RONDEBOSCH BOYS’ HIGH SCHOOL
GRADE 12
MATHEMATICS PAPER 1
1 JUNE 2015
TIME: 3 HOURS
MARKS: 150
EXAMINER: T EDWARDS
MODERATOR: P GHIGNONE
Question 1
1.1.1 x2-4x-21=0 ü
x-7x+3=0 ü
x=7 ü or x=-3 ü Routine (4)
1.1.2 x2-9x=18 ü
x2-9x-18=0 ü
x=--9±-92-4(1)(-18)2(1) ü
x=10,68 or x=-1,68 ü Routine (4)
1.1.3 x2+3x-4≤6 ü
x2+3x-10≤0 ü
(x+5)(x-2)≤0 ü
-5≤x≤2 ü Routine (4)
1.1.4 (x2+2)2+66=17(x2+2) ü OR (x2+2)2+66=17(x2+2) ü
x2+22-17x2+2+66=0 ü x4+4x2+4-17x2-34+66=0 ü
x2+2-11x2+2-6=0 ü x4-13x2+36=0 ü
(x2-9)(x2-4)=0 ü x2-9x2-4=0 ü
x=±3 ü or x=±2 ü x=±3 ü or x=±2 ü
Complex (6)
1.2 6-3x<0 ü
x>2 ü Routine (2)
1.3 x3-22+x4-22=10y ü
x+4x=10y
5x=10y ü
y=12x ü OR 2y=x ü
∴612x=x2-2x-6 ü 6y=2y2-22y-6 ü
3x=x2-2x-6 6y=4y2-4y-6
0=x2-5x-6 ü 2y2-5y-3=0 ü
0=(x-6)(x+1) 2y+1y-3=0
x=6 or x=-1 ü x=6 or x=-1 ü
y=3 ü or y=-12 ü y=3 ü or y=-12 ü
Complex (8)
1.4 20152+2015+12015-1-2015-22015+2-2015-82015+8 üü 20152+20152-1-20152-4-20152-64ü
-1+4+64=67 ü Problem Solving(4)
[32]
Question 2
2.1.1 S20=220+3202ü=1240 ü Routine (2)
2.1.2 T20=S20-S19
T20=1240-219+3192ü=1240-1121ü=119ü
Complex (3)
2.2.1 -1<x-24<1 üü
-4<x-2<4 ü
-2<x<6 ü Routine (4)
2.2.2 S∞=192 ü1-34 ü
S∞=768 ü Routine (3)
2.3 x+20 24 x+ 8 14
4 - x x - 16 6 - x
2x - 20 22 - 2x ü
2x-20=22-2x ü ∴x=212 ü
2a=1 3a+b=-132 a+b+c=612
∴a=12 ü ∴b=-8 üü ∴c=38 üü
∴Tn=12n2-8n+38 ü Complex (9)
2.4 ∴Sn=51 ü2214ü+51-17ü=9639
∴9639+1=9640 ü
OR
Sn=752×532üü-1+1ü=9640ü Complex (4)
[25]
Question 3
3.1 p=3 ü ; q=-1 ü
Sub in D(4;1)
1=a4-3-1 ü
a=2 ü Routine (4)
3.2 y∈R ü ; y≠-1ü Routine (2)
3.3 y=-x+c
Sub in 3;-1
-1=-3+c ü OR y=-x-3-1 ü
c=2 ü y=-x+3-1 ü
y=-x+2 ü y=-x+2 ü Routine (3)
3.4 y=-2üx-1ü+3ü Complex (3)
[12]
Question 4
4.1 hx+12=16x+12 ü
=16x.1612 ü
=4(16x) ü
=4h(x) Routine (3)
4.2 h14=1614=2 ü
h-1x=log16x ü ∴2.h-14=2log164=212ü=1 ü
40=1 ü
∴2+1-1=2 ü Routine (6)
[9]
Question 5
5.1 f1=-21=-2 ü
∴q=2 Routine (1)
5.2 T(1;-2) üü Routine (2)
5.3 x<0 ü or x>1 ü Routine (2)
5.4 y=log2(-x) üü (2 or nothing) Complex (2)
5.5 Shape ü x-intercept ü
another point ü
Routine (3)
5.6 Either x≤1 or x≥1 ü Routine (1)
5.7 kx=2xü-3 ü Complex (2)
[13]
Question 6
6.1 0,3x=x1-0,22n üü
0,3=0,78n
n=4,85 years ü ∴5 years ü Routine (4)
6.2 50000=x[1-1+0.1712-48]0,1712 üü
x=R1442,75üü Complex (4)
6.3 A=865001+0,0541221üü1+0,061417üü-200001+0,061412üü
A=R 98960,44 üü Problem Solving (8)
[16]
Question 7
7.1 f'x=limh→0x+h2+7x+h-x2+7xh ü
=limh→0x2+2xh+h2+7x+7h-x2-7xh ü
=limh→0h(2x+h+7)h ü
=limh→02x+h+7
=2x+7 ü Routine (4)
7.2.1 y=3x5-14x12+3x-1
dydx=15x4-7x-12-3x-2 ü
dydx=15x4ü-7xü-3x2ü OR dydx=15x4ü-7x12 ü-3x2ü
Routine (4)
7.2.2 2πxü+πxπ-1 ü Complex (2)
[10]
Question 8
8.1.1 x3-5x2-8x+12=0 ü
x-1x2-4x-12=0 üü
x-1x-6x+2=0 ü
x=1 or x=6 or x=-2 ü Routine (5)
8.1.2 f'x=3x2-10x-8
0ü=3x2-10x-8 ü
0=3x+2(x-4) ü
x=-23 or x=4 ü
y=40027 ü or y=-36 ü Routine (6)
8.1.3 f''x=6x-10
0ü=6x-10 ü
x=53 ü Routine (3)
8.1.4 f'a=-11 -11=3a2-10a-8 ü
0=3a2-10a+3 ü
0=3a-1a-3 ü
a=13 or a=3
∴a=3 ü
Sub 3;b into y=-11x+3
b=-113+3 ü
b=-30 ü Complex (6)
8.2 Shape ü x-intercepts ü
TP’s ü
Complex (3)
[23]
Question 9
9.1 TSA=2πr2+2πrh ü
Volume: πr2h=330 ü
h=330πr2 ü
∴TSA=2πr2+2πr330πr2 ü
∴TSA=2πr2+660r Problem Solving (4)
9.2 TSA'=4πr-660r2
0ü=4πrü-660r2 ü
4πr3=660 ü
r=3,74cm ü
h=7,49cm ü Problem Solving (6)
[10]