Vector Proofs and Applications

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Vector Proofs and Applications — Free MYP4 Mathematics (Extended) Practice Questions

1QuestionSolving Complex Vector Path ProblemsConcept Practice

8 marks~12 minCriterion A

The diagram shows quadrilateral

ABCD

on a coordinate grid with vertices

A(1, 2)

B(5, 4)

C(7, 1)

, and

D(3, -1)

Calculate the gradient of side

AB

and the gradient of side

DC

. Hence justify whether

AB

is parallel to

DC

. [3]

Find the exact length of diagonal

AC

, giving your answer in the form

\sqrt{k}

where

k

is an integer. [2]

A student claims that

ABCD

is a parallelogram. Verify this claim by finding the midpoints of both diagonals

AC

and

BD

, and justify your conclusion. [3]

Solutions

2QuestionProving Collinearity Using VectorsAssessment Practice

9 marks~14 minCriterion D

A city planner models the monthly profit

P

(in thousands of dollars) from a new market district using the quadratic function

P(x) = -2x^2 + 20x - 32,

where

x

is the number of stalls (in tens) operating each month, and

1 \leq x \leq 8

.

The table below shows the model's predicted profit and the actual recorded profit for three values of

x

$x$ (tens of stalls)	2	5	7
Predicted $P$ (thousands of dollars)	0	18	6
Actual $P$ (thousands of dollars)	2	15	9

Show that the predicted profit when

x = 5

is 18 thousand dollars, and find the value of

x

for which the model predicts maximum profit, stating the maximum profit in thousands of dollars. [3]

The city planner claims the market is profitable (i.e.

P(x) > 0

) for more than half of the valid domain

1 \leq x \leq 8

. By solving the inequality

P(x) > 0

, determine whether this claim is correct. [3]

The percentage error formula is

\text{percentage error} = \frac{|\,\text{predicted} - \text{actual}\,|}{\text{actual}} \times 100\%.

Calculate the percentage error at each value of

x

in the table. Advise the city planner whether the quadratic model should be used to forecast profit across the domain

1 \leq x \leq 8

, and state one reason why the model should not be used for values of

x

outside this domain. [3]

Solutions

3QuestionMidpoints and Parallel VectorsAssessment Practice

10 marks~15 minCriterion B

The diagram shows four line segments on a coordinate grid.

Segment 1:

A_1(1,\ 2)

and

B_1(5,\ 8)

Segment 2:

A_2(3,\ -1)

and

B_2(7,\ 5)

Segment 3:

A_3(-2,\ 4)

and

B_3(4,\ -2)

Segment 4:

A_4(-5,\ -3)

and

B_4(1,\ 1)

Show your working to find the midpoint

M_1

of Segment 1. Write down the midpoints

M_2

M_3

, and

M_4

. [3]

Describe the pattern connecting the coordinates of

A

B

, and

M

for each segment, and hence state a general rule giving the coordinates of

M

in terms of

A(x_A,\ y_A)

and

B(x_B,\ y_B)

. [3]

A fifth segment has midpoint

M_5(2,\ -1)

and one endpoint

A_5(-3,\ 4)

. Use your rule from part (b) to find the coordinates of

B_5

. Then justify whether

M_5

is a valid midpoint of

A_5B_5

by comparing the distances

A_5M_5

and

M_5B_5

. [4]

Solutions

4QuestionSolving Complex Vector Path ProblemsAssessment Practice

11 marks~17 minCriterion C

The diagram shows three figures in a growing pattern made from small equilateral triangles.

Figure 1 contains 1 small triangle. Figure 2 contains 4 small triangles. Figure 3 contains 9 small triangles.

Find the number of small triangles in Figure 5 and Figure 6, and write down the first six terms of the sequence. [2]

A student claims the number of small triangles

T

in Figure

n

is given by

T = n^2.

Show that this rule is consistent with Figures 1, 2, and 3, and find the value of

n

for which

T = 225

. [3]

The total number of small triangles in Figures 1 through

n

is denoted

S_n

. The values of

S_n

for the first four figures are given below.

$n$	1	2	3	4
$S_n$	1	5	14	30

A second student proposes that

S_n

can be written as

S_n = \frac{n(n+1)(2n+1)}{6}.

Verify this formula for

n = 3

and

n = 4

. [2]

Using the formula

S_n = \dfrac{n(n+1)(2n+1)}{6}

, find the smallest value of

n

such that

S_n > 500

. Justify why no smaller value of

n

satisfies this inequality. [4]

Give all answers in exact form.

Solutions

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