2018_maths_alevel_nov_paper1

Extracted Content: 2018_maths_alevel_nov_paper1

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MATHEMATICS NOVEMBER 2018 SESSION Additional materials:

Graph paper
List of Formula
Scientific calculator

TIME

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 each question or part question.

The total number of marks for this paper is 120.

Questions are printed in the order of their mark allocations.

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

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

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res i" ae 210. ~

A curve has equation y? = 3xy + x? — 3.
Find the equation of the tangent to the curve at the point where

y=landx>0.

The function f(x) = —x* + 8x — 12 is defined for 4< x <6.

Express f(x) in the form a(x + b)* +c, where a, b andc are constants. ipa) a

)
(ii) _ State why the function of f(x) has an inverse.

oe
(iii) Find f-1(x), the inverse of f(x). oe
a
Sketch the graphs of f(x) and f~1(x) on the same axes and write down
the domain of f~*(x). Bl

(iv)

The complex number wu is such that

—48 + 56i
u

= 14131.
(a) Find win the forma + ib, where a and b are real numbers.
(i) Write down w*, the conjugate of u,

(ii) Sketch v* on an Argand diagram showing clearly the v.
fut] and arg. u*.

i

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~ Hence or otherwise evaluate {> f(x)dx, giving your answer as single term.

The points A and B have position vectors —4Ai — 34j + 4ak and 3i + 9f — 6k,
respectively relative to the origin O, where A is a scalar. It is given that AB is
perpendicular to OB.
(a) Calculate the
(i)

(ii) angle AOB. {5}

value of A,

(b) Calculate the

(i) coordinates of the centre of the circle passing through O, A
and B,

(ii) _ radius of the circle passing through O, A and B. {4]

il The expansion of (1 + ax)", wheren < 0, has the coefficient of the terms in

x and x? equal to —1 and — 2 respectively.

(a) Show that n satisfies the equation
27n? + 3n—-2=0

(b) (i) Find the value of 7.

(ii) Find the value of a.
ii Hence, state the values of x for v

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n of the 5" and 10" terms of an arithmetic progressi ‘The third term is — 9, (i) __ first term and the common difference,

smallest value of n, for which Sn > 0, where S,, is the sum of

(ii)
the first n terms.

A geometric series has a common ratio r < 0, and its 4" and 6" terms
are 2.56 and 1.6348, respectively.

Find the
(i) first terms and the common ratio,

(ii) ‘the sum to infinity of the series.

14 A circle has centre A(—3, k), where k is a constant, The tangent to the circle
atB(+3,3k), has gradient — 3
(a) _ find the value of k.
(b) Find the
@ equation of the circle in the form
x+y +ax+bytc=0,

values of. x of the points where the circle crosses the
*~ axis, giving your answers correct to 2 decimal places,

area of the minor segment enclosed by the circle and the
x-axis,

—

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