Question 1

Use the figure below. Determine whether the given statement is true or false.  
-G + H = F

Accepted Answer

A)  True 
B)  False 
A)  True 
B)  False

Question 2

Find the distance from P1 to P2.
-$$\mathrm { P } _ { 1 } = ( 2,2 , - 1 ) \text { and } \mathrm { P } _ { 2 } = ( - 1 , - 3,2 )$$

Accepted Answer

A)  5 
B)  $\sqrt { 43 }$ 
C)  $\sqrt { 35 }$ 
D)  $3 \sqrt { 2 }$ 
A)  5 
B)  $\sqrt { 43 }$ 
C)  $\sqrt { 35 }$ 
D)  $3 \sqrt { 2 }$

Question 3

Choose the one alternative that best completes the statement or answers the question.
The polar coordinates of a point are given. Find the rectangular coordinates of the point.
-$\left( - 5 , \frac { 3 \pi } { 4 } \right)$

Accepted Answer

A)  $\left( \frac { - 5 \sqrt { 2 } } { 2 } , \frac { - 5 \sqrt { 2 } } { 2 } \right)$ 
B)  $\left( \frac { - 5 \sqrt { 2 } } { 2 } , \frac { 5 \sqrt { 2 } } { 2 } \right)$ 
C)  $\left( \frac { 5 \sqrt { 2 } } { 2 } , \frac { - 5 \sqrt { 2 } } { 2 } \right)$ 
D)  $\left( \frac { 5 \sqrt { 2 } } { 2 } , \frac { 5 \sqrt { 2 } } { 2 } \right)$ 
A)  $\left( \frac { - 5 \sqrt { 2 } } { 2 } , \frac { - 5 \sqrt { 2 } } { 2 } \right)$ 
B)  $\left( \frac { - 5 \sqrt { 2 } } { 2 } , \frac { 5 \sqrt { 2 } } { 2 } \right)$ 
C)  $\left( \frac { 5 \sqrt { 2 } } { 2 } , \frac { - 5 \sqrt { 2 } } { 2 } \right)$ 
D)  $\left( \frac { 5 \sqrt { 2 } } { 2 } , \frac { 5 \sqrt { 2 } } { 2 } \right)$

Question 4

Use the given vectors to find the indicated expression.
-$$\mathbf { v } = - 3 \mathbf { i } + 5 \mathbf { j } - 4 \mathbf { k } , \quad \mathbf { w } = 2 \mathbf { i } + 4 \mathbf { j } - 2 \mathbf { k } , \quad \mathbf { u } = - 4 \mathbf { i } - 4 \mathbf { j } - 3 \mathbf { k }$$
Find $\mathbf { v } \cdot ( \mathbf { v } \times \mathbf { w } )$.

Accepted Answer

A)  90 
B)  0 
C)  $- 90$ 
D)  130 
A)  90 
B)  0 
C)  $- 90$ 
D)  130

Question 5

Plot the complex number in the complex plane.
-$$6 + \sqrt { 7 } i$$

Accepted Answer

A)  
  
B) 
  
C) 
  
D) 
  
A)  
  
B) 
  
C) 
  
D)

Question 6

Solve the problem. Leave your answer in polar form.
-$$\begin{array} { l } 
z = 5 \left( \cos 35 ^ { \circ } + i \sin 35 ^ { \circ } \right) \
w = 2 \left( \cos 40 ^ { \circ } + i \sin 40 ^ { \circ } \right)
\end{array}$$
Find zw.

Accepted Answer

A)  $10 \left( \cos 50.9 ^ { \circ } + i \sin 50.9 ^ { \circ } \right)$ 
B)  $7 \left( \cos 50.9 ^ { \circ } + i \sin 50.9 ^ { \circ } \right)$ 
C)  $10 \left( \cos 75 ^ { \circ } + i \sin 75 ^ { \circ } \right)$ 
D)  $7 \left( \cos 75 ^ { \circ } + i \sin 75 ^ { \circ } \right)$ 
A)  $10 \left( \cos 50.9 ^ { \circ } + i \sin 50.9 ^ { \circ } \right)$ 
B)  $7 \left( \cos 50.9 ^ { \circ } + i \sin 50.9 ^ { \circ } \right)$ 
C)  $10 \left( \cos 75 ^ { \circ } + i \sin 75 ^ { \circ } \right)$ 
D)  $7 \left( \cos 75 ^ { \circ } + i \sin 75 ^ { \circ } \right)$

Question 7

Solve the problem.
-If $\mathbf { u } = 7 \mathbf { i } - 4 \mathbf { j }$ and $\mathbf { v } = - 2 \mathbf { i } + 9 \mathbf { j }$, find $\mathbf { u } + \mathbf { v }$.

