Mathematics Question MEMO N3 Download PDF

Preview

Summary

Industrial Organisation AND Planning Question MEMO N3 Download...

Description

EKURHULENI TECH COLLEGE. No. 3 Mogale Square, Krugersdorp. Website: www. ekurhulenitech.co.za Email: [email protected] TEL: 011 040 7343 CELL: 073 770 3028/060 715 4529

PAST EXAM PAPER & MEMO N3 ABOUT THE QUESTION PAPERS: THANK YOU FOR DOWNLOADING THE PAST EXAM PAPER AND ITS MEMO, WE HOPE IT WILL BE OF HELP TO YOU. SHOULD YOU NEED MORE QUESTION PAPERS AND THEIR MEMOS PLEASE SEND US AN EMAIL TO [email protected] AND WE WILL SEND YOU A FULL DOWNLOAD VERSION OF THESE. HOWEVER PLEASE NOTE THAT THESE ARE SOLD FOR A SMALL AMOUNT OF R300 PER DOWNLOAD PER SUBJECT.

ABOUT REGISTERING WITH OUR COLLEGE CONSIDER REGISTERING WITH OUR COLLEGE AND WE HAVE THE FOLLOWING TYPES OF LEARNING:  ONLINE OR CORRESPONDENCE LEARNING. THIS IS THE MOST PREFARED ONE BY THOUSANDS  PART-TIME CLASSES DURING SATURDAYS IF NEAR OUR OFFICES IN KRUGERSDORP  FULL-TIME CLASSES IF NEAR OUR CLASSES IN KRUGERSDORP

ABOUT EXTRA TEXTBOOKS IF LOOKING FOR TEXBOOKS FOR CERTAIN SUBJECTS I N1-N6 ENGINEERING STUDIES PLEASE SEND US AN EMAIL ON [email protected]

ABOUT VIDEO MATERIAL WE HAVE VIDEO MATERIAL FOR EXTRA UNDERSTANDING ON CERTAIN ENGINEERING FOR A FEE. REQUEST A QUOTE. SEND US AN EMAIL ON [email protected]

Website: www.ekurhulenitech.co.za

Email: [email protected]

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.

Copyright reserved

Please turn over

(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.

Answer ALL the questions.

2.

Read ALL the questions carefully.

3.

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

4.

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

5.

Show ALL the calculations and intermediary steps.

6.

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

7.

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

8.

Diagrams are NOT drawn to scale.

9.

Write neatly and legibly.

Copyright reserved

Please turn over

(16030143)

-3-

T950(E)(A1)T

QUESTION 1 1.1

1.2

1.3

Factorise the following expressions as far as possible in prime factors: 1.1.1

x(3 x  2)  y(3 y  2)

(4)

1.1.2

4 n4 p  3 n2 p 1

(2)

Factorise the following expression completely: 2x 3  x 2  5x  2

(5)

Simplify the following expression: x  1 2 x  1 2x 2  7x  17   2 x  1 3 x x  2x  3

(6) [17]

QUESTION 2 2.1

Simplify the following: 1 x 2 x 3

3x 2

2.2

2.3

(4)

Use logs to the base 2 and simplify the following WITHOUT using a calculator: log 0,5128

(4)

Solve for x: 2.3.1 2.3.2

Copyright reserved

16  3 2 x  3 x  2 (log x  2)  log( x  2)  0

(2  4)

Please turn over

(8) [16]

(16030143)

-4-

T950(E)(A1)T

QUESTION 3 3.1

3.2

3.3

3.4

Solve for x by completing the square: 4 x  48  2 x 2

(4)

Make ' b ' the subject of the formula: x b D x b

(4)

Make ' w ' the subject of the formula: log e t  log e p  log e w  ds

(3)

Alex paid a deposit of R3x for a computer. He paid the rest in 9 monthly instalments. He paid a total of R33x . What is the payment of each monthly instalment in terms of x .

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

Copyright reserved

Please turn over

(4) [15]

(16030143)

4.2

-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 k  7 .

(3)

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.

4.3

(4)

Consider FIGURE B below. The lines BA and CA with equations y  x  2 and ˆ where B and C y  3x  3 respectively, intersect at A. Determine the size of BAC 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: 3x 2  3 y 2  27

5.2

Given : y  x3  6 x2  9 x 5.2.1

5.2.2

5.3

Determine

(2)

Make use of differentiation to determine the coordinates of the turning points of the given equation.

(5)

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)

dy 1 if y   2 x . Leave the answers with positive indices and in x dx

(4) [14]

surd form.

QUESTION 6 6.1

Prove the following trigonometric identity: sin A  tan A  cos A  sec A

(4)

Calculate the value(s) of  which will satisfy the equation if 0 o    270 o : sin  1 cos 2

(5)

2

6.2

6.3

2

2

2

Consider FIGURE C below. An observer, standing at a point A is watching the top o of a vertical tower BC . The angle of elevation of the top of the tower, BC, is 25 o and the angle of depression of the foot of the tower is 20 . 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)

Copyright reserved

Please turn over

(16030143)

-7-

T950(E)(A1)T

FIGURE C 6.4

Consider FIGURE D below. The sketch represents the graph of f ( x)  a sin px where 0  x   . Determine the values of a and p .

