year 12 hl summer work

Year 12MathematicsSummer Work

Higher Level

Higher Level

In order to prepare affectively for the Higher Level course students should aim to master the follow topics which are covered in their summer preparation pack;

Basic Trigonometry

Be able to find a missing side on a right angled triangle, given an angle and a side.

Be able to find a missing angle in a right angled triangle, given two of its sides.

Sine and Cosine Rule

Be able to apply both the Cosine rule and Sine rule to find missing sides and angles of non-rightangled triangles.

Calculus

Be able to differentiate a given function (up to and including a cubic function), including the useof fractions.

Further research into differentiation would be of great benefit here. It is suggested that you visit the following link or find an alternative source; http://www.dummies.com/how-to/content/the-basic-differentiation-rules.html

Probability

Be able to complete tree diagrams and use them to answers questions on combined events.

Know and be able to apply the and/or rules.

Be able to find the probability of combined events, given a condition.

Know what is meant by independence.

Sets & Venn Diagrams

Rules of Indices

Know and be able to manipulate the rules of indices.

Transformations of graphs

Be able to transform graphs by f(x+a) and af(x) and understand what happens to the transformed graphs.

Reciprocal graphs

Be able to draw reciprocal graphs and find asymptotes.

Rules of Logarithms

To be able to simplify logarithms

Be able to interchange between exponents and logarithms.

Vectors and Vector notation

Be able to add/subtract vectors.

Be able to solve worded vector questions linked to real life problems.

Research topics

In addition, students aiming to complete the Higher Level course should develop a rounded understanding of the following topics by researching them on www.myimaths.com

and www.mathsnetib.com;

Applications of Calculus

Complex Numbers

Students should prepare for their initial assessment by reviewing all aspects of algebra and graphs and some number work from the IGCSE specification as well as some topics from Further Mathematics. Some of these topics are outlined below;

Simultaneous EquationsDimensionsAlgebraic fractionsSolving algebraic equationsCircle theoremsFactorising and expandingFactors and multiplesAreas and sectors of circlesVectorsSurds and RationalisingSine Rule, Cosine Rule and area of a triangleDifferentiation and turning pointsFunctions, including Sine and Cosine graphsUnderstand graphs quadratic and cubicTrigonometryLogarithmsRules of IndicesCompleting the squareStraight line graphsInequalities

In September, you will be given a skills test on the above topics to assess your suitability for the Higher Level course. Thereafter, your HL teacher will provide guidance on your suitability for the course advise taking SL Mathematics or Mathematical Studies.

The skills test will comprise of the following;

Paper 1 Non Calculator paper (45minutes)Paper 2 Calculator paper (45 minutes)

Within your preparation for IB Higher Level Mathematics you should login to

www.mathsnetib.com and become familiar with its usefulness, functionality and features.www.mathsnetib.com login: wellingtonin password: ibhelp

Once you are logged into www.mathsnetib.com you should create an account in your account and create personal account.

COORDINATE GEOMETRY

1. The coordinates of the vertices of a triangle ABC are A (4 , 3) , B (7, 3) and C (0.5, p).

(a) Calculate the gradient of the line AB.(2)

(b) Given that the line AC is perpendicular to the line AB

(i) write down the gradient of the line AC;

(ii) find the value of p.(4)

(Total 6 marks)

2. The equation of the line R1 is 2x + y 8 = 0. The line R2 is perpendicular to R1.

(a) Calculate the gradient of R2.(2)

The point of intersection of R1 and R2 is (4, k).

(b) Find

(i) the value of k;

(ii) the equation of R2.(4)

(Total 6 marks)

3. The straight line L passes through the points A(1, 4) and B(5, 8).

(a) Calculate the gradient of L.(2)

(b) Find the equation of L.(2)

The line L also passes through the point P(8, y).

(c) Find the value of y.(2)

(Total 6 marks)

4. The straight line, L1, has equation y = 21

x 2.

(a) Write down the y intercept of L1.(1)

(b) Write down the gradient of L1.(1)

The line L2 is perpendicular to L1 and passes through the point (3, 7).

