Algebraic Fractions and Rational Expressions
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Algebraic Fractions and Rational Expressions — Free MYP4 Mathematics (Standard) Practice Questions

1QuestionAlgebraic Fraction Word ProblemsConcept Practice
4 marks~6 minCriterion A
A water treatment plant monitors chemical concentration using the rational function

C(t)=2t+4t+1C(t) = \frac{2t + 4}{t + 1}

where CC is the concentration in milligrams per litre and tt is the time in hours after treatment begins. The safe upper limit for the chemical is 2.6 mg/L2.6 \text{ mg/L}.
a
Calculate C(3)C(3), simplifying your answer fully. [2]
b
Advise the plant operator whether the concentration at t=3t = 3 hours requires immediate action, justifying your answer with reference to the safe limit. [2]
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Solutions

2QuestionRestrictions and Undefined Values in ExpressionsAssessment Practice
4 marks~6 minCriterion D
A car engine's fuel efficiency is modelled by the rational expression

E(v)=120vv2+v−12E(v) = \frac{120v}{v^2 + v - 12}

where EE is efficiency in km/L and vv is speed in km/h.
a
Show that the expression is undefined at v=3v = 3 km/h and find any other value of vv for which it is undefined. [2]
b
Discuss the strengths and limitations of this model in the context of engine calibration, referring to the undefined value found in part (a) and at least one other feature of the model. [2]
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Solutions

3QuestionMultiplying and Dividing Algebraic FractionsAssessment Practice
4 marks~6 minCriterion C
A two-stage water filtration system removes contaminants at each stage. The fraction of contaminants removed by stage one is xx+2\dfrac{x}{x+2} and by stage two is x+2x+4\dfrac{x+2}{x+4}, where x>0x > 0 represents the initial contaminant level. The combined removal fraction is the product of the two stage fractions.
a
Deduce a simplified expression for the combined removal fraction after both stages. [1]
b
The initial contaminant level is x=4x = 4. Calculate the combined removal fraction and interpret what this value means for the filtration system. [2]
c
A filtration engineer claims that this two-stage system is sufficient to meet a safety standard requiring a combined removal fraction greater than 34\dfrac{3}{4} for all x>0x > 0. Justify whether the engineer's claim is correct. [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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