Accepted Answer

A)  $- 9 i + 5 j$ 
B)  $5 \mathbf { i } + 5 \mathbf { j }$ 
C)  $9 \mathbf { i } + 4 \mathbf { j }$ 
D)  $4 i + 5 j$ 
A)  $- 9 i + 5 j$ 
B)  $5 \mathbf { i } + 5 \mathbf { j }$ 
C)  $9 \mathbf { i } + 4 \mathbf { j }$ 
D)  $4 i + 5 j$

Question 8

State whether the vectors are parallel, orthogonal, or neither.
-$$\mathbf { v } = 4 \mathbf { i } + 2 \mathbf { j } , \quad \mathbf { w } = 2 \mathbf { i } - 4 \mathbf { j }$$

Accepted Answer

A)  Parallel 
B)  Orthogonal 
C)  Neither 
A)  Parallel 
B)  Orthogonal 
C)  Neither

Question 9

Solve the problem. Leave your answer in polar form.
-$$\begin{array} { l } 
z = 10 \left( \cos 45 ^ { \circ } + i \sin 45 ^ { \circ } \right) \
w = 5 \left( \cos 15 ^ { \circ } + i \sin 15 ^ { \circ } \right)
\end{array}$$
Find $\frac { z } { w }$.

Accepted Answer

A)  $2 \left( \cos 30 ^ { \circ } + i \sin 30 ^ { \circ } \right)$ 
B)  $\frac { 1 } { 2 } \left( \cos 30 ^ { \circ } + i \sin 30 ^ { \circ } \right)$ 
C)  $\frac { 1 } { 2 } \left( \cos 45 ^ { \circ } + i \sin 45 ^ { \circ } \right)$ 
D)  $2 \left( \cos 45 ^ { \circ } + i \sin 45 ^ { \circ } \right)$ 
A)  $2 \left( \cos 30 ^ { \circ } + i \sin 30 ^ { \circ } \right)$ 
B)  $\frac { 1 } { 2 } \left( \cos 30 ^ { \circ } + i \sin 30 ^ { \circ } \right)$ 
C)  $\frac { 1 } { 2 } \left( \cos 45 ^ { \circ } + i \sin 45 ^ { \circ } \right)$ 
D)  $2 \left( \cos 45 ^ { \circ } + i \sin 45 ^ { \circ } \right)$

Question 10

Write the expression in the standard form a + bi.
-$\left[ \sqrt { 2 } \left( \cos \frac { 3 \pi } { 4 } + i \sin \frac { 3 \pi } { 4 } \right) \right] ^ { 4 }$

Accepted Answer

A)  $4 \mathrm { i }$ 
B)  $- 4$ 
C)  $- 4 \mathrm { i }$ 
D)  4 
A)  $4 \mathrm { i }$ 
B)  $- 4$ 
C)  $- 4 \mathrm { i }$ 
D)  4

Question 11

Find the indicated cross product.
-$$\mathbf { v } = 4 \mathbf { i } - 5 \mathbf { j } + \mathbf { k } , \quad \mathbf { w } = 2 \mathbf { i } - 2 \mathbf { j } - \mathbf { k }$$
Find $\mathbf { w } \times \mathbf { v }$.

Accepted Answer

A)  $7 \mathbf { i } + 6 \mathbf { j } + 16 \mathbf { k }$ 
B)  $6 \mathbf { i } + 7 \mathbf { j } - 2 \mathbf { k }$ 
C)  $- 7 \mathbf { i } - 6 \mathbf { j } - 2 \mathbf { k }$ 
D)  $- 2 \mathbf { i } + 7 \mathbf { j } - 7 \mathbf { k }$ 
A)  $7 \mathbf { i } + 6 \mathbf { j } + 16 \mathbf { k }$ 
B)  $6 \mathbf { i } + 7 \mathbf { j } - 2 \mathbf { k }$ 
C)  $- 7 \mathbf { i } - 6 \mathbf { j } - 2 \mathbf { k }$ 
D)  $- 2 \mathbf { i } + 7 \mathbf { j } - 7 \mathbf { k }$

Question 12

Solve the problem. Leave your answer in polar form.
-$$\begin{array} { l } 
z = \sqrt { 3 } \left( \cos \frac { 7 \pi } { 4 } + i \sin \frac { 7 \pi } { 4 } \right) \
w = \sqrt { 6 } \left( \cos \frac { 9 \pi } { 4 } + i \sin \frac { 9 \pi } { 4 } \right)
\end{array}$$
Find $\frac { z } { w }$.

