Gr

							                          ST STITHIANS GIRLS COLLEGE

                                   MATHEMATICS

            ADVANCED PROGRAMME MATHEMATICS: GRADE 12

DATE: 7 August 2010                                                        TIME: 3 hours
                                                                           MARKS: 300



NAME:                                        TEACHER: Mr A


PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY

1.    This question paper consists of 8 pages, including a pink information booklet of 4
      pages.

2.    SECTION A: FINANCE & MODELLING /100
      SECTION B: CALCULUS & ALGEBRA /200

3.    Read the questions carefully.

4.    You may use an approved non-programmable and non-graphical calculator,
      unless otherwise stated.

5.    Unless otherwise stated, round answers to two decimal places where necessary.

6.    All the necessary working details must be clearly shown.

7.    It is in your own interest to write legibly and to present your work neatly.
                          Grade 12: AP Mathematics August 2010                              2

SECTION A – FINANCE /100

QUESTION 1

1.1   Nina has been recording her country’s annual inflation figures for the last five
      years.
      2006: 6,1%
      2007: 7,1%
      2008: 8,1%
      2009: 9,1%
      2010: 10,1%

      What was the effective annual inflation over this time period.                            (8)


1.2   Ginny decides to buy a car for R225 000. She finances the entire cost through a
      bank which charges interest at 12,5% per annum compounded monthly over a 5
      year period. Her first payment is made exactly one month after purchasing the car.

(a)   Calculate her monthly repayments.                                                         (6)

(b)   Determine the balance outstanding on the loan account immediately after her 30th
      payment?                                                                                  (8)

(c)   If Ginny makes a cash deposit of R50 000 immediately after her 30th payment, but
      then doesn’t make any further payments for the next 12 months, determine the
      value of the new monthly instalment if she still wishes to amortize the loan within
      the original loan period.                                                                 (10)
                                                                                                [32]
                          Grade 12: AP Mathematics August 2010                               3

QUESTION 2

Veronica decides to take out a retirement annuity starting on her 40th birthday, the first
payment being at the end of the month. Her premium is fixed at R2 000 per month and
the company offering the retirement annuity guarantees an interest rate of 9% interest per
annum compounded monthly.

2.1    Veronica’s last payment will be on her 60th birthday, at which point she will retire.
      Assuming that she makes all the payments, what is the total amount that she will
      have saved for her retirement?                                                             (8)

2.2   Determine the total amount of interest earned over the duration of the investment.         (4)


It turns out that, after 8 years of paying into the annuity, Veronica runs into financial
difficulties and she misses 5 payments. (Payment numbers 97 to 101). At 10 years
(payment number 120) she has sufficient funds available to make a lump sum payment
which will make up for the missed payment.

2.3   Determine the value of this lump sum payment (excluding her usual monthly
      repayment).                                                                                (12)


On retirement, Veronica withdraws 25% of her annuity in cash and then forms a living
annuity with an interest rate of 12% per annum compounded monthly, with the balance.
She assumes that because she has had a very healthy lifestyle she will live to be 90 years
old.

2.4   How much will Veronica be able to withdraw in cash?                                        (2)

2.5   Determine Veronica’s monthly withdrawals from her living annuity?                          (8)

2.6   If Veronica needs R20 000 per month, determine for how many months her
      retirement savings will last and the value of the final ‘payment’.                         (22)
                                                                                                 [56]
                          Grade 12: AP Mathematics August 2010                            4

QUESTION 3

3.1   The following sequence is given: Tn 1  Tn  0,1Tn , where T1  1000

      Determine the value of T5 showing all your working, and then explain/show why
      this value is equivalent to A in the formula A  P(1  i ) n when P = 1000, n = 4
      and i = 10%.                                                                            (6)


                                                     K  Pt 
3.2   Refer to the given formula: Pt 1  Pt  r.Pt         
                                                     K 
      Determine the value of K if P5  108 , P4  50 and r  1, 2                             (6)
                                                                                              [12]
                               Grade 12: AP Mathematics August 2010                              5

