📄 Exam Questions & Readable Text Transcript
Full OCR-extracted text of questions, syllabus codes, and exam instructions.
Extracted Content: 4027q2Specimen paper
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Surname Forename(s) Centre Number Candidate Number(& }
ZIMBABWE SCHOOL EXAMINATIONS COUNCIL
General Certificate of Education Ordinary L evel
PURE MATHEMATICS 4027/2
PAPER 2
SPECIMEN PAPER 2 hours 30 minutes
Candidates answer on the question paper.
Additional materials: | Data booklet
Mathematical tables/ electronic calculator
\A llow candidates 5 minutes to count pages before the examination.
This booklet should not be punched or stapled and pages should not be removed.
TIME = 2 hours 30 minutes
INSTRUCTIONS TO CANDIDATES
Write your name, Centre number and candidate number in the spaces at the top of this page and
your Centre number and Candidate number on the top right corner of every page of this paper.
Answer all questions in Section A and any four from Section B.
Check that all the pages are in the booklet and ask the invigilator for a replacement if there are
duplicate or missing pages.
Write your answers in the spaces provided on the question paper using black or blue pens.
If working is needed for any question it must be shown in the space below that question.
Omission of essential working will result in loss of marks.
Decimal answers which are not exact should be given correct to three significant figures unless
stated otherwise. Answers in degrees should be given correct to one decimal place.
INFORMATION FOR CANDIDATES
The number of marks is given in brackets [ ] at the end of each question or part question.
Electronic calculators or Mathematical tables may be used to evaluate explicit numerical expressions.
This specimen paper consists of 18 printed pages and 2 blank pages.
Copyright: Zimbabwe School Examinations Council, Specimen.
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Section A [52 marks] Answer all questions in this Section.1 (a) Functions f and gare defined by
1 1
fix7A—, x#=
3x -1 3
gixox-l
Find an expression for
@® = a(x),
Answer [2]
@ f(x),
Answer [2]
(iit) gg(x).
Answer [2]
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(b) Solve the inequality;(x- 2)(x +3) 5
0
(x- 4)
Amswe eee [3]
2 (a) Solve the simultaneous equations;
4x - 3y =15
8x? - 27y =45
Amswer eee [4]
(sinx +c0sx)”
(b) Prove the identity. 2° 2 +secxcosec x.
stnx COSX
Answer
[3]
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4(c) Solve the equation sin’6 + 2sin 8 cos 6 =0 for 0° £ 6 £ 360°.
Answer [3]
“7
x3 in ascending powers of x, up to and including
3 @ @ — Expand ei-
the term in x’.
Answer [3]
di State the set of values of x for which the expansion is valid.
Answer [2]
(b) Find the range of values of pfor which the equation x*- 2px +3p =0 has
real roots.
Answer [4]
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54 (a) ‘The table shows experimental values of two quantities x and y which are
connected by the equationy =
X 1 2 3 4
y 30 75 190 470
(1) Draw an approximate straight line graph on the grid on page 6
di Use the graph to estimate the values of kand b.
Answer k=
Answer on graph
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74 (b) Two points A and B have coordinates (- 2;- 3) and (7:4) respectively.
The perpendicular bisector of the line joining A and B meets the y-axis
at C. Calculate the coordinates of point C.
Answer [4]
5 (a) A particle P travels in a straight line from a fixed point O. Its velocity,
vim/s is given by v =t?- 10¢ +24,where tis the time in seconds after
leaving O.
Find the
@ values of tfor which P is instantaneously at rest,
Answer [2]
di distance OP when t =3,
Answer [2]
(iii) range of values of t for which the acceleration is negative.
Answer [2]
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8(b) 0, P and Q are three points such that OP =3 +4j and OQ =7i+/j.
(i) Find OP + PQ interms of /.
Answer [2]
(ii) Given that OPQ = 90’, calculate the value of /.
Answer [3]
(iii) | Show that triangle OPQ is isosceles.
Answer [3]
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9 Section B [48 marks] Answer any four questions in this section. Each question in this section carries 12 marks6 (a) The remainder when x*- 5x +a is divided by x + 3 is twice the remainder
when it is divided by x- 2. Find the value of a.
Answer [4]
(b) An arithmetic progression has 14 terms. The sum of the odd terms (i.e.
1s, 3"! 5" etc.) is 40 and the sum of the even terms (i.e. 274, 4! 6!
etc.) is 161
Find the
@ first term and the common difference of the progression,
Amswer eee [6]
(ii) 14" term.
Amswer eee [2]
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107 (a) OAB is a sector of a circle with centre O and radius 6 cm. The mid-point
of OA and OB areC and D respectively. CD is a straight line.
Given that angle AOB =0,9 radians, calculate the:
@ length of the arc AB,
Answer [2]
di area of the shaded region.
Answer [4]
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11(b) A rectangular block is such that the sides of its base are of length
xcm and 3xcm. The sum of the lengths of all its edges is 200 cm.
Find the
(@) volume, v cm? in terms of x,
Answer [3]
di value of x for which v has a maximum value.
Answer [3]
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128 (a) The diagram shows part of the curve y- = 10x— x and the straight
line y= 2x.
A
y
y=2x
y = 10x-x"
x
Calculate the
@ area of the shaded part,
Answer [4]
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13di volume of the solid generated when the shaded region is
rotated through 360° about the x-axis.
Answer [4]
(b) Find the gradient of the curve x? +y* +5x- 4y +7=0 at the point
(-4; 3).
Amswer eee [4]
9 @ Expand (9-4) in ascending powers of x up to and including the term
inx’,
Amswer [4]
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14di State the range of values of x for which the expansion is valid.
Answer [2]
| P
(ii) By using x= 5 in the expansion, find an approximation for ¥77
leaving the answer in the form .
q
Amswer ee [6]
x3 12
10 a Evaluate Q—— a
® Qa
Answer [3]
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15(b) A curve is defined parametrically by the equations.
x =t 41 , y =t- 1
t t
Find,
@ ay in terms of t,
dx
Answer [3]
di the values of t at the points on the curve where - =3,
X
Answer [2]
(iii) the equation of the tangent to the curve at the point S ; =
Answer [4]
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1611 (a) _—‘ Solvethe equation 4°" = 16%.
Answer [3]
(b) (@ — Express 2 cos 6 — 5 sin @ in the form rcos(q-+a).
Answer [3]
di Hence solve the equation
2 cos 8@—5 sin @ =2 for 0° <@ < 360°.
Answer [3]
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17() __ Solve the equation [2x - 3 = 3x.
Answer [3]
12 (a) ‘The time of swing T, of a simple pendulum of length x units is given by
T =kVx, where kis a constant. Find in terms of aand xan
expression for the approximate value of
i) aT,
Amswer eee [4]
a 7.
T
Amswer [3]
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18 (b) A geometric progression is such that the 4" term is twice the 3 term. Given that the 7" term is 512, find@ the first term and the common ratio,
Answer [3]
di the sum of the first 10 terms of the progression.
Answer [2]
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