IB Questionbank

User interface language: English | Español

Date	November Example questions	Marks available	3	Reference code	EXN.2.AHL.TZ0.7
Level	Additional Higher Level	Paper	Paper 2	Time zone	Time zone 0
Command term	Find	Question number	7	Adapted from	N/A

Question

Consider the identity $\frac{2 + 7 x}{(1 + 2 x) (1 - x)} \equiv \frac{A}{1 + 2 x} + \frac{B}{1 - x}$ , where $A, B \in ℤ$ .

Find the value of $A$ and the value of $B$ .

[3]

Hence, expand $\frac{2 + 7 x}{(1 + 2 x) (1 - x)}$ in ascending powers of $x$ , up to and including the term in $x^{2}$ .

[4]

Give a reason why the series expansion found in part (b) is not valid for $x = \frac{3}{4}$ .

[1]

Markscheme

* This sample question was produced by experienced DP mathematics senior examiners to aid teachers in preparing for external assessment in the new MAA course. There may be minor differences in formatting compared to formal exam papers.

$2 + 7 x \equiv A (1 - x) + B (1 + 2 x)$

EITHER

substitutes $x = 1$ and attempts to solve for $B$ and substitutes $x = - \frac{1}{2}$ and attempts to solve for $A$ (M1)

$9 = 3 B \Rightarrow B = 3; \frac{3 A}{2} = - \frac{3}{2} \Rightarrow A = - 1$

forms $A + B = 2$ and $- A + 2 B = 7$ and attempts to solve for $A$ and $B$ (M1)

THEN

$A = - 1$ and $B = 3$ A1A1

[3 marks]

uses the binomial expansion on either $3 {(1 - x)}^{- 1}$ or ${(1 + 2 x)}^{- 1}$ M1

$3 {(1 - x)}^{- 1} = 3 (1 + x + x^{2} + \dots)$ A1

${(1 + 2 x)}^{- 1} = (1 - 2 x + \frac{(- 1) (- 2)}{2!} {(2 x)}^{2} + \dots) (= 1 - 2 x + 4 x^{2} + \dots)$ A1

$3 + 3 x + 3 x^{2} - (1 - 2 x + 4 x^{2})$

so the expansion is $2 + 5 x - x^{2}$ (in ascending powers of $x$ ) A1

[4 marks]

${(1 + 2 x)}^{- 1}$ (is convergent) requires $|x| < \frac{1}{2}$ and $x = \frac{3}{4}$ is outside this so the expansion is not valid R1

[1 mark]

Examiners report

[N/A]

Syllabus sections

Topic 1—Number and algebra » AHL 1.10—Perms and combs, binomial with negative and fractional indices

Show 42 related questions

22M.2.AHL.TZ1.9a:
Find the number of ways these five people can be seated in this row.
22M.2.AHL.TZ1.9b:
Find the number of ways these five people can now be seated in this row.
22M.2.AHL.TZ2.9b:
the digits are distinct and are in increasing order.
19M.1.AHL.TZ1.H_11d:
Write down $\overrightarrow {{\text{PA}}}$ and $\overrightarrow {{\text{PB}}}$ .
18N.1.AHL.TZ0.H_2a:
Find the number of different possible teams that could be chosen.
19M.1.AHL.TZ1.H_11a.i:
Find how many sets of four points can be selected which can form the vertices of a quadrilateral.
19M.1.AHL.TZ1.H_11c:
Write down the value of $\lambda$ corresponding to the point ${\text{A}}$ .
19M.1.AHL.TZ1.H_11a.ii:
Find how many sets of three points can be selected which can form the vertices of a triangle.
EXM.3.AHL.TZ0.1i:
Hence, write down the first four terms in what is called the Extended Binomial Theorem for ${\left( {1 + x} \right)^q}{\text{,}}\,\,q \in \mathbb{Q}$ .
EXM.3.AHL.TZ0.1d:
Repeat this process to find the first four terms in a power series for ${\left( {1 + x} \right)^{ - 3}}$ .
EXM.3.AHL.TZ0.1f:
By substituting $x = 0$ , find the value of ${a_0}$ .
EXM.3.AHL.TZ0.1a:
Expand ${\left( {1 + x} \right)^5}$ using the Binomial Theorem.
21M.1.AHL.TZ1.9a:
Each pen is large enough to contain five sheep. Amber and Brownie must not be placed in the same pen.
21M.1.AHL.TZ1.9b:
Each pen may only contain one sheep. Amber and Brownie must not be placed in pens which share a boundary.
17N.1.AHL.TZ0.H_10b.i:
Determine the mean of X.
EXN.2.AHL.TZ0.7c:
Give a reason why the series expansion found in part (b) is not valid for $x = \frac{3}{4}$ .
21M.2.AHL.TZ2.7a:
in the position immediately after Andrea.
21M.2.AHL.TZ2.7b:
in any position after Andrea.
21M.2.AHL.TZ2.9a:
Write down the first three terms of the binomial expansion of $(1 + t)^{- 1}$ in ascending powers of $t$ .
17M.2.AHL.TZ2.H_8:
In a trial examination session a candidate at a school has to take 18 examination papers including the physics paper, the chemistry paper and the biology paper. No two of these three papers may be taken consecutively. There is no restriction on the order in which the other examination papers may be taken.

