Pure Math Paper 1s 2013-2020
Pure Math Paper 1s 2013-2020
Pure Math Paper 1s 2013-2020
ii
coDE 02134010
FORM TP 2012230
li
MAY/JUNE 20lP
CARIBBEAN EXAMINATIONS COUN CIL
ADVANCED PROFICIENCY EXAMINATION
PURE MATHEMATICS
ALGEBRA, GEOMETRYAND CALCULUS
Unitl-Paper0l
90 minutes
tl
il
Each item in this test has four suggested answers lettered (A), (B), (C), (D). Read each itelh
t-t
I.-l
4.
you are about to answer and decide which choice is best. ri
l;l il
t_t 5. On your answer sheet, find the number which corresponds to your item and shade the spajp
having the same letter as the answer you have chosen. Look at the sample item below.
ll
Sample Item
The expression (l +..6 )t ir equivalent to Sample Answer
(A)
(B)
4
10
@@@ o
(C) I + 3..6
(D) 4+ 2Jt
The best answer to this item is "4 + 2 .f ", so answer space (D) has been shaded.
6. If you want to change your answer, be sure to erase it completely before you fill in your nery
choice.
ii
1. When you are told to begin, turn the page and work as quickly and as carefully yo., cu4.
If you cannot answer an item, omit it and go on to the next one. You can retum "s
later to tlib
item omitted. Your score will be the total number of correct answers.
8. You may do any rough work in this booklet.
9. The use of silent, non-programmable scientific calculators is allowed.
-
I
I
Examrnation Materials:
- A list of mathematical formulae and tables. (Revised 2012)
-
-
I
DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO.
I
1. In the real number system the inverse of x-2isafactorof
addition is represented by
(A) 4x1 -Zx2
(A) x*0-r (B) 4xi +2x2 -16
(B) x+(-:)=9 (C) 2xi +2xz -4x-8
(C) 0*x=r*0 (D) 3.rs -10x3 -5x2 +4
(D) x$t+z):xy*xz
ji
2n ='E'
(A)
o ln 12
(B) Ir'=lIr
Lt lLl
I
I
r-l L r-l I
L.n
(c) I(z*,')=2*2,'
.=l rl
(D)
S" g.
/r'=
r=l
/r'
r=0
(A) -l I
(B) -l
(c) I
(D) ll
s
GO ON TO THE NEXT PAGS
n"rt?an1 nlc aDE ?ntt ti
ti
vv vrr
i'|a7 a AAl rllft A DE tnl .) !
H
-3-
8. The graph ofJ(r) : lr - 2l + | is BEST 9. If log,4 + log" x -log,7 = 2, then the
illustrated by value ofx is
(A) IF) 7
(A) v
1t)
(B) -a-
4
4"
(c) -a-
I
7"o
(D)
4
(A) (2,-3)
(B) (4,4)
(c) (4, -l)
(D) (8, -3)
(A) 4
(B) 5
)7
(C) T
)\
(D)
;
(A) x<3
(B) x)5
(C) 3<x<4
(D) 4<x<5
T
13. An arch may be modelled by the Cartesian 15. The expression 2 - 4x + 3*can be Jitn"n
equation ! = -2x2+ 4x + l,where x and as ij
y represent, respectively, horizontal and
vertical distances. The coordinates ofthe
HIGHEST point on the arch are (A) z(,-?\' *?
3/
\ 3
(A) (r,2)
(B)
(c)
(1,3) (B) ,(,_3)'_;
(2,2)
(D) (2,3)
(c) 3( *-?\' -?
\ 3/ 3
14. The general quadratic equation with roots
o and p may be written as (D) r[,-1)'*?
(A) x'-(o+F)x-oF=0 \ 2) 3
(B) -r2-(a+B)x+0F=0
(C) f +(a+F)r-eF:0
(D) f+(a+p)-r+aF=0
sind(l - sin2 d;
t7.
"*?d:;'1i:
(A) cot 0
(B) tan 0
(C) cot2 0
(D) tan2 e
vector f _l
/I
The retationship between p
and q i5\ -'
(A) p+3q:o
(B) 3p+q:s
{c) p-3q:o
{D) 3p*q:o
p isapoint thatFfi=
such
i\_-/:,)./ The posi-
tion vector of p is
(A) r o)
(-12l
(B)
(+)
II
I l2J
(zt
(c) tl
\2)
(-z)
(D) ti
\.-2l
, are, respectivelY
(B) -sina)
;!t"o'o (A) 2 and -1
(c) -cosa)
(B) 2 and I
|trt"" (C) -2 and I
(D) -2 and -l
(D) t"o.o + sina)
f
25. Given that cr is an acute angle ard
Item22 refers to the following graph. tan ct=-3 then sin (90" - ct ; :
v
(A) ?)
