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T950(E)(A)T

APRIL EXAMINATION

NATIONAL CERTIFICATE

MATHEMATICS N3

(16030143)

1 April 2016 (X-Paper)

09:00–12:00

This question paper consists of 6 pages and 1 formula sheet of 2 pages.

(16030143) -2- T950(E)(A1)T


DEPARTMENT OF HIGHER EDUCATION AND TRAINING

REPUBLIC OF SOUTH AFRICA NATIONAL CERTIFICATE

MATHEMATICS N3

TIME: 3 HOURS

MARKS: 100

INSTRUCTIONS AND INFORMATION

1.

2.

3.

4.

5.

6.

7.

8.

9.

Answer ALL the questions.

Read ALL the questions carefully.

Number the answers according to the numbering system used in this question paper.

Questions may be answered in any order but subsections of questions must NOT be

separated.

Show ALL the calculations and intermediary steps.

ALL final answers must be accurately approximated to THREE decimal places.

ALL graph work must be done in the ANSWER BOOK. Graph paper is NOT

supplied.

Diagrams are NOT drawn to scale.

Write neatly and legibly.

(16030143) -3- T950(E)(A1)T


QUESTION 1

1.1 Factorise the following expressions as far as possible in prime factors:

1.1.1 (3 2) (3 2)x x y y (4)

1.1.2 4 24 3 1p pn n (2)

1.2 Factorise the following expression completely: 3 22 5 2x x x

(5)

1.3 Simplify the following expression:

2

2

1 2 1 2 7 17

1 3 2 3

x x x x

x x x x

(6)

[17]

QUESTION 2

2.1 Simplify the following:

3

2

1

2

3

xx

x

(4)

2.2 Use logs to the base 2 and simplify the following WITHOUT using a calculator:

0,5log 128

(4)

2.3 Solve for x:

2.3.1 216 3 3 2x x

2.3.2 (log 2) log( 2) 0x x (2 4) (8)

[16]

(16030143) -4- T950(E)(A1)T


QUESTION 3

3.1 Solve for x by completing the square: 24 48 2x x

(4)

3.2 Make 'b ' the subject of the formula:

x bD

x b

(4)

3.3 Make ' w ' the subject of the formula:

log log loge e et p w ds

(3)

3.4 Alex paid a deposit of 3R x for a computer. He paid the rest in 9 monthly

instalments. He paid a total of 33R x . What is the payment of each monthly

instalment in terms of x .

(4)

[15]

QUESTION 4

4.1 Consider FIGURE A below. ABC is an isosceles triangle with AB BC and

vertices A(2;1), B(4;5) and C(0;k).

FIGURE A

(16030143) -5- T950(E)(A1)T

4.1.1 Find the length of AB. (2)

4.1.2 Determine the value(s) of k. (4)

4.1.3 Show that AB is perpendicular to BC if k 7 . (3)

4.1.4 Calculate the area of ABC when 7k . (3)

4.2 P ( 2; 1) and Q (4;7) are points in the plane with M as the midpoint of PQ.

Determine the equation of the line parallel to the y-axis and passing through the

point M.

(4)

4.3 Consider FIGURE B below. The lines BA and CA with equations 2y x and

3 3y x respectively, intersect at A. Determine the size of ˆBAC where B and C

are the intercepts on the x-axis as shown.

FIGURE B

(5)

[21]

(16030143) -6- T950(E)(A1)T


QUESTION 5

5.1 Draw the graph defined by the equation: 2733 22 yx (2)

5.2 Given : 3 26 9y x x x

5.2.1 Make use of differentiation to determine the coordinates of the turning

points of the given equation.

(5)

5.2.2 Draw the graph of the given function. Show ALL values at the points of

intersection with the system of axes and the co-ordinates of the turning

points.

(3)

5.3 Determine

dy

dx if

12y x

x . Leave the answers with positive indices and in

surd form.