Accepted Answer

A)  $3 \sqrt { 2 } \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$ 
B)  $\frac { \sqrt { 2 } } { 2 } \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$ 
C)  $\frac { \sqrt { 2 } } { 2 } \left( \cos \frac { 3 \pi } { 2 } + i \sin \frac { 3 \pi } { 2 } \right)$ 
D)  $3 \sqrt { 2 } \left( \cos \frac { 3 \pi } { 2 } + i \sin \frac { 3 \pi } { 2 } \right)$ 
A)  $3 \sqrt { 2 } \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$ 
B)  $\frac { \sqrt { 2 } } { 2 } \left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$ 
C)  $\frac { \sqrt { 2 } } { 2 } \left( \cos \frac { 3 \pi } { 2 } + i \sin \frac { 3 \pi } { 2 } \right)$ 
D)  $3 \sqrt { 2 } \left( \cos \frac { 3 \pi } { 2 } + i \sin \frac { 3 \pi } { 2 } \right)$

Question 13

Find the unit vector having the same direction as v.
-$$\mathbf { v } = - 3 \mathbf { i } + 4 \mathbf { j }$$

Accepted Answer

A)  $\mathbf { u } = - \frac { 5 } { 3 } \mathrm { i } + \frac { 5 } { 4 } \mathrm { j }$ 
B)  $\mathbf { u } = - \frac { 4 } { 5 } \mathrm { i } + \frac { 3 } { 5 } \mathrm { j }$ 
C)  $u = - \frac { 3 } { 5 } i + \frac { 4 } { 5 } j$ 
D)  $\mathbf { u } = - 15 \mathrm { i } + 20 \mathrm { j }$ 
A)  $\mathbf { u } = - \frac { 5 } { 3 } \mathrm { i } + \frac { 5 } { 4 } \mathrm { j }$ 
B)  $\mathbf { u } = - \frac { 4 } { 5 } \mathrm { i } + \frac { 3 } { 5 } \mathrm { j }$ 
C)  $u = - \frac { 3 } { 5 } i + \frac { 4 } { 5 } j$ 
D)  $\mathbf { u } = - 15 \mathrm { i } + 20 \mathrm { j }$

Question 14

Test the equation for symmetry with respect to the given axis, line, or pole.
-$\mathrm { r } ^ { 2 } = \sin ( 2 \theta )$; the pole

Accepted Answer

A)  Symmetric with respect to the pole 
B)  May or may not be symmetric with respect to the pole 
A)  Symmetric with respect to the pole 
B)  May or may not be symmetric with respect to the pole

Question 15

Match the point in polar coordinates with either A, B, C, or D on the graph.
-$\left(-3, \frac{\pi}{3}\right)$

Accepted Answer

A)  A 
B)  B 
C)  C 
D)  D 
A)  A 
B)  B 
C)  C 
D)  D

Question 16

Use the figure below. Determine whether the given statement is true or false.  
-H + I + J = B

Accepted Answer

A)  True 
B)  False 
A)  True 
B)  False

Question 17

Use the vectors in the figure below to graph the following vector.  
-$\mathbf { z } - \mathbf { v }$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 18

Write the expression in the standard form a + bi.
-$$( 1 - \mathrm { i } ) ^ { 10 }$$

Accepted Answer

A)  32 
B)  $- 32 \mathrm { i }$ 
C)  $- 32 + 32 i$ 
D)  $32 - 32 \mathrm { i }$ 
A)  32 
B)  $- 32 \mathrm { i }$ 
C)  $- 32 + 32 i$ 
D)  $32 - 32 \mathrm { i }$

Question 19

Solve the problem.
-If $\mathbf { v } = - \mathbf { i } - \mathbf { j }$, find $\| v \|$.

