Substitution and Evaluation of Expressions

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Substitution and Evaluation of Expressions — Free MYP4 Mathematics (Extended) Practice Questions

1QuestionMultiple Substitutions in Complex ExpressionsConcept Practice

4 marks~6 minCriterion A

The graph shows a parabola modelled by

y = ax^2 + bx + c

, opening upwards, with vertex at

(0, -1)

. The parabola passes through

(-2, 3)

(0, -1)

, and

(2, 3)

State the value of

c

by identifying the

y

-intercept from the graph. [1]

Substitute the points

(-2, 3)

and

(2, 3)

into

y = ax^2 + bx + c

to construct two equations in

a

and

b

. [1]

Deduce the values of

a

and

b

by solving the system of equations from part (b). [1]

The parabola models the cross-sectional profile of a satellite dish. The dish is functional only if its profile is symmetric about the central axis. Justify whether this dish meets the symmetry requirement, using the values of

a

b

, and

c

. [1]

Solutions

2QuestionEvaluating with Negative Numbers and FractionsAssessment Practice

4 marks~6 minCriterion D

A meteorologist uses the following formula to calculate wind chill temperature

T_{wc}

(in °C), based on air temperature

T_a

(in °C) and wind speed

v

(in km/h):

T_{wc} = 13.12 + 0.6215T_a - 11.37v^{0.16} + 0.3965T_av^{0.16}

Calculate

T_{wc}

when

T_a = -15

°C and

v = 30

km/h. [1]

Explain why

T_{wc}

is lower than

T_a

when wind speed increases, referring to the structure of the formula. [1]

A public safety advisory warns of dangerous conditions whenever

T_{wc} < -35

°C. A forecast gives

T_a = -25

°C and

v = 50

km/h. Calculate

T_{wc}

for these conditions, then advise whether the advisory should be issued, justifying your decision with reference to the threshold and the risk to public safety. [2]

Solutions

3QuestionEvaluating with Negative Numbers and FractionsAssessment Practice

6 marks~9 minCriterion B

A signal-processing engineer models the attenuation (reduction in strength) of a radio signal at stage

n

of a transmission network using the expression

\left(n + \dfrac{1}{n}\right)^{-1}

.

The first five values are:

\left(1 + \frac{1}{1}\right)^{-1} = \frac{1}{2}, \quad \left(2 + \frac{1}{2}\right)^{-1} = \frac{2}{5}, \quad \left(3 + \frac{1}{3}\right)^{-1} = \frac{3}{10}, \quad \left(4 + \frac{1}{4}\right)^{-1} = \frac{4}{17}, \quad \left(5 + \frac{1}{5}\right)^{-1} = \frac{5}{26}

Deduce a general formula for the attenuation at stage

n

, expressing your answer in the form

\dfrac{n}{f(n)}

. [2]

Calculate the attenuation at stage

10

using your formula. [2]

The engineer claims that the attenuation will never fall below

0.09

for any stage in this network (

n \leq 12

). Justify whether this claim is correct. [2]

Solutions

4QuestionMultiple Substitutions in Complex ExpressionsAssessment Practice

12 marks~18 minCriterion C

A structural engineer uses the following values to model load-bearing stress in a beam:

a = 2, \quad b = -3, \quad c = \tfrac{1}{2}

Evaluate

2a^2 - 3b + 4c

. [2]

Evaluate

\dfrac{a^3 + b^2}{c}

. [2]

The two expressions give different results. Justify which expression produces the greater stress value and explain how the structure of each expression — including the effect of negative values and division by a fraction — leads to that outcome. [8]

Solutions

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