Kalfree Only

Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.

3x – 2y + 2z – w = 2
4x + y + z + 6w = 8
-3x + 2y – 2z + w = 5
5x + 3z – 2w = 1

	A. {(2, 0, – , )}
	B. {(1, – , , 6)}
	C. ∅
	D. {( , 0, – , )}

Solve the system of equations using matrices. Use Gauss-Jordan elimination.

3x – 7 – 7z = 7
6x + 4y – 3z = 67
-6x – 3y + z = -62

	A. {( 7, 1, 7)}
	B. {( 14, 7, -7)}
	C. {( -7, 7, 14)}
	D. {( 7, 7, 1)}

Find the product AB, if possible.

A = , B =

	A.
	B.
	C.
	D.

Use Cramer’s rule to solve the system. 2x + 4y – z = 32 x – 2y + 2z = -5 5x + y + z = 20

	A. {( 1, -9, -6)}
	B. {( 2, 7, 6)}
	C. {( 9, 6, 9)}
	D. {( 1, 9, 6)}

Find the products AB and BA to determine whether B is the multiplicative inverse of A.
A = , B =

	A. B = A-1
	B. B ≠ A-1

Let A = and B = . Find A – 3B.

	A.
	B.
	C.
	D.

Find the inverse of the matrix, if possible.

A =

	A.
	B.
	C.
	D.

Let B = [-1 3 6 -3]. Find -4B.

	A. [-4 12 24 -12]
	B. [-3 1 4 -5]
	C. [4 -12 -24 12]
	D. [4 3 6 -3]

Evaluate the determinant.

	A.
	B.
	C.
	D.

Give the order of the matrix, and identify the given element of the matrix.

; a12

	A. 4 × 2; -11
	B. 4 × 2; 14
	C. 2 × 4; 14
	D. 2 × 4; -11

Find the product AB, if possible.
A = , B =

	A.
	B. AB is not defined.
	C.
	D.

Use Gaussian elimination to find the complete solution to the system of equations, or state that none exists.

x + y + z = 9
2x – 3y + 4z = 7
x – 4y + 3z = -2

	A. {(-+ , + , z)}
	B. {( + , – , z)}
	C. {(- + , – , z)}
	D. {( + , + , z)}

Find the products AB and BA to determine whether B is the multiplicative inverse of A.

A = , B =

	A. B = A-1
	B. B ≠ A-1

Solve the matrix equation for X.

Let A = and B = ; 4X + A = B

	A. X =
	B. X =
	C. X =
	D. X =

Find the product AB, if possible.

A = , B =

	A.
	B.
	C.
	D. AB is not defined.

Use Cramer’s rule to determine if the system is inconsistent system or contains dependent equations.

2x + 7 = 8
6x + 3y = 24

	A. system is inconsistent
	B. system contains dependent equations

Find the product AB, if possible.

A = , B =

	A.
	B.
	C.
	D. AB is not defined.

Evaluate the determinant.

	A. 60
	B. -30
	C. -60
	D. 30

Determinants are used to show that three points lie on the same line (are collinear). If
= 0,
then the points ( x₁, y₁), ( x₂, y₂), and ( x₃, y₃) are collinear. If the determinant does not equal 0, then the points are not collinear. Are the points (-2, -1), (0, 9), (-6, -21) and collinear?

	A. Yes
	B. No

Solve the system of equations using matrices. Use Gaussian elimination with back-substitution.

3x + 5y – 2w = -13
2x + 7z – w = -1
4y + 3z + 3w = 1
-x + 2y + 4z = -5

	A. {(-1, – , 0, )}
	B. {(1, -2, 0, 3)}
	C. {( , -2, 0, )}
	D. {( , – , 0, )}

Halley’s comet has an elliptical orbit with the sun at one focus. Its orbit shown below is given approximately by In the formula, r is measured in astronomical units. (One astronomical unit is the average distance from Earth to the sun, approximately 93 million miles.) Find the distance from Halley’s comet to the sun at its greatest distance from the sun. Round to the nearest hundredth of an astronomical unit and the nearest million miles.

	A. 12.13 astronomical units; 1128 million miles
	B. 91.54 astronomical units; 8513 million miles
	C. 5.69 astronomical units; 529 million miles
	D. 6.06 astronomical units; 564 million miles

Use the center, vertices, and asymptotes to graph the hyperbola.