(c) Write down the gradient of the line L2.(1)

(d) Find the equation of L2. Give your answer in the form ax + by + d = 0 where a, b, d .

(3)(Total 6 marks)

5. The diagram below shows the line PQ, whose equation is x + 2y = 12. The line intercepts the axes at P and Q respectively.

diagram not to scale

(a) Find the coordinates of P and of Q.(3)

(b) A second line with equation x y = 3 intersects the line PQ at the point A. Find the coordinates of A.

(3)(Total 6 marks)

6. The following diagrams show six lines with equations of the form y = mx + c.

In the table below there are four possible conditions for the pair of values m and c.Match each of the given conditions with one of the lines drawn above.

Condition Line

m > 0 and c > 0

m < 0 and c > 0

m < 0 and c < 0

m < 0 and c < 0(Total 6 marks)

7. The points A(4, 1), B(0, 9) and C(4, 2) are plotted on the diagram below. The diagram also shows the lines AB, L1 and L2.

(a) Find the gradient of AB.(2)

L1 passes through C and is parallel to AB.

(b) Write down the y-intercept of L1.(1)

L2 passes through A and is perpendicular to AB.

(c) Write down the equation of L2. Give your answer in the form ax + by + d = 0where a, b and d .

(3)

(d) Write down the coordinates of the point D , the intersection of L1 and L2.(1)

There is a point R on L1 such that ABRD is a rectangle.

(e) Write down the coordinates of R.(2)

The distance between A and D is 45 .

(f) (i) Find the distance between D and R.

(ii) Find the area of the triangle BDR.(4)

(Total 13 marks)

8. A and B are points on a straight line as shown on the graph below.

(a) Write down the y-intercept of the line AB.(1)

(b) Calculate the gradient of the line AB.(2)

The acute angle between the line AB and the x-axis is .

(c) Show on the diagram.(1)

(d) Calculate the size of .(2)

(Total 6 marks)

9. A line joins the points A(2, 1) and B(4, 5).

(a) Find the gradient of the line AB.

Let M be the midpoint of the line segment AB.

(b) Write down the coordinates of M.

(c) Write down the gradient of the line perpendicular to AB.

(d) Find the equation of the line perpendicular to AB and passing through M.

(Total 8 marks)

CALCULUS

\1. Let f(x) = 2x2 + x 6

(a) Find f(x).(3)

(b) Find the value of f(3).(1)

(c) Find the value of x for which f(x) = 0.(Total 6 marks)

2. The figure shows the graphs of the functions f(x) = 41

x2 2 and g(x) = x.

(a) Differentiate f(x) with respect to x.(1)

(b) Differentiate g(x) with respect to x.(1)

(c) Calculate the value of x for which the gradients of the two graphs are the same.(2)

(d) Draw the tangent to the parabola at the point with the value of x found in part (c).(2)

(Total 6 marks)

3. A function is represented by the equation

f (x) = ax2 + x4

3.

(a) Find f (x).(3)

The function f (x) has a local maximum at the point where x = 1.

(b) Find the value of a.(3)

(Total 6 marks)

4. (a) Differentiate the following function with respect to x:

f (x) = 2x 9 25x1

(b) Calculate the x-coordinates of the points on the curve where the gradient of the tangent to the curve is equal to 6.

(Total 6 marks)

5. (a) Differentiate the function y = x2 + 3x 2.

(b) At a certain point (x, y) on this curve the gradient is 5. Find the co-ordinates of this point.

(Total 6 marks)

6. The curve y = px2 + qx 4 passes through the point (2, 10).

(a) Use the above information to write down an equation in p and q.(2)

The gradient of the curve y = px2 + qx 4 at the point (2, 10) is 1.

(b) (i) Find xy

dd

.

(ii) Hence, find a second equation in p and q.(3)

(c) Solve the equations to find the value of p and of q.(3)

(Total 8 marks)

7. A function is defined by f(x) = 25x + 3x + c, x 0, c .

(a) Write down an expression for f(x).(4)

Consider the graph of f. The graph of f passes through the point P(1, 4).