3
(B) 5
22. Which of the following equations best
,1
represents the graph?
(c) ;
(A) y=sinx
(B) y: sin 2x (D) 4
(C) y: 2 sin x s
.x
(D) ./ = sin - a 26. sin (30' - l) is equal to
x-3 t;
(B)
'2
v- (D) !1"o.1- I sin ,l
(c) 2
v=x+3 21. The line through the points P(k,D lhd
(D) 2 Q(6, 8) is parallel to the line with equaftn
Y--;1 ix+ y-21= 0. The value of & is
I
(A) I
(B) 4
(c) 8
(D) 24
21
:
GO ON TO THE NEXT PAGp
t il
trtrr'A UL
'ttt.,
- t-
28. The vector u has magnitude a..6 r,nits and 31. The first derivative of *x' with resoect
is parallel to the vector v : i - 2j. A unit to.r is -l
vector parallel to u is
(A) -2x
(A) €rt-ril (r'-t)'
2x
_:.-.:
(B)
(B)
(x' - l)'
fri-zir (c) x
--.-----.:--
(c) 2(x'-l\
fri-rir (D)
-x
(D) 4.6 (i - 2i) 2(x2 -t)
lpx+2, x> 3
distance, in metres, from the starting point
is For the function to be continuous at x : 3,
the value of p should be
(A) s
(B) 12 (A) -3
(c) 13 (B) -l
(D) r7 (c) 4
(D) 12
30. The point (2, 3) is at one end of a diameter
of the circle whose equation is d, ?\.
*Vr- )
ts equal to i
f +f-lOx +2y+1:9.
The coordinates of the other end of the (A) lim n(r +h)2 - rr2
diameter are h-+0-- I
(A) (-12, -s) (B) lim (nr
0----j-
+ h)2 - nr2
(B) (-12, -r) h -+
(c) (8, -s)
(D) (8, -l) (c) lim r(r-h)2 - nr2
h -+0---- I
-
(D) lim nr2 - z(r + nr2)
n -+O--n
-8-
34. civenu* [t l71x)& =9, the value of The value of lim sin3r ir -
x-+0 x
l,t nl a i" (A) sin0
0
(A) i (B) +
(B) i (c) 3
(D) sin 3
(c) i
(D) 27 s7. rf Y='lT,+t nen fr is
4
(A)
Item 35 refers to the diagram below which (2x +l)
shows fr1e curve.f + f : 4, 0 < x < 2.
(B) #
I +f:t
1zx+r)(.,f r)
(A) "f;{+- f) a,
(B) "ll{a-"'> *
(c) "f;{a* y') dr
(D) "!i{+* *') * In the graph showing f : x, ! is NOT
defined for
(A) x: o
CB) ;<0
(C) x>0
(D) :r>0
-9- .
JI
it
ji
39. GivenY:3;+ 5 sin 2r, then
dtf i, The displacement, s metres, of a mar$e
il
(C) 6 + 20 sin 2x
(D) 6 + l0 sin 2x (A) t <2
(B) t> 3
,f
(C) 2<t<3
40. [; secz x dx: (D) t<2ort>3
Jo
(A) -l
(B) + Given that limsinx_1, where
x"
x fis
measured in radians, tn* l$-tUf ir
(c) 1 ]i
2
(D) I (A) .i";
sin3x
(B)
dv.
41. lfy = tan 6x then :ls 2x
dx 2
(A) 6tafi 6x
(c)
3
(B)
(C)
(D)
sec2 6x
6 sec2 6x
sec 6x tan 6x (D)
I
2
END OF TEST
IF YOU FINISH BEFORE TIME IS CALLED, CHECKYOUR WORr( ON THIS TES*
021340t0/cAPE 2012
2013¢
l -J6
Jg *J-zz can be simplified as Rationalisins - liji gives
.12+l -
(A) 1J'
(B) 4J? (A) t-2J'
(B) 3-2Ji
(c) 4J' (c) r+Ji
(D) -6J' (D) t+2Jj
5. If a remainder of 7 is obtained when
lf p and q are positive integers such that :f - 3.r + k is divided by.r - 3, then k equals
p < q, then which of the following statements
iVare conect? (A) -ll
(B) -r0
l' -p>-q (c) lo
ll. f>pq (D) ll
III. p-l<q-l
(A) I only Which of the following are factors of
(B) II only 4 x4 +8x3 -2x2 -6x-4?
