Paper 1A
Section A
Question 1
(a) Solve for x in each case
(log x - 2).log(x - 2) = 0
(1)
3
(2)
𝑥
−
64
3
𝑥+1
=
1
2
𝑥
(3)
3
2
(3)
1
2
(𝑥 3 ) × ( 4 ) = 3
(3)
52𝑥 −3
(4)
53𝑥 −3(5𝑥 )
(3)
10𝑥
= 10𝑥+2
(3)
(b) Solve for p if 2𝑝2 − 3 ≥ 0. Leave answers in surd form.
(3)
(c) Determine the value(s) of k for which the roots of
(𝑥 + 𝑘)(𝑥 + 2𝑘) = 6(𝑥 + 𝑘)
are equal.
(Hint: Find the roots first)
(4)
[19]
Question 2
(a)
John buys a house for R2 000 000.
He puts down a 20% deposit and gets a loan from the bank for the balance.
The interest charged on the loan is a fixed rate of 9% p.a. compounded
monthly.
Payments are made monthly starting one month after the loan is granted.
(1) Determine the monthly payment if the loan is paid back over 20 years?
(3)
(2) What is total amount of interest that John will pay over the 20 year
period.
(2)
(b)
John decides to pay R15 000 per month to pay back the loan.
(1) After how many months will the loan be repaid?
(4)
(2) Assuming John has to make 216 full payments of R15 000, how much
will John save in total interest as a result of increasing his payment to
R15 000.
(3)
[12]
Question 3
(a) Solve for a if
12

i 1
j 1
 i (a  4)   a(2)  j  230
(6)
(b) John is making a pattern using triangular tiles.
If John has 200 tiles in total,
(1)
how many complete rows of patterns can John make?
(2)
how many tiles will he have spare?
(c)
(5)
(3)
John and Luis start jobs at the beginning of the same year. John’s annual
salary in the first year is R 300 000 and increases by 5% at the beginning of
each subsequent year3
Luis’s annual salary in the first year is R330 000 and increases by R15 000 at
the beginning of each subsequent year.
(1)
Show that in the 10th year John’s annual salary is higher than Luis’s
annual salary.
(6)
(2)
In the first 10 years how much, in total, does John earn?
(3)
(3)
Every year, Luis saves one third of his annual salary. How many years
does it take him to save R875 000?
(3)
[26]
Question 4
(a)
(b)
Determine 𝑓′(𝑥) by first principles if 𝑓 (𝑥 ) = −𝑥 2 − 𝑥 − 1
4
1
𝑑ℎ
If 𝑥 5 = 𝑦 and ℎ = 𝑥 10 + 𝑥 15 + 𝜋, determine 𝑑𝑦
(5)
(4)
(c)
The gradient of the curve 𝑦 = 2𝑥 3 + 𝑥 at the point where 𝑥 = 1 is 8.
√
Determine the value of a .
[13]
(4)
𝑎
QUESTION 5
The graph of f(x) is sketched below.
3
Sketch on the same set of axes the graph of 𝑦 = 𝑓 −1 (𝑥).
(2)
Write down the Domain and Range of 𝑦 = 𝑓 −1 (𝑥)
(2)
Complete the graph for 𝑥 > 0, so that the inverse of the completed
graph is a function.
(1)
[5]
(a)
(b)
(c)
Total For Section A=75 Marks
SECTION B
Question 6
3
and g(x) = (b)x+c + d
x -3
The graphs given have a common asymtote.
The graphs have the same x and y intercepts.
(1)
Write down the equation of the common asymptote.
(2)
Explain why b >1.
(3)
Determine the values of b, c and d.
(a) Given: f (x) = -3-
(1)
(2)
(3)
2
(b) Given: y = x +1and y = x + bx + c .
The graphs intersect at A(-1;0) and B.
The y-intercepts differ by 4.
(1) Sketch the graph of 𝑦 = 𝑥 + 1.
(2) Determine the two values of c.
(2)
(3)
(3) On the same set of axes, draw a possible graph of y = x + bx + c for
each value of c.