(b) Find the value of c.(2)

(c) There is a local minimum at the point Q.

(i) Find the coordinates of Q.

(ii) Find the set of values of x for which the function is decreasing.(7)

Let T be the tangent to the graph of f at P.

(d) (i) Show that the gradient of T is 7.

(ii) Find the equation of T.(4)

(e) T intersects the graph again at R. Use your graphic display calculator to find the coordinates of R.

(2)(Total 19 marks)

8. A dog food manufacturer has to cut production costs. She wishes to use as little aluminium as possible in the construction of cylindrical cans. In the following diagram, h represents the height of the can in cm, and x represents the radius of the base of the can in cm.


The volume of the dog food cans is 600 cm3.

(a) Show that h = 2600x .

(2)

(b) (i) Find an expression for the curved surface area of the can, in terms of x.Simplify your answer.

(ii) Hence write down an expression for A, the total surface area of the can, in terms of x.

(4)

(c) Differentiate A in terms of x.(3)

(d) Find the value of x that makes A a minimum.(3)

(e) Calculate the minimum total surface area of the dog food can.(2)

(Total 14 marks)

RIGHT ANGLED TRIGONOMETRY

1. In the diagram, AD = 4 m, AB = 9 m, BC = 10 m, ADB = 90 and CBD = 100.


(a) Calculate the size of CBA .(3)

(b) Calculate the length of AC.(3)

(Total 6 marks)

2. The diagram represents a small, triangular field, ABC, with BC = 25 m, angle BAC = 55 and angle ACB = 75.


(a) Write down the size of angle ABC.(1)


(c) Calculate the area of the field ABC.(3)

N is the point on AB such that CN is perpendicular to AB. M is the midpoint of CN.

(d) Calculate the length of NM.(3)

A goat is attached to one end of a rope of length 7 m. The other end of the rope is attached to the point M.

(e) Decide whether the goat can reach point P, the midpoint of CB. Justify your answer.(5)

(Total 15 marks)

3. Jos stands 1.38 kilometres from a vertical cliff.

(a) Express this distance in metres.(1)

Jos estimates the angle between the horizontal and the top of the cliff as 28.3 and uses it to find the height of the cliff.


(b) Find the height of the cliff according to Joss calculation. Express your answer in metres, to the nearest whole metre.

(3)

(c) The actual height of the cliff is 718 metres. Calculate the percentage error made by Jos when calculating the height of the cliff.

(2)(Total 6 marks)

4. The diagram shows triangle ABC. Point C has coordinates (4, 7) and the equation of the line AB is x + 2y = 8.


(a) Find the coordinates of

(i) A;

(ii) B.(2)

(b) Show that the distance between A and B is 8.94 correct to 3 significant figures.(2)

N lies on the line AB. The line CN is perpendicular to the line AB.

(c) Find

(i) the gradient of CN ;

(ii) the equation of CN.(5)

(d) Calculate the coordinates of N.(3)

It is known that AC = 5 and BC = 8.06.

(e) Calculate the size of angle ACB.(3)

(f) Calculate the area of triangle ACB.(3)

(Total 18 marks)

5. The diagram shows a point P, 12.3 m from the base of a building of height h m. The angle measured to the top of the building from point P is 63.

1 2 . 36 3 P

h m

(a) Calculate the height h of the building.

Consider the formula h = 4.9t2, where h is the height of the building and t is the time in seconds to fall to the ground from the top of the building.

(b) Calculate how long it would take for a stone to fall from the top of the building to theground.

(Total 6 marks)

6.

A

B C

D

3 c m

4 . 5 c m

2 5

3 c m

In the diagram, AB = BC = 3 cm, DC = 4.5 cm, angle CBA = 90 and angle DCA = 25.

(a) Calculate the length of AC.

(b) Calculate the area of triangle ACD.

