Copyright reserved Please turn over

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

Copyright reserved Please turn over

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

Copyright reserved Please turn over

QUESTION 1 1.1 Factorise the following expressions as far as possible in prime factors: 1.1.1 (4) 1.1.2 (2) 1.2 Factorise the following expression completely:

(5) 1.3 Simplify the following expression:

(6) [17]

QUESTION 2 2.1 Simplify the following:

(4) 2.2 Use logs to the base 2 and simplify the following WITHOUT using a calculator:

(4) 2.3 Solve for x: 2.3.1 2.3.2 (8) [16]

(3 2) (3 2)x x y y- - +

4 24 3 1p pn n+ -

3 22 5 2x x x+ - +

2

2

1 2 1 2 7 171 3 2 3

x x x xx x x x- - - -

- ++ - - -

32

12

3

xx

x

-

0,5log 128

216 3 3 2x x+ = +

(log 2) log( 2) 0x x- ´ - = (2 4)´

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

Copyright reserved Please turn over

QUESTION 3 3.1 Solve for by completing the square:

(4) 3.2 Make ' ' the subject of the formula:

(4)

3.3 Make ' w ' the subject of the formula:

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

instalments. He paid a total of . What is the payment of each monthly instalment in terms of .

(4) [15]

QUESTION 4 4.1 Consider FIGURE A below. is an isosceles triangle with and

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

FIGURE A

x24 48 2x x= -

bx bDx b+

=-

log log loge e et p w ds- + =

3R x33R x

x

ABCD AB BC=

(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 is perpendicular to if . (3) 4.1.4 Calculate the area of when . (3) 4.2 P and Q 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 and with equations and

respectively, intersect at A. Determine the size of where B and C are the intercepts on the x-axis as shown.

FIGURE B

(5) [21]

AB BC k 7=

ABCD 7k =

( 2; 1)- - (4;7)

BA CA 2y x= +

3 3y x= - ˆBAC

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

Copyright reserved Please turn over

QUESTION 5 5.1 Draw the graph defined by the equation: (2) 5.2 Given : 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 if . Leave the answers with positive indices and in

surd form.

(4) [14] QUESTION 6 6.1 Prove the following trigonometric identity:

(4) 6.2 Calculate the value(s) of which will satisfy the equation if :

(5) 6.3 Consider FIGURE C below. An observer, standing at a point A is watching the top

of a vertical tower . The angle of elevation of the top of the tower, BC, is and the angle of depression of the foot of the tower is . 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 and the point B. (3) 6.3.2 The distance between the two towers. (3)

2733 22 =+ yx

3 26 9y x x x= + +

dydx

1 2y xx

= +

2 2 2 2sin tan cos secA A A A+ + =

q o o0 270q£ £2sin 1 cosq q= -

BC o25o20

A

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

Copyright reserved Please turn over

FIGURE C

6.4 Consider FIGURE D below. The sketch represents the graph of

where . Determine the values of and .

FIGURE D

(2) [17]

TOTAL: 100

( ) sinf x a px=0 x p£ £ a p

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

Copyright reserved Please turn over

FORMULA SHEET Any applicable formula may also be used. 1. Factors 2. Logarithms a3 - b3 = (a - b)(a2 + ab + b2) a3 + b3 = (a + b)(a2 - ab + b2)

3. Quadratic formula

4. Parabola

5. Circle 6. Straight line

Perpendicular: Parallel lines:

Distance:

Midpoint:

Angle of inclination:

blogalogablog +=

blogalogbalog -=

blogalog

alogc

cb =

2a4acbb

x2 -±-

= alogmalog m =

blog1aloga

b =

cbxaxy 2 ++=

4ab4acy2-

=

11∴ == e n1aloga

meta m1ntloga == ∴

2abx -

=

222 ryx =+

h4hxD2+=

24h4Dhx -=

)xm(xyy 11 -=-

1mm 21 -=×

21 mm =

212

212 )y(y)x(xD -+-=

÷ø

öçè

æ ++=

2yy

;2xx

P 2121

mtanθ -1=

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

Copyright reserved

7. Differentiation

Max/Min For turning points: 8. Trigonometry

Area of ∆ABC = ½ ac sin B

( ) ( )h

xfhxf0h

limdxdy

→-+

=

( ) 1-nn nxxdxd

=

( ) 0xf ' =

cosecθ1

rysinθ ==

secθ1

rxcosθ ==

cotθ1

xytanθ ==

1=+ θcosθsin 22

θsecθtan1 22 =+

θcosecθcot1 22 =+

cosθsinθtanθ =

sinθcosθcotθ =

csinC

bsinB

asinA

==

cosA2bccba 222 -+=