa from the ball. 8

84 .9 Problems Involving Bearings - EXAM QUESTIONS. The diagram below, which is not drawn to scale, represents the positions of three mobile phone masts. Mast Q is on a bearing of o from mast P and is km awa. The bearing of mast R from mast Q is o. Masts P and R are 66km apart. N N P o km Q o 66 km N R Use the information in the diagram to establish the size of angle PQR. Hence find the bearing of mast P from mast R.. A par hole on a golf course the tee is a distance of metres due west from the pin. On his first shot, Bruce hits the ball metres but not at the correct angle. On his second shot he hits the ball metres and gets it in the hole. On what bearing, a o, did he hit his first stroke? N a o m m m 8

85 . A helicopter sets out from its base P and flies on a bearing of o to point Q where it changes course to 6 o and flies 8 km to point R. Rel When the helicopter is at point R it is km from its starting point. N N km R o P N 8km 6 o Q Find the size of angle PQR. Calculate the bearing on which the helicopter must fl to return directl to its base i.e. the shaded angle in the diagram. Give answers to the nearest whole number throughout our calculations.. Brampton is 7 kilometres due east of Abbott. The bearing of Corwood from Abbott is o and from Brampton is 9 o. N Corwood N N Abbott 7 km Brampton Make a neat cop of the diagram and fill in all three angles inside the triangle. Calculate the distance between Corwood and Brampton, to the nearest kilometre. 8

86 . The diagram shows part of a golf course where plaers have to get the ball from the tee (T) to the pin (P). The can either pla one stroke across the lake or pla stroke from T to B then another from B to P which avoids the lake. Harr decides to take the stroke option and hits his first shot on a bearing of 6 o or a distance of 7metres. For his second shot he hits the ball on a bearing of 7 o from B to P. N Rel B 7 o N 7m T 6 o 8m P Calculate the size of angle TBP. The distance TP is 8 metres. David decided to attempt to hit his ball across the lake. Calculate the bearing on which he would have to hit the ball to achieve this. 6. N Two ships, the Argent and the Gelt leave port Banco at the same time. Banco Argent The Argent follows a course of o for km and the Gelt travels on a course of 8 o for km. Calculate the distance between the two ships. Gelt 86

87 7. A ship s mate is planning the course for a voage. The course is shown in the diagram below. He knows that he has to sail from Port A to Port B on a bearing of 77 o and from there to Port C on a bearing of 7 o for km. In order to return to port A the ship has to sail on a bearing of 8 o. Rel N B 7 o N 77 o A km N C 8 o Calculate how far the ship will have to sail to return to its starting point. 8. Three oil platforms, Alpha, Gamma and Delta are situated in the North Sea as shown in the diagram below. The distances between the oil platforms are shown in the diagram. N N 9 km Gamma Alpha 7 km N 6 km Delta If the bearing of Delta from Alpha is o, what is the bearing of Gamma from Alpha? 87

88 9. Two coastguard stations, P and Q, are km apart. Q is due East of P. A ship, S, is at a distance of 8 km from P and km from Q. Rel S N 8km km N P km Q Calculate the size of angle SPQ. Hence calculate the bearing of the ship S from station P.. The diagram below shows the positions of three radar stations Alpha, Beta and Delta. The bearing of Beta from Alpha is o. N BETA N o km 7km ALPHA N 8km Calculate the bearing of Delta from Alpha. DELTA 88

89 Section 6 Trigonometric Functions 6. Drawing Graphs of Trig Functions. With the help of a calculator, cop and complete the table below. o sin o Plot the points from our table. Join the points with a smooth curve. Write down the equation of the curve.. With the help of a calculator, cop and complete the table below. o cos o Plot the points from our table. Join the points with a smooth curve. Write down the equation of the curve... With the help of a calculator, cop and complete the table below. o tan o Plot the points from our table.(be careful with the scale on the -ais) Join the points with a smooth curve. Write down the equation of the curve. 89

90 6.a Equations of Trig Functions (). The graphs represent the functions a sin o and a cos o. Write down the equation for each o o o (f) ¼ o o o ¼ (g) (h) (i) o o o (j) (k) (l) o o o ¾ 6 (m) (n) (o) o o o 8 ¾ 6 9

91 . The graphs represent trigonometric functions. Write down the equation for each o o o o 7 (f) o o 7 (g) 6 (h) o o 6 (i) (j) o o 9 9