SECTION B: CALCULUS & ALGEBRA /200

QUESTION 4

4.1       Prove by induction that 9n + 3 is divisible by 4, n                                       (9)


4.2       The function f is defined by

                 ( x  1) 2   if x  0
 f ( x)         2p  k      if x  0
                      1
                 ( x  k ) 2 if x  0
                      2

          Determine the values of k and p for which f is continuous at x = 0                         (12)



4.3       The graph of h'(x), a linear function, is shown:
                                                                  h' ( x )

(a)       Sketch h ''( x)                (4)

(b)       Sketch h(x)                    (4)
                                                                                                 x
                                                                             2
                                                                6




4.4                            B

      A
                                               Points A and B lie on the circle with centre O.

                                               The area of the minor sector AOB is 72 cm2
                                               The perimeter of the minor sector is 36 cm.
                         O
                                               Calculate the radius of the circle.


                                                                                                     (14)
                                                                                                     [43]
                               Grade 12: AP Mathematics August 2010                                     6

QUESTION 5

5.1   Determine the following limits

             2 x  12
a)    lim                                                                                                   (8)
      x 3     3 x

          5sin 2 x
b)    lim                                                                                                   (6)
      x 0 sin 4 x



               x
c)    lim                                                                                                   (8)
      x 
              x2  x


                              1  sec 2A
5.2   Prove the identity:                 2sec 2A                                                          (10)
                                cos 2 A
                                                                                                            [32]


QUESTION 6

                                                                                        
6.1   Find the gradient of the tangent to the curve y  tan 2 x at x                     without the use
                                                                                        8
      of a calculator.                                                                                       (9)



6.2   Find the gradient of the tangent to the curve defined by the equation
         3         2
      xy  3x  xy  12 at the point (2; 1).                                                               (10)



6.3   If f ( x)  ( x  2) 1 find D x [ f ( x)], D 2 x [ f ( x)], D 3 x [ f ( x)], D 4 x [ f ( x)].
      Now write down a formula for D n x [ f ( x)].                                                          (7)
                                                                                                            [26]
                             Grade 12: AP Mathematics August 2010                    7

QUESTION 7

7.1   Determine the required derivatives:

       d 2  sin x 
(a)                                                                                    (10)
       dx 2  x 

       dy
(b)       if y  ( x  1) tan x                                                          (10)
       dx



7.2   Given the equation 2cos    2 .

(a)   Show that the equation 2cos    2 has a solution in the interval  3;5       (6)

(b)   Now solve the above equation using the Newton-Raphson method of
      approximation. Solve, rounded to 5 decimal places if necessary.                    (12)
                                                                                         [38]


QUESTION 8

Consider f ( x)  3x 5  5x 3

8.1   Find the intervals where f is strictly increasing or strictly decreasing.          (7)

8.2   Find the intercepts of the graph with the x -axis.                                 (4)

8.3   Sketch the graph of y  f (x).                                                     (7)
                                                                                         [18]


QUESTION 9

                        2x 1
The function f ( x)           is given:
                        3x  1

9.1   Show that the curve y  f ( x) has no turning points.                              (5)

9.2   Find the asymptotes and intercepts with the axes and sketch the graph of
      y  f ( x) . Show all calculations.                                                (11)
                                                                                         [16]
                            Grade 12: AP Mathematics August 2010                    8



QUESTION 10

You may not just type the question into your calculator. Show appropriate working!

10.1 Use a Riemann Sum to determine the area under the
       curve of f ( x)   x 2  4 between the positive
         and positive axes.                    (10)




10.2 Calculate the area enclosed by the graphs f ( x)  x 2 and g ( x)  2x  x 2       (9)
                                                                                        [19]



QUESTION 11

Evaluate the following integrals:


       x       x3  1 dx
            2
11.1                                                                                    (9)

11.2    cos ecx.cot x dx                                                               (4)
                                                                                        [13]

                                                                                    MAX 205