2019_maths_alevel_nov_paper1

Extracted Content: 2019_maths_alevel_nov_paper1

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Poe Perrone Mramarewant

ZIMBABWE SCHOOL EXAMINATIONS COUNCIL

General Certificate of Education Advanced Level

PURE MATHEMATICS 6042/1

PAPER 1

3 hours

NOVEMBER 2019 SESSION

Additional materials:
Answer paper
Graph paper

List of Formulae MF7
Scientific calculator

TIME 3 hours

INSTRUCTIONS TO CANDIDATES

Write your Name, Centre number and Candidate number in the spaces provided on the answer
paper/answer booklet.

Answer all questions.

Ifa numerical answer cannot be given exactly, and the accuracy required is not specified in the
question, then in the case of an angle it should be given correct to the nearest degree, and in other
cases it should be given correct to 2 significant figures.

INFORMATION FOR CANDIDATES

The number of marks is given in brackets [ ] at the end of each question or part question.

The total number of marks for this paper is 120.

The use of a scientific calculator is expected, where appropriate.

You are reminded of the need for clear presentation in your answers.

his question paper consists of 5 printed pages and 3 blank pages.

T

Copyright: Zimbabwe School Examinations Council, N2019.

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©ZIMSEC N2019

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=) e ” % Solve the equation 3-1 + 3*-? = 12. (3] % a 2 46 Find the term independent of y in the expansion of ( = ‘) 5 [3] Given that y = — Y > 3]q@zay find the approximate percentage change in y if x is increased by 10% when x=7, (4) A complex number is given by u = ~ (a) Express vin the form a + ib where a and b are real numbers. [2] (b) Find the modulus and argument of u. [2] (c) Show the complex number wu on an Argand diagram. a8) Cc

The diagram above shows a circle centre O and radius r, divided into two
regions R; and Rz. AOB = 6 radians. The perimeter of the region R, is equal
to the length of the major arc ACB.

(a) Show that@=2—1. {2]
(b) Given that the area of region R, is 21 cm?, find the area of region Ry

correct to 3 significant figures. (4]
(a) Express 3Cos@ — 2Sin6 in the form RCos(6 + a) where R>O and

O0<a < 90°, {2\
(b) — Hence or otherwise solve the equation 3Cos@ — 2Sin@ ==. 3) for

—180° < @ < 180°. Bl
(c) Determine the maximum and mini ———— 2

nd minimum values of Scorocasine [2]

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OABCDEFG is a perfect cube of edge 10 units. The unit vectors i,j and k are along OA, OC and OG, respectively. X and Y are the mid-points on ED and GD respectively.

Find
(a)
(b)
(c)
(a)

(b)

(c)

OX in terms of i,j and k, 1]
a unit vector parallel to OE, (2]
XOoyY. [4]
Functions f and g are defined by:

fx 7 Inx; x >0

g:x > (x+2);x>-2

Find

@) fax), 1]
(ii) the inverse of fg(x). (21
Given that h(@) = ksin@;k < 0.

Sketch the graph of y = h(@) for0 <4 s 2n. (31
Describe the geometrical transformations which maps f(x) = Sinx onto

y = 6Sin(x - 1). BI

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10

11

12

4

; . . th th
(a) Find the sum to infinity of the geometric progression whose 4” and 6

25 i

terms are 64 and = respectively, where the common ratio, r > 0. 15]
. th is—

(b) Inn arithmetic progression the first term is - 20, the 13" term is — 5 and

the last term is 15.

Find the sum of the terms of the progression. [6]
E : = —4*t5 __ in parti ions. (5)
(a) Express f(x) = anoxthe partial fractions
(b) Hence find the exact value of i f(x)dx. [4]
The diagram above shows points A, B and C lying on the line whose equation is
3y =x +9. The point C lies on the y-axis and AB = BC. The line from
D(—9; 20) to B is perpendicular to AC.
Calculate the
(a) co-ordinates of A and B, (6)
(b) exact area of triangle ADC. ; [3]
Acurve C has equation y = V4x + 9.
5 Ay.
(a) Find when x= 4, (3]
(b) A particle P moves along the curve C in such a way that the rate of
increase of x is 0.20 units /second.
Find the rate of increase of y at the instant when x = 4, [2]
(c) _ Find the exact area enclosed by the curve, the y axis, the x — axis and the
line x =4, 14

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15

5

(a) The e :
‘mation x* + ax3 + bx? — 2x — 4 = 0 has roots — 2 and 1.
(i) Find the values of @ and b, [4]
ii
(ii) Hence find the other roots, (4]
(b) (i) spe
Express 4x° — 8x — 5 in the form a(x + b)* +c where a, b and
are constants. (2]
(ii) Fj F
ee Co-ordinates of the stationary point of 4x7 — 8x — 5
etermine its nature. [2]
The heigl .
hi ht A metres of a tree at time ¢ years after being planted assumes that the
al i . . “ 1
nee ia of its height is directly proportional to (10 — h)2. When r = 0,
1=1 and ae 7 4.
3 e Z
(a) Show that this Satisfies the differential equation = = = (10 —h)2. (3]
(b) Solve this differential €quation to obtain an expression for h in terms of t. [6]
(c) Find the maximum height of the tree and the time taken to reach this
height after planting. 12]
(d) Find the time taken to reach : of its maximum height. [2]
(a) Find the exact value of i xlnxdx. [4]
(b) (i) Show by sketching a suitable pair of graphs, that the equation
4 = x+Inx has only one real root. [3]
(ii) Verify by calculation that this root lies between 2.9 and 3, 3]
(iii) | Show also that the equation 4 = x + Inx reduces to the equation
Xner = et, (1)
(iv) Use the iteration x41 = e<em>~</em>m twice with initial value x, = 2.91 .
to determine this root correct to 4 decimal places. (2)

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