Maths Module F4 2011b

8/3/2019 Maths Module F4 2011b

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Probability: Experiments, Sample space and events

Experiment → a process or an operation with outcomes

Sample space → set of all possible outcomes of an experiment

Event → set of outcomes that fulfill a given condition.

1. The cards shown below are put into a box. A card is picked at random from the box. Determine which ofthe following is a possible outcomes.

(a) A letter "C". (b) A consonant. (c) A letter "Z"

2. List the sample space of the following experiment.

(a) A two digits numbers is formed from thedigits 1, 3, 5 and 7.

(b) Box A contains the cards labelled with A, B andC. Box B contains the cards labelled with 1 and2. A card is picked at random from each box.

3. Ten cards labelled with numbers 21 to 30 are placed in a box. A card is picked at random from the box.

(a) Determine if the following event are possible.

(i) Getting a prime number. (ii) Getting a perfect square. (iii) Getting a fraction.

(b) List all the possible outcomes for the following event.

(i) Getting a even number. (ii) Getting a multiple of 3. (iii) Getting a number with thesum of digits is 3

E ECREX I S



Probability: Probability of an events

Number of successful outcomesP(E)

Number of trials

All answers should be express as a fraction in lowest terms

=

1. Five cards labelled with the numbers 1, 2, 3, 4 and 5 are placed in a box. A card is drawn at random, thenumber on the card is recorded and the card is returned to the box. The process is repeated 84 timesand the result is shown as below.

1 4 5 1 5 5 4 3 3 5 1 2 4 4

2 2 1 1 5 1 2 5 1 1 4 3 4 3

1 3 5 5 1 1 4 3 3 2 4 1 3 1

2 1 1 1 1 3 4 3 5 3 2 2 4 12 2 4 2 5 3 3 4 1 4 5 1 1 1

3 5 4 4 3 5 4 2 1 3 2 3 5 4

Find the probability of the following events.

(a) Getting the number "1". (b) Getting the number "2".

2. Table shows the number of members for three sports clubs.

Sports Club Soccer Hockey Volleyball

Number of Members 56 34 20

If a students is chosen at random. Find the probability that the students is

(a) Soccer club members. (b) Volleyball club members.

3. There are red, yellow and blue marbles in a box. The number of red and yellow marbles are 12 and 14

respectively. If a marble is chosen at random, the probability of choosing a red marbles is3

7. Find

(a) The total numbers of marbles in the box. (b) the probability of choosing a yellow marble.



4. There are x red cards, 16 yellow cards and 10 blue cards in a box. If a card is drawn at random, the

probability that a blue card is picked is2

7. Find the value of x.

5.The probability of Ali scoring a goal from penalty kick is

3

4. If Ali is given 120 penalty kicks. Find the

expected number of goals scored by Ali.

6. A fair dice is rolled 540 times. How many times will it likely get the number "5".

7. A bag contains red and white marbles. If a marble is drawn at random, the probability of getting a red

marble is3

8

. When a marble is drawn at random for k times, 125 times of red marbles is obtained.

Find the value of k.

8. Table shows the number of questions on health, entertainment, sport and history in a question bank.

Category Health Entertainment Sport History

Number of Questions 16 20 24 40

(a) If a question is chosen at random, find the probability of choosing a question on Entertainment.(b) If the number of questions in the question bank is increased to 250 question, how many question

on spot should placed into the question bank.



