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CORNWALL HILL COLLEGE
DEPARTMENT OF MATHEMATICS
Advanced Programme Mathematics
GRADE: 11 TIME: 2hours
DATE: November 2012 MARKS: 130 marks
EXAMINER: Mrs. M. van Niekerk
MODERATORS: Mrs S. Hickling, Mr. P. van Schalkwyk
INSTRUCTIONS |
1. The question paper consists of 6 pages. Please check that your test is complete.
2. Read the questions carefully.
3. Its in your own interest to write legibly and to present your work neatly.
4. allthe necessary working details must be clearly shown.
5. Approved calculators may be used except where otherwise stated.
6. Goop.UCck! ‘Advanced Programme Mathematics November 2012 Page 2 of 6
QUESTION 4
a) Solve for x:
7 14
1) fle 145 = 5
(1) y2x a 6)
Sx
— <4
2 T= (6)
(3) || @)
I
b) Functions f and ¢ are defined by and g(x) = 5 - 2
) Fu f and g y F(x) Jraa 0 86) *
Determine
(1) (g © f(x) (leave your answer in simplified form) (4)
(2) The domain of (g © fx) (2)
@ (g > C2) (2)
(4) ¢"'(x) in the form g (3)
Bo
QUESTION 2
Given p(x) = x° + ax” + bx - 6 witha zeroatx =147
a) Determine the vales of a and b 8)
b) Give the real root of the equation p(x) = 0 (2)
c) Hence, or otherwise, solve for x if x7 — 5x7 + 8x -6>0. (2) ‘Advanced Programme Mathematics November 2012 Page 3 of 6
QUESTION 3
a) Factorise x3 —1 2)
b) Hence solve x3-1=0 for xEC 4)
i
QUESTION 4
ff) = oy » decompose f(x) into partial fractions ©
et
QUESTION §
Solve the following inequality by making use of a sketch, showing all relevant points
on the graphs: |e - 5] <4r
112) ‘Advanced Programme Mathematics November 2012 Page 4 of 6
QUESTION 6
The sketch shows the graphs of g(x) =x +6 and f (x) = x?+ 4x — 12,
which intersects at A(-6; 0) and B(3; 9)
a) Use the information given on the sketch to draw, on the answer sheet provided,
graphs of of A(x) = |x +6] andj (x) = |x?+ 4x — 12 6)
b) These two graphs (h and) intersect at the same points as g(x) and f(x) as well
as at a third point. Calculate, without the use of a calculator, the coordinates of the
other point of intersection of h(x) and / (x) (6)
12) Advanced Programme Mathematics November 2012 Page 5 of 6
UESTION 7
a) Determine the value of the following limits if they exist. All relevant working must
be shown
oy lim (: E 2) 6)
el Sea
2) tim (+1 4
wie Qe mae
b) Given that sin A.cosB + cos 4.sin B = sin(4 + B)
(1) Simplify: sin 4x.cos x + cos 4x. sin x 2
4x. si
(2) Hence evaluate spies teste sae) ©
x
116)
QUESTION 8
The sketch shows the graph of y = f(x).
a) Discuss, with motivation, the continuity of y = f(x) at:
GQ) @)
(2x =5 (3)
b) Give values of x for which f(x) > 0 @) ‘Advanced Programme Mathematics November 2012 Page 6 of 6
QUESTION 9
px - 10 ;x
1
Determine the values of p and q which make g(x) continuous. (8)
BB)
QUESTION 10
In the diagram, OCD is an isosceles triangle with OC = OD = 10cm
COD = 08 radians. The points 4 and B, on OC and OD respectively are joined
by an arc of a circle with centre O and radius 6 cm
2 6om B 4am D
Find (rounded off to two decimal places):
a) The area of the shaded region @”
b) The perimeter of the shaded region ®
15]
THE END Nemo
AP Mots November 2012
ae
ozs |
a) ext +s = te
2-1
Ce Joe
lu
Cee oe
kitsk-1% = 0
(k+7ylk-2) =o
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jae =-7
N/A
=
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sx
y+ —% £0
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yr #0
ea
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sx u(t) ae a
aed
on
k=2
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ee
tx =F 3) fx| = xt-2
2 fr
x =r 2 7 Ge xr
O=K = K-d ~~
oe
X =2 of xe-1 ~~
MA
, eet
a
b) 1) (4 of) (x)
= 400)
9 (35)
ct |i
3
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s(v#3)- 1
Snes
= SLASH!
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= Stu
xr
= 5-
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ade Obert ches
(xr-mere \Oc =
‘ a-sy> eer —b
Y% Bane
@)
b) y=3 uw .
¢) yi-ry2 ex -b>o
(x2-re4e)( x-3) 70
1>3 @) Cuaron
a)
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= (x DC x? 4x47) 6)
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z
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a
2 2 EBs
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4 (x) = =
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toy
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core Ache
MED
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fore ux) ~ @(et)
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on atl j=
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= /
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us -
bL) xvte = -x* 4x42
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2,LS|
eed
(i;7) «) Quartion cn
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ayy) Lem eee
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e fw yan ta
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wa.
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2
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2 x .
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2
be or
ire z st ZA
ee
= be L = z al
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SuGh CAL + LY swx
sin (UX tx) <=
Sin FH
uw,
y2 0
ect @)
Sin OK ceil
wu
Ze
6 Suse
s«u
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ere
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2) @ Lim, f Od = aoe
hue fa) =lovw
t-F-j
Low iG) de vot exit
(Pp com haw ty at sy (3)
7 hy) =
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Lew i k { r)} = 3
woe uv
fer f(x) cloep exit
xe od
But f Co) door vot exat
2
fomovale es at yes 6)
? b) fo) >0 when LYLE of UB
ea “= “
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poe =p
-3r-to =O,
-S(p+2 =o
oye! i pr-a —
a
Bet sr) ee
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: pene
wy bs y (3) Question to
