Vector Representation and Notation
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Vector Representation and Notation — Free MYP4 Mathematics (Extended) Practice Questions

1QuestionUnit Vectors and Position Vectors IntroductoryAssessment Practice
6 marks~9 minCriterion D
A right-angled triangle has legs of length aa cm and bb cm, and hypotenuse of length cc cm, where aa, bb, cc are positive integers satisfying a2+b2=c2a^2 + b^2 = c^2.

The triangle below has legs a=5a = 5 cm and b=12b = 12 cm.
a
Write down the value of cc. [1]
b
A second right-angled triangle is similar to the first, with hypotenuse c′=39c' = 39 cm. Find the lengths of its two legs. [2]
c
Show that the perimeter of the second triangle is 33 times the perimeter of the first triangle, and hence justify whether the second triangle could be used to represent the first triangle at a scale of 1:31:3 on an architectural plan. [3]
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Solutions

2QuestionNotation and Labeling for Displacement VectorsAssessment Practice
8 marks~12 minCriterion B
The diagram shows four figures in a growing pattern made from small equilateral triangles.

Figure 1 has 1 shaded triangle. Figure 2 has 4. Figure 3 has 9. Figure 4 has 16. In each figure, some triangles point upward (△) and some point downward (▽).

Figure nn1234
UU13610
DD0136
TT14916
a
Write down the number of upward-pointing triangles UU and downward-pointing triangles DD in Figure 5. [2]
b
Find a rule for the total number of triangles TT in Figure nn. Hence find the figure number in which the total number of triangles first exceeds 200. [3]
c
The number of downward-pointing triangles in Figure nn is given by
D(n)=(n−1)n2.D(n) = \frac{(n-1)n}{2}.
Verify this formula for n=3n = 3, and justify why D(n)D(n) must always be a whole number for any positive integer nn, by referring to the structure of the expression. [3]
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Solutions

3QuestionUnit Vectors and Position Vectors IntroductoryAssessment Practice
8 marks~12 minCriterion A
The diagram shows the graph of f(x)=x2+kf(x) = \sqrt{x^2 + k} for a positive constant kk, passing through the point A(4,5)A(4, 5).
a
Show that k=9k = 9. [2]
b
Find the coordinates of the two points where f(x)=34f(x) = \sqrt{34}. [3]
c
A second function is defined by g(x)=f(x−3)+1g(x) = f(x - 3) + 1. Determine the coordinates of the image of point AA under this transformation, and hence state the equation of g(x)g(x) in the form g(x)=x2+bx+c+1g(x) = \sqrt{x^2 + bx + c} + 1, identifying the values of bb and cc. [3]
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Solutions

4QuestionNotation and Labeling for Displacement VectorsAssessment Practice
8 marks~12 minCriterion C
The diagram shows triangle PQRPQR on a coordinate grid with vertices P(1,1)P(1, 1), Q(7,1)Q(7, 1), and R(7,9)R(7, 9).
a
Calculate the length of PRPR, giving your answer in exact form. [2]
b
Find the bearing of RR from PP, giving your answer to the nearest degree. [3]
c
A point SS lies on PRPR such that QSQS is perpendicular to PRPR. The length QSQS represents the shortest distance from vertex QQ to the opposite side of the triangle. Justify whether QSQS is sufficient to confirm that QQ lies closer to PRPR than to either PQPQ or QRQR. [3]
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