Circle : Revisions

1. Find the values of x, y and z.


Circle : Tangents of circle

1. Find the values of x, y and the length ofAB.


85 °°°°

O ••••

z °°°°

y °°°°x °°°°

80 °°°°

50 °°°°

25 °°°°

Oz °°°°

y °°°°

x °°°°

35 °°°°

••••

A

By °°°°

x °°°°O

60 °°°°

C

4 cm

A

Ox °°°°z °°°°

T

5 cm

y °°°°

12 cm

B

C




4. Find the values of p and q.


Circle : Angle between Tangent and Chord

1. Find the values of a and b.

p °°°°

30 °°°°

100 °°°°

q °°°°

E

O

H

G

F

140 °°°°

Oz °°°°

y °°°°x °°°°

B

C

A

D

30 °°°°

x °°°°

80 °°°°

z °°°°

A

D

C

B

FE

y °°°°

b °°°°

80 °°°°

60 °°°°

AD

C

B

E

a °°°°



2. Find the values of a and b.

3. Find the values of a, b and c.



b °°°°25 °°°°

35 °°°°

a °°°°

A

C

B

F ED

a °°°°

b °°°°38 °°°°

28 °°°°

AD

C

B

F E G

c °°°°

78 °°°°

58 °°°°

100 °°°°

b °°°°

a °°°°

30 °°°°

P

R

Q

ST

Q

40 °°°°

P

b °°°°

T30 °°°°

RS

c °°°°

a °°°°




Circle : Common tangents


2. Find the value of x, y and the length of AB.

80 °°°°

50 °°°°

A

C

B

FE D

35 °°°° b °°°°

a °°°°

c °°°°

35 °°°°

BA

F

E

DG

I

H

y °°°°x °°°°

z °°°°

E

C10 cm y °°°°

A

x °°°°

B

6 cm



Trigonometry : Value of sine, cosine and tangent

1. Determine the quadrant of the following angles.

(a) 142 °

(b) 273 °

(c) 259 °

2. Determine the angle for the following diagrams.

(a) (b) (c)

3. Find the value of sine, cosine and tangent for the following angle.

(a) (b)

4. Find the value of sine, cosine and tangent for the following angle.(a) (b) (c)

QuadrantIV

QuadrantIII

QuadrantII QuadrantI

37 °°°°

56 °°°° 61 °°°°

(0.75, 0.66)

41 °°°°

••••

(0.56, 0.83)

θθθθ °°°°

••••

(−−−−0.57, 0.82)

θθθθ °°°°

••••

(−−−−0.97, −−−−0.24)

θθθθ °°°°

••••

(0.44, −−−−0.90)

θθθθ °°°°

••••



5. By using the answers from Questions 3 and 4, complete the following table by fill in "+" or " −" and

make the brief conclusion.

Quadrant I Quadrant II Quadrant III Quadrant IV

sin θ

cos θ

tan θ

conclusion

6. Determine whether the following is positive or negative.

(a) sin 138° (b) cos 175° (c) tan 166° (d) cos 187° (e) tan 264°

(f) sin 238° (g) cos 275° (h) tan 266° (i) sin 287° (j) tan 304°

7. State the quadrants that θ is likely to be in.

(a) sin θ = − 0.71 (b) cos θ = − 0.48 (c) tan θ = 1.32

(d) cos θ = 0.18 (e) tan θ = − 2.55 (f) cos θ = 0.84

8. Complete the following table by fill in the values.

35 145 215 325 53 127 233 303

sin θ sin θ

cos θ cos θ

tan θ tan θ

Relationship to

the angle in Q I180 − 35 180 +35 360 − 35 Relationship to

the angle in Q I180 − 53 180 +53 360 − 53

α 180 - α 180 + α 360 − α α 180 - α 180 + α 360 − α

sin θ sin α sin α −sin α −sin α sin θ sin α sin α −sin α −sin α

cos θ cos α −cos α −cos α cos α cos θ cos α −cos α −cos α cos α

tan θ tan α −tan α tan α −tan α tan θ tan α −tan α tan α −tan α

Only tangent+

Only cosine+

Only sine

+

All +



9. Find the corresponding acute angle in the first quadrant for each angle below.

(a) 98° (b) 298° (c) 260.4° (d) 340°25'

10. Complete teh following

(a) sin 148° = sin (b) sin 248° = sin (c) cos 148° = cos

(d) tan 288° = tan (e) tan 188° = tan (f) cos 348° = cos

Trigonometry : Problems Solving

1. Find the value of θ in degrees and minutes, if 0° < θ < 360° :

(a) sin θ = − 0.866 (b) cos θ = 0.4955

(c) cos θ = − 0.866 (d) tan θ = 0.4955

(e) sin θ = sin 27° (f) cos θ = − tan 32°

2. Given sin x = − 0.7071 and 180° < x < 270°, findthe value of x.

3. Given cos x = − 0.7071 and 0° < x < 180°, find thevalue of x.

4. Given tan x = 1.1918 and 90° < x < 270°, find thevalue of x.

5. Given cos x = 0.7071 and 180° < x < 360°, findthe value of x.

6. If sin x° = sin 37° and 90° < x° < 270°, find thevalue of x.

7. If cos x° = −cos 42° and 90° < x° < 270°, find thepossible values of x..



8. Find the value of tan θ and cos α

9.Given cos ∠ABC =

5

13, Find

(a) the length of CD.(b) tan ∠ADB

10. In the diagram, ∠Q = 90° and PQ = 10 cm. Given S is the midpoint for QR and ∠SPQ = 30°. find(a) The length of SQ,(b) ∠RPQ.

