RONDEBOSCH BOYS’ HIGH SCHOOL
SENIOR CERTIFICATE
GRADE 12
MATHEMATICS (PAPER 1)
29 AUGUST 2017
MARKS: 150 EXAMINERS: T EDWARDS
S CARLETTI
E DU TOIT
S VERSTER
TIME: 3 HOURS MODERATOR: P GHIGNONE
This question paper consists of 9 pages, a formula sheet and a diagram sheet.
Mathematics/P1 RBHS September 2017
INSTRUCTIONS & INFORMATION
Read the following instructions carefully before answering the questions.
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. Start each question on a clean side of paper.
5. Answers only will NOT necessarily be awarded full marks.
6. You may use an approved scientific calculator (non-programmable and non-
graphical), unless stated otherwise.
7. If necessary, round off answers to TWO decimal places, unless stated otherwise.
8. Diagrams are NOT necessarily drawn to scale.
9. An information sheet with formulae is included at the end of the question paper.
10. Numbers the answers correctly according to the numbering system used in this
question paper.
11. Write neatly and legibly.
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Mathematics/P1 RBHS September 2017
QUESTION 1
1.1 Solve for x:
1.1.1 3 x 2−5 x−1=0 (correct to TWO decimal places) (3)
1.1.2 2 x2 + x−6 ≤ 0 (3)
1.1.3 √ 11 x+20=2−x (5)
1.2 Solve for x and y:
2 x+ y =5 and 2 x2 −xy−4 y 2=8 (6)
1.3 Given: 3 a x 2+ bx−3 a=0 where a ; b ∈ R.
1.3.1 Prove that the roots of the equation are real. (3)
1.3.2 If b=8 a, determine, with a reason, whether the roots will be rational or
irrational. (4)
1.4 You are given four statements. Write down if they are always true, sometimes
true or never true. In each example, you are told what type of number the
variables are.
x
1.4.1 =1 where x ∈ R (1)
x
1.4.2 If x < y <0 then x 2< y 2< 0 where x ; y ∈ R (1)
1.4.3 2 xy ≤ x 2+ y2 where x ; y ∈ R (1)
a+ x a
1.4.4 > where a ; x ∈ N (1)
a+2 x a+ x
[28]
QUESTION 2
Given the quadratic sequence: 34 ; 42; 48 ;52 ; 54 ; …
2.1 Determine an expression for the general term, T n, of the quadratic sequence. (4)
2.2 Jack gets the 20th term to be −155. Without doing any calculations, explain why
this answer has to be incorrect. (1)
2.3 The general term of the first row of differences is T n=−2 n+10. For a certain
value of p, the pth term of the quadratic sequence is equal to the pth term in the
first row of differences. Determine the value of p. (4)
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Mathematics/P1 RBHS September 2017
[9]
QUESTION 3
3.1 Evaluate:
∞
∑ 2.7 1−r (3)
r=1
3.2 CALCULATORS MAY NOT BE USED IN THIS QUESTION
log 2 and 2 log 2 are the first two terms of an arithmetic as well as a geometric
sequence.
3.2.1 Find the n th term of the arithmetic sequence. (2)
The n th term of the geometric sequence is T n=2n−1 × log 2
3.2.2 Calculate the difference between the fifth term of the geometric sequence
and the fifth term of the arithmetic sequence. (3)
3.2.3 If S is the sum of the first six terms of the geometric sequence and T is
the sum of the first six terms of the arithmetic sequence, calculate the
ratio S :T. (5)
3.3 In an arithmetic progression, T 2016 +T 2017= p . Find, in terms of p, the value of
T 2013 +T 2014 +T 2015 +T 2016 +T 2017 +T 2018 +T 2019 + T 2020 (2)
[15]
QUESTION 4
9
Given f ( x )= +5
x +3
4.1 Calculate the y-intercept of f . (1)
4.2 Determine the equation of the positive axis of symmetry off . (2)
4.3 Write down the equations of the asymptotes of f −1. (2)
[5]
4
Mathematics/P1 RBHS September 2017
QUESTION 5
The graph of k is defined by the equation k ( x )=√ ax. The point (32 ; 4) lies on k.
5.1 Calculate the value of a. (2)
5.2 For what values of x will k be defined? (1)
5.3 Write down the equation of k −1, the inverse of k, in the form y=¿ … (2)
5.4 If p ( x ) =x−32, determine algebraically the point(s) of intersection of k and p. (5)
5.5 Hence, or otherwise, determine the values of x for which k ( x ) > p(x). (2)
[12]
QUESTION 6
6.1 Given f ( x )=−x 2−2 x +3 and g ( x )=−2.2 x−1 +1.
6.1.1 Draw neat graphs of f (x) and g( x ) on the same set of axes.
Clearly label ALL important information. (8)
6.1.2 Write down the equation of the axis of symmetry of f . (1)
6.1.3 Determine the value(s) of k for which −x 2−2 x+ k=0 will have no
real roots. (2)
6.2 Given h ( x + y )=h ( x ) +h ( y )+ xy and h ( 1 )=3. Calculate h( 4). (4)
[15]
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Mathematics/P1 RBHS September 2017
QUESTION 7
7.1 The school buys a Toyota Hilux Single Cab for R 242 000 and sets up a sinking fund
to replace it in 6 years’ time.
