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Cambridge International Examinations
CambridgeInternationalGeneralCertificateofSecondaryEducation
* 7 0 2 4 7 0 9 2 3 8 *
0606/22
ADDITIONAL MATHEMATICS
Paper2
May/June 2014
2 hours
CandidatesanswerontheQuestionPaper.
AdditionalMaterials:
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2
Mathematical Formulae
1. ALGEBRA
Quadratic Equation
For the equation ax2 + bx + c = 0,
x=
Binomial Theorem
()
()
−b
b 2 − 4 ac
2a
()
()
n
n
n
(a + b)n = an + 1 an–1 b + 2 an–2 b2 + … + r an–r br + … + bn,
n
n!
where n is a positive integer and r =
(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
∆=
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1
bc sin A
2
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3
1 Without using a calculator, express
constants to be found.
2 Find the values of k for which the line
© UCLES 2014
^2 + 5h2
5-1
in the form a + b 5 , where a and b are
y + kx - 2 = 0
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[4]
is a tangent to the curve y = 2x 2 - 9x + 4 .
[5]
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4
3 (i) Given that x + 1 is a factor of
(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
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3x 3 - 14x 2 - 7x + d , show that d = 10.
3x 3 - 14x 2 - 7x + 10 = 0 .
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[1]
[2]
5
in the form p ^x - qh2 + r , where p, q and r are constants to be found.
[3]
4 (i) Express
(ii) Hence find the greatest value of
occurs.
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12x 2 - 6x + 5
1
and state the value of x at which this
12x - 6x + 5
2
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[2]
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6
5 (i) Find and simplify the first three terms of the expansion, in ascending powers of x, of ^1 - 4xh5 .
(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]
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[2]
7
6 (a) (i) State the value of u for which lg u = 0 .
[1]
[2]
(b) Express 2 log 3 15 - ^loga 5h^log 3 ah,
(ii) Hence solve lg 2x + 3 = 0 .
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where a > 1, as a single logarithm to base 3.
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[4]
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8
7 Differentiate with respect to x
(i) x 4 e 3x ,
[2]
(ii) ln ^2 + cos xh,
[2]
(iii) sin x
.
1+ x
[3]
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0606/22/M/J/14
9
8 The line y = x - 5
exact length of AB.
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meets the curve
x 2 + y 2 + 2x - 35 = 0
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at the points A and B. Find the
[6]
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10
1
dy
= ^2x + 1h2 . The curve passes through the point (4, 10).
dx
9 A curve is such that
(i) Find the equation of the curve.
(ii) Find
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[4]
y ydx and hence evaluate y0
1.5
ydx .
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[5]
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.
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.
[2]
ln y
6
5
4
3
2
1
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.
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[2]
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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]
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13
(iii) Find an expression for gf (x) and explain why the equation gf (x) = 0 has no solutions.
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[4]
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14
12 A curve has equation y = x 3 - 9x 2 + 24x .
dy
H 0.
dx
(i) Find the set of values of x for which
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.
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[4]
[5]
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© UCLES 2014
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0606/22/M/J/14