Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 TOTALmaths.stithian.com/New CAPS 2019 Prelim Papers... · Q1 Q2 Q3...

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Transcript of Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 TOTALmaths.stithian.com/New CAPS 2019 Prelim Papers... · Q1 Q2 Q3...

  • 1

    PRELIMINARY EXAMINATION

    2019

    MATHEMATICS DEPARTMENT

    Mathematics Paper I

    Time: 3 Hours 150 Marks

    PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY

    1. This question paper consists of 20 pages and 10 questions. Please check that your

    paper is complete. A separate formula sheet is given.

    2. Answer all questions in the spaces provided.

    3. Read the questions carefully.

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

    specific question prohibits the use of a calculator.

    5. Round your answer to two decimal digits 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.

    Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 TOTAL

  • 2

    SECTION A

    QUESTION 1

    a. Solve for 𝑥 ∈ ℝ

    i. 2𝑥2 = 5𝑥 (3)

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    ii. 𝑥 + √−7𝑥 − 6 = 0 (5)

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    iii. (𝑥 + 1)2 ≤ 𝑥 + 1 (5)

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  • 3

    b. Solve for 𝑥 and 𝑦 simultaneously.

    (𝑥 − 3)2 + 𝑦2 = 25

    3𝑦 − 4𝑥 = 13 (5)

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    c. 4𝑥2 + 8𝑥 + 𝑛 = 0

    i. Write down an expression in terms of n which will represent the roots

    of the equation. (2)

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    ii. For which value(s) of n will the roots be non-real? (2)

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  • 4

    d. Simplify: 3𝑥+1−3𝑥−1

    3𝑥−2 (4)

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    [26]

    QUESTION 2

    a. Consider ∑ 3𝑘 − 1𝑛𝑘=1

    i. Show that 𝑆𝑛 =3

    2𝑛2 +

    1

    2𝑛. (3)

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    ii. How many terms must be added for the sum to equal 345? (5)

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  • 5

    b. A certain virus has the property that it can replicate itself every week. If two

    viruses are placed in a petri dish answer the questions that follow.

    i. Write an expression for the number of viruses in the petri dish after

    n weeks. (2)

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    ii. The maximum number of viruses that the petri dish can accommodate is

    8 192. After how many weeks will the petri dish reach its capacity? (3)

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    [13]

    QUESTION 3

    a. A car worth R120 000 decreases in value, on a straight line basis,

    to a value of R80 400 over a period of 3 year.

    i. Calculate the rate of depreciation. (3)

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  • 6

    ii. If the rate of depreciation is 11% p.a., how long will it take for the car to

    lose half of its value, on a reducing-balance basis? (3)

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    b. Jason starts investing R2 500 a month at the end of each month as a nest

    egg for his retirement. He starts his investment at the end of the month of his 25th

    birthday and will retire at the end of the month of his 55th birthday.

    i. Calculate the total value of the investment, to the nearest Rand,

    if the interest rate is calculated at 6,75% p.a. compounded monthly. (4)

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    ii. When he eventually retires, Jason re-invests his investment into a

    living annuity which he intends living off for 12 years. Calculate the

    amount he can expect to receive each month if interest is calculated

    at 9% p.a. compounded monthly. Assume he received R 2 903 665

    at retirement. (4)

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    [14]

  • 7

    QUESTION 4

    a. If 𝑓(𝑥) =−2

    𝑥+3− 4, draw a rough sketch of 𝑓(𝑥), clearly showing all intercepts with

    the axes and the asymptotes.

    (4)

    b. A parabola has 𝑥-intercpets 𝐴(−3; 0) and 𝐵(4; 0). It also passes through the

    point 𝐶(2; −20). Determine the equation of the parabola in the form

    𝑦 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐. (4)

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  • 8

    c. 𝑔(𝑥) = 𝑎𝑥 and 𝑔(−2) = 16.

    i. Show that the value of 𝑎 is 1

    4. (2)

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    ii. Determine 𝑔−1(𝑥) in the form 𝑦 = ⋯. (2)

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    iii. Sketch 𝑔(𝑥) and 𝑔−1(𝑥) on the set of axes below. Clearly label your graphs. (4)

    [16]

  • 9

    QUESTION 5

    a. Given: 𝑓(𝑥) = −3𝑥2 + 𝑥

    i. Determine 𝑓(𝑥 + ℎ) − 𝑓(𝑥). (3)