(x – 1)² – 9(y – 2)²= 9

	A.
	B.
	C.
	D.

Find the standard form of the equation of the ellipse and give the location of its foci.

	A. + = 1 foci at (- , 0) and ( , 0)
	B. – = 1 foci at (- , 0) and ( , 0)
	C. + = 1 foci at (- , 0) and ( , 0)
	D. + = 1 foci at (-7, 0) and ( 7, 0)

Rewrite the equation in a rotated x’y’-system without an x’y’ term. Express the equation involving x’ and y’ in the standard form of a conic section.

31x² + 10xy + 21y²-144 = 0

	A. x‘2 = -4 y’
	B. y‘2 = -4x’
	C. + = 1
	D. + = 1

Find the standard form of the equation of the ellipse satisfying the given conditions. Foci: (0, -2), (0, 2); y-intercepts: -5 and 5

	A. + = 1
	B. + = 1
	C. + = 1
	D. + = 1

Find the vertices and locate the foci for the hyperbola whose equation is given.

49x² – 100y²= 4900

	A. vertices: ( -10, 0), ( 10, 0) foci: (- , 0), ( , 0)
	B. vertices: ( -10, 0), ( 10, 0) foci: (- , 0), ( , 0)
	C. vertices: ( -7, 0), ( 7, 0) foci: (- , 0), ( , 0)
	D. vertices: (0, -10), (0, 10) foci: (0, – ), (0, )

Write the equation in terms of a rotated x’y’-system using θ, the angle of rotation. Write the equation involving x’ and y’ in standard form. xy +16 = 0; θ = 45°

	A. +  = 1
	B. y‘2 = -32x’
	C. + = 1
	D. – = 1

Write the appropriate rotation formulas so that in a rotated system the equation has no x’y’-term.

10x² – 4xy + 6y²– 8x + 8y = 0

	A. x = -y’; y = x’
	B. x = x’ – y’; y = x’ + y’
	C. x = (x’ – y’); y = (x’ + y’)
	D. x = x’ – y’; y = x’ + y’

Find the location of the center, vertices, and foci for the hyperbola described by the equation.

– = 1

	A. Center: ( -4, 1); Vertices: ( -10, 1) and ( 2, 1); Foci: and (
	B. Center: ( -4, 1); Vertices: ( -9, 1) and ( 3, 1); Foci: ( -3 + , 2) and ( 2 + , 2)
	C. Center: ( -4, 1); Vertices: ( -10, -1) and ( 2, -1); Foci: ( -4 – , -1) and ( -4 + , -1)
	D. Center: ( 4, -1); Vertices: ( -2, -1) and ( 10, -1); Foci: and

Sketch the plane curve represented by the given parametric equations. Then use interval notation to give the relation’s domain and range.

x = 2t, y = t²+ t + 3

	A. Domain: (-∞, ∞); Range: -1x, ∞)
	B. Domain: (-∞, ∞); Range: [ 2.75, ∞)
	C. Domain: (-∞, ∞); Range: [ 3, ∞)
	D. Domain: (-∞, ∞); Range: [ 2.75, ∞)

Use vertices and asymptotes to graph the hyperbola. Find the equations of the asymptotes.

y = ±

	A. Asymptotes: y = ± x
	B. Asymptotes: y = ± x
	C. Asymptotes: y = ± x
	D. Asymptotes: y = ± x

Graph the ellipse.

16(x – 1)² + 9(y + 2)²= 144

	A.
	B.
	C.
	D.

Is the relation a function?

y = x²+ 12x + 31

	A. Yes
	B. No

Determine the direction in which the parabola opens, and the vertex.

y²= + 6x + 14

	A. Opens upward; ( -3, 5)
	B. Opens upward; ( 3, 5)
	C. Opens to the right; ( 5, 3)
	D. Opens to the right; ( 5, -3)

Match the equation to the graph.

x²= 7y

	A.
	B.
	C.
	D.

y²= -2x

	A.
	B.
	C.
	D.

Convert the equation to the standard form for a hyperbola by completing the square on x and y.

