m11ac Full 2013

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M11Ac

M11Ac Front Cover

M11Ac

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ADEC Examinations

2012-2013

Required

Ruler, Pencil, Protractor

Scientic calculator

(not graphic display)

Read these instructions rst:

1. Complete the box above

2. Write in blue or black pen

3. The paper consists of (18) questions4. Read each question carefully; attempt every one

5. The number of marks is in [ ]

6. Show appropriate working to arrive at a solution

7.Any diagrams/shapes are not drawn to scale

- Mathematics Academic: Grade 11

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]

]

]

P(x) = 5x4+ 2x3 7x+ 8. .1 .A

.B

.C

.D .2

? .3.A

.B

.C

.D

M11Ac page 2 of 24

8

5

4

3

f (x) = 2x2 + 8x 6

geQ M IFGO SQGh d dG HLEG NG 10``1:e SCG

A SQG.A HLEG fc GPEG :e

C A dG HLEG M IFGO SQGh TG CNCG GPEG

.

-5 -4 -3 -2 -1 1

-1

1

2

3

4

x

yB.

-4 -3 -2 -1 1 2 3 4

1

2

3

4

x

y

C.

-4 -3 -2 -1 1 2 3 4

-4

-3

-2

-1

1

2

3

4

5

6

7

x

y

D.

-1 1 2 3 4

-8

-6

-4

-2

2

x

y

f (x) = 6x2 2

f (x) = 6x

f (x) = 6x 2

f (x) = 3x

f (x) = 3x2 2

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For questions 1-10, choose one answer and circle it : e.g. A

If you make a mistake, cross out the rst answer and circle

the correct one: e.g. A C

1. What is the degree of the polynomialP(x) = 5x4+ 2x3 7x+ 8 ?

A.

B.

C.D.

2. Which graph represents the following function?

3. What is the derivative of f (x) = 3x2 2 ?

A.

B.

C.

D.

M11Ac page 3 of 24

[1]

[1]

[1]

8

5

43

f (x) = 6x2 2

f (x) = 6x

f (x) = 6x 2

f (x) = 3x

f (x) = 2x2 + 8x 6

3

.

-5 -4 -3 -2 -1 1

-1

1

2

3

4

x

yB.

-4 -3 -2 -1 1 2 3 4

1

2

3

4

x

y

C.

-4 -3 -2 -1 1 2 3 4

-4

-3

-2

-1

1

2

3

4

5

6

7

x

y

D.

-1 1 2 3 4

-8

-6

-4

-2

2

x

y

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.4.A

.B

.C

.D

? .5 .A

.B

.C

.D

? .6.A

.B

.C

.D

? log4 16 =xx .7.A

.B

.C

.D

? . 8.A

.B

.C

.D

? log5 .9

.A

.B

.C

.D

M11Ac page 4 of 24

]

]

]

]

]

]

log5

a + log5

b

log5a log

5b

log5a log

5b

blog5a

2

2

4

5

27

15

3

5

y= 4x5+ 3x4 x 2

(3 2a2) + ( 4a3 + 5a2 9)

7a7 6

4a3+ 7a2 12

4a3+ 3a2 6

6+ 7a

= 7x9 x 2

= 20x5+ 12x4 x 2

= 20x4+ 12x3 x

= 20x4+ 12x3 1

dy

dx

dy

dx

dy

dx

dy

dx

P(x)=x4 3x + 5 P(2)

a

b

dy

dx

f (x)d

dxf (x)

f (x)

dy

dx

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4. Which of the following notation does notrepresent the derivative?

A.

B.

C.

D.

5. Simplify

A.

B.

C.

D.

6. What is for the following function?

A.

B.

C.

D.

7. What is the value ofxif log4 16 =x ?

A.

B.

C.

D.

8. What is the value ofP(2), if P(x)=x4 3x + 5 ?

A.

B.

C.

D.

9. Which of the following is the equivalent statement for log5

?

A.

B.

C.

D.

M11Ac page 5 of 24

[1]

[1]

[1]

[1]

[1]

[1]

log5

a + log5

b

log5a log

5b

log5a log

5b

blog5a

2

2

4

5

27

15

3

5

6

y= 4x5+ 3x4 x 2

(3 2a2) + ( 4a3 + 5a2 9)

7a7 6

4a3+ 7a2 12

4a3+ 3a2 6

6+ 7a

= 7x9 x 2

= 20x5+ 12x4 x 2

= 20x4+ 12x3 x

= 20x4+ 12x3 1

dy

dx

dy

dx

dy

dx

dy

dx

dydx

a

b

dy

dx

d

dx

f (x)

f (x)

f (x)

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F(x)(x 2) .10 .A

.B

.C

.D

x 4 + (a) .11

2x3 + 15x2 + 33x+ 20 (b)

a3x f3 (x) =x3 + ax2 + 8x 9 .12

M11Ac page 6 of 24

]

]

]

1

2x3 + 15x2 + 33x+ 20

.d MG MG HLEG cG (18-11) :SCd

F(2) = 0

F( 2)= 0

F( 2)=x

F(x)= 0

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M11Ac page 7 of 24

[4]

[2]

[4]

[1]10. If (x 2) is a factor ofF(x), which of the following is true?

A. F(2) = 0

B. F( 2)= 0

C. F( 2)=x

D. F(x)= 0

11. (a) Divide 2x3 + 15x2 + 33x+ 20 byx + 4

(b) Use your results to factorise 2x3 + 15x2 + 33x+ 20 completely

12. Find the value of agiven that: f (x) =x3 + ax2 + 8x 9 has remainder 3 when divided byx 3

11

For questions 11 - 18, write your answers in the spaces provided.

