8/13/2019 cape math unit 1 2012

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FORM TP 20L2230

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08 JUNE 2012 (p.m.)

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coDE 02134016rii1MAY/JLINE2O1D,:iall

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CARIBBEAN EXAMINATIONS COUNCILADVANCED PROFICIENCY EXAMINATIONitPURE MATHEMATICS i

ALGEBRA, GEOMETRYAND CALCULUS JililiUnit I - Paper 01 tiI - Paper 01 ii'ia90 minutes 1

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1.2.3.

4.

5.

READ THE FOLLOWING INSTRUCTIONS CAREFULLY. ]iThis test consists of 45 items. You will have 90 minutes to answer them. ]i',;In addition to this test booklet, you should have an answer sheet. fiDo not be concerned that the answer sheet provides spaces for more answers than there aipitems in this test. ]i1iEach item in this test has four suggested answers lettered (A), (B), (C), (D). Read each iteihyou are about to answer and decide which choice is best. ilOn your answer sheet, find the number which corresponds to your item and shade the spa(Shaving the same letter as the answer you have chosen. Look at the sample item below. li

Sample ItemThe expression (1 +,.6 )'is equivalent to(A) 4(B) l0(c) 1+3\6(D) 4 + 2JtThe best answer to this item is 0,4 + z./5',, so answer space (D) has been shaded.6. If you want to change your answer, be sure to erase it completely before you filI in yourchoice.

7. When you are told to begin, turn the page and work as quickly and as carefully as youitem omitted. Your score will be the total number of correct answers.You may do any rough work in his booklet.The use of silent, non*programmable scientific calculators is allowed.Examination Materials :A list of mathematical formulae and tables. (Revised 2012)

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1. In the real number system the inverse ofaddition is represented by x-2isafactorof4x4 -2xz4x3 +2x2 -162x3 +2x2 -4.r-83xs -10x3 -5x2 +4

-a+b

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,iiiiltll:iiii1iIf log" 4 * 1og, x - 1og,7 : 2, then the iivaluebfx is . lili7'ji(A) 7 : :iiri7, i:(B) 4a' liii4, ,j,(c) -a- ;:i1"i(D) -20 :4 |uiiThe coordinates of the point P are (4, -3).Under a one-way stretch by scale factor:ipin they-direction with the x-axis invariadt,

the image of P would be .iili(A) (2,-3) :li(B) (4,4) . ii(c) (4, -1) , ;il(D) (8, -3) .ii

9..-3-

I is BESThe graph of{r) : h - 2l +illustrated by(A) f@)

10.

Given that the roots of :f - 5x + a: 0equal, then a:(A)4:(B) s)1 e(c) =(D) 254

11.(c)

(D)

] ::rlliiiirlJi4l;'.ll1lGo oN ro rHE NExr PAGEI}:t., I a.in1 n/ra A DE rn t,)

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Itern 12 refers to the graph below.

12. The functionflx) is decreasing for the range

13.

(A)(B)(c)(D)

(A)(B)(c)(D)

An arch may be modelled by the Cartesianequation : -2x2* 4x * l,where x andy represent, respectively, horizontal andvertical distances. The coordinates of theHIGHESTpoint on the arch are

x(3r)53

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-5-16. The o@a ofthe circle(r-l)P+O-2Y:l6is

(A) ('l ' -2)(B) (-l, 2)(c) (r,2)(D) (l, -2)sin d(l - sin2 d;t7' ""rd(l -"*,a;(A) cot 0(B) tan 0(C) cot2 0(D) tan2 0

Which of the following sketchdsrepresents the curvey: cos I ,, (O < x

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sin(a +450) is equal tol .(A) ,5.(sina + cosa)1(B) ,fu-t.oto -sina)(c) l(rin, -cosa)2'(D) l("o., +sina)2'

Itetr:zz refers to the following graph'v

Which of the followingrepresents the graPh?(A) y = sinx(B) y: sinZx(C) y:2sinx.x(D) : srrr -

23. A curve is defined by the parametric1equations x : 3 * 2t and Y =1'

The Cartesian equation of the curve is

A vector equation is given'(-i).'[l =( ;) rhe varues ors, are, respectivelY(A) 2 and -1(B) 2 and I(C) -2 and 1(D) -2 and*lGivcn that a is an acute angle"ltan ct= j thensin(90"-ct):4(A)(B)(c)(D)

sin (30' - l) is equal to

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25.

