Blathnaid's copy of Partial Fractions: Simple Linear Factors

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Blathnaid Sheridan 6 years, 4 months ago

Created this as a copy of Partial Fractions: Simple Linear Factors.

Name	Status	Author	Last Modified
Partial Fractions	draft	Bill Foster	06/08/2020 08:42
Partial Fractions 1	Ready to use	joshua boddy	01/06/2016 09:47
Partial Fractions 2	Ready to use	joshua boddy	01/06/2016 09:47
Partial Fractions 3 - double root	Ready to use	joshua boddy	01/06/2016 09:47
Integration with Partial Fractions 2	Ready to use	joshua boddy	01/06/2016 09:47
Integration by Partial Fractions 1 - double root	Has some problems	joshua boddy	23/02/2021 13:55
Partial Fractions: Simple Linear Factors	Ready to use	Michael Proudman	07/12/2021 10:59
Partial Fractions: Repeated Factor	Ready to use	Michael Proudman	07/12/2021 10:59
CF Maths In class test three mick paper question 7	Ready to use	Jinhua Mathias	28/08/2024 12:15
CF Maths January test mock paper Partial Fractions 3 - double root	Ready to use	Jinhua Mathias	28/08/2024 12:15
CF Maths Januaray test mock paper Partial Fractions 1	Ready to use	Jinhua Mathias	28/08/2024 12:15
Partial Fractions: Simple Linear Factors [L5]	draft	Abbi Mullins	25/07/2018 14:02
cormac's copy of CF Maths January test mock paper Partial Fractions 3 - double root	draft	cormac breen	30/10/2018 15:53
Blathnaid's copy of Partial Fractions: Simple Linear Factors	draft	Blathnaid Sheridan	13/11/2018 10:42
Blathnaid's copy of Michael's copy of Partial Fractions 3 - double root	draft	Blathnaid Sheridan	13/11/2018 10:43
Laura's copy of Partial Fractions: Repeated Factor	draft	Laura Horton	12/02/2019 16:22

There are 43 other versions that do you not have access to.

Question statement

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	8
c	integer	-7
b	integer	-8
d	integer	1
nb	string	taking away
a1	integer	1
a2	integer	1
s1	integer	-1
a_	integer	2
c_	integer	3
b_	integer	-6
d_	integer	1
nb_	string	adding
a1_	integer	1
a2_	integer	1
s1_	integer	1

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random(1..9)

JME function reference

Description

Describe what this variable represents, and list any assumptions made about its value.

Can an exam override the value of this variable?

Generated value: `integer`

→ Used by:

Advice
"Unnamed part" - prompt
"Unnamed part" → "Unnamed gap" - Correct answer

Parts

Gap-fill
1. Gap Mathematical expression
  Add

Gap-fill

Ask the student a question, and give any hints about how they should answer this part.

Split

$\simplify{({a*a2 + c*a1} * x + {a * d + c * b})/ (({a1}*x + {b}) * ({a2}*x + {d}))}$ into partial fractions.

Input the partial fractions here: Unnamed gap.

Advice

We use partial fractions to find $A$ and $B$ such that:

$\simplify[std]{({a*a2+c*a1}*x+{c*b+a*d})/(({a1}x +{b})*({a2}x+{d}))} \;\;\;=\simplify[std]{ A/({a1}x+{b})+B/({a2}x+{d})}$

Dividing both sides of the equation by $\displaystyle \simplify[std]{1/( ({a1}x+{b})({a2}x+{d}) )}\;\;$ we obtain:

$\simplify[std]{A*({a2}x+{d})+B*({a1}x+{b}) = {a*a2+c*a1}*x+{a*d+c*b}} \Rightarrow \simplify[std]{({a2}A+{a1}B)*x+{d}*A+{b}*B={a*a2+c*a1}*x+{a*d+c*b}}$

Identifying coefficients:

Constant term: $\simplify[std]{ {d}*A+{b}*B={a*d+c*b} }$

Coefficent $x$ : $\simplify[std]{ {a2}A+{a1}B = {a*a2+c*a1} }$

On solving these equations we obtain $A = \var{a}$ and $B=\var{c}$

Which gives:

$\simplify[std]{({a*a2+c*a1}*x+{c*b+a*d})/(({a1}x +{b})*({a2}x+{d}))}\;\;= \simplify[std]{{a}/({a1}x+{b})+{c}/({a2}x+{d})}$

We use partial fractions to find $A$ and $B$ such that:

$\simplify[std]{({a_*a2_+c_*a1_}*x+{c_*b_+a_*d_})/(({a1_}x +{b_})*({a2_}x+{d_}))} \;\;\;=\simplify[std]{ A/({a1_}x+{b_})+B/({a2_}x+{d_})}$

Dividing both sides of the equation by $\displaystyle \simplify[std]{1/( ({a1_}x+{b_})({a2_}x+{d_}) )}\;\;$ we obtain:

$\simplify[std]{A*({a2_}x+{d_})+B*({a1_}x+{b_}) = {a_*a2_+c_*a1_}*x+{a_*d_+c_*b_}} \Rightarrow \simplify[std]{({a2_}A+{a1_}B)*x+{d_}*A+{b_}*B={a_*a2_+c_*a1_}*x+{a_*d_+c_*b_}}$

Identifying coefficients:

Constant term: $\simplify[std]{ {d_}*A+{b_}*B={a_*d_+c_*b_} }$

Coefficent $x$ : $\simplify[std]{ {a2_}A+{a1_}B = {a_*a2_+c_*a1_} }$

On solving these equations we obtain $A = \var{a_}$ and $B=\var{c_}$

Which gives:

$\simplify[std]{({a_*a2_+c_*a1_}*x+{c_*b_+a_*d_})/(({a1_}x +{b_})*({a2_}x+{d_}))}\;\;= \simplify[std]{{a_}/({a1_}x+{b_})+{c_}/({a2_}x+{d_})}$

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This question is used in the following exam:

MATH1601 CA2 Semester 1 by Blathnaid Sheridan in Blathnaid's workspace.

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Blathnaid's copy of Partial Fractions: Simple Linear Factors

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Taxonomy: mathcentre

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Blathnaid Sheridan 6 years, 4 months ago

Error in variable testing condition

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Generated value: `integer`

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Gap-fill

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Blathnaid's copy of Partial Fractions: Simple Linear Factors

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Blathnaid Sheridan 6 years, 4 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`