(4)

[14]

QUESTION 6

6.1 Prove the following trigonometric identity:

2 2 2 2sin tan cos secA A A A

(4)

6.2 Calculate the value(s) of which will satisfy the equation if o o0 270 :

2sin 1 cos

(5)

6.3 Consider FIGURE C below. An observer, standing at a point A is watching the top

of a vertical tower BC . The angle of elevation of the top of the tower, BC, is o25

and the angle of depression of the foot of the tower is o20 . If the height of the

tower BC is known to be 30 m, determine the following:

6.3.1 The distance between the observer at A and the point B. (3)

6.3.2 The distance between the two towers. (3)

(16030143) -7- T950(E)(A1)T


FIGURE C

6.4 Consider FIGURE D below. The sketch represents the graph of ( ) sinf x a px

where 0 x . Determine the values of a and p .

FIGURE D

(2)

[17]

TOTAL: 100

(16030143) -1- T950(E)(A1)T


FORMULA SHEET

Any applicable formula may also be used.

1. Factors 2. Logarithms

a3 - b

3 = (a - b)(a

2 + ab + b

2)

a3 + b

3 = (a + b)(a

2 - ab + b

2)

blogalogablog

blogalogb

alog

3. Quadratic formula

blog

alogalog

c

cb

2a

4acbbx

2

alogmalog m

4. Parabola

blog

1alog

ab

cbxaxy 2

4a

b4acy

2

11∴ e n1aloga

meta m1ntloga ∴

2a

bx

5. Circle 6. Straight line

222 ryx

h4h

xD

2

24h4Dhx

)xm(xyy 11

Perpendicular: 1mm 21

Parallel lines: 21 mm

Distance: 212

212 )y(y)x(xD

Midpoint:

2

yy;

2

xxP 2121

Angle of inclination: mtanθ -1

(16030143) -2- T950(E)(A1)T

Copyright reserved

7. Differentiation

h

xfhxf

0h

lim

dx

dy

→

1-nn nxxdx

d

Max/Min

For turning points: 0xf '

8. Trigonometry

cosecθ

1

r

ysinθ

secθ

1

r

xcosθ

cotθ

1

x

ytanθ

1 θcosθsin 22

θsecθtan1 22

θcosecθcot1 22

cosθ

sinθtanθ

sinθ

cosθcotθ

c

sinC

b

sinB

a

sinA

cosA2bccba 222

Area of ∆ABC = ½ ac sin B


NATIONAL CERTIFICATE

APRIL EXAMINATION

MATHEMATICS N3

1 APRIL 2016

This marking guideline consists of 10 pages.

MARKING GUIDELINE

MARKING GUIDELINE -2- T950(E)(A1)T

MATHEMATICS N3


QUESTION 1

1.1 1.1.1

2 2

2 2

(3 2) (3 2)

3 2 3 2

3( ) 2( )

3( )( ) 2( )

( )[3( ) 2]

( )[3 3 2]

x x y y

x x y y

x y x y

x y x y x y

x y x y

x y x y

(4)

1.1.2 4 2

2 2

4 3 1

( 1)(4 1)

p p

p p

n n

n n

(2)

1.2 3 2

3 2

2

3 2

3 2

2

2

( ) 2 5 2

(1) 2(1) (1) 5(1) 2

=2 1 5 2

=0

-1 is a factor of f ( )

2 3 2

x-1 2 5 2

2 2

3 5

3 3

f x x x x

f

x x

x x

x x x

x x

x x

x x

2

- 2 2

-2 2

. .

