Precalculus Practice Final Andrew Dynneson
(1) Find two positive numbers whose difference equals 4 and whose product equals 25.
[2.2;16]
(2) Find all real numners x such that x6 − 4x3 − 45 = 0
[2.4;20]
(3) Suppose:
r(x) =
x+1
x2 + 1
Find two distinct numbers x such that r(x) = 41 .
[2.5;25]
(4) Find two complex numbers in the form a + bi whose squares equal 16 − 30i
[2.6;38]
(5) Use Gaussian elimination to find all solutions to the given system of equations.
2y + 3z = 6
−x + 4y + 3z = −3
2x + 5y − 3z = 2
(6) [Calculator] Suppose a country’s population increases by a total of 3 percent over a threeyear period. What is the continuous growth rate for this country?
[4.5; 10]
(7) Using the fact that the area( x1 , 1, ct ) = t · area( x1 , 1, c), show that the area( x1 , 1, c) = ln c
[page 326]
(8) Find the four smallest positive numbers θ such that sin θ =
[5.3;19]
(9) Use the right triangle below, which is not drawn to scale:
8
.
13
Suppose c = 3 and cos u =
[5.5;50]
(10) Given that cos
Evaluate a [Exact Value].
√
π
12
=
√
2+ 3
,
2
evaluate cos 25π
[Exact Value].
12
1
1
2
[Exact answers]
[5.6; 15]
(11) Find the area of a regular
dodecagon with sides of length s, using the fact that the area
p
√
of a dodecagon with side 2 − 3 equals 3.
[6.1;36]
(12) [Calculator] Suppose a = 19, b = 18 and B = 60◦ . Use the following figure to evaluate:
A (where A > 90◦ , C and c.
[6.2;16]
(13) [Calculator] Suppose a = 4, b = 6 and C = 2 radians. Use the figure (not drawn to scale)
to evaluate c, A and B. Evaluate in both degrees and radians. Round radian measurements to
three decimal places, and degree measurements to one decimal place.
[6.2; 12]
(14) Assuming that u is in the interval ( π2 , π), and that sin u = 16 , evaluate sin
u
2
[6.3;39;page 486]
(15) Find the formula for cos θ +
π
4
.
[6.4;31;page 497]
(16) By what fraction of 6 cos
of 6 cos π2 x + 6π
?
5
π
x
2
has the graph been shifted to the left to obtain the graph
[6.5;53]
(17) Use the law of cosines to find a formula for the distance (in the usual rectangular coordinate
plane) between the point with polar coordinates r1 and θ1 and the point with polar coordinates
r2 and θ2
[6.6;31]
√
(18) Evaluate (−5 + 5 3i)333 [Possibly worth 15 points!!!]
[6.7;18;De Moivre’s Theorem on page 545]
2
(19) Use the dot product to find the angle between the vectors (2, −3) and (0, 4)
[6.7;12; Use theorem on page 540]
The theorem is: u · v = |u||v| cos θ.
3