Algebraic Fractions and Rational Expressions
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Algebraic Fractions and Rational Expressions — Free MYP5 Mathematics (Standard) Practice Questions

1QuestionAdding and Subtracting with Common DenominatorsConcept Practice
4 marks~6 minCriterion A
A water treatment plant monitors chemical concentration using the function

y=2x+1+3x+1y = \frac{2}{x+1} + \frac{3}{x+1}

where yy is the concentration (mg/L) and xx is the time elapsed (hours) after treatment begins, x>0x > 0.
a
Show that 2x+1+3x+1=5x+1\dfrac{2}{x+1} + \dfrac{3}{x+1} = \dfrac{5}{x+1}. [1]
b
Deduce the concentration at x=3x = 3 hours. [1]
c
The plant requires a minimum concentration of 1.5 mg/L to be effective. Advise the plant operator whether the treatment is effective at x=3x = 3 hours, justifying your answer with reference to the threshold. [2]
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Solutions

2QuestionRestrictions and Undefined Values in ExpressionsAssessment Practice
4 marks~6 minCriterion C
A water treatment plant monitors flow rate using the model y=1x−3y = \dfrac{1}{x - 3}, where xx is time in hours after midnight and yy is flow rate in megalitres per hour. The graph of this function is shown.
a
State the value of xx for which 1x−3\dfrac{1}{x-3} is undefined. Explain why the denominator determines this restriction. [2]
b
Identify the feature on the graph that corresponds to this restriction, and explain what it indicates about the behaviour of yy near that value of xx. [1]
c
Advise the plant operators whether this model can be used to determine flow rate at 3:00 am, justifying your answer with reference to the model's behaviour at x=3x = 3. [1]
Question diagram

Solutions

3QuestionMultiplying and Dividing Algebraic FractionsAssessment Practice
6 marks~9 minCriterion D
A biologist models the population of a bacterial colony (in thousands) using

P(t)=5t2+3tt+1P(t) = \frac{5t^2 + 3t}{t + 1}

where tt is time in hours. Recorded populations are:

t=1t = 1: actual 4.04.0 thousand
t=2t = 2: actual 8.28.2 thousand
t=3t = 3: actual 13.513.5 thousand
a
Calculate P(1)P(1), P(2)P(2), and P(3)P(3). [3]
b
Deduce, for each value of tt, whether the model overestimates, underestimates, or matches the actual population, and by how much. [2]
c
Assess whether this model is suitable for predicting the colony's population over time, using the pattern of errors as evidence. [1]
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Solutions

4QuestionAdding and Subtracting with Common DenominatorsAssessment Practice
6 marks~9 minCriterion B
A chemical engineer models the net concentration of a reagent in a reactor using the expression

2x2+3x−5+5x−1x−5−x2+2x−5\frac{2x^2+3}{x-5} + \frac{5x-1}{x-5} - \frac{x^2+2}{x-5}

where xx is temperature in °C and x≠5x \neq 5.
a
Deduce the general rule for adding and subtracting algebraic fractions that share a common denominator QQ. [2]
b
Apply your rule to simplify the expression above, showing all working. [3]
c
The reactor operates safely only when the net concentration expression yields a value greater than 30 for integer temperatures in the range 6≤x≤86 \leq x \leq 8. Justify whether the reactor operates safely across this entire range. [1]

Solutions

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