# Mathematics Grade 11 P2 Gauteng November 2023 QP (Eng)

> Grade 11 Mathematics Paper 2 question paper from the Gauteng November Exam 2023 (English), 16 pages. Read online or download the free PDF.

- Page: https://www.hlayiso.com/library/gr-11-mathematics-p2-english-2023-question-paper/b8f6a489-3cb4-476d-8c5a-3c2de035fcf8
- Updated: 2026-09-27
- Subject: Mathematics
- Grade: Grade 11
- Language: English
- Document type: Question paper
- Year: 2023
- Exam period: Gauteng November Exam
- Paper: 2
- Curriculum: CAPS
- Pages: 16

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## About this document

Mathematics · Grade 11 · Gauteng November Exam · 2023 · English. Question paper, 16 pages. Read online or download the PDF.

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## Document text

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          PROVINCIAL EXAMINATION
                        NOVEMBER 2023
                                GRADE 11

                                MATHEMATICS

                                     PAPER 2

TIME:       3 hours

MARKS:      150

12 pages + 1 information sheet and 3 answer sheets

                                                     P.T.O.

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                                               MATHEMATICS
                                                                                         2
                                               (PAPER 2)                  GRADE 11

INSTRUCTIONS AND INFORMATION

1.    This question paper consists of 11 questions.

2.    Answer ALL the questions.

3.    Clearly show ALL calculations, diagrams, graphs, et cetera that you have used in
      determining your answers.

4.    Answers only will NOT necessarily be awarded full marks.

5.    You may use an approved scientific calculator (non-programmable and non-
      graphical), unless stated otherwise.

6.    If necessary, round-off answers to TWO decimal places, unless stated otherwise.

7.    Answer sheets for answering QUESTIONS 1.1, 1.2 and 9.1 are provided at the
      end of the question paper. Write you name in the spaces provided on the
      ANSWER SHEETS and submit them together with your ANSWER BOOK.

8.    Diagrams are NOT necessarily drawn to scale.

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

10.   Write neatly and legibly.

                                                                                         P.T.O.

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                                                             MATHEMATICS
                                                                                                        3
                                                             (PAPER 2)                   GRADE 11

QUESTION 1

A small tuck shop displayed a record of daily sales in rands, for the past two months
(60 days) using the following histogram.

                      25

                      20

  Freqency(in days)
                      15

                      10

                       5

                       0
                           300     400
                                 300       500
                                          400      600
                                                 500         700
                                                           600         800
                                                                     700      900
                                                                             800      1000
                                                                                    900
                                                  Daily sales (in rands)

1.1                   Complete the following table. Use the table provided on ANSWER SHEET A.

                             Class interval      Frequency         Cumulative frequency
                             300 < x  400           4                      4
                             400 < x  500
                             500 < x  600
                             600 < x  700
                             700 < x  800
                             800 < x  900                                                                (2)

1.2                   Draw a cumulative frequency curve for the sales over the past two months. Use
                      the graph sheet provided on ANSWER SHEET B.                                         (3)

1.3                   Use the graph in QUESTION 1.2 and determine the estimated median value for
                      the daily sales.                                                                    (2)

1.4                   The tuck shop must make R475 in sales per day to break-even. On how many
                      days did the tuck shop make a profit?                                               (2)

1.5                   On the first day of the following month, the tuck shop made R725 in sales. Does
                      this day lie within the top 25% of sales from the previous two months?             (2)
                                                                                                        [11]

                                                                                                        P.T.O.

