Mathematics Grade 12 Exemplar Paper 2 June 2011 Memo - DOC by hPU3zn

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									       MEMORANDUM


            GRADE 12




  MATHEMATICS PAPER 2
       JUNE 2011




This memo paper consists of 8 pages.
Grade 12 Mathematics Paper 2             Memorandum                                     June 2011




Question 1

1.1.1    A reflection about the y-axis                         1 for reflection
                                                               1 for y-axis                  (2)
1.1.2    A rotation through 900 in an anti clockwise           1 for rotation
         direction                                             1 for size and direction      (2)
1.1.3    a rotation through θ in an anti clockwise direction   1 for rotation
                                                               1 for size and direction      (2)
1.1.4    An enlargement by a scale factor of 2                 1 for enlargement
                                                               1 for the scale factor        (2)
1.2      1.1.4 ( the enlargement). It changes the size.        1 for the correct
                                                               transformation
                                                               1 for the reason              (2)
1.3                                                            Accept calculator
                                                               answers as well
                                                               1 for substitution
                                                               1 for values of special
                                                               angles
                                                               1 for answer
                                                               1 for substitution
                                                               1 for values of special
                                                               angles
                                                               1 for answer                  (6)



                                                                                             [16]




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Grade 12 Mathematics Paper 2             Memorandum                                       June 2011




Question 2

2.1      A reflection in the line y=-x                            1 for reflection             (2)
                                                                  1 for line of reflection
2.2                                                                                            (2)
2.3      A rotation through 1800                                  1 for rotation               (2)
                                                                  1 for angle of rotation
2.4                                                                                            (2)
2.5.1    k = 1,5                                                                               (2)
2.5.2    F is ( -4,5 ; -1,5)                                      1 per coordinate             (2)
2.6      Area ΔABC                                                                             (2)
                                                                                               [14]


Question 3

3.1.1                                                             1 for substitution           (2)
                                                                  1 for answer

3.1.2    the equation of AC will be:                              1 for substitution of
                                                                  gradient
                                                                  1 for substitution of
                                                                  point
                                                                                               (3)
                                                                  1 for answer


3.1.3                                                             1 for tan…
                                                                  1 for gradient
                                                                  1 for answer                 (3)
3.1.4                              ( product of gradients = -1)   1 for gradient
         Midpt. of DB is (3;-2).                                  1 for conclusion
         Substitute this to check if it is on line AC             1 for reason
         LHS = -2                                                 2 for midpoint

         RHS =                                                    2 for showing midpoint
                                                                  on AC                        (7)
         So AC bisect DB perpendicularly.



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Grade 12 Mathematics Paper 2            Memorandum                                    June 2011




3.1.5                                     =6,48              2 for AM
                                                             2 for BD
                                                             2 for area
                                                                                           (6)

3.2.1    M is                                                1 per substitution for
                                                             each co-ordinate.
                                                                                           (2)
3.2.2    M is (1;4)                                          1 per gradient                (3)
                                                             1 for conclusion

         Therefore the gradients are equal
         So the points are collinear
3.2.3    Midpoint of AC is (1;4). So diagonals bisect each   1 for midpoint
         other.                                              1 for conclusion
                                                             1 for gradient
         So diagonals are perpendicular                      1 for conclusion              (4)
         Hence diagonals of ABCD bisect each other
         perpendicularly.
                                                                                           [30]


Question 4

4.1                                                            1 for LHS ( completing
                                                               the squares)

                                                               1 for RHS
                                                               1 for final equation
         So M is (-2 ; 6) and r = 6                            2 for centre
                                                                                           (6)
                                                               1 for radius
4.2.1    When y=2,                                             1 for recognising the
                                                               value of y
         x = -5 ( from the equation of the line)
                                                               1 for the value of x
         And the radius r= 5
                                                               1 for the value of r
                                                                1 for equation in
         Hence the circle is
                                                               centre radius form
         Therefore:                                                                        (6)
                                                               1 for expanding
         So:


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Grade 12 Mathematics Paper 2           Memorandum                                   June 2011




4.2.2    Let D be

         Then                                                  1 for the calculation
                                                               of x
         And                                                   1 for the calculation
                                                               of y
                                                                1 for the gradient of
                                                               the radius
         Thus the equation of the tangent is given by:         1 for the gradient of
                                                               the tangent
                                                               1 for substitution
                                                               1 for the answer.         (6)


                                                                                         [18]


Question 5

5.1
                                                             1 for simplification
                                                             1 for each answer
                                                             1 for each answer
                                 = 0,78 or
                                                                                         (5)
                                  = 0,37
5.2.1                                                        2 per diagram
                                               17            1 for expansion
                           α
                  -4                       8             β   1 for substitution
             -3                                              1 for answer                (5)
                       5                       -15




         =


5.2.2                                                        1 for expansion
                                                             1 for substitution          (3)
                                                             1 for answer.

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Grade 12 Mathematics Paper 2   Memorandum                          June 2011


5.3                                         1 per reduction             (6)


                                                                        [19]
Question 6

6.1
                                            1 for getting all to LHS
                                            1 for grouping
                                            1 for factors
                                            1 for roots
                                            1 for working with sinx
                                            only                        (7)
                Therefore                   1 for each answer


6.2.1                                                                   (4)
6.2.2


                                            2 for numerator
                                            1 for denominator
                                            1 for simplification
                                            1 for simplification


                                            1 for conclusion            (6)



                                                                        [17]


Question 7

7.1.1           1330                                                    (1)
7.1.2                                       1 for Substitution          (2)
                                            1 for answer
         So AC = 183,12

7.1.3                                       1 for substitution          (2)
         So                                 1 for answer



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Grade 12 Mathematics Paper 2   Memorandum                          June 2011


7.1.4                                       1 substitution
                                            1 for the value of θ
                                            2 for the answer            (4)


7.2.1                                                                   (1)
7.2.2                                                                   (2)

7.2.3                                       1 for substitution


                                            1 for expanding
                                            compound angle
                                            1 for special angle
                                            values


                                            1 for isolating p
                                            1 for simplification
                                                                        (5)


                                                                        [17]




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Grade 12 Mathematics Paper 2    Memorandum                            June 2011




Question 8

8.1.1                                                                      (1)
8.1.2                                                                      (1)
8.1.3    (300 ; 0)                                                         (1)
8.1.4                                                                      (1)




8.2.1    cos graph: (4)                      sin graph: (3)
         1 for turning points                1 for turning points
         1 for y-intercept                   1 for x-intercept
         1 for endpoints                     1 for endpoints              (8)
         1 for x-intercepts                  1 for y-intercept


8.2.2                                                                     (1)
8.2.3                                        1 for each endpoint
                                             1 for correct notation       (3)
8.2.4                                        1 for each value
                                             1 for the answer             (3)
                                                                          [19]]

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Grade 12 Mathematics Paper 2   Memorandum          June 2011




                                            Total: 150 marks




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