2020_maths_alevel_mj_paper1

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ZIMBABWE SCHOOL EXAMINATIONS COUNCIL

General Certificate of Education Advanced Level

PURE MATHEMATICS 6042/1

PAPER 1
JUNE 2020 SESSION 3 hours

Additional materials:
Answer paper
Graph paper
List of Formulae MF7
Non-programmable electronic 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 non-programmable electronic scientific calculator is expected, where appropriate.

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

This question paper consists of 4 printed pages.
Copyright: Zimbabwe School Examinations Council, J2020.

@ZIMSEC 12020

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By means of the substitution y = x3, or otherwise, find the values of x for which

—3xF =2, [4]
{a) Find the value of & for which the line kx + (k — 2)y + 10 = 0 is parallel

to the line 3x + 2y—16=0. [3]

(b) Find the gradienit of the line perpendicular to both lines. (1)

(a) Express 6x? — 24x — 25 in the form A(x + BY’ | C, giving the numerical
values of A, B and C. [2]

(b) Hence, or otherwise, solve exactly the inequality 6x*-24x-25>0. [3]

Solve the inequality |2x + 2| > 1—4x. 5]

(a) Expand (p — x)~ as a series of ascending powers of x up to the term in
x where p is a positive constant. {5]

(b) Given that the coefficient of x* in the expansion is = find the value ofp. [2]

(c) Hence state the set of yalues of x for which the expansion is valid. [1]

The points A, B and C have position vectors a = i— 27 + pk,b = we +5j + 6k
and c = 5i + 7j respectively relative to the origin.

IfAB = —5i + 7j + 3k find the

(a) values of p and q, 2]
(b) ‘exact length of AC, [2]
(c) acute angle BAC. [3]

The population of a city is 500 000. The population grows at a rate of 5% every
year.

Find

(a) —_ interms of 7, the population at the end of the nth year. (2]
(b) the population to the nearest thousand after ten years. [2
(ce) the year in which the population first exceeds 1 000 000. 3]

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12

The function f(x) = 2 =22 >0.

(a) Sketch the graph of f(x) and state the range. [3]
(b) Find £~!(), the inverse of ffx). [2]
(©) — Calculate the value of x for which f(x) = f71(x). [3]

A circle with centre (2, —5) touches the line x + 6y — 9 = 0. Find the equation
of the circle in the forms (x — a)? + (y — b)* =r? where a, A and rare
constants. [8]

(a) | Complex numbers w and v are such that ¢= 1+ 2iand v = —2—i.
Find w = =, leaving the answer in the form a + ib, where a and bh are
integers. [3]

(b) Hence, or otherwise, find
@ = [wi [2]
Gi} arg w.

The polynomial 2x3? — 11x? + ax + b is exactly divisible by (x — 2) and leaves a
remainder of -36 when divided by (x + 1).

(a) Find the values of the constants @ and A. [6]
(b) —_ Hence factorise the polynomial completely. {3]
F(2) = sageeg x ER

(a) — Express /(x) in the form
Ax+B she — See aa? where A, B, C andD sa ies [5]

(b) Hence find f, ° f(x)dx. Leave the answer in exact form. [4]

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(a)

(b)

©

{a)

(b)

(a)

(b)

Express 2cosx — 5sinx in the form Rcos(x + @), where R > 0 and
0°< 6 < 90°.

Hence, or otherwise, solve the equation 2cos2x — Ssin2x = 2.5 for
0° <x < 360°.

Give values of x between 0° and 360° at which the maximum and
minimum values of 2cos2x — 5sin2x occur.

F : é 15 :
Use the trapezium rule with 4 ordinates to evaluate f i x? sin? xdx,
giving the answer correct to 3 significant figures.

@ Find the area of the region bounded by the curve y? = 4x and
the line y =.

(ii) The region in (i) is rotated through 360° about the y-axis. Find
the volume generated giving the answer in terms of x.

It is given that y = Trcosx*
Find

2

ny fone
eh a when x = 0,

(ii) the Maclaurin’s series of y up to the term inx’.

‘Variables x and y are related by the equation y = ab*, where a and b are

constants. The graph of /ny against x is a straight line of gradient 0.7
and /ny intercept at 2.3.

Find the values of a and b.

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[2]

[4]

4]

[3]

[5]

[5]

[5]
[4]

[6]

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