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                                                 MATHEMATICS
                                                                                                4
                                                 (PAPER 2)                     GRADE 11

QUESTION 2

Five data values are represented as follows: 2k ; k + 1 ; k + 2 ; k – 3 ; 2k – 2

2.1   If the mean of the data set is 15, show that k = 11.                                        (3)

2.2   Calculate the standard deviation ()for this data, rounded-off to one decimal place.        (2)

2.3   If t units are subtracted from each data value in the set, without further calculation,
      explain how the mean and standard deviation would be affected in terms of t.                (2)
                                                                                                  [7]

QUESTION 3

In the diagram below, RPQ is drawn, with P(0 ; 6), Q(4 ; –2). M(–2 ; 0) is the midpoint
of RQ.
                                                     y
                                                             P(0 ; 6)

          R

                                                                               x
                                                         O
                                  M(–2 ; 0)

                                                                            Q(4 ; –2)

3.1   Determine the gradient of the line MQ.                                                      (2)

3.2   Determine the equation of the line MP, in the form y = mx + c.                              (3)

3.3   Determine the coordinates of R.                                                             (3)

3.4   Calculate the length of PQ, in simplified surd form.                                        (2)

3.5   Given that RPQT is a parallelogram, determine the coodinates of T if point T is in
      the third quadrant.                                                                         (2)

3.6   Explain why RPQT is a rhombus.                                                             (2)
                                                                                                [14]

                                                                                                P.T.O.

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                                                MATHEMATICS
                                                                                            5
                                                (PAPER 2)                    GRADE 11

QUESTION 4

In the diagram below, lines AC and BD intersect at B, where AC ⊥ BD. C and D lie on
the y-axis, while A lies on the x-axis. The equation of AC is py – x – 5p = 0, while  is
the angle of inclination for AC, with CD̂B = .
                                                        y
                                                            C
                                         B

                                                            O            x
                A

4.1   Determine the coordinates of C.                                                         (3)

4.2                              1
      If the gradient of AC is     , show that p = 2.
                                 2                                                            (2)

4.3   Calculate the coordinates of B.                                                         (5)

4.4   Determine the size of .                                                                (2)

4.5   Hence, or otherwise, prove that ABOD is a cyclic quadrilateral.                         (3)

4.6   Determine the coordinates of the centre of the circle which passes through D, B
      and C.                                                                                 (2)
                                                                                            [17]

                                                                                            P.T.O.

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                                               MATHEMATICS
                                                                                        6
                                               (PAPER 2)                  GRADE 11

QUESTION 5

5.1 If 3sin  = 2. and cos  < 0, determine with the aid of a diagram and without the
    use of a calculator, the value of:

     5.1.1     3cos2  – 1                                                                (4)

     5.1.2     tan (– – 180)                                                            (3)

5.2 Given: t cos 15 = 4

     Determine the following in terms of t, without the use of a calculator:

     5.2.1     sin 15                                                                    (3)

     5.2.2     sin 75                                                                    (2)

     5.2.3     1 – tan2 15 (Give the answer as a single fraction.)                       (3)

5.3 Simplify the following to a single trigonometric function, without the use of a
    calculator.

      cos (90 − α) sin ( −α − 540 )
      tan 225 + sin α.sin (180 + α)                                                     (5)

5.4 Given: 1 – cos = 2 sin2

     5.4.1 Show that the equation can be written as: (2cos + 1)(cos – 1) = 0.           (2)

     5.4.2 Hence, determine the general solution of (2cos + 1)(cos – 1) = 0.           (5)
                                                                                        [27]

                                                                                        P.T.O.

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                                                     MATHEMATICS
                                                                                               7
                                                     (PAPER 2)                     GRADE 11

QUESTION 6

                                                                         1
In the diagram below, the graphs of f (x) = tan x – 1 and g(x) = –         cos 2x are drawn,
                                                                         2
where x [–90 ; 180].