(4)
(4) For each value of c , give the corresponding value of b
(2)
2
(c)
The diagram below shows the graphs of y = 2 x and y = 4 x .
KM is parallel to the y axis.
K
⦁
⦁M
O
(1)
(2)
(3)
Determine the value of q. Justify
Determine the length of KM
K remains fixed. M is a variable point on the graph.
Determine the coordinates of M when OKˆ M  97,1 .
Give the answer correct to the nearest whole number.
(2)
(2)
(5)
[26]
Question 7
The point 𝑃(𝑡; 𝑡 2 + 3) lies on the curve 𝑦 = 𝑥 2 + 3.
The perpendicular distance from P to the line 𝑦 = 𝑥 − 1 is 𝐷(𝑡).
y  x 1
R
(a) Show that 𝐷(𝑡) =
𝑡 2 −𝑡+4
(3)
√2
(b) Determine the value of t when P is closest to the line = 𝑥 − 1 .
(c) Show that, when P is closest to the line 𝑦 = 𝑥 − 1, the tangent to the curve
at P is parallel to the line 𝑦 = 𝑥 − 1.
[10]
(3)
(4)
Question 8
(a)
The cubic 𝑦 = 𝑎𝑥 3 + 𝑏𝑥 2 + 𝑐𝑥 has a point of inflection at x = p .
𝑏
Show that 𝑝 = − 3𝑎
(b)
(3)
The graph of 𝑓 (𝑥 ) = 𝑥 3 − 2𝑎𝑥 2 + 𝑎2 𝑥 is given where 𝑎 < 0
(1) Determine the coordinates of the point of inflection in terms of a .
(1)
(2) The point of inflection has integral coordinates. Determine two
possible values of a.
(2)
(3) Draw a sketch graph of 𝑓(𝑥) .
Indicate the intercepts with the axes as well as the coordinates of the
stationary points.
(5)
(4) Use your graph to solve for k in terms of a if
(i)
x 3 - 2ax 2 + a2 x = k has three different real roots
(ii)
(2)
( x - k ) - 2a ( x - k ) + a (x - k) = 0 has positive roots only. (2)
3
2
2
[16]
Question 9
(a)
A group of 150 people was surveyed and the results recorded.
Determine whether the events “ male” and “own a mobile” are
independent or not.
(b)
(4)
John is designing a website that requires unique logins to be generated.
He plans to generate the logins using two capital letters from the alphabet
followed by a series of numerals from 0 to 9 inclusive.
All logins will have the same number of numerals.
Repetition of letters and numerals is allowed.
(1) Determine the minimum number of numerals required for each login
so that John can generate at least 3 million logins.
(3)
(2) How many of the logins will start with the letter A and end with the
numeral 9?
(3)
(c)
In the diagram below, a square target board for a dart game is shown.
The dimensions of the outer square are 100 cm by 100 cm.
The small squares are numbered 1 to 9 and each of them have dimensions
20 cm by 20 cm.
4
9
2
3
5
7
8
1
6
The number shown in each square is the score obtained by a dart hitting
the square.
A dart hitting the shaded region scores 0.
If two darts are thrown, calculate the probability of scoring a total of 9
points.
(5)
[15]
Question 10
æ
t2 ö
A tap releases liquid A into a tank at a rate of ç 2 +
÷ litres per minute,
è t +1 ø
where t is time in minutes.
A second tap releases liquid B into the same tank at a rate of
æ
1 ö
ç1+
÷ litres per minute.
è t +1 ø
The taps are opened at the same time and release the liquids into an empty
circular cone.
The lines representing the height of the water surface and the slant height
from the vertex of the cone make an angle of a .
See Diagram below.
1
(1 litre = 1000 cm3 and V = p r 2 h )
3
(a) If the ratio of liquid A to liquid B in the tank is 13 : 3 after t minutes.
Determine t
(4)
(b) If a = 30° , determine the radius of the circular water level after 4 minutes
of the taps running.
(4)
[8]
TOTAL FOR SECTION B = 75 Marks
THE END