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,O .13t (a)

. (b)

t (c)

(d)

M11Ac page 8 of 24

]

]

]

]

s(t) = (t + 2)(t 2)(t 4)

s(t)

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

-8

-6

-4

-2

2

4

6

8

10

12

14

16

18

t

s(t)

s(t)

s(1)

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13. The position, from a xed point O, of a toy train racing in a straight line and timed in seconds tfrom the

blowing of a whistle is given by

s(t) = (t + 2)(t 2)(t 4)

(a) Sketch the graph ofs(t) on the axes below

(b) s(t) is a function. By using the vertical line test, show on the graph that s(t) is a function.

(c) Explain what the vertical line test tells us about the position of the object at any time t

(d) Finds(1) and in words, explain what this answer means

M11Ac page 9 of 24

[4]

[1]

[1]

[2]

8

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

-8

-6

-4

-2

2

4

6

8

10

12

14

16

18

t

s(t)

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. .15

(a)

(b)

. (c)

M11Ac page 10 of 24

]

]

]

y = logax

-1 1 2 3 4

-2

-1

1

2

3

4

x

y

(3, 0.5)

y = log

ax a

y = ax

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14. The graph y = logax is given below.

(a) Using the co-ordinates given on the graph, nd the value of ain the equationy = logax

(b) On the axes above, sketch the graph of y = axshowing intercepts and relevant features.

(c) Compare twoimportant differences between the log graph and the exponential graph.

M11Ac page 11 of 24

[2]

[3]

[2]

7

-1 1 2 3 4

-2

-1

1

2

3

4

x

y

(3, 0.5)

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.f (x) = 6x2f (x) = limh0

.15

.x .16

M11Ac page 12 of 241

]

]

f(x + h) f(x)

h

32x 10 (3x)+ 9 = 0

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15. Using f (x) = limh0

nd the derivative of f (x) = 6x2from rst principles.

16. Solve the following equation forxby changing to a quadratic rst.

32x 10 (3x)+ 9 = 0


[5]

[6]

f(x + h) f(x)

h

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.17

( a)

( b)

(b) ( c)

]

]

]

M11Ac page 14 of 24

P(x) =x4

-6 -5 -4 -3 -2 -1 1 2 3 4

-4

-3

-2

-1

1

2

3

4

x

y

y= P(x)

y=P (x + 4)

(x + 4)4 = 3

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17. The graph of P(x) =x4is shown below

(a) On the axes above sketch the graph y= P(x)

(b) On the same axes also sketch the graph y=P (x + 4)

(c) Using your sketch from (b) determine how many solutions the equation

(x + 4)4 = 3

would have. Show the solutions on your graph.

[2]

[2]

[2]

M11Ac page 15of 246

-6 -5 -4 -3 -2 -1 1 2 3 4

-4

-3

-2

-1

1

2

3

4

x

y

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h 1.2. .18h= 1.2 1.1t

t 20 t 2.. ( a)

]

M11Ac page 16 of 24

h= 1.2 1.1t

-5 5 10 15 20 25 30 35

-1

1

2

3

4

5

6

7

8

9

10

11

t

h

)18(

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18. Ali planted a tree that was 1.2 m high.

The height hof the tree, in metres, can be modeled by the equation

h= 1.2 1.1t

where tis the time in years since the tree was planted and t 20

(a) Determine when the tree will reach a height of 2 metres. Give your answer to one

decimal place.

The graph of h= 1.2 1.1tis shown below

[5]


-5 5 10 15 20 25 30 35

-1

1

2

3

4

5

6

7

8

9

10

11

t

h

(Question 18 continues on the next page)

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(b)

.)b( (c)

P P (d)PQ

. 15x (e)

]

]

]

]


t[5, 10]

Q (15, 5.0127)

t= 15

15.1 5.0607 = 0.485.0607 5.0127

5.0127 4.9651

15.1 15

15 14.9

15.01 5.017514.9 4.9651 = 0.476

14.99

Q x yQ(4dp) PQ

)18(

(18(

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M11Ac page 19 of 24

(b) Determine the average rate of change over the domain t[5, 10]

(c) Explain your answer from (b) above giving the average growth of the tree during that period

in centimetres per year.

(d) Pis the point (15, 5.0127) Qis a point on the curve close toP. Complete the table

below to nd the gradient of the secantPQand determine the instantaneous growth

of the tree when t= 15

(e) Asxapproaches 15, what does the gradient approach? Give your answer to 1 dp.

[4]

[2]

[3]

[1]

10

xcoordinate of Q Q y coordinate of PQ(4dp) Gradient of

15.1 5.0607 = 0.485.0607 5.0127

5.0127 4.9651

15.1 15

15 14.9

15.01 5.017514.9 4.9651 = 0.476

14.99

(Question 18 continues on the next page)

(Question 18 continued)

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(f)

(g)]

]

M11Ac page 20 of 24

t= 35h= 1.2 1.1t

h= 1.2 1.1t, t 20

SCG fG

(18(

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M11Ac page 21 of 24

(f) Explain why t= 35 years might not t the model h= 1.2 1.1t

(g) State an appropriate domain for the given function h= 1.2 1.1t, t 20 [1]

[2]

3

End of Questions

(Question 18 continued)

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M11Ac page 22 of 24

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There are no questions on this page

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Mark ChecksheetMathematics: Grade 7 Mark ChecksheetMathematics: Grade 11Ac Mark Checksheet

Name of Marker: Signature:

Name of Reviewer: Signature:

1 3

4 9

10 12

13

14

15 16

17

18 (a)

18 (b) 18 (e)

18 (f) 18 (g)

3

6

11

8

7

11

6

5

10

3

Question/s Max. Marker Reviewer

70 70 70TOTAL

Exam Mark % %

Documents

m11ac Full 2013