(c) Calculate the area of quadrilateral ABCD.(Total 8 marks)

SETS AND VENN DIAGRAMS

1. U is the set of all the positive integers less than or equal to 12.A, B and C are subsets of U.

A = {1, 2, 3, 4, 6,12}B = {odd integers}C = {5, 6, 8}

(a) Write down the number of elements in A C.(1)

(b) List the elements of B.(1)

(c) Complete the following Venn diagram with all the elements of U.

(4)(Total 6 marks)

2. One day the number of customers at three cafs, Alans Diner (A), Sarahs Snackbar (S) and Petes Eats (P) was recorded and are given below.

17 were customers of Petes Eats only27 were customers of Sarahs Snackbar only15 were customers of Alans Diner only10 were customers of Petes Eats and Sarahs Snackbar but not Alans Diner8 were customers of Petes Eats and Alans Diner but not Sarahs Snackbar

(a) Draw a Venn Diagram, using sets labelled A, S and P, that shows this information.(3)

There were 48 customers of Petes Eats that day.

(b) Calculate the number of people who were customers of all three cafs.(2)

There were 50 customers of Sarahs Snackbar that day.

(c) Calculate the total number of people who were customers of Alans Diner.(3)

(d) Write down the number of customers of Alans Diner that were also customers of Petes Eats.

(1)

(e) Find n[(S P) A].(2)

(Total 11 marks)

3. The sets P, Q and U are defined as

U = {Real Numbers}, P = {Positive Numbers} and Q = {Rational Numbers}.

Write down in the correct region on the Venn diagram the numbers

722

, 5 102 , sin(60) , 0 , 3 8 ,

(Total 6 marks)

4. A fitness club has 60 members. 35 of the members attend the clubs aerobics course (A) and 28 members attend the clubs yoga course (Y). 17 members attend both courses.A Venn diagram is used to illustrate this situation.

(a) Write down the value of q.(1)

(b) Find the value of p.(2)

(c) Calculate the number of members of the fitness club who attend neither the aerobics course (A) nor the yoga course (Y).

(2)

(d) Shade, on your Venn diagram, A Y.(1)

(Total 6 marks)

5. 100 students are asked what they had for breakfast on a particular morning. There were three choices: cereal (X), bread (Y) and fruit (Z). It is found that

10 students had all three17 students had bread and fruit only15 students had cereal and fruit only12 students had cereal and bread only13 students had only bread8 students had only cereal9 students had only fruit

(a) Represent this information on a Venn diagram.(4)

(b) Find the number of students who had none of the three choices for breakfast.(2)

(c) Write down the percentage of students who had fruit for breakfast.(2)

(d) Describe in words what the students in the set X Y had for breakfast.(2)

(e) Find the probability that a student had at least two of the three choices for breakfast.(2)

(f) Two students are chosen at random. Find the probability that both students had all three choices for breakfast.

(3)(Total 15 marks)

6. A group of 30 students were asked about their favourite topping for toast.

18 liked peanut butter (A)10 liked jam (B)6 liked neither

(a) Show this information on the Venn diagram below.

(2)

(b) Find the number of students who like both peanut butter and jam.(2)

(c) Find the probability that a randomly chosen student from the group likes peanut butter, given that they like jam.

(2)(Total 6 marks)

SINE AND COSINE RULE

1. In the diagram, AD = 4 m, AB = 9 m, BC = 10 m, ADB = 90 and CBD = 100.


(a) Calculate the size of CBA .(3)


(Total 6 marks)

2. The diagram shows a triangle ABC in which AC = 17 cm. M is the midpoint of AC.Triangle ABM is equilateral.


(a) Write down

(i) the length of BM in cm;

(ii) the size of angle BMC;

(iii) the size of angle MCB.(3)

(b) Calculate the length of BC in cm.(3)

(Total 6 marks)

3. In the diagram below A , B and C represent three villages and the line segments AB, BC and CA represent the roads joining them. The lengths of AC and CB are 10 km and 8 km respectively and the size of the angle between them is 150.


