Honors Math Analysis Test 1 – 2
Express all decimal answers to the nearest one-thousandth (x.xxx).
For problems 1  4, let u  4, 1 , v = 2,5 , and
w  1,3 be vectors. Find the indicated expression.
1. w  2u
w  2u  1,3  2 4, 1  1,3  8, 2
  1  8  ,  3  2   7,5
2. u w
u w   4   1   1 3  4  3  1
3. w  3v  2u
w  3v  2u  1,3  3 2,5  2 4, 1
 1,3  6,15  8, 2  13, 20


4. u w  v


13   20 

u w  v  4, 1
 569
2
2
 1,3
 2,5

 4, 1 1,8  4  8  12
Honors Math Analysis Test 1 – 2
Find the angle between u and v.
5. u  2, 4 , v  6, 4
u  2, 4 , v  6, 4
  cos
1
uv
 u v  2, 4
6, 4  12  16  4
u v
u 
 2    4 
  cos
2
1
2
 20, v 
6   4
2
2
 52
4
 82.875  1.446 rad
20 52
Convert the polar coordinates to rectangular
coordinates.
6. 5.4, 7
6
x  r cos   5.4 cos 7  2.7 3
6
y  r sin   5.4sin 7  2.7
6
5.4, 7  2.7 3, 2.7
6



 

Honors Math Analysis Test 1 – 2
Convert the rectangular coordinates to polar
coordinates with 0 ≤ θ ≤ 2π.
7. P   10, 5 
r
 10    5  125  5 5
2
2
5
  tan
 0.463    3.605
10
1

P   10, 5   5 5,3.605

Eliminate the parameter t and identify the graph.
8. x  5sin t , y  5cos t
x 2  52 sin 2 t , y 2  52 cos 2 t  x 2  y 2  25  sin 2 t  cos 2 t 
x 2  y 2  25  circle of radius 5 centered at the origin
Write the complex number in standard form.
9. 4  cos 330  i sin 330 
4  cos330  i sin330   4cos330  4i sin330  2 3  2i
Honors Math Analysis Test 1 – 2
Write the complex number in trigonometric form
where 0    2 .
10. 2  i
r
 2
  tan
1
2
  1  3
2
1
 0.615  2  5.668
2
2  i  3  cos 5.668  i sin 5.668 
Write the complex numbers z1 z2 and
z1
z2
in trigonometric form.
z1  14  cos140  i sin140  and
11.
z2  7  cos125  i sin125 
z1 z2  14  7   cos 140    i sin 140   
 98  cos 265  i sin 265 
z1 14

cos 140    i sin 140   

z2 7
 2  cos15  i sin15 
Honors Math Analysis Test 1 – 2
Use De Moivre’s theorem to find the indicated power
of the complex number. Write your answer in both
(a) trigonometric and (b) standard form.
12
 

 
12. 3  cos  i sin  
24
24  
 
12
 

 
 
  
12 
3
cos

i
sin

3
cos12

i
sin12

 
 

 
24
24
24

 
 24  
 




 531,441 cos  i sin 
2
2

 531,441i
Honors Math Analysis Test 1 – 2
Find and graph the nth roots of the complex number
for the specified value of n.
13. 512, n  5
z  512  cos 0  i sin 0 
0
0 

z1  512  cos  i sin 
5
5

5
 5 512  cos 0  i sin 0 
0  360
0  360 

z2  512  cos
 i sin

5
5


5
 5 512  cos 72  i sin 72 
z3  5 512  cos144  i sin144 
z4  5 512  cos 216  i sin 216 
z5  5 512  cos 288  i sin 288 
y




x













Honors Math Analysis Test 1 – 2
Convert the polar equation to rectangular form and
identify the graph.
14. r  4sec 
4
r  4sec  
 r cos   4  x  4
cos 
Vertical line through x  4
Convert the rectangular equation to polar form and
graph the polar equation.
15. 2 x  3 y  12
y




x






2r cos   3r sin   12
r  2 cos   3sin    12
12
r
2 cos   3sin 







Honors Math Analysis Test 1 – 2
16. A force of 120 lb acts on an object at an angle of
120º. A second force of 300 lb acts at an angle of
300º, and a third force of 200 lbs acts at 210º. Find
the direction and magnitude of the resultant force.
F1  120 cos120,sin120  60,103.923
F2  300 cos 300,sin 300  150, 239.807
F3  200 cos 210,sin 210  173.205, 100
FR  83.205, 235.884
FR 
 83.205   235.884 
2
2
 250.129lb
235.884
  tan
 70.570  180  250.570
83.205
1
17. Find the work done by a force F of 78 lb acting
in the direction given by the vector 3,5 in moving
an object 15 ft along the line x = y.
1 5
1 1
 F  tan
 59.036,  D  tan
 45
3
1
  59.036  45  14.036
W  F D cos   78 15cos14.036  1135.067 ft-lbs
Honors Math Analysis Test 1 – 2
18. Brian hits a baseball at a velocity of 130 ft per sec
toward a 15 ft high fence 400 ft away. The ball was
hit at a point 2.5 ft above the ground at an angle of
30º above the horizontal. Does the ball clear the
fence? If so, by how much does it clear the fence to
the nearest quarter of a foot?
x  t   130  cos30 t , y  t   2.5  130  sin 30  t  16t 2
Ball clears fence by 16.25 or 16.50 ft
Honors Math Analysis Test 1 – 2
19. Extra credit. Lucinda is on a Ferris wheel of
radius 35 ft that turns at the rate of one revolution
every 20 sec. The lowest point on the Ferris wheel (6
o’clock) is 15 ft above level ground. If a point on the
ground, directly below the center of the wheel, is the
origin of a rectangular coordinate system, find
parametric equations for the position of Lucinda as a
function of time t in seconds, if at time t = 0 she is at
the lowest point of travel.
2 2 
b


p
20 10
x  t   35sin

10
t , y  t   50  35cos

10
t