GRADE 12 EXAMINATION

JUNE 2017

ADVANCED PROGRAMME MATHEMATICS

ALGEBRA AND CALCULUS

Time: 2 hours

200 marks

Examiner: Mrs Thorne

_____________________________________________________________________

PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY

1. This question paper consists of 5 pages and an Information Booklet.

2. Please check that your paper is complete.

3. Read the questions carefully.

4. Answer all the questions.

5. Number your answers exactly as the questions are numbered. 

6. You may use an approved, non-programmable, and non-graphical calculator,

unless otherwise stated.

7. Round off your answers to TWO DECIMAL PLACES, unless otherwise indicated.

All the necessary working details must be clearly shown.

8. It is in your own interest to write legibly and to present your work neatly.

9. Diagrams are not drawn to scale.

10. Trigonometric calculations should be done using radians and answers should be given in radians.

Name:_________________________________ Teacher:______________________

QUESTION 1

1.1 Solve for x, showing your working:

a) eπx−3=0 (3)

b) ; (MAKE SURE YOU CALCULATOR IS IN RADIANS) (5)

c) (5)

1.2 The half-life of radium-226 is 1590 years. The formula for the mass, m (in milligrams), of radium that remains after t years is given by:

m(t)=100 e−( ln 2)t

1590

a) Find the mass after 100 years correct to the nearest milligram. (2)

b) After how many years will the mass be reduced to 30mg? (5)

1.3 Given that x=2 i is a solution to the equation: a x3+b x2+8 x+20=0;

determine the other solutions as well as the values of a and b. (7)

[27]

QUESTION 2

Prove by Mathematical Induction that:

(1)2+(2)2+(3)2+…+¿ for all n∈N . [12]

QUESTION 3

3.1 Evaluate the following limit:

a) limx→ ∞ ( 1−x2

3 x2−x )❑

(4)

b) limx →1 ( 2 x−2x2

√x−1 )❑

(4)

3.2 Say whether the following are true or false. If true, give a reason. If false, give a counterexample.

a) If f is continuous at a, then so is |f|. (2)

b) If f (x)<g(x ) for all x in an interval around a, and if limx→ a

f (x ) and limx→ a

g (x)

both exist then limx→ a

f (x )<limx→ a

g(x ). (2)

[12]

QUESTION 4

4.1 Given f ( x )={ 243 x−1

,∧x<3

−x2+2ax−6 ,∧x≥ 3

a) Determine a if f is continuous at x=3. (4)

b) Use the value of a calculated above to determine whether f is differentiable at x=3. (6)

4.2 Sketch the possible graph of a function f(x) that satisfies the following conditions:

f ' (x)>0 for x∈(−∞;1)

f ' (x)<0 for x∈(1 ;∞)

f ' ' (x)<0 for x∈(−2 ;2)

f ' ' (x)>0 for x∈(−∞;−2)∨(2 ;∞ )

limx→−∞

f (x)=−2

limx→ ∞

f (x)=0

(7)

[17]

QUESTION 5

5.1 Determine dydx

:

a) y=x . tanx (4)

b) y= x2+5√2x+3

(7)

5.2 Determine the equation of the tangent to the curve x2 y+ y2=8+2 x

at the point (-1:2). (10)

[21]

QUESTION 6

A portion of the graph f (x)=x4+5 x3−2x−1is shown below:

Use the Newton-Rhapson method with an initial approximation of 1 to

find the x-coordinate of the local minimum shown above. Give your answer correct to 4 decimal places.

[10]

QUESTION 7

Determine the following integrals:

7.1 ∫7 x2 √5 x3−13 dx (7)

7.2 ∫sin 5θ sin 2 θ dθ

(use the identity sinAsinB=1

2(cos( A−B)−cos (A+B))

(7) 7.3

∫3 π4

5 π4

x cos2 xdx

(show all your working) (13)

[27]

QUESTION 8

The sketch shows the graph of f (x)= x2+6x2+x−6

. The graph has vertical and horizontal asymptotes

and two turning points of which the maximum turning point is at A.