(a) Find the length of the road AB.(3)

(b) Find the size of the angle CAB.(3)

Village D is halfway between A and B. A new road perpendicular to AB and passing through D is built. Let T be the point where this road cuts AC. This information is shown inthe diagram below.


(c) Write down the distance from A to D.(1)

(d) Show that the distance from D to T is 2.06 km correct to three significant figures.(2)

A bus starts and ends its journey at A taking the route AD to DT to TA.

(e) Find the total distance for this journey.(3)

The average speed of the bus while it is moving on the road is 70 km h1.The bus stops for 5 minutes at each of D and T.

(f) Estimate the time taken by the bus to complete its journey. Give your answer correct to the nearest minute.

(4)(Total 16 marks)

4. The diagram shows triangle ABC in which angle CAB = 30, BC = 6.7 cm and AC = 13.4 cm.


(a) Calculate the size of angle BCA .(4)

Nadia makes an accurate drawing of triangle ABC. She measures angle CAB and finds it to be 29.

(b) Calculate the percentage error in Nadias measurement of angle CAB .(2)

(Total 6 marks)

5. A farmer has a triangular field, ABC, as shown in the diagram.AB = 35 m, BC = 80 m and BC =105, and D is the midpoint of BC.


(a) Find the size of BA.(3)

(b) Calculate the length of AD.(5)

The farmer wants to build a fence around ABD.

(c) Calculate the total length of the fence.(2)

(d) The farmer pays 802.50 USD for the fence. Find the cost per metre.(2)

(e) Calculate the area of the triangle ABD.(3)

(f) A layer of earth 3 cm thick is removed from ABD. Find the volume removed in cubicmetres.

(3)(Total 18 marks)

6. Triangle ABC is such that AC is 7 cm, angle CBA is 65 and angle BCA is 30.

(a) Sketch the triangle writing in the side length and angles.(1)

(b) Calculate the length of AB.(2)

(c) Find the area of triangle ABC.(3)

(Total 6 marks)

PROBABILITY

1. Amy has 10 CDs in a CD holder.Amys favourite group is Edex.She has 6 Edex CDs in the CD holder.

Amy takes one of these CDs at random.She writes down whether or not it is an Edex CD.She puts the CD back in the holder.Amy again takes one of these CDs at random.

(a) Complete the probability tree diagram.

0 . 6

. . . . . . . . . .

S e c o n d c h o i c e

. . . . . . . . . .

F i r s t c h o i c e

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

E D E XC D

E D E XC D

E D E XC D

N O T - E D E XC D

N O T - E D E XC D

N O T - E D E XC D

(2)

Amy had 30 CDs.The mean playing time of these 30 CDs was 42 minutes.

Amy sold 5 of her CDs.The mean playing time of the 25 CDs left was 42.8 minutes.

(b) Calculate the mean playing time of the 5 CDs that Amy sold.

......................... minutes(3)

(Total 5 marks)

2. Amy is going to play one game of snooker and one game of billiards.

The probability that she will win the game of snooker is 43

The probability that she will win the game of billiards is 31

Complete the probability tree diagram.

(Total 2 marks)

3. Jacob has 2 bags of sweets.

Bag P Bag Q

Bag P contains 3 green sweets and 4 red sweets.Bag Q contains 1 green sweet and 3 yellow sweets.

Jacob takes one sweet at random from each bag.

(a) Complete the tree diagram.

37

B a g P B a g Q

g r e e n

r e d

g r e e n

y e l l o w

g r e e n

y e l l o w(2)

(b) Calculate the probability that Jacob will take 2 green sweets.

.(2)

(Total 4 marks)

4. William has two 10-sided spinners.The spinners are equally likely to land on each of their sides.

B L U E R E D

R E D

R E D

R E D

R E D

B L U EG R E E N

B L U E G R E E N

B L U E R E D

R E DB L U E

B L U E

B L U E B L U E

B L U E

B L U E

A B

G R E E N

Spinner A has 5 red sides, 3 blue sides and 2 green sides.Spinner B has 2 red sides, 7 blue sides and 1 green side.

William spins spinner A once.He then spins spinner B once.

