Math Teacher:
Name:
Mathematical Investigations IV
The Semester in Review
This is a beginning. This is not intended as a complete review. It is a reminder of many of the
types of problems we have done this semester. Please review problem set, worksheets, and
quizzes in addition to doing these problems. Pace yourself while studying this material during
the days before the exam. Topics that caused difficulty earlier in the semester deserve extra time
now. Also, even those topics which you understood well must still be reviewed to refresh your
memory for formulas and specific methods. Many problems should be able to be done without
the aid of a calculator. These problems are marked with an NC.
NC
(1)
Given 5 + 8 + 11 + 14 + ..., find both a32 and S32.
NC
(2)
Given 3 +
NC
(3)
In an arithmetic sequence, the third term is 12 and the sum of the first 20 terms is 30.
Find the first term and the fifth term of the corresponding harmonic sequence.
NC
(4)
(a)
3
3
3
+
+
+ ..., find a15 and S.
4
64
16
50
  3k  2 
Find
k 1

(b)
2
5
k 1
k 1
Semester Review P 1
Math Teacher:
(5)
(6)
Name:
A plane traveling at 475 mph is flying with a bearing of 70°. There is a wind of 100 mph
from the South. If no correction is made for the wind, what are the final bearing and
ground speed of the plane? Draw a well labeled diagram to explain your answer.
Let v1  4,  2 and v2  3, 2 . Determine each of the following.
(a) v2 – 3v1
(d) the angle between v1 and v2
NC
(7)
(e) the projv1 v 2
Given  a, a  3, 4  and  2a  1, a,  2  , find all a such that the two
(a)
(8)
(c) 3v1
(b) v1 · v2
vectors are orthogonal
(b)
vectors are parallel
Let v1  4, 1, 3 and v2  6,  1, 2 . Find each of the following.
(a) 3v1  6v2 
(b) the angle between v1 and v2
(c) the projv1 v 2
Semester Review P 2
Math Teacher:
Name:
(9)
In MNP, M = 14°, m = 12, and n = 16. Solve the triangle and find its area. State your
answers clearly.
NC
(10)
A regular pentagon is inscribed in a circle of radius 32. Find the perimeter and the area
of the pentagon.
NC
(11)
Find: sin() =
cos() =
tan() =
cos(2) =
tan(2) =
(-4,3)

NC
(12)
sin(2) =
Find:
sin( + )
sin( – )
(-2,4)


cos( + )
cos( – )
(-8,-6)
tan( + )
tan( – )
Semester Review P 3
Math Teacher:
NC
NC
(13)
(14)
Name:
Solve for x  [–, 2]:
(a) 2sin2(x) + sin(x) = 0
Prove: (a)
sec  A
sin  A
–
= cot(A)
sin  A
cos  A
sin  x 
1  cos  x 
=
1  cos  x 
sin  x 
(b)
NC
(b) sin(2x) = tan(x)
(15)
Solve for x  [0, ]: sin(3x) = –0.2
(16)
 1 
Find (exact): sin 1  
 2 

tan 1  3

 2
cos 1 

 2 
Semester Review P 4
Math Teacher:
NC (17)
Name:
x
Sketch ƒ(x) = 3cos 1   .
2

 2  
Find tan  sin 1    .
 3 

NC
(18)
NC
(19)
Sketch and label the graphs:
(a) r = 3sin(4)
(20)
(a)
Express (24.6)cis 198° in a + bi form.
(b)
Express –6 – 2i in cis form.
(b) r = 1 + 2cos(2)
Semester Review P 5
Math Teacher:
Name:
NC
(21)
Write the four fourth roots of 16cis96° in cis form.
NC
(22)
Simplify:
(23)
Find the measure of the angle between 3x + 2y = 12 and 4 x  2 y  2 .
(24)
Find the area of the circle given by x2 – 8x + y2 – 10y – 3 = 0.
(25)
If $150 is deposited into an account at beginning of each month that pays an annual
interest of 6% compunded monthly, what is the account’s value after 10 years?
NC
NC
(26)
(5cis12)2 (2cis13) 4
.
8cis70
 x  3t  2
a) Sketch: 
 y  2t  3
b) Determine the slope and y-intercept of the graph.
Semester Review P 6
Math Teacher:
NC
NC
(27)
(28)
NC (29)
Name:
Let v1  4, a,  3 and v2  6, 2, a  1 .
Find a if the vectors are orthogonal.
Substitute t  4sin( ) into
and simplify the result.
.
16  t 2
Is it possible to find a such that the
vectors are collinear? Show/Explain.
Substitute t  3 tan( ) into
6
9  t2
and simplifythe result.

1
 2  
Find the exact value of tan  sin 1    cos1    .
 3
 3 

(30)

(0,6)
Find the angle  between the two
lines by three methods.
(-4,1)
(0,-1)
(a) Use vectors.
(b) Find the angle of
inclination of each line.
Semester Review P 7
(c) Use tan( – ).
Math Teacher:
Name:
120
NC
(31)
Find the sum:
  k  2  k  3 . (Use techniques developed in the sequence unit.)
k 1
(32)
NC
Solve, approximate, in radians.
(a) tan(4x + 1) = 6
(33)
(b) sin(x/3) = –0.85
Write a polar equation describing each of the following graphs.
(a)
r=
(b)
r=
4
5
2
2
2
2
4
6
8
10
1
1
2
12
2
5
4
NC
(34)
Rewrite the rectangular equation x 2  3x  y 2  0 in simplest polar form.
Semester Review P 8
Math Teacher:
NC
Name:
x 
Describe the transformtation that occurs when you multiply a vector   by each of the
 y
transformation matrices below (putting the transformation matrix on the left). Give a one or two
sentence description as modeled for the first matrix.
(35)
0 1
b. 

1 0 
1 0 
a. 

0 1
This matrix reflects vectors over the
1 0   x   x 
x-axis. Note: 
    
0 1  y    y 


c. 



 2

2 
2 

2 
2
2
2
2


d. 



1
2
3
2
 3

2 
1 

2 
NC (36) Prove the following statements using the Princiople of Mathematical Induction:
I.
For all positive integers n, 3  11  19 
II.
For all positive integers n, 11! 2  2! 3  3! ...........  n  n!  ( n  1)! 1
III.
For all integers n  0, n3  5n  6 is divisible by 3.
Semester Review P 9
 (8n  5)  4n2  n