8.1 Determine the equations of vertical and horizontal asymptotes. (4)

8.2 Determine the coordinates of the turning point A. (8)

8.3 Given that f (x)= x2+6x2+x−6

=1− x−12x2+x−6

.Decompose f (x)into partial fractions. (5)

The following sketch shows a portion of the same graph of y=f (x ), with the lines x=−1 and

x=a. The region included by the lines, the graph of f and the x-

axis is shaded.

8.4 Assume that f (x)=1+ 2x−2

− 3x+3

. Write an expression

that can be used to calculate this area. (2)

8.5 Hence determine this area in terms of a. (6)(5)

[25]

QUESTION 9

Below is a picture of the graph for . The area bounded by the graph of and the -axis between the -intercepts marked A and B has been shaded.

9.1 Determine the coordinates of A and B,leaving your answers in terms of . (6)

9.2 Determine the area of the shadedregion. (6)

9.3 Prove that . (4)

9.4 The volume generated when the graph of rotates around the -axis between

and is equal to . Use Question 9.3, or otherwise to calculate the smallest possible value of . (14)

[30]

QUESTION 10

Along the edge of the Durban promenade runs an unusual fence designed to resemble waves. The

pattern is made up of many arcs of a circle as shown below.

`θ

B

A

r

a) A developer needs to make fencing that spans 4 arcs. If she lets the radius be r metres and the distance from A to B is 2r metres, calculate the angle at the centre used to create each sector ( θ ). (8)

b) If she has enough materials to make 16 metres of fencing, calculate r. (4)

c) The landscaper wants to plant flowers in the shaded segments below to make a flower bed. Calculate the area to be covered by flowers. (7)

[19]

200

QUESTION 5

The functions p and q are defined, for , by:

5.1 John was asked to find , the derivative of .

His first line of working was:

(a) Using John’s first line of working determine implicitly. (6)

(b) If John’s first line of working was determine

implicitly and using the product rule of differentiation. (5)

(c) Prove that your answer’s to Question 5.1 (a) and 5.1 (b) are equivalent. (2)

5.2 Express the following as composite functions of p and q:

(a) (2)

(b) (4)

5.3 Given that , determine the coordinates of the stationary points of g. (8)

[27]

QUESTION 3

A pattern on a square tile is created from 7 equal sectors as shown below. The circle is inscribed in

the square. The total shaded area in the tile measures 8,6 cm2.

3.1 Determine the radius, r, of the circle to one decimal. (8)

3.2 Hence, determine the area of the square tile to

one decimal. (3)

3.3 Determine minor arc length AB to one decimal. (3)

[14]

1.1.

An old satellite dish is lying abandoned next to John’s house, flat on its base and completely filled with water from the previous night’s deluge. How much water is contained in the dish if this dish was created

by revolving the parabola y=10√ x about the x-axis, and it’s diameter across the top of the dish is 80cm.

Give your answer correct to the nearest litre (1cm3 = 1ml).

[9]

Question 4

4.1 Given that x= ln a is a solution to the equation 10 e2 x−7 ex=26. Find, without using a calculator, the value of a. (6)

B

A

r

4.2 Euler’s formula states that:

e iy=cos y+isin y

4.2.1 Find the value of e iπ, showing your working. (3)

4.2.2 Prove that: cos x= e ix+e−ix

2 [Hint: think of e−ix as e i (− x )] (4)

[13]The diagram shows part of the curve y2=x+1.

14.1 Find the co-ordinates of A. (2)

14.2 The region AOB is rotated through 360 ° about the x-axis to generate a volume V. Calculate the value of V.(4)

14.3 The region ACD, with CD parallel to the y-axis, is rotated through 360 ° about the x-axis to generate a volume 9V. Calculate the value of k . (6)

[12]1.

AB is a tangent to the circle centre O. radians. OA = 7 cm.

a) Write down the size of . (2)

Find the area of the shaded region. (9)

[11]

Now take your calculator OUT OF RADIANS