Work out the probability that spinner A and spinner B do not land on the same colour.

..........................(Total 4 marks)

INDICES

1. (a) Expand and simplify 3(x + 4) + 5(2x + 1)

.....................................(2)

(b) Simplify t4 t6

.....................................(1)

(c) Simplify p8 p5

.....................................(1)

(d) Simplify (x4)3

.....................................(1)

(Total 5 marks)

2. (a) Simplify t6 t2

.............................(1)

(b) Simplify3

8

mm

.............................(1)

(c) Simplify (2x)3

........................................(2)

(d) Simplify 3a2h 4a5h4

........................................(2)

(Total 6 marks)

RECIPROCAL GRAPHS

1.

Write down the letter of the graph which could have the equation

(i) y = 1 3x

......................................

(ii) y = x1

......................................

(iii) y = 2x2 + 7x + 3

......................................(Total 3 marks)

2. (a) Complete the table of values for y = x1

x 0.2 0.4 0.8 1.0 2.0 4.0

y 5.0 1.25 1.0(2)

(b) On the grid, draw the graph of y = x1

for x > 0.2

6

5

4

3

2

1

1 2 3 4O

y

x

(2)(Total 4 marks)

TRANSFORMATIONS OF GRAPHS

1. (a) Expand and simplify 3(x + 4) + 5(2x + 1)

.....................................(2)

(b) Simplify t4 t6

.....................................(1)

(c) Simplify p8 p5

.....................................(1)

(d) Simplify (x4)3

.....................................(1)

(Total 5 marks)

2. (a) Simplify t6 t2

.............................(1)

(b) Simplify3

8

mm

.............................(1)

(c) Simplify (2x)3

........................................(2)

(d) Simplify 3a2h 4a5h4

........................................(2)

(Total 6 marks)

3. The graph of y = f(x) is shown on the grids.

(a) On this grid, sketch the graph of y = f(x) + 2

(2)

(b) On this grid, sketch the graph of y = f(x)

(2)(Total 4 marks)

4. x2 8x + 23 = (x p)2 +q for all values of x.

(a) Find the value of p and the value of q.

p = ....................................

q = ....................................(3)

Here is a sketch of the curve with equation y = x2 8x + 23

x

y

O

B

B is the minimum point on the curve.

(b) Find the coordinates of B.

(.............. , .............)(1)

The equation of the curve can be written in the form y = f (x),where f (x) = x2 8x + 23

(c) On the diagram below, draw a sketch of the curve y = f (x).

x

y

y = f ( x )

O

(1)(Total 5 marks)

5.

y = f ( x )

y

xO

( 2 , 3 )

The diagram shows part of the curve with equation y = f(x).The coordinates of the maximum point of this curve are (2, 3).

Write down the coordinates of the maximum point of the curve with equation

(a) y = f(x 2)

(......... , ..........)(1)

(b) y = 2f(x)

(......... , ..........)(1)

(Total 2 marks)

6.

yy = f ( x )

2 4 6 2 O x

The curve with equation y = f(x) is translated so that the point at (0, 0) is mapped onto the point (4, 0).

(a) Find an equation of the translated curve.

.....................................(2)

4y

x

4

2

2

O1 8 0 3 6 0 5 4 0

The grid shows the graph of y = cos x for values of x from 0 to 540

(b) On the grid, sketch the graph of y = 3 cos (2x) for values of x from 0 to 540(2)

(Total 4 marks)

7. The graph of y = f(x) is shown on the grids.

(a) On this grid, sketch the graph of y = f(x) 4

y

2

4

6

8

1 0

1 2

2

4

6

8

1 0

1 2

1 4

1 6

1 8

2 4 6 8 1 0 2 4 6 8 1 0 O x

(2)

(b) On this grid, sketch the graph of y = f( 21

x).

y

2

4

6

8

1 0

1 2

2

4

6

8

1 0

1 2

1 4

1 6

1 8

2 4 6 8 1 0 2 4 6 8 1 0 O x

(2)(Total 4 marks)

