Final Workshop 9th Grade

8/13/2019 Final Workshop 9th Grade

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FINAL WORKSHOP 9th GRADE

1. If A = and det A = 14, find the possible values of p.

Working:

Answers :

....……………………………………..........

(Total 4 marks)

2. Let M = , where a .

(a) Find M 2 in terms of a . (4)

(b) If M 2 is equal to , find the value of a .

(2)

Using this value of a , find M – l and hence solve the system of equations:

– x + 2 y = – 32 x – y = 3

(6)(Total 12 marks)

p p

p 3

4

2

1

2

2

a

5

4

4

5



3. A and B are 2 × 2 matrices, where A = and BA = . Find B.

Working:

Answers :

....……………………………………..........

(Total 4 marks)

4. Given the matrix A = find the values of the real number k for which det( A

– kI ) = 0

where I = .

Working:

Answers :

....……………………………………..........

(Total 4 marks)

5. (a) Find the values of a and b given that the matrix A = is the

inverse of the matrix B =

0

2

2

5

8

2

44

11

0

2

1

3

1

0

0

1

4

7

6

3

5

4

5

8

a

.

3

1

2

1

2

1

3

1

b



(b) For the values of a and b found in part (a), solve the system of linearequations

x + 2 y – 2 z = 53 x + by + z = 0

– x + y – 3 z = a – 1.

Working:

Answers :

(a) ………………………………………….. (b) ..................................................................

(Total 4 marks)

6. Find the values of the real number k for which the determinant of the matrix

is equal to zero.

Working:

Answers :

....……………………………………..........

(Total 3 marks)

7. (a) Given matrices A , B , C for which AB = C and det A 0, express B in termsof A and C .

(2)

1

3

2

4

k

k



(b) Let A = , D = and C = .

(i) Find the matrix DA ;

(ii) Find B if AB = C .(3)

(c) Find the coordinates of the point of intersection of the planes x + 2 y + 3 z = 5, 2 x – y + 2z = 7 and 3 x – 3 j + 2 z = 10.

(2)(Total 7 marks)

1. The vectors , are unit vectors along the x-axis and y-axis respectively.

The vectors = – + and v = 3 + 5 are given.

(a) Find + 2 in terms of and .

A vector has the same direction as + 2 , and has a magnitude of 26.

(b) Find in terms of and .

Working:

Answers :

(a) …………………………………………..

(b) ……………………………………..........

(Total 4 marks)

2

2

3

3

1

2

3

2

1

5

4

7

9

7

13

3

2

4

10

7

5

i j

u i j i j

u v i j

w u v

w i j



2. The circle shown has centre O and radius 6. is the vector , is the

vector and is the vector, .

(a) Verify that A, B and C lie on the circle.(3)

(b) Find the vector .(2)

(c) Using an appropriate scalar product, or otherwise, find the cosine of angleOAC .

(3)

(d) Find the area of triangle ABC , giving your answer in the form a , wherea .

(4)(Total 12 marks)

OA

0

6OB

0

6OC

11

5

A B

C

O x

y

AC

11



3. The vectors u , v are given by u = 3 i + 5 j , v = i – 2 j .

Find scalars a , b such that a (u + v ) – 8i + (6 – 2) j .

Working:

Answers :

....……………………………………..........

(Total 4 marks)

4. The triangle ABC is defined by the following information

= , = , = 0, , is parallel to

.

(a) On the grid below, draw an accurate diagram of triangle ABC.

OA

3

2AB

4

3BCAB AC

1

0

O

–1

–2

–3

–4

–2 –1

4

3

2

1

1 2 3 4 5 6

y

x



(b) Write down the vector .

Working:

Answers :

(b) ..................................................................

(Total 4 marks)

1. The perimeter of this rectangular field is 220 m. One side is x m as shown.

(a) Express the width ( W ) in terms of x.

(b) Write an expression, in terms of x only, for the area of the field.

(c) If the length ( x) is 70 m, find the area.

Working:

Answers :(a) ………………………………………….. (b) ………………………………………….. (c) …………………………………………..

(Total 4 marks)

OC

x m

W m



2. The diagram shows the graph of y = x2 – 2 x – 8. The graph crosses the x-axis at the point A, and has a vertex at B.

(a) Factorise x2 – 2 x – 8.

(b) Write down the coordinates of each of these points

(i) A;

(ii) B.

Working:

Answers :(a) ………………………………………….. (b) (i) ……………………………………..

(ii) ……………………………………..

(Total 4 marks)

y

x

B

A O



3. The diagram below shows a path x m wide around a rectangular lawn whichmeasures 10 m by 8 m.

(a) Write down an expression in terms of x for the area of the path.

(b) What is the width of the path when its area is 208 m 2?

Working:

Answers :(a) ………………………………………….. (b) …………………………………………..

(Total 4 marks)

4. (a) Solve the equation x2 – 5 x + 6 = 0.

8 m

10 m

LAWN

x m

x m





(b) the number of bacteria present after 3 hours;

(c) the number of hours it would take for the number of bacteria to reach 19200.

Working:

Answers :(a) …………………………………………..

(b)…………………………………………..

(c) …………………………………………..

(Total 4 marks)

7. In the diagram, the lines L1 and L2 are parallel.

x

y

L

L

1

2

6

4

2

–2

–4

–6

–6 –4 –2 0 2 4 6



(a) What is the gradient of L1?