                                   y
                                                 f
                                   2

                                   1
                                                                            g     g

                       -45º         O      45º        90º         135º     180º
          -90º                                                                        x

                                   -1

                                   -2
                                                              f

6.1   Write down the period of g.                                                                (1)

6.2   Determine the range of g(x).                                                               (2)

6.3   Use the graphs to determine graphically the values of x where:

      6.3.1       f (x)  0                                                                      (2)

      6.3.2       f (x).g(x) > 0                                                                 (2)

      6.3.3       2tan x + cos 2x = 2                                                            (3)

6.4               sin x + cos x
      If h(x) =                 , describe the vertical translation of h from f.
                      cos x                                                                      (2)

6.5   Determine the maximum value of p(x) = 4g(x).                                              (2)
                                                                                               [14]

                                                                                               P.T.O.

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                                                 MATHEMATICS
                                                                                         8
                                                 (PAPER 2)                   GRADE 11

QUESTION 7

In the figure below, ABC is drawn where AB⊥ BC and AĈB = p, with CB = 3AB.
DCB is drawn such that DĈB = 135, and BD = 121 m

                                                                 A

                        C          p
                                                                  B
                            135

                                         121 m

            D

7.1   Determine the value of p, correct to 3 decimal places.                               (2)

7.2   If p = 18,4, determine the length of CD.                                            (3)
                                                                                           [5]

QUESTION 8

The diagram below represents an open tank with a square base (side dimensions of x cm)
and a height of h cm. The tank has a volume of 490 cm3.

                                                 h cm

                                                        V = Area of base x Height
                                          x cm

                 x cm

8.1   Determine the height (h) of the tank in terms of x.                                  (2)

                                                                                         P.T.O.

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                                                MATHEMATICS
                                                                                               9
                                                (PAPER 2)                     GRADE 11

8.2   Show that A, the external surface area of the tank, is given by the formula:
      A = x2 + 1 960 cm2
                  x                                                                              (2)

8.3   Given that the tank is 10 cm high, calculate the external surface area of the tank.        (4)
                                                                                                 [8]

QUESTION 9

9.1   In the diagram below, O is the centre of the circle passing through P, Q and R.
      Chords PQ and PR are drawn, with OQ and OR joined.

                                 Q

                                                O

                                                                   P
                          R

      Use the diagram provided in ANSWER SHEET C to prove the theorem that
      states that the angle subtended by an arc at the centre of a circle is double the size
      of the angle subtended by the same arc at the circle, that is RÔQ = 2 P̂                  (6)

                                                                                               P.T.O.

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                                                    MATHEMATICS
                                                                                        10
                                                    (PAPER 2)              GRADE 11

9.2   In the diagram below, DF is a diameter of the circle with centre O. Chord EG
      intersects DF at H such that DF ⊥ EG. Chords EF and GF are drawn. EĜF = 55.
                                            E
                                        1
                                            2 3

                          D

                                   32
                               H
                                   1
                                            1
                                                2
                                                O

                                                             1
                                                         2
                          G                                      F

      9.2.1 Determine, giving reasons, the size of:

             (a)   D̂                                                                     (2)

             (b) Ô 2                                                                     (2)

             (c)   Ê 2                                                                   (2)

             (d)   Ê 3                                                                   (3)

      9.2.2 Determine the length of OH, if the diameter of the circle is 10 units and
            GE = 9,4 units.                                                              (4)
                                                                                        [19]

                                                                                        P.T.O.

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                                             MATHEMATICS
                                                                                     11
                                             (PAPER 2)                   GRADE 11

QUESTION 10

In the diagram below, AT is a tangent to the circle at T. Chords BT, BV and VC are
drawn. CB is extended to A, such that AC || TV. BC = CV and BV̂T = x.

                                A

                                                                 T
                                                         2
                                                             1
                              B       3
                                      2
                                  1

                          C
                                                     x

                                                 1 2

                                                         V

10.1 Determine, with reasons, 3 angles equal to x.                                     (6)

10.2 If ATVC is a parallelogram, prove that AT = BT.                                   (5)

10.3 Determine the size of x.                                                         (2)
                                                                                     [13]

                                                                                     P.T.O.

