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T980(E)(A6)T APRIL EXAMINATION NATIONAL CERTIFICATE

MATHEMATICS N6

(16030186) 6 April 2016 (X-Paper)

09:00?12:00

Calculators may be used.

This question paper consists of 5 pages and 1 formula sheet of 7 pages.

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T980(E)(A6)T

DEPARTMENT OF HIGHER EDUCATION AND TRAINING REPUBLIC OF SOUTH AFRICA

NATIONAL CERTIFICATE MATHEMATICS N6 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 be kept

together.

5.

Show ALL the intermediate steps.

6.

ALL the formulae used must be written down.

7.

Questions must be answered in BLUE or BLACK ink.

8.

Write neatly and legibly.

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T980(E)(A6)T

QUESTION 1

1.1

If z = 5x3 y2 y4 3x2 y , determine 2z

xy

(2)

1.2

Given: I V

R

Calculate the change in I if V decreases with 5 volts and R with 8 ohms. The original (4)

value of V is 30 volts and of R is 10 ohms.

[6]

QUESTION 2

Determine y dx if:

2.1

y = sin4 5x cos3 5x

(5)

2.2

y= 1

16x x2

(3)

2.3

y = sin4 mx

(4)

x

2.4

y = e2 .cos3x

(6)

[18]

QUESTION 3

Use partial fractions to calculate the following integrals:

3.1

x2 3x 4 x(1 2x)2

dx

(6)

3.2

10x2 7x 1 (2x2 1)(4x 1)

dx

(6)

[12]

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T980(E)(A6)T

QUESTION 4

4.1

Calculate the particular solution of:

2 sin x dy y(sin2x) 2 sin x at (0;1)

dx

sec x

(5)

4.2

Calculate the particular solution of:

d2y dx2

7 dy + 6 y dx

=

2x 3,

if

y 1 when x 0 and dy 2 when x 0 . dx

(7) [12]

QUESTION 5

5.1

5.1.1

5.1.2

5.2

5.2.1

5.2.2 5.2.3

5.3

5.3.1

Calculate the points of intersection of the two curves y 3 and x

y x 4 0 . Make a neat sketch of the curves and show the area, in the

first quadrant, bounded by the curves. Show the representative

strip/element that you will use to calculate the volume (use the SHELL

method only) generated if the area bounded by the curves rotates about

the y -axis.

(3)

Use the SHELL method to calculate the volume generated if the area,

described in QUESTION 5.1.1, bounded by the two curves y 3 and x

y x 4 0, rotates about the y-axis.

(5)

Make a neat sketch of the graph y tan x . Show the representative

strip/element that you will use to calculate the volume generated if the

area bounded by the graph, the ordinates y 0 and x rotates about 3

the x-axis.

(2)

Calculate the volume generated if the area, described in

QUESTION 5.1.1, rotates about the x -axis.

(3)

Calculate the volume moment about the y -axis as well as the distance of

the centre of gravity from the y -axis.

(6)

Calculate the points of intersection of the two curves y = 2x2 and

x y . Make a neat sketch of the curves and show the area bounded by 3

the curves. Show the representative strip/element, PERPENDICULAR to

the x -axis, that you will use to calculate the area bounded by the curves.

(3)

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T980(E)(A6)T

5.3.2

Calculate the area described in QUESTION 5.3.1, bounded by the two

curves y = 2x2 and x y .

3

(3)

5.3.3

Calculate the second moment of area of the area described in

QUESTION 5.3.1 about the y-axis.

(4)

5.3.4

Express the answer in QUESTION 5.3.3 in terms of the area.

(1)

5.4

5.4.1

A weir in the form of a trapezium is 2 m high, 10 m wide at the top and

4 m wide at the bottom. The top of the weir is in the water surface.

Sketch the weir and show the representative strip/element that you will use to calculate the depth of the centre of pressure on the retaining wall.

Calculate the relation between the two variables x and y.

(3)

5.4.2

Calculate, by using integration, the area moment of the weir about the

water level.

(3)

5.4.3

Calculate, by using integration, the second moment of area of the weir

about the water level, as well as the depth of the centre of pressure on the

weir.

(4)

[40]

QUESTION 6

6.1

Calculate the arc length of the curve described by the parametric equations,

x 5(cost t sint) and y 5(sint t cost) , between the points t 0 and t .

(6)

6.2

Calculate the surface area generated when the curve of y 16x , over the interval

1 x 4, is rotated about the x-axis.

(6)

[12]

TOTAL: 100

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MATHEMATICS N6

FORMULA SHEET Any other applicable formula may also be used.

Trigonometry sin2 x + cos2 x = 1 1 + tan2 x = sec2 x 1 + cot2 x = cosec2 x sin 2A = 2 sin A cos A cos 2A = cos2A - sin2A

tan

2A

=

2 tan A 1 tan2 A

sin2 A = ? - ? cos 2A cos2 A = ? + ? cos 2A

sin (A ? B) = sin A cos B ? sin B cos A

cos (A ? B) = cos A cos B sin A sin B

tan (A ? B) = tan A tan B 1 tan Atan B

sin A cos B = ? [sin (A + B) + sin (A - B)]

cos A sin B = ? [sin (A + B) - sin (A - B)] cos A cos B = ? [cos (A + B) + cos (A - B)] sin A sin B = ? [cos (A - B) - cos (A + B)]

tan x sin x ; sin x 1 ; cosx 1

cosx

cosec x

secx

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_________________________________________________________________

f(x)

d f (x)

dx

f(x)dx

_________________________________________________________________

xn

nxn - 1

xn1 C n 1

(n - 1)

T980(E)(A6)T

axn

a d xn

dx

a xn dx

e ax b

eaxb. d (ax + b) dx

eax b

d

C

ax b

dx

a dx+e ln(ax) e f (x) a f (x) ln f(x)

adxe. ln a. d (dx + e) dx

1 . d ax ax dx e f (x) d f (x)

dx a f (x). ln a. d f (x)

dx 1 . d f (x) f (x) dx

adx e

C

ln a. d dx e

dx

xln ax - x + C

-

-

-

sin ax cos ax tan ax cot ax sec ax

a cos ax -a sin ax a sec2 ax -a cosec2 ax a sec ax tan ax

- cosax C a

sinax C a

1 ln [sec (ax)] C a 1 ln [sin (ax)] C a 1 ln [sec ax tan ax] C a

cosec ax

-a cosec ax cot ax

1 a

ln

tan

ax 2

C

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