2
Mathematical Formulae
1. ALGEBRA
Quadratic Equation
For the equation ax2 + bx + c = 0,
x=
- b ! b 2 - 4ac
.
2a
Binomial Theorem
()
()
()
()
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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6
1.4
You must not use a calculator in Question 4.
For
Examiner’s
Use
In the triangle ABC, angle B = 90°, AB = 4 + 2 2 and BC = 1 + 2 .
(i) Find tan C, giving your answer in the form k 2 .
[2]
(ii) Find the area of the triangle ABC, giving your answer in the form p + q 2, where p and q
are integers.
[2]
(iii) Find the area of the square whose side is of length AC, giving your answer in the form
s + t 2, where s and t are integers.
[2]
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0606/12/M/J/11
11
2.9
The figure shows a circle, centre O, radius r cm. The length of the arc AB of the circle is 9π cm.
Angle AOB is θ radians and is 3 times angle OBA.
O
r cm
For
Examiner’s
Use
A
θ
rad
9π cm
B
3π
(i) Show that θ = 5 .
[2]
(ii) Find the value of r.
[2]
(iii) Find the area of the shaded region.
[3]
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0606/12/M/J/11
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7
3.5
A
r cm
O θ rad
B
The diagram shows a circle with centre O and radius r cm. The minor arc AB is such that angle AOB is
i radians. The area of the minor sector AOB is 48 cm2.
96
.
r2
[2]
(i)
Show that i =
(ii)
Given that the minor arc AB has length 12 cm, find the value of r and of i.
[3]
(iii)
Using your values of r and i, find the area of the shaded region.
[2]
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14
11
4.
P
Q
r cm
i rad
O
The diagram shows the sector OPQ of a circle, centre O, radius r cm, where angle POQ = i radians. The
perimeter of the sector is 10 cm.
(i)
Show that area, A cm2, of the sector is given by
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A=
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50i
.
(2 + i) 2
[5]
15
It is given that i can vary and A has a maximum value.
(ii)
Find the maximum value of A.
[5]
Question 12 is printed on the next page.
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