Accepted Answer

A)  0 
B)  $\sqrt { 2 }$ 
C)  2 
D)  1 
A)  0 
B)  $\sqrt { 2 }$ 
C)  2 
D)  1

Question 20

Find v .
-$$\mathbf { v } = - 5 i - 3 j + 4 k$$

Accepted Answer

A)  $\sqrt { 41 }$ 
B)  $5 \sqrt { 2 }$ 
C)  $\sqrt { 34 }$ 
D)  $2 \sqrt { 3 }$ 
A)  $\sqrt { 41 }$ 
B)  $5 \sqrt { 2 }$ 
C)  $\sqrt { 34 }$ 
D)  $2 \sqrt { 3 }$

Use the figure below. Determine whether the given statement is true or false. -G + H = F

Find the distance from P1 to P2. - $\mathrm { P } _ { 1 } = ( 2,2 , - 1 ) \text { and } \mathrm { P } _ { 2 } = ( - 1 , - 3,2 )$

Choose the one alternative that best completes the statement or answers the question. The polar coordinates of a point are given. Find the rectangular coordinates of the point. - $\left( - 5 , \frac { 3 \pi } { 4 } \right)$

Plot the complex number in the complex plane. - $6 + \sqrt { 7 } i$ $Plot the complex number in the complex plane. - 6 + \sqrt { 7 } i$

Solve the problem. Leave your answer in polar form. - z=5 3+i3 w=2 4+i4 Find zw.

Solve the problem. -If $\mathbf { u } = 7 \mathbf { i } - 4 \mathbf { j }$ and $\mathbf { v } = - 2 \mathbf { i } + 9 \mathbf { j }$ , find $\mathbf { u } + \mathbf { v }$ .

State whether the vectors are parallel, orthogonal, or neither. - $\mathbf { v } = 4 \mathbf { i } + 2 \mathbf { j } , \quad \mathbf { w } = 2 \mathbf { i } - 4 \mathbf { j }$

Solve the problem. Leave your answer in polar form. - z=10 4+i4 w=5 1+i1 Find $\frac { z } { w }$ .

Write the expression in the standard form a + bi. - $\left[ \sqrt { 2 } \left( \cos \frac { 3 \pi } { 4 } + i \sin \frac { 3 \pi } { 4 } \right) \right] ^ { 4 }$

Find the indicated cross product. - $\mathbf { v } = 4 \mathbf { i } - 5 \mathbf { j } + \mathbf { k } , \quad \mathbf { w } = 2 \mathbf { i } - 2 \mathbf { j } - \mathbf { k }$ Find $\mathbf { w } \times \mathbf { v }$ .

Solve the problem. Leave your answer in polar form. - z= +i w= +i Find $\frac { z } { w }$ .

Find the unit vector having the same direction as v. - $\mathbf { v } = - 3 \mathbf { i } + 4 \mathbf { j }$

Test the equation for symmetry with respect to the given axis, line, or pole. - $\mathrm { r } ^ { 2 } = \sin ( 2 \theta )$ ; the pole

Match the point in polar coordinates with either A, B, C, or D on the graph. - $\left(-3, \frac{\pi}{3}\right)$ $Match the point in polar coordinates with either A, B, C, or D on the graph. - \left(-3, \frac{\pi}{3}\right)$

Use the figure below. Determine whether the given statement is true or false. -H + I + J = B

Write the expression in the standard form a + bi. - $( 1 - \mathrm { i } ) ^ { 10 }$

Solve the problem. -If $\mathbf { v } = - \mathbf { i } - \mathbf { j }$ , find $\| v \|$ .

Find v . - $\mathbf { v } = - 5 i - 3 j + 4 k$

Functions and Their Graphs

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Analytic Trigonometry

Applications of Trigonometric Functions

Analytic Geometry

Systems of Equations and Inequalities

Sequences; Induction; the Binomial Theorem

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A Preview of Calculus: the Limit, Derivative, and Integral of a Function

Foundations: a Prelude to Functions

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Exam 8: Polar Coordinates; Vectors

Use the figure below. Determine whether the given statement is true or false. -G + H = F

Find the distance from P1 to P2. - P1=(2,2,−1) and P2=(−1,−3,2)\mathrm { P } _ { 1 } = ( 2,2 , - 1 ) \text { and } \mathrm { P } _ { 2 } = ( - 1 , - 3,2 )P1​=(2,2,−1) and P2​=(−1,−3,2)

Choose the one alternative that best completes the statement or answers the question. The polar coordinates of a point are given. Find the rectangular coordinates of the point. - (−5,3π4)\left( - 5 , \frac { 3 \pi } { 4 } \right)(−5,43π​)

Plot the complex number in the complex plane. - 6+7i6 + \sqrt { 7 } i6+7​i