7.1.1 If the rate of depreciation is 14,3 % p.a. compounded annually, calculate
how much the truck will be worth after 6 years. (2)
7.1.2 If the rate of inflation is 11,2% p.a. compounded annually, calculate how
much a new truck will cost at the end of 6 years. (2)
7.1.3 What is the total required value for the sinking fund if the current truck
will be traded in? (1)
7.1.4 Determine the monthly payments into the sinking fund if interest is
calculated at 9,8 % p.a. compounded monthly and payments start in one
months’ time. (3)
7.2 John retired at the end of 2016 and received his pension of R 1 400 000.
7.2.1 How many full months can he live off the fund if he withdraws R 13 500
every month? Interest is calculated at 10,7% p.a. compounded monthly. (4)
7.2.2 How much can he withdraw in the 292nd month? (4)
[16]
QUESTION 8
−3
8.1 Given f ( x )= find f ' ( x) by first principles. (5)
x
8.2 Evaluate, leaving your answers with positive exponents:
3
8.2.1 f ' ( x ) if f ( x )= + 4 √ x (4)
x
2 a2
8.2.2 Dt ( )
9 t3
where a is a constant. (3)
[12]
6
Mathematics/P1 RBHS September 2017
QUESTION 9
9.1 The turning points of g ( x )=−2 x3 + a x 2 +bx +c are at x=2 and x=5. The
point (2 ;−9) lies on g. Determine the value of a , b∧c . (6)
9.2 ANSWER THIS QUESTION ON THE DIAGRAM SHEET
The graph of y=f ( x ) is given below.
9.2.1 Use the diagram sheet to draw a possible graph of y=f ' ( x ) on the same set
of axes. (2)
9.2.2 Indicate the point A, on the x-axis, where f ' ' (x )=0. (1)
9.3 If h ( x )=x 3 +3 x2−9 x , determine the value(s) of kfor which y=−9 x+ k will be a
tangent to the curve of h . (7)
[16]
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Mathematics/P1 RBHS September 2017
QUESTION 10
A is a town 30 km west of town B. Two athletes start walking simultaneously from the two
towns.
The athlete who starts from town A, walks due east in the direction of B at a constant speed
of 6 km/h. He reaches point P after x hours.
The athlete who starts at B, walks due north in the direction of another town C at a constant
speed of 8 km/h. He reaches point Q after x hours.
hours x hours
x hours hours
A P B
30 kilometres
10.1 Find the distance, i.e. PQ, between the two athletes after x hours. (3)
10.2 Given P Q 2=100 x2 −360 x +900 . How many hours will it take for the athletes to be
at a minimum distance from each other? (2)
10.3 What was this minimum distance between them? (2)
[7]
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Mathematics/P1 RBHS September 2017
QUESTION 11
11.1 A main course and dessert are chosen from the following menu:
Main Course Dessert
Beef Stew Apple Crumble
Fish of the Day Ice Cream
Lamb Shank Malva Pudding
Pork Chops Peppermint Crisp Tart
What is the probability that the choice will contain:
11.1.1 Lamb Shank and Peppermint Crisp Tart? (2)
11.1.2 Lamb Shank or Peppermint Crisp Tart? (2)
11.2 The following Venn diagram refers to probabilities. If X and Y are independent
events, find the values of a and b. Show all calculations.
(4)
11.3 The letters of the word PROBABILITY are rearranged to form different words.
Assume that all the words have meaning and repeated letters are treated as identical.
11.3.1 How many different words can be formed? (2)
11.3.2 What is the probability that the word will start and end with the same
letter? (2)
11.4 There are 23 students in a class. What is the probability (correct to 3 decimal
places) that at least two will have the same birthday? Assume there are exactly
365 days in the year. (3)
[15]
[TOTAL: 150 MARKS]
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Mathematics/P1 RBHS September 2017
INFORMATION SHEET: MATHEMATICS
−b ± √ b2−4 ac
x=
2a
n n
A=P ( 1+¿ ) A=P ( 1−¿ ) A=P ( 1−i ) A=P ( 1+i )
n
T n=a+ ( n−1 ) d Sn= ( 2 a+ ( n−1 ) d )
2
a ( r n−1 ) a
T n=a r n−1 Sn= ; r ≠1 S∞ = ;−1<r <1
r−1 1−r
x [ ( 1+i )n−1 ] x [ 1−( 1+ i )−n ]
F= P=
i i
f ( x +h )−f ( x )
f ' ( x )=lim
h→ 0 h
2 2
√
d= ( x 2−x 1 ) + ( y 2− y1 )
M ( x +2 x ; y +2 y )
1 2 1 2
y 2− y 1
y=mx+c y− y1 =m ( x−x 1 ) m= m=tan θ
x 2−x 1
( x−a )2+ ( y −b )2=r 2
In ΔABC:
a b c 1
= = a 2=b2 +c 2−2 bc .cos A area ∆ ABC= ab . sin C
sin A sin B sin C 2
sin ( α + β )=sin α . cos β +cos α .sin β sin ( α −β )=sin α .cos β−cos α . sin β
cos ( α + β )=cos α . cos β−sin α . sin β cos ( α−β ) =cos α . cos β +sin α . sin β
cos2 α −sin 2 α
{
cos 2 α = 1−2 sin 2 α
2 cos2 α −1
sin 2 α =2sin α . cos α
n
2
x́=
∑ fx 2 ∑ ( x i−x́ )
σ = i=1
n
ń
n(A)
P ( A )= P ( A∨B )=P ( A ) + P ( B )−P ( A∧B )
n( S )
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Mathematics/P1 RBHS September 2017
^y =a+bx b=
∑ ( x−x́ )( y− ý )
∑ ( x−x́ )2
11
