Cambridge International Examinations

CambridgeInternationalGeneralCertificateofSecondaryEducation
* 7 0 2 4 7 0 9 2 3 8 *

ADDITIONAL MATHEMATICS 0606/22

Paper2 May/June 2014
 2 hours
CandidatesanswerontheQuestionPaper.
AdditionalMaterials: Electroniccalculator

READ THESE INSTRUCTIONS FIRST

WriteyourCentrenumber,candidatenumberandnameonalltheworkyouhandin.
Writeindarkblueorblackpen.
YoumayuseanHBpencilforanydiagramsorgraphs.
Donotusestaples,paperclips,glueorcorrectionfluid.
DONOTWRITEINANYBARCODES.

Answerallthequestions.
Givenon-exactnumericalanswerscorrectto3significantfigures,or1decimalplaceinthecaseof
anglesindegrees,unlessadifferentlevelofaccuracyisspecifiedinthequestion.
Theuseofanelectroniccalculatorisexpected,whereappropriate.
Youareremindedoftheneedforclearpresentationinyouranswers.

Attheendoftheexamination,fastenallyourworksecurelytogether.
Thenumberofmarksisgiveninbrackets[ ]attheendofeachquestionorpartquestion.
Thetotalnumberofmarksforthispaperis80.

Thisdocumentconsistsof14printedpagesand2blankpages.

DC(NF/SW)73391/3
©UCLES2014 [Turn over
 2

Mathematical Formulae

1. ALGEBRA

Quadratic Equation

For the equation ax2 + bx + c = 0,

−b b 2 − 4 ac
x=
2a

Binomial Theorem

()
n n
() n
()
(a + b)n = an + 1 an–1 b + 2 an–2 b2 + … + r an–r br + … + bn,

()
n
where n is a positive integer and r =
n!
(n – r)!r!

2. TRIGONOMETRY

Identities

sin2 A + cos2 A = 1
sec2 A = 1 + tan2 A
cosec2 A = 1 + cot2 A

Formulae for ∆ABC

a b c
sin A = sin B = sin C

a2 = b2 + c2 – 2bc cos A
1
∆= bc sin A
2

© UCLES 2014 0606/22/M/J/14

 3

^2 + 5h2
1 Without using a calculator, express in the form a + b 5 , where a and b are
5-1
constants to be found. [4]

2 Find the values of k for which the line y + kx - 2 = 0 is a tangent to the curve y = 2x 2 - 9x + 4 .
[5]

© UCLES 2014 0606/22/M/J/14 [Turn over

 4

3 (i) Given that x + 1 is a factor of 3x 3 - 14x 2 - 7x + d , show that d = 10. [1]

(ii) Show that 3x 3 - 14x 2 - 7x + 10 can be written in the form ^x + 1h^ax 2 + bx + ch, where a, b
and c are constants to be found. [2]

(iii) Hence solve the equation 3x 3 - 14x 2 - 7x + 10 = 0 . [2]

© UCLES 2014 0606/22/M/J/14

 5

4 (i) Express 12x 2 - 6x + 5 in the form p ^x - qh2 + r , where p, q and r are constants to be found.
[3]

1
(ii) Hence find the greatest value of and state the value of x at which this
12x - 6x + 5
2

occurs. [2]

© UCLES 2014 0606/22/M/J/14 [Turn over

 6

5 (i) Find and simplify the first three terms of the expansion, in ascending powers of x, of ^1 - 4xh5 .
[2]

(ii) The first three terms in the expansion of ^1 - 4xh5 ^1 + ax + bx 2h are 1 - 23x + 222x 2 . Find the
value of each of the constants a and b. [4]

© UCLES 2014 0606/22/M/J/14

 7

6 (a) (i) State the value of u for which lg u = 0 . [1]

(ii) Hence solve lg 2x + 3 = 0 . [2]

(b) Express 2 log 3 15 - ^loga 5h^log 3 ah, where a > 1, as a single logarithm to base 3. [4]

© UCLES 2014 0606/22/M/J/14 [Turn over

 8

7 Differentiate with respect to x

(i) x 4 e 3x , [2]

(ii) ln ^2 + cos xh, [2]

sin x
(iii) . [3]
1+ x

© UCLES 2014 0606/22/M/J/14

 9

8 The line y = x - 5 meets the curve x 2 + y 2 + 2x - 35 = 0 at the points A and B. Find the
exact length of AB. [6]

© UCLES 2014 0606/22/M/J/14 [Turn over

 10

dy 1

9 A curve is such that = ^2x + 1h2 . The curve passes through the point (4, 10).
dx
(i) Find the equation of the curve. [4]

y ydx and hence evaluate y0

1.5
(ii) Find ydx . [5]

© UCLES 2014 0606/22/M/J/14

 11

10 Two variables x and y are connected by the relationship y = Ab x , where A and b are constants.

(i) Transform the relationship y = Ab x into a straight line form. [2]

An experiment was carried out measuring values of y for certain values of x. The values of ln y and x
were plotted and a line of best fit was drawn. The graph is shown on the grid below.

ln y

0 1 2 3 4 5 6 x
–1

(ii) Use the graph to determine the value of A and the value of b, giving each to 1 significant figure.
[4]

(iii) Find x when y = 220. [2]

© UCLES 2014 0606/22/M/J/14 [Turn over

 12

11 The functions f and g are defined, for real values of x greater than 2, by

f (x) = 2 x - 1,
g (x) = x ^x + 1h.

(i) State the range of f. [1]

(ii) Find an expression for f -1 (x) , stating its domain and range. [4]

© UCLES 2014 0606/22/M/J/14

 13

(iii) Find an expression for gf (x) and explain why the equation gf (x) = 0 has no solutions. [4]

© UCLES 2014 0606/22/M/J/14 [Turn over

 14

12 A curve has equation y = x 3 - 9x 2 + 24x .

dy
(i) Find the set of values of x for which H 0. [4]
dx

The normal to the curve at the point on the curve where x = 3 cuts the y-axis at the point P.

(ii) Find the equation of the normal and the coordinates of P. [5]

© UCLES 2014 0606/22/M/J/14

 15

BLANK PAGE

© UCLES 2014 0606/22/M/J/14

 16

BLANK PAGE

Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every
reasonableefforthasbeenmadebythepublisher(UCLES)totracecopyrightholders,butifanyitemsrequiringclearancehaveunwittinglybeenincluded,the
publisherwillbepleasedtomakeamendsattheearliestpossibleopportunity.

CambridgeInternationalExaminationsispartoftheCambridgeAssessmentGroup.CambridgeAssessmentisthebrandnameofUniversityofCambridgeLocal
ExaminationsSyndicate(UCLES),whichisitselfadepartmentoftheUniversityofCambridge.

© UCLES 2014 0606/22/M/J/14