24.21.

?535'344:5

22.

(B)(c)(D)

ii11*,{r{i?1liitii{di;i,

*

26.

(A)

(A)(B)(c)(D)

l"o. r -{ sio,l1')1fr-cos A+- stn a"o*,r+lsinl,,)"or,a-I sin,a', ')

Jv-' x-2x-3V=-22' x+3 27..,

v- Jr-J

iiThe line through the points P(k,z) d$d0(6, 8) is parallel to'the line with equatbn3x + -21:0. The value of k is '(A) I(B) 4(c) 8(D) 24

GO ON TO THENEXT PA

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-7 -The vector u has magnitude +.6 units andis parallel to the vector y : i - 2j. A unitvector parallel to u is

The first derivative ", * with respecttoxis31.

(c)(D)

(A)(B)

&lG(i-2i)ft,-rir4.6 (i - 2i)

The distance, d metres, of an arm ofa shaping machine from its startingposition can be modelled by the equationd: L2 cos d + 5 sin 0. The MAXIMUMdistance, in metres, from the starting pointis(A) s(B) t2(c) 13(D) t7The point (2, 3) is at one end of a diameterof the circle whose equation is*+f-lOx +2y+1:0.The coordinates of the other end of thediameter are(A)(B)(c)(D)

(-12, -5)(-12, *t)(8, -5)(8, -1)

(A)(B)(c)(D)

-2x(r'-l)'2xa?4xG)-x------=-2(x" -l)

The function g is defned as[3-x+5. x < 3s(x): lp*+2, *>z

For the function to be continuous at x:3,the value of p should be(A) -i(B) -1(c) 4(D) t2d.- (rr" ) is equal todr-{

tim n(r +h)2 - nrzn-+O nlim (nr + h)2 - nr2lr-+O n

29. 32.

30.33.

(A)

(B)

(c)

(D)

limh-+0limh -+0

n(r-h)2 -rr2hnrz -n(r+nr2)

i.t1.r.Arl1n /rr A DE 'n1') GO ON TO THE NEXT PAGE

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34. Given that ls 4f (x)dx =9, the value oflit rul a' i"The value 61 limx-+0(A) sin00

36.

, d2vthen -;, tsctx

(B)(c)(D)

(A) i(B) iI(c) i(D) 274133sin 3

37. If .v = J2-AItem 35 refers to the diagrambelow whichShows the curye i + f : 4,4 < x < 2.

An expression for obtaining the volumegenerated by rotating the bounded, shadedregi,on through 360'about the x-axis is(A) of g - v') dx

p2(B) rlo(4- x') dx(c) "f G* y') dx(D) ,f;{+*x') &

(A)(B)(c)(D)

(2x+t)I

Jzx+t

(z"r+r)(.,8+r)Item 3E refers to the graph below.

In the graph showing f : *, y is NOldefined for(A) x=0(B) x0(D) x>0

GO ON TO THE NEXT.PAGE

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ii-g-lii;39- Giveny:3a:+ 5 sin2r, then 44 is equal 43. The displacement, s metres, of a m#eto &2 --r-'--- .-'^-t--1t^--;:;;;;;;i-,;-"'-oving along a board at time r minutesilsgiven by s(r) : 4f - 30P+72t+ 7 for r ) Xi.(A) 6 - 20 sin 2x For what values of r is the displacemenr &f(B) 6 *10 cos 2x the marble increasing? |i(C) 6 + 20 sin 2x - ,ii(D) 6 + 10 sin 2x (A) t 3 1i" (C) 2