Find the number of different orders in which these 18 examination papers may be taken.
18M.2.AHL.TZ2.H_5a:
Express the binomial coefficient $\left( \begin{gathered} 3n + 1 \hfill \\ 3n - 2 \hfill \\ \end{gathered} \right)$ as a polynomial in $n$ .
18M.2.AHL.TZ2.H_5b:
Hence find the least value of $n$ for which $\left( \begin{gathered} 3n + 1 \hfill \\ 3n - 2 \hfill \\ \end{gathered} \right) > {10^6}$ .
EXM.2.AHL.TZ0.1a:
Write down and simplify the first three terms, in ascending powers of $x$ , in the Extended Binomial expansion of ${\left( {1 - x} \right)^{\frac{1}{3}}}$ .
19M.1.AHL.TZ1.H_11e:
Let ${\text{C}}$ be the point on ${l_1}$ with coordinates (1, 0, 1) and ${\text{D}}$ be the point on ${l_2}$ with parameter $\mu = - 2$ .

Find the area of the quadrilateral ${\text{CDBA}}$ .
17N.1.AHL.TZ0.H_10a:
Show that the probability that Chloe wins the game is $\frac{3}{8}$ .
EXM.3.AHL.TZ0.1k:
Hence, using integration, find the power series for ${\text{arctan}}\,x$ , giving the first four non-zero terms.
18N.1.AHL.TZ0.H_2b:
Find the number of different possible teams that could be chosen, given that the team must include at least one girl and at least one boy.
EXM.3.AHL.TZ0.1h:
Repeat this procedure to find ${a_2}$ and ${a_3}$ .
21N.1.AHL.TZ0.9a:
Find the value of $a$ and the value of $b$ .
22M.1.AHL.TZ1.6:
Consider the expansion of ${(8 x^{3} - \frac{1}{2 x})}^{n}$ where $n \in ℤ^{+}$ . Determine all possible values of $n$ for which the expansion has a non-zero constant term.
22M.2.AHL.TZ2.9a:
the digits are distinct.
19M.1.AHL.TZ1.H_11b:
Verify that ${\text{P}}$ is the point of intersection of the two lines.
17N.1.AHL.TZ0.H_10b.ii:
Determine the variance of X.
EXM.3.AHL.TZ0.1c:
Differentiate the equation obtained part (b) and hence, find the first four terms in a power series for ${\left( {1 + x} \right)^{ - 2}}$ .
17M.1.AHL.TZ1.H_4:
Three girls and four boys are seated randomly on a straight bench. Find the probability that the girls sit together and the boys sit together.
EXM.3.AHL.TZ0.1e:
Hence, by recognising the pattern, deduce the first four terms in a power series for ${\left( {1 + x} \right)^{ - n}}$ , $n \in {\mathbb{Z}^ + }$ .
EXM.3.AHL.TZ0.1g:
By differentiating both sides of the expression and then substituting $x = 0$ , find the value of ${a_1}$ .
EXM.2.AHL.TZ0.1b:
By substituting $x = \frac{1}{9}$ find a rational approximation to $\sqrt[3]{9}$ .
EXN.2.AHL.TZ0.7b:
Hence, expand $\frac{2 + 7 x}{(1 + 2 x) (1 - x)}$ in ascending powers of $x$ , up to and including the term in $x^{2}$ .
EXM.3.AHL.TZ0.1b:
Consider the power series $1 - x + {x^2} - {x^3} + {x^4} - ...$

By considering the ratio of consecutive terms, explain why this series is equal to ${\left( {1 + x} \right)^{ - 1}}$ and state the values of $x$ for which this equality is true.
EXM.3.AHL.TZ0.1j:
Write down the power series for $\frac{1}{{1 + {x^2}}}$ .
21N.1.AHL.TZ0.9b:
State the restriction which must be placed on $x$ for this expansion to be valid.

Hide related questions

Topic 1—Number and algebra » AHL 1.11—Partial fractions

Topic 1—Number and algebra

Question

Markscheme

Examiners report

Syllabus sections

View options