11.Given cos ∠CED =

5

8and tan ∠ABC =

5

4. Find the length of AE.

12. Giveb AE = 16 cm, AB = 12 cm and AD = 4 cm. Find the value of tan y and value of x

θθθθ αααα

5 cm

4 cm 8 cm

5 cm

10 cm

B

A

DC

30 °°°°

P10 cmQ

S

R

8 cm

15 cm B

A

D C

E

B

A

C

D

E

16 cm

4 cm

12 cmx °

y °



Trigonometry : Graph of sine, cosine and tangent

y = sin x y = cos x y = tan x

1. The diagram shows the graph of sine and cosine. Find the coordinates of all the points.

2. Identify each of the following graph.

(a) (b) (c)

1

−−−−1

90 °

180 °

360 °270 °

1

−−−−1

90 ° 180 °

360 °270 °

270 °

180 °

90 °

360 °

•

• • •0.5

270 °180 ° 360 °

90 °

•••−−−−0.5

•−−−−1

1

A

B

C

D

E F

M

H

G

•

142 °

1

−−−−1

90 ° 180 °

360 °270 °

−−−−1

90 °

180 ° 270 °

270 °

180 °

90 °



Angles of Elevation and Depression

1. Find the angle of elevation of point P from

point Q.

2. Find the angle of depression of point Q from

point P.

3. Given the angle of elevation of point P from

point Q is 33.69°. Find the value of x.

4. Given the angle of depression of point Q from

point P is 39.29°. Find the value of y.

5. Given the angle of elevation of point P (top of a

flagpole with the height of 8 m) from a man with the

height 1.7 m is aproximately 35°. Estimate the distance

of the flagpole from the man.

6. Given the angle of depression of point R from point P

and point Q are 35 and 25 respectively°. Find the

distance between point P and point Q.

P•

•Q

12 m

10 m

2 m

P•

•Q

15 m

x m

4 m

12 m

3 m

y m

Q•

P•

1.7 m

8 m

P•

4 m

15 m

Q •

P •

R•

P•

•Q

18 m

15 m

4 m



7. Given the angle of depression of point Q and R from

point P are 30 and 22 respectively°. Find the distance

between Q and R..

8. Given the angle of depression of point Q from point P

is 44°. Find the angle of elevation of point R from

point Q.

Lines and Planes in 3-Dimensions : Angle between Lines and Planes

1. Diagram shows a cuboid.

(a) Name the angle between the straight line DG and thebase ABCD.

(b) Find the angle between the straight line DG and thebase ABCD.

2. Diagram shows a cuboid.(a) Name the angle between the straight line DG and the

plane BCHG.(b) Find the angle between the straight line DG and the

plane BCHG.

P•

Q•

12 m

3 m

R•

3 m

•P

•R12 m

•Q

7 m

4 m9 m

H

GF

E

D C

BA8 cm

4 cm

4 cm

HE

4 cm

CD

GF

4 cm

8 cmA B



3. Diagram shows a prism.(a) Name the angle between the straight line BD and the

plane ADEF.(b) Find the angle between the straight line BD and the

plane ADEF.

4. Diagram shows a prism.(a) Name the angle between the straight line BD and the

plane CDE.(b) Find the angle between the straight line BD and the

plane CDE.

5. Diagram shows a pyramid with the vertex E is verticallyabove the midpoint of the diagonals of the base.(a) Name the angle between the straight line CE and the

base ABCD(b) Find the angle between the straight line CE and the

base ABCD.

Lines and Planes in 3-Dimensions : Angle between Two Planes

1. Diagram shows a cuboid.(a) Name the angle between the plane CDFG and the

base ABCD(b) Find the angle between the plane CDFG and the

base ABCD.

A

F

E

C

B

D

8 cm

4 cm 10 cm

E

D

FC

10 cm4 cm

8 cm BA

E

A

B

CD

6 cm

7 cm

7 cm

H

GF

E

DC

BA8 cm

4 cm

4 cm



2. Diagram shows a cuboid.(a) Name the angle between the plane BDEG and the

plane ABGF.(b) Find the angle between the plane BDEG and the

plane ABGF.

3. Diagram shows a prism.(a) Name the angle between the plane BDE and the

plane ADEF.(b) Find the angle between the plane BDE and the plane

ADEF.

4. Diagram shows a prism.(a) Name the angle between the plane BCEF and the

plane ADEF.

(b) Find the angle between the plane BCEF and theplane ADEF.

5. Diagram shows a pyramid with the vertex E is verticallyabove the midpoint of the diagonals of the base..(a) Name the angle between the plane CDE and the

base ABCD(b) Find the angle between the plane CDE and the base

ABCD.

E

E

D C

A B

F

10 cm4 cm

8 cm

7 cm

7 cm

6 cm

D C

B

A

E

HE

4 cm

CD

GF

4 cm8 cm

A B

4 cm

A8 cm B

DC

F

10 cm

Maths Module F4 2011b

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