st. david’s marist inanda

of 28

ST. DAVID’S MARIST INANDA

MATHEMATICS

PAPER 1 PRELIMINARY EXAMINATION GRADE 12

10 SEPTEMBER 2021

EXAMINER: MS N VAZZANA MARKS: 150 MODERATOR: MRS S RICHARD TIME: 3 HOURS

NAME:_____________________________________________________________ PLEASE PUT A CROSS NEXT TO YOUR TEACHER’S NAME

Mrs Black Mrs Kennedy Mrs Nagy Mrs Richard Mr Sokana Mr Vicente Ms Vazzana

INSTRUCTIONS:

✓ This paper consists of 28 pages. Please check that your paper is complete.

✓ Please answer all questions on the Question Paper,

✓ You may use an approved non-programmable, non-graphical calculator unless otherwise

stated.

✓ Answers must be rounded off to two decimal places, unless otherwise stated.

✓ It is in your interest to show all your working details.

✓ Work neatly. Do NOT answer in pencil. Write using a dark pen, preferably black.

✓ Diagrams are not drawn to scale.

SECTION A Q1 [20]

Q2 [18]

Q3 [6]

Q4 [9]

Q5 [5]

Q6 [8]

Q7 [12]

TOTAL [78]

LEARNER’S MARKS

SECTION B Q8 [8]

Q9 [8]

Q10 [12]

Q11 [10]

Q12 [15]

Q13 [7]

Q14 [12]

TOTAL [72]

LEARNER’S MARKS

TOTAL: /150

of 28

SECTION A QUESTION 1 [20 Marks]

a) Solve for 𝑥 in each of the following equations:

i. 𝑥2 − 4𝑥 = 12

(3)

ii. 𝑥+2

𝑥+1−

3

𝑥−2=

1

𝑥+1

(4)

of 28

iii. 𝑙𝑜𝑔(𝑥+1)(2𝑥 − 3) = 1, stating restrictions where necessary.

(4)

iv. log(𝑥 + 1) + 𝑙𝑜𝑔𝑥 = 2, round off your answers to TWO decimal

places.

(5)

of 28

b) Determine the value(s) of p so that 3𝑥2 + 2𝑥 − 𝑝 + 1 = 0 has real roots.

(4)

of 28

QUESTION 2 [18 Marks]

a) The 4th term of an arithmetic series is 108 and the 11th term is 80. Find the

common difference and the first term of the series.

(4)

b) Calculate the value of: ∑ (−3)𝑝

(4)

𝑝 = 4

21

of 28

c) For what value(s) of x will the following infinite geometric series converge?

(1+3x)+(1-9x2)+…

(4)

of 28

d) All the terms of a geometric series are positive. The sum of the first two terms

is 34 and the sum to infinity is 162. Determine the common ratio.

(6)

of 28


Given 𝑃(𝐴) = 0,3 and 𝑃(𝐵) = 0,5. Calculate 𝑃(𝐴 𝑜𝑟 𝐵) if:

a) A and B are mutually exclusive events.

(2)

b) A and B are independent events.

(4)

of 28


a) Jackie deposited R25 000 into a savings account with an interest rate of 18%

p.a. compounded quarterly. Jackie withdrew R8000 from the account 2 years

after depositing the initial amount. She deposited another R4000 into this

account 3,5 years after the initial deposit. How much money will Jackie have 5

years after the initial deposit was made?

(4)

of 28

b) Mohammed has about 10 years to go to his retirement and he decides to set

up a savings annuity to enhance his pension. He decides to pay R10 000 on a

monthly basis. He plans to make his first payment on 1 November 2021 and

his final payment will be on 1 October 2031. The interest rate is 8% p.a

compounded monthly. Calculate the amount of money that will be available to

Mohammed on 1 November 2031 when he retires.

(5)

of 28


The graph of a parabola ℎ has 𝑥 intercepts 𝑥 = −3 and 𝑥 = 5. The line 𝑘(𝑥) = 6 is a

tangent to ℎ at P, the turning point of ℎ.

Sketch the graph of ℎ on the axes below, clearly showing ALL intercepts

with axes and the turning point of h.

(5)

of 28


The diagram below shows the graphs of the following functions:

𝑓(𝑥) =1

4𝑥2; 𝑥 ≥ 2 and 𝑔(𝑥) = (

1

3)

𝑥

a) Write down the range of 𝑓−1

(2)

b) Determine the inverse of 𝑓 in the form 𝑦 =

(2)

of 28

c) Determine the inverse of 𝑔 in the form 𝑦 =

(1)

d) Sketch the graph of 𝑔−1 on the axes below

(3)

of 28


a) Determine 𝑓′(𝑥) by first principles if 𝑓(𝑥) = −5𝑥2 − 𝑥 + 1

(5)

of 28

b) Determine 𝐷𝑥[ (2𝑥 − 1)(𝑥 + 5)]

(3)

c) The gradient of the curve 𝑦 = 2𝑥3 +𝑎

√𝑥 at the point where 𝑥 = 1 is 8.

