half multiple choice half solve need by tommorw 20 questions 🙂
Pre-Calculus
Final Exam
Name: _________________________
Score: ______ / ______
Multiple Choice: Type your answer choice in the blank next to each question number.
_____1. Find the indicated sum.
A. 2
B. 54
C. 46
D. -54
Pre-Calculus Final Exam
_____2. Graph the ellipse and locate the foci.
𝑥
36 +
𝑦
11 = 1
A.
foci at (0, 6) and (0, -6)
C.
foci at (√11, 0) and (- √11, 0)
B.
foci at ( 5, 0) and (-5, 0)
D.
foci at (0, 5) and (0, -5)
_____3. Solve the system by the substitution method.
2y – x = 5
x2 + y2 – 25 = 0
A. 0, , 4,
B. 0, , 0, − 4,
C. {( 5, 0), ( -5, 0), ( 3, 4)}
D. {( -5, 0), ( 3, 4)}
Pre-Calculus Final Exam
_____4. Graph the function. Then use your graph to find the indicated limit.
f(x) = 5x – 3, f(x)
A. 5
B. 25
C. 2
D. 22
_____5. Use Gaussian elimination to find the complete solution to the system of equations, or state
that none exists.
4x – y + 3z = 12
x + 4y + 6z = -32
5x + 3y + 9z = 20
A. {(8, -7, -2)}
B. {(-8, -7, 9)}
C. ∅
D. {(2, -7, -1)}
_____6. Solve the system of equations using matrices. Use Gaussian elimination with back-
substitution.
x + y + z = -5
x – y + 3z = -1
4x + y + z = -2
A. {( 1, -4, -2)}
B. {( -2, 1, -4)}
C. {( 1, -2, -4)}
D. {( -2, -4, 1)}
Pre-Calculus Final Exam
_____7. A woman works out by running and swimming. When she runs, she burns 7 calories per
minute. When she swims, she burns 8 calories per minute. She wants to burn at least 336
calories in her workout. Graph an inequality that describes the situation. Let x represent the
number of minutes running and y the number of minutes swimming. Because x and y must be
positive, limit the graph to quadrant I only.
A.
C.
B. D.
Pre-Calculus Final Exam
Short Answer Questions: Type your answer below each question. Show your work.
8 A statement Sn about the positive integers is given. Write statements S1, S2, and S3, and show that
each of these statements is true.
Sn: 12 + 42 + 72 + . . . + (3n – 2)2 =
( )
9 A statement Sn about the positive integers is given. Write statements Sk and Sk+1, simplifying
Sk+1 completely.
Sn: 1 ·∙ 2 + 2 ·∙ 3 + 3 ·∙ 4 + . . . + n(n + 1) = [n(n + 1)(n + 2)]/3
10 Joely’s Tea Shop, a store that specializes in tea blends, has available 45 pounds of A grade tea and
70 pounds of B grade tea. These will be blended into 1 pound packages as follows: A breakfast
blend that contains one third of a pound of A grade tea and two thirds of a pound of B grade tea
and an afternoon tea that contains one half pound of A grade tea and one half pound of B grade
tea. If Joely makes a profit of $1.50 on each pound of the breakfast blend and $2.00 profit on
each pound of the afternoon blend, how many pounds of each blend should she make to
maximize profits? What is the maximum profit?
Pre-Calculus Final Exam
11 Your computer supply store sells two types of laser printers. The first type, A, has a cost of $86
and you make a $45 profit on each one. The second type, B, has a cost of $130 and you make a
$35 profit on each one. You expect to sell at least 100 laser printers this month and you need to
make at least $3850 profit on them. How many of what type of printer should you order if you
want to minimize your cost?
12 A statement Sn about the positive integers is given. Write statements S1, S2, and S3, and show that
each of these statements is true.
Sn: 2 + 5 + 8 + . . . + ( 3n – 1) = n(1 + 3n)/2
13 Use mathematical induction to prove that the statement is true for every positive integer n.
2 is a factor of n2 – n + 2
Pre-Calculus Final Exam
14 A statement Sn about the positive integers is given. Write statements S1, S2, and S3, and show that
each of these statements is true.
Sn: 2 is a factor of n2 + 7n
15
(i.) f(x)
(ii.) f(x)
(iii.) What can you conclude about f(x)? How is this shown by the graph?
(iv.) What aspect of costs of renting a car causes the graph to jump vertically by the same amount
at its discontinuities?
Pre-Calculus Final Exam
16 Use mathematical induction to prove that the statement is true for every positive integer n.
8 + 16 + 24 + . . . + 8n = 4n(n + 1)
17 The following piecewise function gives the tax owed, T(x), by a single taxpayer on a taxable
income of x dollars.
T(x) =
(i) Determine whether T is continuous at 6061.
(ii) Determine whether T is continuous at 32,473.
(iii) If T had discontinuities, use one of these discontinuities to describe a situation where it might
be advantageous to earn less money in taxable income.
Pre-Calculus Final Exam
18 A statement Sn about the positive integers is given. Write statements Sk and Sk+1, simplifying
Sk+1 completely.
Sn: 1 + 4 + 7 + . . . + (3n – 2) = n(3n – 1)/2
19 An artist is creating a mosaic that cannot be larger than the space allotted which is 4 feet tall and
6 feet wide. The mosaic must be at least 3 feet tall and 5 feet wide. The tiles in the mosaic have
words written on them and the artist wants the words to all be horizontal in the final mosaic. The
word tiles come in two sizes: The smaller tiles are 4 inches tall and 4 inches wide, while the large
tiles are 6 inches tall and 12 inches wide. If the small tiles cost $3.50 each and the larger tiles cost
$4.50 each, how many of each should be used to minimize the cost? What is the minimum cost?
20 The Fiedler family has up to $130,000 to invest. They decide that they want to have at least
$40,000 invested in stable bonds yielding 5.5% and that no more than $60,000 should be
invested in more volatile bonds yielding 11%. How much should they invest in each type of bond
to maximize income if the amount in the stable bond should not exceed the amount in the more
volatile bond? What is the maximum income?