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.
This review is your entry ticket to the final exam. It is to be completed and turned in when you
come to take the exam.
(1)
Given 5 + 8 + 11 + 14 + ..., find both a32 and S32.
(2)
Given 3 +
(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.
(4)
(a)
3
3
3
+
+
+ ..., find a15 and S.
4 16
64
50
Find
  3k  2 
k 1

(b)
5
k 1
2
k 1
Semester Review P 1
MI-4 F12
Math Teacher:
Name:
1
1
1
n
1
+
+
+ ... +
=
1·2 2·3 3·4
n 1
n(n  1)
(5)
Prove by math induction:
(6)
Let v1  4,  2 and v2  3, 2 . Determine each of the following.
(a) v2 - 3v1
(b) v1 · v2
(d) the angle between v1 and v2
(7)
What makes two vectors orthogonal? parallel?
(8)
Let v1  4,1,3 and v2  6, 1, 2
(c) 3v1
(e) the projv1  v 2 
(a) 3v1  6v2 
(b) the angle between v1 and v2
(c) the projv1  v 2 
Semester Review P 2
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Math Teacher:
Name:
(9)
In MNP, M = 14°, m = 12, and n = 16. Solve the triangle and find its area.
(10)
A regular pentagon is inscribed in a circle of radius 32. Find the perimeter and the area of
the pentagon.
(11)
Find:
(-4,3)
sin()
cos(), tan()

sin(2), cos(2), tan(2)
(12)
(13)
Find:
sin( + )
sin( - )
cos( + )
cos( - )
tan( + )
tan( - )
Solve for x  [-, 2]:
(a) 2sin2(x) + sin(x) = 0
(-2,4)


(-8,-6)
(b) sin(2x) = tan(x)
Semester Review P 3
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Math Teacher:
(14)
Name:
Prove: (a)
(b)
sec  A 
sin  A 
sin  x 
1  cos  x 
=
-
sin  A 
cos  A 
= cot(A)
1  cos  x 
sin  x 
(15)
Solve for x  [0, ]: sin(3 x)  0.2
(16)
 1 
Find (exact): sin 1  
 2 
(17)
x
Sketch f ( x)  3cos 1   .
2
(18)

 2  
Find tan tan  sin 1    .
 3 


tan 1  3
Semester Review P 4

 2
cos 1 

 2 
MI-4 F12
Math Teacher:
Name:
(19)
Sketch the graphs:
(a) r = 3sin(4),
(20)
(a)
Express 24.6cis(198°) in a + bi form.
(b)
Express -6 - 2i in cis form.
(b) r = 1 + 2cos(2)
(21)
Write the four fourth roots of 16cis(96°) in cis form.
(22)
(5cis(12))2 (2cis(13)) 4
Simplify:
.
8cis(70)
(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.
Semester Review P 5
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Math Teacher:
(25)
(26)
Name:
$150 is deposited into an account at beginning of each month that pays 6% compunded
monthly. What is the account’s value after 10 years?
 x  3t  2
a) Sketch: 
 y  2t  3
b) Determine the slope and y-intercept of the graph.
 x  6t  4
c) Explain the difference between 
and the above.
 y  4t  6
(27)
(28)
Let v1  4, a,  3
v2  6,2, a  1 .
Find a if the vectors are orthogonal.
Let t = 4sin(). Substitute and simplify
16  t .
2
(29)
Is it possible to find a such that the
vectors are collinear? Show/Explain.
Let t = 3tan(). Substitute and
6
simplify
.
9  t2

1
 2 
Find the exact value of tan  sin 1    cos 1     .
3
 3 

Semester Review P 6
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Math Teacher:
Name:
Find the angle  between the two
lines by three methods.
(30)

(0,6)
(-4,1)
(0,-1)
(a) Use vectors
(b) Find the angle of
inclination of each line.
(c) Use tan(   ) .
120
(31)
Find the sum:
  k  2  k  3 . (Use techniques developed in the sequence unit.)
k 1
(32)
Solve, approximate, in radians.
(a) tan(4x + 1) = 6
x
(b) sin    0.85
3
Semester Review P 7
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