(2) [17]

FIGURE D TOTAL: Copyright reserved

100

(16030143)

-1-

T950(E)(A1)T

FORMULA SHEET Any applicable formula may also be used. 1. Factors

2. Logarithms

a3 - b3 = (a - b)(a2 + ab + b2)

log ab  log a  log b

a  log a  log b b logc a log b a  logc b log

a3 + b3 = (a + b)(a2 - ab + b2) 3. Quadratic formula

 b  b 2  4ac x 2a

log a m  m log a

log b a 

4. Parabola

1 loga b

y  ax 2  bx  c

loga a  1 ∴ 1n e  1

4ac  b2 4a b x 2a

alogat  t ∴ e1n m  m

y

5. Circle x2  y 2  r 2

D

x2 h 4h

6. Straight line

y  y1  m(x  x1 ) Perpendicular: m1  m 2  1 Parallel lines: m1  m2

x  4Dh  4h2 Distance: D  (x 2  x1 )2  (y 2  y1 ) 2

 x  x 2 y1  y 2  Midpoint: P   1 ;  2 2   Angle of inclination: θ  tan-1m

Copyright reserved

Please turn over

(16030143)

7. Differentiation

lim f  x  h  f  x dy  dx h → 0 h

 

d n x dx

 nx n - 1

Max/Min For turning points: f '  x  0

8. Trigonometry sinθ 

y 1  r cosecθ

cosθ 

x 1  r secθ

tanθ 

y 1  x cotθ

sin 2 θ  cos 2 θ  1 1  tan 2θ  sec 2 θ 1  cot 2θ  cosec 2 θ

tanθ 

sinθ cosθ

cotθ 

cosθ sinθ

sinA sinB sinC   a b c

a 2  b2  c2  2bc cosA Area of ∆ABC = ½ ac sin B

Copyright reserved

-2-

T950(E)(A1)T

MARKING GUIDELINE NATIONAL CERTIFICATE APRIL EXAMINATION MATHEMATICS N3

1 APRIL 2016

This marking guideline consists of 10 pages.

Copyright reserved

Please turn over

MARKING GUIDELINE

-2MATHEMATICS N3

T950(E)(A1)T

QUESTION 1 1.1

x (3x  2)  y (3y  2)

1.1.1

 3 x 2 x3 y  2 y 2

2



  3( x 2  y 2)  2( x  y)  3( x  y)( x  y)  2( x  y)  (x  y )[3(x  y ) 2]  (x  y )[3x  3y  2]   4 n4 p  3 n2 p 1

1.1.2



 (n 2 p  1)(4n 2p  1)

1.2

(4)



(2)

f ( x)  2 x 3  x 2  5 x  2  f (1)  2(1)3  (1)2  5(1)  2 =2 1  5  2  =0  x-1 is a factor of f ( x)  2x 2  3x  2 x-1 2 x3  x2  5 x 2 2 x  2 x2 3



3x 2  5 x 3x 2  3 x - 2x  2 -2 x  2 .

.

 f ( x)  ( x  1)(2 x2  3 x  2) = ( x 1)(2 x 1)( x  2)



 (5)

Copyright reserved

Please turn over

MARKING GUIDELINE

-3MATHEMATICS N3

T950(E)(A1)T

1.3



x  1 2 x  1 2 x2  7 x  17    ( x  1)(x  3) x 1 x 3

=

( x  1)( x  3)  (2 x  1)( x  1)  2 x 2  7 x  17  ( x  1)( x  3)

=

x2  4 x  3  2 x2  x  1  2 x2  7 x  17  (x  1)(x  3)



5( x 2  2x  3)  ( x 1)(x  3)

 (6) [17]

Copyright reserved

Please turn over

MARKING GUIDELINE

T950(E)(A1)T

-4MATHEMATICS N3

QUESTION 2 2.1

x

1 2 x 3

 

3x 2 2x  1 1

2 x2 2x  1 1 2

3

 3x 2 

1



3

2x 3 x2 2x  1  6x 2 2.2



 (4)



log 0,5 128 log2 128 1 log 2 2 log2 27  log2 21





 

7 log2 2 1log 2 2  7 

2.3

2.3.1



(4)

x x 16  32  3  2



16  32 x  3 x  2



2

16  32 x  32 x  4.3 x  4 12  4.3 x

 

  3x  3   x 1 TEST : If x  1 then LHS=RHS=5 2.3.2

(4)

(log x  2)  log(x  2)  0 log(x  2)  0  log x  2  0 or  log x  2 

x  2=10 0

 

x  2 1 x 3 

x  10 x  100 2





(4) [16]