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                                                             MATHEMATICS
                                                                                            12
                                                             (PAPER 2)         GRADE 11

QUESTION 11

DE is a tangent to the larger circle at E. DH is a tangent to the smaller circle at H.
Chord HK is extended to meet the larger circle at E. F and K are the points of
intersections between the circles, with FK produced to D. GK is a chord of the smaller
circle with FE a chord of the larger circle. HF, GH and GF are joined.

                                                     D 3
                                                     1       2

                                                                           1   E
                                                 K 4                       2
                           H 1               1           3
                                 2               2
                           4 3

                          2
                           1
                      G
                                             2 1
                                         3

                                             F

11.1 Complete the following:

       D̂ 3 = ... + ...   (ext  )                                                          (1)

11.2 Prove that DEFH is a cyclic quadrilateral.                                              (4)

11.3 Prove that DF bisects HF̂E .                                                            (3)

11.4 If K̂1 = Ê1 , prove that GK is a tangent to the larger circle at K.                    (7)
                                                                                            [15]

                                                                                   TOTAL:   150

                                                         END

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                                            INFORMATION SHEET

    − b  b 2 − 4ac
x=
          2a
A = P(1 + ni)                     A = P(1 − ni)                     A = P(1 − i ) n          A = P(1+ i ) n
Tn = a + (n − 1)d
                                                    n
                                                  Sn = 2a + (n − 1)d 
                                                     2
Tn = ar n −1              Sn =
                                  (     )
                                 a rn − 1
                                             ;r 1     S =
                                                                a
                                                                     ; −1  r  1
                                   r −1                      1− r

F=
      
   x (1 + i )n − 1                              P=
                                                         
                                                    x 1 − (1 + i )
                                                                  −n
                                                                        
          i                                                i
               f ( x + h) − f ( x)
f ' ( x ) = lim
          h→ 0          h
                                                 x + x2 y1 + y 2 
d = ( x2 − x1 ) 2 + ( y2 − y1 ) 2              M 1     ;         
                                                 2          2 
                                                          y − y1
y = mx + c                y − y1 = m( x − x1 )      m= 2                              m = tan
                                                          x2 − x1
(x − a )2 + ( y − b)2 = r 2
                    a        b       c
In ABC:                =       =                            a 2 = b2 + c 2 − 2bc.cosA
                  sinA sinB sinC
                                  1
                  area ΔABC = ab.sinC
                                  2
sin (α + β ) = sin α.cos β + cosα.sin β           sin (α − β ) = sin α.cos β − cosα.sin β
cos(α + β ) = cosα.cos β − sin α.sin β            cos(α − β ) = cosα.cos β + sin α.sin β
        cos 2 α − sin 2 α
        
cos2α = 1 − 2sin 2 α                             sin2 α = 2sin α.cos α
        
        2cos α − 1
               2

                                                                            n

         fx                                                                 ( xi − x ) 2
     x=                                                             σ 2 = i =1
          n                                                                       n
            n (A)
     P(A) =                                              P(A or B) = P(A) + P(B) – P(A and B)
            n (S)

yˆ = a + bx                                       b=
                                                        (x − x )( y − y )
                                                         ( x − x )2

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Name and Surname:                                    Grade:

ANSWER SHEET A

QUESTION 1

1.1

          Class interval   Frequency   Cumulative frequency

          300 < x  400        4                4

          400 < x  500

          500 < x  600

          600 < x  700

          700 < x  800

          800 < x  900
                                                              (2)

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Name and Surname:                                                                   Grade:

ANSWER SHEET B

1.2                           70

                              60

                              50

       Cumulative Frequency
                              40

                              30

                              20

                              10

                              0
                                   0    100   200   300   400    500    600   700     800    900   1000
                                                          Daily Sales

                                                                                                          (3)

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Name and Surname:                  Grade:

ANSWER SHEET C

QUESTION 9

9.1
                        Q

                            O

                                   P
                    R

                                            (6)

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