(C) I and III only
(D) II and III only I. .r+ 1
II. x- |
IIL x+2
Two roots of the cubic equation IV. x-2
?j + 3x2 - 5x - 6 are -l and -2. The
THIRD root is (A) I and II only
(B) II and III only
(C) I and III only
(A)
t (D) I and IV only
I
(B)
t 7. a5-b5:
(A) (a - b)(d - alb + dbz - abi + ba)
3 (B) (a - b)(d + a3b + db2 + ab3 + b4)
(c) (C) (a + b)(d- a3b + 6t6z - a$3 + ba)
, (a + b)(d + a,b + a2$2 4 a$3 * ba)
(D)
(D)
7
(D)
10. Which of the following is true if c, / 13. The valuesofr that satisry the inequality
and 7 are roots of the cubic equation l2x-al>lxl,a>0,are
3tr-4f-7x-10=0?
(A) ,.9 ort ro
a+ p+y
33aB+By+ya=-
J
(A) =-,
(B) -3 aB+BY+Ya=-
d+ P+Y=--:, -7 (B) *.-4 o, , , o
43 J
lA (A) p^-q
logl5 -log6+-log-= (B) p +-q
(c) -p^-q
(D) -p+-q
(A)
l. 36
2 -25
)\
(B)
(c) 0
(D) I
(A)
5
/ are, respectively
l3
(A) -2 and -l (B) t2
(B) -2 and I l3
(C) 2 and I
(c) l3
(D) 2 and-r
t2
(D) l3
t7. sin (30" - l) is equal to 5
,t;
(A) I cosl - v' sinl Thdpoint (2, 3) is at one end of diameter
22( 21. a
of the circle whose equation is
(A) f+f+8x+/-49:0
(B) x2+f-8x-2y-32=0
(C) f+f-Bx-Y+49=0
x2+f+8x*2y+66=0
I
23. What value of 0, 0 < 0 < zr, satisfies the 27. Ifp=2i+j and q = i+ 6j are perpendicular
)',
0- 2 = 0?
equation 2 cos2 0 + 3 cos vectors, then the value of l" is
I
(A)
(A) -3
l
o (B) -l
(B) tr (c) 0
(D) 2
[L] ;J 28. The general solution for sin 20 = stn- ls
(D) ::-
z
lznt+-
IA
o=1
(A)
l0n+l\a
24. With respect to an origin O, A has |.' '16
coordinates (3, -2). The position vector
of3OA is lnr
I
+L
It
(B)
(A) (3, -6)
|tnft+-5tt
t12
(B) (e,-2)
(*
/o\ lnt +!
(L) | -l
|\-L) "l
(c) ^t t2
l(2nttl:
L' '12
(D) re)
|!'-o,l.l
lo
lnv+-
(D) v=1
Jtt
|, ,,lr-;
t< The expression sin 64, + sin 4A may be l(r+
written as
to
29' The cosine ofthe angle between the vectors
(A) sin l0A
(B) -2 cosZA -6 j and i +j is
(C) 2 cos 5A sin A
-l
(D) 2 sin 5,A cos A (A)
6
(D)
( (A) - -3
(B) -l
30. In the diagram above showing f=x,yis (c) 4
NOT defined for
(D) 12
x=0
If y= x-6
(A)
(B) xZ0 35. tnen {dx is
(B) 0
(c)
--2l -Bx
(c) 6 --.-
(3 - 4x)'
(D) @
(D) -27 -8x
(3- 4x)'
Given that hT
sinr
32. -1, where x is meas-
x
ured in radians, ttrenl'S# tt 36. If y =".,ffiT ttt"n d'y is
2
dx3
.3
(A) stn-2
sin 3x
(B)
2x (B)
2
(c) ;J
(D)
i
a
J (c) ffi
2
I
(D)
(2x + l)
(A) -k
[/= '7
,lt
dV -k
(B)
dt
m.
ry.
2
(A) -/t
)
a
(B) :ft
)
(C) 2n
(D)( 4n
:.