( ) ( 1)(2 3 2)

= ( 1)(2 1)( 2)

x

x

f x x x x

x x x

(5)


MATHEMATICS N3


1.3

)3)(1(

1772

3

12

1

1 2

xx

xx

x

x

x

x

=)3)(1(

1772)1)(12()3)(1( 2

xx

xxxxxx

=)3)(1(

17721234 222

xx

xxxxxx

)3)(1(

)32(5 2

xx

xx

(6)

[17]


MATHEMATICS N3


QUESTION 2

2.1

3

2

3

21

2

1 3

2 2

2

1

2

3

2 13

2

2 1 1

2 3

2 1

6

xx

x

xx

x

x

x x

x

x

(4)

2.2 0,5

2

2

7

2

1

2

2

2

log 128

log 128

1log

2

log 2

log 2

7 log 2

1log 2

7

(4)

2.3 2.3.1

2

22

2 2

16 3 3 2

16 3 3 2

16 3 3 4.3 4

12 4.3

3 3

1

TEST : If 1 then LHS=RHS=5

x x

x x

x x x

x

x

x

x

(4)

2.3.2

0

2

(log 2) log( 2) 0

log 2 0 or log( 2) 0

log 2 2=10

10 2 1

100 3

x x

x x

x x

x x

x x

(4)

[16]


MATHEMATICS N3


QUESTION 3

3.1

(4)

3.2

bx

bxD

2

(4)


MATHEMATICS N3


3.3

dsp

twe log

dsep

tw

dset

pw

(3)

3.4 Deposit is R3x

Total price is R33x

Total amount for 9 instalments= R30x

Each monthly instalment =30 10

R9 3

x xR

(4)

[15]


MATHEMATICS N3

Copyright reserved

QUESTION 4

4.1 4.1.1

2 2

2 1 2 1

2 2

AB

= 4 2 5 1

= 4 16

= 20

=2 5

x x y y

(2)

4.1.2

2 2

2

2

BC=2 5

0 4 5 20

16 5 = 20

5 = 4

5 = 2

5 2 5 2

7 3

k

k

k

k

k or k

k k

(4)

4.1.3 5 12

4 2

7 5 1

0 4 2

1

therefore CB AB

AB

CB

AB CB

M

M

M M

(3)

4.1.4

2

1Area of ΔABC = × base × height

2

1 =

2

1 = 20 20

2

=10 units

AB BC

(3)


MATHEMATICS N3


4.2

1 2 1 2coordinates of M ;2 2

2 4 1 7 = ;

2 2

=(1;3)

Equation =1

x x y y

x

(4)

4.3

o o

o o o

1 3

tan 1 tan =3

=45 =71,565

ˆBAC = = 71,565 45 = 26,565

AB ACM M

(5)

[21]

QUESTION 5

5.1 2733 22 yx

Y

3

-3 0 3 X

-3 (2)


MATHEMATICS N3


5.2.1

5.2.2

3 2

2

2

2

3 2

3 2

6 9

3 12 9

Let 0

3 12 9 0

3( 4 3) 0

( 1)( 3) 0

1 or 3

( 1) ( 1) 6( 1) 9( 1) 4

( 3) ( 3) 6( 3) 9( 3) 0

Turningpoints ( 3;0) and ( 1;4)

y x x x

dyx x

dx

y

x x

x x

x x

x x

f

f

(5)

(3)

5.3

1

1 2

1

2 2

2

12

2

2(0,5)

1 1

y xx

y x x

dyx x

dx

dy

dx x x

(4)

[14]


MATHEMATICS N3


QUESTION 6

6.1

(4)

6.2 2

2

2

o o

o

sin 1 cos

sin sin

sin sin 0

sin (sin 1) 0

sin 0 or sin 1

0 or 90

or 180

(5)

6.3 6.3.1

6.3.2

o o

o

o

o

o

o

:

30

sin 70 sin 45

30sin 70

sin 45

39,868

:

cos 25

cos 25

39,868 cos 25

36,133

In ABC

AB

AB

m

In ABD

AD

AB

AD AB

m

(3)

(3)


MATHEMATICS N3


6.4 2

2

a

p

(2)

[17]

TOTAL: 100

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FULL-TIME CLASSES IF NEAR OUR CLASSES IN

KRUGERSDORP

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ENGINEERING FOR A FEE. REQUEST A QUOTE. SEND US AN EMAIL ON

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Website: www. ekurhulenitech.co.za

Email: [email protected] TEL: 011 040 7343 CELL: 073 770 3028/060 715 4529



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