Copyright reserved

Please turn over

MARKING GUIDELINE

-5MATHEMATICS N3

T950(E)(A1)T

QUESTION 3 3.1 







(4)

3.2

D2 

x b  x b





 (4)

Copyright reserved

Please turn over

MARKING GUIDELINE

T950(E)(A1)T

-6MATHEMATICS N3

3.3

log e

tw  ds  p

tw  e ds  p w

3.4

p ds e  t

Deposit is R3x  Total price is R33x  Total amount for 9 instalments= R30x 30x 10x R  Each monthly instalment = R 9 3

Copyright reserved

(3)





(4) [15]

Please turn over

MARKING GUIDELINE

-7MATHEMATICS N3

T950(E)(A1)T

QUESTION 4 4.1

4.1.1

AB  =

 x2  x1    y2  y1  2

2

 4  2 2   5  1 2 

= 4  16 = 20



=2 5

4.1.2

(2)

BC=2 5 

 0  4 2   k  5 2

 20

16  k  5 

= 20

2



 k  5 = 4 2

 k  5 =  2  k  5  2 or k  5   2 k  7  k3  4.1.3

4.1.4

5 1 2 4 2 75 1  M CB   04 2 1 M AB  M CB   therefore CB  AB  M AB 

Area of ΔABC =

  

(3)

1 × base × height 2

1 AB  BC 2 1 = 20  20 2  =10 units 2 =

(4)

  (3)

Copyright reserved

MARKING GUIDELINE

T950(E)(A1)T

-8MATHEMATICS N3

4.2

xx y y  coordinates of M   1 2 ; 1 2  2   2  2  4 1  7  =  ; 2   2 =(1;3)   x =1 Equation   (4) 4.3

M AB  1 tan  1



M AC  3



tan  =3

  =45o  =71,565o ˆ =   = 71,565o  45o = 26,565 o BAC

 

(5) [21]

QUESTION 5 5.1

3x 2  3 y 2  27

Y 3 

-3

0

3 

X

-3 Copyright reserved

(2) Please turn over

MARKING GUIDELINE

5.2.1

T950(E)(A1)T

-9MATHEMATICS N3

y  x3  6 x2  9 x dy  3x 2  12x  9 dx Let y  0



  3 x 2 12 x  9  0 3( x2  4 x  3)  0 ( x  1)( x  3)  0 x   1 or x   3





f ( 1)  (1)3  6( 1) 2  9( 1)  4 f ( 3)  (3)3  6(3)2  9( 3)  0



Turningpoints ( 3; 0) and ( 1; 4)

(5)



5.2.2





 (3) 5.3

y

1 2 x x 1

y  x 1  2 x2



1  dy   x  2  2(0,5) x 2 dx dy 1 1  2  dx x x 

Copyright reserved





(4) [14]

Please turn over

MARKING GUIDELINE

-10MATHEMATICS N3

T950(E)(A1)T

QUESTION 6 6.1    6.2

(4)

sin  1 cos2  sin   sin 2 

 sin   sin   0 sin  (sin  1)  0 2



sin   0 or sin   1 or   90o  or  180o  

  0o

6.3

6.3.1

6.3.2

Copyright reserved

In ABC : AB 30  o sin 70 sin 45 o 30 sin 70 o  AB  sin 45 o  39,868m

(5)

 

(3)



In ABD :  AD  cos 25o AB  AD  AB cos 25 o  o  39,868 cos 25  36,133m 

(3)

Please turn over

MARKING GUIDELINE

6.4

a2 p2

-11MATHEMATICS N3

T950(E)(A1)T

 (2) [17]

 TOTAL:

Copyright reserved

100

EKURHULENI TECH COLLEGE. No. 3 Mogale Square, Krugersdorp. Website: www. ekurhulenitech.co.za Email: [email protected] TEL: 011 040 7343 CELL: 073 770 3028/060 715 4529

REGISTERING WITH OUR COLLEGE CONSIDER REGISTERING WITH OUR COLLEGE AND WE HAVE THE FOLLOWING TYPES OF LEARNING:  ONLINE OR CORRESPONDENCE LEARNING. THIS IS THE MOST PREFARED ONE BY THOUSANDS  PART-TIME CLASSES DURING SATURDAYS IF NEAR OUR OFFICES IN KRUGERSDORP  FULL-TIME CLASSES IF NEAR OUR CLASSES IN KRUGERSDORP

ABOUT EXTRA TEXTBOOKS IF LOOKING FOR TEXBOOKS FOR CERTAIN SUBJECTS I N1-N6 ENGINEERING STUDIES PLEASE SEND US AN EMAIL ON [email protected]

ABOUT VIDEO MATERIAL WE HAVE VIDEO MATERIAL FOR EXTRA UNDERSTANDING ON CERTAIN ENGINEERING FOR A FEE. REQUEST A QUOTE. SEND US AN EMAIL ON [email protected]

Website: www.ekurhulenitech.co.za

Email: [email protected]...