END OF TEST
IF YOU FINISII BEFORE TIME IS CALLED, CIIECK YOUR WORK ON TIIIS TEST'
2014
CAPE
CAPE June
20142014
Pure Pure Mathematics
Mathematics U1 P1U1 P1
/uee rflt+fls eo/5
2015¢
-2-
3 log;2q -Zlogr3q + I exPressed asa
d-b5= rrrtE G"rittui'in itt stuplesr form is
+ b)
(A) @-b)(d -db+ db2-ab'
i'ti i"-ujia + db + dbz + ab'+
+
bt)
(A) bs,(g
i.i io * ulix- a1t6 + d8 - "b' bl)
bi i"*aiia* r1t6+ dbz+ ab3 + ba) (B)
"rr[9
when
lf a remainder of 3 is obtained (c) ros,(4i*r
il; ;; i is divided bY x l' then k
-
equals
(A) -11
(D)
1"*,(t, ).'
(B) -e
(c) -l
(D) ls Which of the following are factors
of
4t + 8f -?t -6x-4?
Two roots of the cubic equation I. x+ I
Z*-* 3i - 5x - 6 are -1 and -2'
The
il. x-l
THIRD root is ilI. x+2
.J IV. x-2
(A)
T (A) I and lI onlY
I iB) II and III onlY
(B)
2 (Cl I and III onlY
J
(D) I and IV onlY
(c)
t (in thousands) of
(D) J 7. The annual growth, g(x)'
for x years ts
tt ooput"tion in a country
"
,"pr"t"nt a by g(x)=T' Inhowmanyyears
eve d?
*tnil edtt. ":t tztho usand be ach i
ici 't''+(ct+F)x+aF=o
t' +(o+ F)r-oP=9
iD)
,r.-rt
(c) l(z+r'z)=2*Zr'
r=l r=l
(B)
(D)
i,'=f,'
r=l r=0
12. logr5-ro96+
i^tr!r=
JU)
(A)
l, 36
2 "2s
.25
rI
(c) (B) lo8
--4
(c) 0
(D) I
(B) ,.4o, ,, a o
J
(A) p
(B) p ^-q
:+-q
(c) -p
^-q
(D) -p+-q
nAr^ln tar^r hr
^
tF COON O THE NEXT PAGE
.4-
17 ' sin (30" - l) is equal to
14. 3roca5 -
(A) I
I
(A) ,cosA-|sinA
'fg
(B) s
rJt
(c) e (B) ,cosA+|sin A
(D) 27
t;1
(c) {r cosl+ i sinl
and f(g(x))=x ' then g(x)
22
15. If r(x) = 3x -4
(D) {tl^t cosl-isinl
is \-, 2 Z
(A)
3x-4
18' The centre of the circle
x+4
(B)
J
(x - 1)' + O/ -2)2: 16 is
(c) 3-4x (A) (-r,-2)
4x-3 (B) (-r,2)
(D)
(c) (r,-2)
(D) (1,2)
2
(D)
- Y=-
x-J
E
5 (A)
(A) 6
l3
7t
l2 (B)
(B) 4
l3
1t
r3 (c)
(c) 3
12
1'
l3 (D)
(D) 2
5
and g is
(B) (9,1)
(A) P+3q:O
(c) (-l (B) 3p+q=g
(C) p -3q:0
(D) (.:)
(D) 3P-q:0
(D) l, t, 6 z
(c) ! = 1x-3
(D) | =2x-3
I +cos14-sin14=
(A) I +cos4,{
(B) 2 coszA
(C) cos 2A
(D) 2 cos2A sin2A
29. The cosine ofthe angle between the vectors (A) .f (cos x + 3 sin x)
-6jandi+jis (B) :i (x cos:r - 3 sin x)
(C) :f (3 cos x + sin x)
(A) -l (D) :f (x cos x + 3 sin x)
42
I
(B)
^12
(c) -5
.12
6
(D)
^12
(A) 200
(B) 20s
5tr
(c) 200 +
V
(D) 200 + 5n
p4
tt l, f @)dr =12 ,what is the value of
e3 p4
lrf(x)dx + J, (/(x) -r)dx2 37. The gradient of the normal to the curve
y = lnx atx=2 is
(A) 2
(B) 4 (A) -2
(c) ll
(D) 12 (B) -: I
(c) : I
lf y-",12..1then dtl is
2
dxz
(D) 2
I
(A) If the rate of change of y with respect to x
(r,.@T) is
^4
3x'*1 theny is equal to
x-
(B)
(A) .12
OX---
I x'
,lZx+l (B) ^2
x- *-;-*Constant
I
(D)
@ (c)
-)
x'-j+constant
(D) Ar-4
x-
G9
(B)
I no =J(5+cosr)ar
732 I
Gr0
(c) I oo=Jts*
732 I
cost)dt
(t l0
(D)
! ac=Jts*cosr)dr
732 0
(c) 2n
(D) 4n
u.
m.
rv.