Determine the value of 𝑎.

(4)

of 28

SECTION B


Grant can get a loan of R40 000 from a friend, Ameer, who allows him to make 8

half-yearly installments, starting only in two years’ time, making the first instalment

at the end of the second year. The interest charged is 9,5% p.a. compounded

monthly.

a) Convert the monthly interest rate of 9,5% p.a. to an annual interest rate

compounded semi-annually.

(3)

b) Calculate the half-yearly repayments he must make.

(5)

of 28


A rectangular box has a length of 5𝑥 metres, breadth of 9 − 2𝑥 metres and its height

of 𝑥 metres.

a) Show that the volume of the box is given by 𝑉 = 45𝑥2 − 10𝑥3

(2)

b) Determine the maximum volume of the box.

(6)

of 28


a) Determine: lim 𝑥→0

4+

3

𝑥1

𝑥−7

(4)

b) If 𝑦 = (𝑥3 − 1)2 , show that 𝑑𝑦

𝑑𝑥= 6𝑥2√𝑦

(4)

of 28

c) Given 𝑓(𝑥) = 2𝑥3 − 2𝑥2 + 4𝑥 − 1. Determine the interval on which f is

concave up.

(4)

of 28


a) The letters of the words KINGDOM are rearranged to form other

arrangements of all the letters.

i. How many arrangements are possible?

(2)

ii. How many arrangements are possible if the letters K and G can’t be

next to each other

(3)

iii. What is the probability that if an arrangement is chosen at random, the

letters K and G will be separated?

(1)

of 28

b) In a class there are 15 girls and 7 boys. Two students are chosen to represent

the class in a meeting. Determine the probability that if two students are

chosen at random one will be a boy and the other will be a girl.

(4)

of 28


a) Graphs 𝑓(𝑥) = −𝑥2 − 4𝑥 and 𝑔(𝑥) = 2𝑥 − 6 are sketched below. Point D

is the turning point of 𝑓. A and O are the x intercepts of 𝑓 and point C is

the x intercept of 𝑔 respectively. Point B is the y intercept of 𝑔.

i. Determine the area of ∆ 𝐴𝐷𝑂

(4)

of 28

ii. For what value(s) of k will −𝑥2 − 4𝑥 = 2𝑥 + 𝑘 have two real roots that

are opposite in sign?

(2)

b) The graph of ℎ(𝑥) = 𝑥3 − 4𝑥2 + 4𝑥 has a local minimum at (𝑎; 0) and a

local maximum at (𝑏; 𝑐).

Determine the values of 𝑎, 𝑏 and 𝑐

(6)

of 28

c) The sketch below shows the graph of 𝑦 = 𝑔′(𝑥). It is further given that

𝑔(0) = −5 and 𝑔(−2) = 0.

i. Write down the co-ordinates of the point of inflection of g

(1)

ii. Write down the co-ordinates of the point where g cuts the x axis

(1)

iii. For which values of x is the graph of g decreasing?

(1)

of 28


Rectangles are cut from a strip of fabric of constant width and arranged in a row.

The first rectangle has a length of 10cm. The length of each subsequent rectangle is

85% of the length of the previous rectangle.

a) Determine the length of fabric required if the strip consists of 20 rectangles.

(4)

b) Determine the longest strip of fabric that can be used.

(3)

10𝑐𝑚

of 28


The path of a slide can be modelled by the equation 𝑓(𝑥) =1

2𝑥2 − 2𝑥 + 2 between

0 ≤ 𝑥 ≤ 𝑎. A child comes off the slide and into the water following the path modelled

by 𝑔(𝑥) = −1

2𝑥2 + 4𝑥 + 𝑘 for 𝑥 ≥ 𝑎. ( the graph represented by the dotted line)

The child leaves the slide at an angle of 45° to the horizontal.

a) Determine the value of 𝑎

(3)

of 28

b) Determine the value of 𝑘

(3)

c) How high above the water does the child reach before entering the water?

(3)

of 28

d) Determine the horizontal distance the child travels by the time she starts on

the slide (at 𝑃) before entering the water.

(3)

TOTAL: 150 Marks

st. david’s marist inanda

Documents