Binomial extended [59 marks]
1.
[Maximum mark: 8]
EXN.2.AHL.TZ0.7
Consider the identity
2+7x
≡
( 1+2x ) ( 1−x )
A
1+2x
+
(a)
Find the value of A and the value of B.
(b)
Hence, expand
2+7x
( 1+2x ) ( 1−x )
B
1−x
, where A,
B ∈ Z
[3]
in ascending powers of x, up to
[4]
and including the term in x .
2
(c)
Give a reason why the series expansion found in part (b) is not
valid for x
2.
=
3
4
.
[1]
[Maximum mark: 6]
(a)
Write down and simplify the first three terms, in ascending
EXM.2.AHL.TZ0.1
1
powers of x, in the Extended Binomial expansion of (1 − x) .
[3]
By substituting x
[3]
3
(b)
.
=
1
9
find a rational approximation to √9.
3
3.
[Maximum mark: 27]
EXM.3.AHL.TZ0.1
This question will investigate power series, as an extension to the Binomial
Theorem for negative and fractional indices.
A power series in x is defined as a function of the form
f (x) = a0 + a1 x + a2 x
2
3
+ a3 x +. . .
where the a
i
∈ R
.
It can be considered as an infinite polynomial.
(a)
Expand (1 + x) using the Binomial Theorem.
[2]
5
This is an example of a power series, but is only a finite power series, since only a
finite number of the a are non-zero.
i
(b)
Consider the power series 1 − x + x
2
− x
3
4
+ x −. . .
By considering the ratio of consecutive terms, explain why this
series is equal to (1 + x)
−1
and state the values of x for which
this equality is true.
(c)
[4]
Differentiate the equation obtained part (b) and hence, find the
first four terms in a power series for (1 + x)
(d)
.
[2]
Repeat this process to find the first four terms in a power series
for (1 + x)
(e)
−2
−3
[2]
.
Hence, by recognising the pattern, deduce the first four terms in
a power series for (1 + x)
−n
,n ∈
Z
+
.
[3]
We will now attempt to generalise further.
Suppose (1 + x)
a0 + a1 x + a2 x
(f )
q
2
, q ∈ Q
can be written as the power series
3
+ a3 x +. . .
By substituting x
= 0
.
, find the value of a .
0
[1]
(g)
By differentiating both sides of the expression and then
substituting x = 0, find the value of a .
[2]
(h)
Repeat this procedure to find a and a .
[4]
(i)
Hence, write down the first four terms in what is called the
Extended Binomial Theorem for (1 + x) , q ∈ Q.
[1]
(j)
Write down the power series for
[2]
(k)
Hence, using integration, find the power series for arctan x,
giving the first four non-zero terms.
1
2
3
q
4.
1
2
1+x
.
[Maximum mark: 5]
[4]
22M.1.AHL.TZ1.6
Consider the expansion of (8x
3
−
n
1
2x
)
where n ∈
Z
+
. Determine all
possible values of n for which the expansion has a non-zero constant
term.
5.
[Maximum mark: 7]
[5]
21N.1.AHL.TZ0.9
Consider the expression
1
− √1 − x
√1+ax
where a ∈
Q, a ≠ 0
.
The binomial expansion of this expression, in ascending powers of x, as far as the
term in x is 4bx + bx , where b ∈
2
2
Q
.
(a)
Find the value of a and the value of b.
[6]
(b)
State the restriction which must be placed on x for this
expansion to be valid.
[1]
6.
[Maximum mark: 6]
18M.2.AHL.TZ2.H_5
(a)
3n + 1
) as a polynomial in
Express the binomial coefficient (
3n − 2
.
[3]
n
(b)
Hence find the least value of n for which (
3n + 1
) > 10
3n − 2
© International Baccalaureate Organization, 2023
6
.
[3]