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    ii. Hence, determine 𝑓′(𝑥), from first principles. (2)

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    b. Determine the derivative in each case

    i. 𝑑𝑦

    𝑑𝑥 if 𝑦 = (𝑥 − 1)2 (4)

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    ii. 𝐷𝑥 [2𝑥3−𝑥

    √𝑥] (4)

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  • 10

    c. Determine the equation of the tangent to 𝑓(𝑥) = −1

    2𝑥2 + 2𝑥 at 𝑥 = −2. (5)

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    [18]

    TOTAL FOR SECTION A: 87 MARKS

  • 11

    SECTION B

    QUESTION 6

    a. 1; 𝑥; 15; 𝑦; 45 are the first five terms of a quadratic number pattern.

    Determine the values of 𝑥 and 𝑦. (7)

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  • 12

    b. A square with side lengths 1 cm is divided into 4 equal squares and 1 square is

    then shaded. Each of the remaining unshaded squares is then also divided into 4

    and 1 square of each of these smaller squares are also shaded. This process

    continues indefinitely.

    Step 1 Step 2 Step 3

    i. Below is the series representing the areas of the shaded squares

    is given. Complete the series for Step 3. (1)

    Area of shaded squares after step 1: 1

    4

    Area of shaded squares after Step 2: 1

    4+

    3

    16

    Area of shaded squares after Step 3: ____________________________

    ii. What fraction of the unit square will be shaded by step 11? (5)

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    [13]

  • 13

    QUESTION 7

    Below you will find the sketch of 𝑓(𝑥) = log1𝑘

    𝑥 and 𝑔(𝑥) = 𝑥2 − 𝑘𝑥 + 𝑐. Answer the

    questions that follow.

    a. Determine the value of:

    i. 𝑘 (3)

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    ii. 𝑐 (2)

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  • 14

    b. Rewrite 𝑔(𝑥) in the form 𝑔(𝑥) = 𝑎(𝑥 − 𝑝)2 + 𝑞. (3)

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    c. For which value(s) of 𝑥 is 𝑓(𝑥) ≥ 𝑔(𝑥)? (3)

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    d. Write down the range of 𝑓−1(𝑥) + 3. (2)

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    [13]

    QUESTION 8

    𝑔(𝑥) = −(𝑥 − 1)2(𝑥 + 2) = −𝑥3 + 3𝑥 − 2

    a. Determine the co-ordinates of the stationary points of 𝑔(𝑥). (5)

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  • 15

    b. Hence, sketch 𝑔(𝑥). (3)

    c. For which value(s) of 𝑥 is 𝑔(𝑥) both increasing and concave up? (3)

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    d. For which value(s) of 𝑘 will 𝑔(𝑥) + 𝑘 = 0 have three distinct roots? (3)

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    e. For which value(s) of 𝑝 will 𝑔(𝑥 + 𝑝) = 0 have two negative roots? (2)

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    [16]

  • 16

    QUESTION 9

    a. Consider the word LIVERPOOL.

    i. Calculate the total number of arrangements of the letters if repeated

    letters are identical. (3)

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    ii. How many of these arrangements start with the letter O? (2)

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    iii. Hence, calculate the probability that an arrangement will start with the

    letter O. (2)

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  • 17

    b. A school has the following subject groupings. All students in grade 12 take all 7

    subjects and all subjects are taken on a particular day.

    Group A: English and German

    Group B: Mathematics, Physical Science and Biology

    Group C: Accounting and Psychology

    i. In how many ways can the subjects occur during a particular day? (2)

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    ii. In how many ways can the subjects occur during a particular day if the

    subjects in Group B must happen together, in any order? (2)

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    iii. What is the probability that the day will start with English and end with

    Mathematics? (2)

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    [13]

  • 18

    QUESTION 10

    A square based pyramid has the property that the sum of its perpendicular height (ℎ)

    and side length (𝑟) of the base is 22 units. Determine the maximum volume of the pyramid.

    𝑉𝑜𝑙 =1

    3𝑎𝑟𝑒𝑎 𝑜𝑓 𝑏𝑎𝑠𝑒 × 𝑝𝑒𝑟𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟 ℎ𝑒𝑖𝑔ℎ𝑡

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    [8]

    TOTAL FOR SECTION B: 63 MARKS

    𝒉

    𝒓

  • 19

    ADDITIONAL WRITING SPACE

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