2020_maths_alevel_nov_paper1

Extracted Content: 2020_maths_alevel_nov_paper1

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te Prope y Vere

ZIMBABWE SCHOOL EXAMINATIONS COUNCIL

General Certificate of Education Advanced Level

PURE MATHEMATICS 6042/1

PAPER 1

NOVEMBER 2020 SESSION 3 hours
Additional materials:
Answer paper
Graph paper
List of Formulae MF7
Scientific calculator (Non-programmable)

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.

If a 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 cach question or part question.

The total number of marks for this paper is 120,

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

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

This question paper consists of 5 printed pages and 3 blank pages.
Copyright: Zimbabwe Schoo! Examinations Council, N2020.

@ZIMSEC N2020 (Turn over

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Find in the form ax + by + c = 0 the equation of the line passing through (2; -3)

and parallel to the line 3x — 4y + 2 = 0.

Solve the equation 2**~* = 6 leaving the answer correct to three significant
figures,

A curve has parametric equations x = cos2t and y = sin2t.
Find the gradient function of the curve in terms of f,

Given that 27,7? = =n(n + 1)(2n + 1).

Evaluate 32,77.

Simplify

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4

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125a°\ 3
(Fe)
The equation of a circle is x* + y? -2x —6y +1=0.
Find the gradient of the circle at the point (1; 0).

(a) Express f(x) = V3sinx + cosx in the form Rcos(x — a) where R is
positive constant and0O < @ < 7.

(b) Sketch the graph the graphs f (x) = V3 Sinx + Cosx for 0 < x < 3r.

Solve the equation cos@ = 2cos26@ + 1 giving solution in the interval
0° < @ < 360° to the nearest 0,1°.

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Find in the form ax + by + c = 0 the equation of the line passing through (2; -3)

and parallel to the line 3x — 4y + 2 = 0. [3]
Solve the equation 23<em>-</em> = 6 leaving the answer correct to three significant
figures. [3]
A curve has parametric equations x = cos2t and y = sin2t.
Find the gradient function of the curve in terms of f, [3]
Given that }"_,r? = =n(n + 1)(2n + 1),
Evaluate })2°,, 77. [3]
— 3

a2zxaé

(a) — (1)
a 4
—
125a 3

(Se) 2
The equation of a circle is x<em> + y</em> —2x —6y +1=0.
Find the gradient of the circle at the point (1; 0). [5]
(a) Express f(x) = V3sinx + cosx in the form Rcos(x — a) where R is

positive constant andO < a< q, [3]
(b) Sketch the graph the graphs f(x) = V3 Sinx + Cosx forO<x< 37, [2]
Solve the equation cos@ = 2cos2@ + 1 giving solution in the interval
0° < @ < 360° to the nearest 0,1°. [6]

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Vv

On separate diagrams, sketch the graphs showing clearly the co-ordinates of the
marked points.

(a) y=2f(x), [2]
(b) y=f(x-3), (2]
(c) y=f(-~x). [2]

10

, KW )

The diagram shows a circle centre O, radius 9 cm and POQ = - PQ is

a chord to the circle.
Find the

+=

(a) length of the minor arc PQ in terms of 7, [2]

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(b) area of the triangle POQ, [2]
(c) area of the shaded segment in terms of 7. [2]

The function fis defined by f:x > x* — 4x, where x € R.

(a) Sketch the graph of fshowing the intercepts and turning points. [3]
(b) State the range of £ [1]
(c) = fi) Ifx > k, fis a one to one function. State the value of k. [1]

(ii) Using this value of k find the inverse of fstating its domain. [3]

The polynomial p(x) = 6x? — 11x? + ax + b where a and b are constants.
When p(x) is divided by (x + 1), it leaves a remainder of —24. Given that
(x — 1) is a factor of p(x),

(a) find the values of a and h. [5]
(b) — factorise p(x) completely, [2]
(c) find the roots of p(x) = 0. [3]
(a) Sketch on the same axes the graphs of y = |x? — 2| and y = |x]. [3]
(b) Hence or otherwise solve the equation |x? — 2] = |x]. [5]
(c) State the range of values of x for which |x? — 2] < |x], [1]

The functions fand g are defined as follows:

f:x—x*-2x+2, xER

g:x 3x +3, x ER

(a) Find the set of values of x for which f(x) > 10, [4]
(b) ‘Find the range of fand state with a reason whether fhas an inverse [2]
(c) Show that the equation gf (x) = 0 has no distinct roots. [3]
(a) Obtain the first three terms in the expansi —

pansion of ra [4]

b Taki = 2 a a = v10

(b) aking y : show that fay 10 ° [4]

(c) Calculate the value of ¥ 10, using y = : in the expansion

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17

18

(a) Solve the equation In(S + e~2*) = 3 giving the answer to 3 significant
figures,

(b) Two variables x and / are related by the equation x = mn where m and
n are constants. The values of [nx are plotted against the values of /,
and the points lic ona straight line with gradient —2. 3 and crossing the
vertical axis at (0; 3).
Find the value of m and the valuc of n.

(c) Solve |x —3| <5.

It is given that f(x) = V9 —x,

(a) Find the inverse of the function f{x) stating its domain.

(b) Sketch the graphs of y = f~*(x) and y = f(x) on the same axes.

(c) State the relationship between the graphs of y = f(x) and y = f~1(x).

(d) Find the x — coordinate of the point where the graphs of y = f (x) and
y = f~*(x) intersect.

. _ 3x+4
It is given that f(x) = Gna) ata
(a) Express f(x) in partial fractions,

(b) Hence or otherwise find the series expansion of f(x) when x is small
such that terms in x" and higher are neglected.

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