(b) Write down the equation of L1.

(c) Write down the equation of L2 in the form ax + by + c = 0.

Working:

Answers :

(a)…………………………………………..

(b) ………………………………………….. (c) …………………………………………..

(Total 4 marks)

8. In the diagram, triangle ABC is isosceles. AB = AC , CB = 15 cm and angle ACB is23°.

Diagram not to scale

Find

(a) the size of angle CAB;

A

C B23º

15 cm



(b) the length of AB.

Working:

Answers :(a) ………………………………………….. (b) …………………………………………..

(Total 4 marks)

9. The gradients of several lines are as follows:

Line a b c d e f g h

Gradient – 3 0.5 0.4

(a) Find two pairs of lines that are parallel to each other.

(b) Find any two pairs of lines that are at right angles to each other.

Working:

Answers :(a) ………………………………………….. (b) …………………………………………..

(Total 4 marks)

25 –

3

1

63

52 –

2 – 5



10. The diagram shows the plan of a playground with dimensions as shown.

Calculate

(a) the length BC ;

(b) the area of triangle ABC .

Working:

Answers :(a) ………………………………………….. (b) …………………………………………..

(Total 4 marks)

C

B

A

48 m

57 m

117º



11. The diagram shows a water tower standing on horizontal ground. The height of thetower is 26.5 m.

From a point A on the ground the angle of elevation to the top of the tower is 28°.

(a) On the diagram, show and label the angle of elevation, 28°.

(b) Calculate, correct to the neatest metre , the distance x m.

Working:

Answers :

(b) …………………………………………..

(Total 4 marks)

12. ABCD is a trapezium with AB = CD and [BC] parallel to [AD]. AD = 22 cm,BC = 12 cm, AB = 13 cm.

Diagram not to scale

(a) Show that AE = 5 cm.(2)

(b) Calculate the height BE of the trapezium.(2)

A x m

B C

A E D



(c) Calculate

(i) ;

(ii) .

(3)

(d) Calculate the length of the diagonal [CA].(3)

(Total 10 marks)

13. The mean of the ten numbers listed below is 5.5.

4, 3, a , 8, 7, 3, 9, 5, 8, 3

(a) Find the value of a.

(b) Find the median of these numbers.

Working:

Answers :(a) ………………………………………….. (b) …………………………………………..

(Total 4 marks)

14. The table shows the number of children in 50 families.

Number ofchildren

Frequency Cumulativefrequency

1 3 3

2 m 22

3 12 34

4 p q

5 5 48

6 2 50

T

(a) Write down the value of T .

EAB

DCB



(b) Find the values of m, p and q.

Working:

Answers :(a) ………………………………………….. (b) …………………………………………..

(Total 4 marks)

15. The following table shows the age distribution of teachers who smoke at Laughlin High School.

Ages Number of smokers

20 ≤ x < 30 5

30 ≤ x < 40 4

40 ≤ x < 50 3

50 ≤ x < 60 2

60 ≤ x < 70 3

(a) Calculate an estimate of the mean smoking age.



(b) On the following grid, construct a histogram to represent this data.

Working:

Answers :

(a) …………………………………………..

(Total 4 marks)



16. The graph below shows the cumulative frequency for the yearly incomes of 200 people.

Use the graph to estimate

(a) the number of people who earn less than 5000 British pounds per year;

(b) the median salary of the group of 200 people;

(c) the lowest income of the richest 20% of this group.

Working:

Answers :

(a) ………………………………………….. (b) ..................................................................(c) ……………………………………..........

(Total 4 marks)

200

180

160

140

120

100

80

60

40

20

00 5000 10 000 15 000 20 000 25 000 30 000 35 000

Annual income in British pounds

Cumulativefrequency



1. The table below shows the number of left and right handed tennis players in asample of 50 males and females.

Left handed Right handed TotalMale 3 29 32

Female 2 16 18

Total 5 45 50

If a tennis player was selected at random from the group, find the probability thatthe player is

(a) male and left handed;

(b) right handed;(c) right handed, given that the player selected is female.

Working:

Answers :(a) ………………………………………….. (b) ………………………………………….. (c) …………………………………………..

(Total 4 marks)

2. Note : For this question, it is important that you show your working and explain

your method clearly.

A box contains 10 coloured light bulbs, 5 green, 3 red and 2 yellow. One light bulbis selected at random and put into the light fitting of room A.

(a) What is the probability that the light bulb selected is

(i) green?(1)

(ii) not green?(1)



A second light bulb is selected at random and put into the light fitting in room B.

(b) What is the probability that

(i) the second light bulb is green given the first light bulb was green?(l)

(ii) both light bulbs were not green?(2)

(iii) one room had a green light bulb and the other room does not have agreen light bulb?

(3)

A third light bulb is selected at random and put in the light fitting of room C.

(c) What is the probability that

(i) all three rooms have green light bulbs?(2)

(ii) only one room has a green light bulb?(3)

(iii) at least one room has a green light bulb?(2)

(Total 15 marks)

3. It is known that 5% of all AA batteries made by Power Manufacturers aredefective. AA batteries are sold in packs of 4.

Find the probability that a pack of 4 has

(a) exactly two defective batteries;(3)

(b) at least one defective battery.(2)

(Total 5 marks)

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Final Workshop 9th Grade