Special Identities and Common Patterns
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Special Identities and Common Patterns — Free MYP5 Mathematics (Standard) Practice Questions

1QuestionPerfect Square IdentitiesConcept Practice
2 marks~3 minCriterion B
A square tile has side length (x+4)(x + 4) cm. The tile is divided into four rectangular regions with areas x2x^2, 4x4x, 4x4x, and 1616 cm² respectively.

Show that the total area of the tile can be written as (x+4)2=x2+8x+16(x + 4)^2 = x^2 + 8x + 16. [2]
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2QuestionDifference of Squares IdentityAssessment Practice
4 marks~6 minCriterion C
A square tile of side length a=5a = 5 cm has a smaller square of side length bb cm removed from one corner, leaving a shaded L-shaped region. The graph shows the area of this shaded region, AA cm2^2, as a function of bb, for 0≤b≤50 \leq b \leq 5.
a
Write down an expression for AA in terms of bb when a=5a = 5, and state the value of AA when b=3b = 3. [1]
b
Show that a2−b2=(a−b)(a+b)a^2 - b^2 = (a-b)(a+b), and verify this identity using a=5a = 5, b=3b = 3. [2]
c
A manufacturer requires the shaded region to have area at least 9 cm2^2. Using your result from (b), advise the manufacturer on the largest value of bb that meets this requirement, justifying your answer in context. [1]
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3QuestionPerfect Square IdentitiesAssessment Practice
4 marks~6 minCriterion D
A landscape architect models the cross-section of a decorative water channel using the function y=x2+6x+9y = x^2 + 6x + 9, where xx is the horizontal distance (metres) from a reference point and yy is the height (metres) above ground level.
a
Show that x2+6x+9=(x+3)2x^2 + 6x + 9 = (x + 3)^2. [1]
b
Hence deduce the coordinates of the xx-intercept of the graph. [1]
c
The channel is designed so that water flows only where y=0y = 0. Justify whether this design is suitable for carrying water along the channel. [2]
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4QuestionRecognizing and Using Identities in ProblemsAssessment Practice
4 marks~6 minCriterion A
An architect models the curved profile of a pedestrian bridge using a parabola. The lowest point of the arch is at coordinates (3,−2)(3, -2) relative to a reference point, and the arch passes through (4,−1)(4, -1). The yy-intercept of the model represents the height of the bridge support at the reference edge.
a
Deduce the equation of the parabola in vertex form y=a(x−h)2+ky = a(x - h)^2 + k, showing how the value of aa is determined. [2]
b
Show that the standard form of the equation is y=x2−6x+7y = x^2 - 6x + 7, applying the perfect square identity in your expansion. [1]
c
Interpret the yy-intercept of the model, and justify whether this value confirms the arch is structurally above the reference level at the support edge. [1]
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