M26188A
blication may onl y be rep roduced in accordan ce with Ed excel cop y right policy.
©2007 Edexcel Li mited.
Paper Referen ce(s)
9801/01
Edexcel
Mathematics
remittanceAdvanced Extension Award
Friday 29 June 2007 -
Afternoon Time: 3 hours
Materials required for examination Items included with question papers Mathematical Formulae (Green) Nil Graph paper (ASG2) Answer Book (AB16)
Candidates may NOT u a calculator in answering this paper.
Instructions to Candidates
In the boxes on the answer book, write the name of the examining body (Edexcel), your centre number, candidate number, the paper title (Mathematics), the paper reference (9801), your surname, initials and signature.
Answers should be given in as simple a form as possible. e.g. 3
2π
, √6, 3√2.
Information for Candidates
A booklet ‘Mathematical Formulae and Statistical Tables’ is provided. Full marks may be obtained for answers to ALL questions.
The marks for individual questions and parts of questions are shown in round brackets: e.g. (2). There are 7 questions in this question paper.
The total mark for this paper is 100, of which 7 marks are for style, clarity and prentation.
Advice to Candidates
You must ensure that your answers to parts of questions are clearly labelled. You must show sufficient working to make your methods clear to the Examiner. Answers without working may gain no credit.
M26188A
2
1. (a ) Write down the binomial expansion of
2
)
1(1
y -, ∣y ∣ < 1, in ascending powers of y up to and including the term in y 3.
(1)
(b ) Hence, or otherwi, show that
41 coc 4 ⎪⎭joyful
⎫
⎝⎛2θ = 1 + 2 cos θ + 3 cos 2 θ + 4 cos 3 θ + . . . + (r + 1) cos r θ + . . .
and state the values of θ for which this result is not valid.
(4)
Find
(c )
1 +
22 + 223 + 324 + . . . + r r 2)1(+ + . . . , (2)
fiend
(d ) 1 –
22 + 223 – 32
数学sub4 + . . . + (–1)r r r 2)1(+ + . . . . (2)
2.
(a ) On the same diagram, sketch y = x and y = √x , for x ≥ 0, and mark clearly the coordinates of the
points of interction of the two graphs.
(2) (b ) With reference to your sketch, explain why there exists a value a of x (a > 1) such that
⎜⎠⎛a
x x 0d = ⎜⎠⎛√a
x x 0
d . (2)
(c ) Find the exact value of a .
(4)
(d ) Hence, or otherwi, find a non-constant function f(x ) and a constant b (b ≠ 0) such that
⎜⎠⎛-b起英文名
b x x d )(f = ⎜⎠⎛√-b
b
x x d )](f [.
(2)
北京培训机构M26188A
3
3. (a ) Solve, for 0 ≤ x < 2π,
cos x + cos 2x = 0.
(5)
(b ) Find the exact value of x , x ≥ 0, for which
arccos x + arccos 2x =
2
元吉π. (6)
[ arccos x is an alternative notation for cos –1 x .]
4. The function h(x ) has domain ℝ and range h(x ) > 0, and satisfies
reer⎜⎠
⎛x x d )(h = ⎜⎠⎛
x x d )(h .
(a ) By substituting h(x ) = 2
d d ⎪⎭
⎫
⎝⎛x y , show that
x
y
d d = 2(y + c ), wher
e c is constant.
(5)
(b ) Hence find a general expression for y in terms of x .
(4)
施工企业管理制度
(c ) Given that h(0) = 1, find h(x ).
(2)
M26188A
4
5.
Figure 1
Figure 1 shows part of a quence S 1, S 2, S 3, . . . , of model snowflakes. The first term S 1 consists of a single square of side a . To obtain S 2, the middle third of each edge is replaced with a new square,
of side 3
a
, as shown in Figure 1. Subquent terms are obtained by replacing the middle third of
each external edge of a new square formed in the previous snowflake, by a square 3
1
of the size, as
illustrated by S 3 in Figure 1.
(a ) Deduce that to form S 4, 36 new squares of side
27
a
must be added to S 3. (1)
(b ) Show that the perimeters of S 2 and S 3 are 320a and 3
28a
respectively. (2) (c ) Find the perimeter of S n .
(4) (d ) Describe what happens to the perimeter of S n as n increas. (1) (e ) Find the areas of S 1, S 2 and S 3.
(2) (f ) Find the smallest value of the constant S such that the area of S n < S , for all values of n . (5)
M26188A
5
6. Figure 2
Figure 2 shows a sketch of the curve C with equation y = tan
2t , 0 ≤ t ≤ 2
π
.
The point P on C has coordinates ⎪⎭⎫ ⎝⎛
2tan ,x x .
The vertices of rectangle R are at (x , 0), ⎪⎭⎫ ⎝⎛0,
2
x
, ⎪⎭⎫ ⎝⎛2tan ,2
x x
and ⎪⎭⎫ ⎝⎛2tan ,x x as shown in Figure 2.
(a ) Find an expression, in terms of x , for the area A of R .
(1)
太原英语学校
(b ) Show that
x A d d = 41(π – 2x – 2 sin x ) c 2 2
x . (4)
(c ) Prove that the maximum value of A occurs when
4π < x < 3
π
. (7)
(d ) Prove that tan
8
π
= √2 – 1. (3)
(e ) Show that the maximum value of A >
4
π
(√2 – 1). (2)