LOGARITHMS

Writing an Expression as a Log

1. Rewrite the following as a logarithm

a)

72 = 49

72 = 49 c)

26 = 64

26 = 64

b)

34 = 81

34 = 81 d)

53 = 125

53 = 125

2. Rewrite the following as a power

a) Log 2 x = 24 c) Log 5 14 = x

b) Log 3 81 = 4 d) Log 2 16 = 4

3. Rewrite the following as a power and then solve.

a) Log 5 125 = x c) Log 7 343 = x

Wellington International School IB Higher Preparation

logarithms - Using your CalculatorFind the value of the following using your calculator, give your answer to 3sf

1. Log 10 50

2. Log 10 145

3. Log 10 170

4. Log 10 180

5. Log 10 75

6. Log 10 400

7. 10x = 780

8. 10x = 55

9. 10x = 900


Laws of LogarithmsWrite as a log of a single number

1.

log 4 + log 7

log 4 + log 7 2.

log 8 log 2

log 8 log 2

3.

log 3 + log 5

log 3 + log 5 4.

log 25 log 5

log 25 log 5

5.

log 4 + log 3 + log 2

log 4 + log 3 + log 2 6.

log 60 log 10 + log 2

log 60 log 10 + log 2

7.

6log 2

6log 2 8.

3log 5

3log 5

9.

12

log 36

12log 36

10.

34log 81

34log 81

Vectors 1

8 a

7 ~

b ~

6

5 c

4 ~

d ~

3 f e ~~2

1

00 1 2 3 4 5 6 7 8 9 10 11 12

Write the following vectors in the form ( x ).

1 a~

1 .......................

2 b 2 ....................... ~

3 c 3 .......................~

4 d 4 ....................... ~

5 e 5 ....................... ~

6 f 6 ....................... ~

7 3a 7 ....................... ~

8 -2e 8 .......................~

9 -d~

9 .......................

10 a + b 10 ..................... ~ ~

11 3a 2b 11 ..................... ~ ~

12 e + f 12 ..................... ~ ~

13 e f 13 ..................... ~ ~

14 a + b + c 14 ..................... ~ ~ ~

15 2a 3e 15 .....................~ ~

y


Vectors 21 DE is parallel to BC

Vector = x A~

Vector AB = y ~

1 AE = 2 AC D E

B CExpress these vectors in terms of x and y. ~ ~

a AE 1a .....................

b BC b..................... c

c .....................

d ED d ....................

e BE e .....................

f f .....................

2 What is the value of x and y?

a ( 3 )+( x )=(7 )

b ( 2 ) +( x )= ( -3)

c ( 8 )( x )= (11)

2a x = ............... y = ...............

b x = .............. y = ...............

c x = .............. y = ..........................

AC

AD

DC

5 6 y

8 y 10

-1 y -3


Vectors 3

1 A ship can sail at 20 km/h in still water. The ship heads due south.

The current is flowing at 8 km/h due west.

Ship 20 km/h South

Current

8 km/h West

a What is the actual velocity of the ship? 1a ..............................

b What is the direction the ship actually takes? (Give the bearing.) b ..............................

2 A ship needs to sail due east from A to B.

The current is flowing at 2 m/s due north.

The ship sails at 8 m/s.

The distance from A to B is 5 km.

Current A 2 m/s B

Due North

a In which direction must the ship head? 2a ..............................

b How far does the ship actually sail in one second? b ..............................

c How long will the journey take? c ..............................

Give your answer in minutes and seconds. ...............................


Vectors 4

1 Two forces are pulling an object.

N20 N

15 N

a Calculate the resultant force. 1a .....................

b Calculate the direction of that force. b.....................

2 A plane flies from Calder airport to Deacon airport.

The plane flies at 500 km/h in still air.

The wind is blowing at 60 km/h in the direction shown.

NDeacon

Wind direction

70o 55o

Calder

a Find the direction in which the plane must fly. 2a .....................

b Find the actual speed of the plane. b.....................

Year 12 HL Summer Work 52

year 12 hl summer work

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