x² – y²+ 6x – 4y + 4 = 0

	A. (x + 3)2 + (y + 2)2 = 1
	B. – = 1
	C. (x + 3)2 – (y + 2)2 = 1
	D. (y + 3)2– (x + 2)2 = 1

Eliminate the parameter t. Find a rectangular equation for the plane curve defined by the parametric equations.

x = 6 cos t, y = 6 sin t; 0 ≤ t ≤ 2π

	A. x2 – y2 = 6; -6 ≤ x ≤ 6
	B. x2 – y2 = 36; -6 ≤ x ≤ 6
	C. x2 + y2 = 6; -6 ≤ x ≤ 6
	D. x2 + y2 = 36; -6 ≤ x ≤ 6

Convert the equation to the standard form for a parabola by completing the square on x or y as appropriate.

y²+ 2y – 2x – 3 = 0

	A. (y + 1)2 = 2(x + 2)
	B. (y – 1)2 = -2(x + 2)
	C. (y + 1)2 = 2(x – 2)
	D. (y – 1)2 = 2(x + 2)

Convert the equation to the standard form for a hyperbola by completing the square on x and y.

y² – 25x²+ 4y + 50x – 46 = 0

	A. – (x – 2)2 = 1
	B. – (y – 1)2 = 1
	C. (x – 1)2 – = 1
	D. – (x – 1)2 = 1

The finite sequence whose general term is a_n = 0.17n² – 1.02n + 6.67 where n = 1, 2, 3, …, 9 models the total operating costs, in millions of dollars, for a company from 1991 through 1999.

Find

	A. $21.58 million
	B. $27.4 million
	C. $23.28 million
	D. $29.1 million

Use the formula for the sum of the first n terms of a geometric sequence to solve. Find the sum of the first 8 terms of the geometric sequence: -8, -16, -32, -64, -128, . . . .

	A. -2003
	B. -2040
	C. -2060
	D. -2038

Find the probability. What is the probability that a card drawn from a deck of 52 cards is not a 10?

	A. 12/13
	B. 9/10
	C. 1/13
	D. 1/10

Find the common difference for the arithmetic sequence. 6, 11, 16, 21, . . .

	A. -15
	B. -5
	C. 5
	D. 15

Find the indicated sum.

	A. 28
	B. 16
	C. 70
	D. 54

Evaluate the expression.

1 –

	A.
	B.
	C.
	D.

Find the sum of the infinite geometric series, if it exists. 4 – 1 + – + . . .

	A. – 1
	B. 3
	C.
	D. does not exist

Find the probability. One digit from the number 3,151,221 is written on each of seven cards. What is the probability of drawing a card that shows 3, 1, or 5?

	A. 5/7
	B. 2/7
	C. 4/7
	D. 3/7

A game spinner has regions that are numbered 1 through 9. If the spinner is used twice, what is the probability that the first number is a 3 and the second is a 6?

	A. 1/18
	B. 1/81
	C. 1/9
	D. 2/3

Use the formula for the sum of the first n terms of a geometric sequence to solve. Find the sum of the first four terms of the geometric sequence: 2, 10, 50, . . . .

	A. 312
	B. 62
	C. 156
	D. 19

Write a formula for the general term (the nth term) of the geometric sequence.

, – , , –, . . .

	A. an = n – 1
	B. an = –  (n – 1)
	C. an = n – 1
	D. an = n – 1

Does the problem involve permutations or combinations? Do not solve. In a student government election, 7 seniors, 2 juniors, and 3 sophomores are running for election. Students elect four at-large senators. In how many ways can this be done?

	A. permutations
	B. combinations

Solve the problem. Round to the nearest hundredth of a percent if needed. During clinical trials of a new drug intended to reduce the risk of heart attack, the following data indicate the occurrence of adverse reactions among 1100 adult male trial members. What is the probability that an adult male using the drug will experience nausea?

	A. 2.02%
	B. 1.73%
	C. 27.59%
	D. 2.18%

The general term of a sequence is given. Determine whether the given sequence is arithmetic, geometric, or neither. If the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. a_n= 4n – 2

	A. arithmetic, d = -2
	B. geometric, r = 4
	C. arithmetic, d = 4
	D. neither

Evaluate the factorial expression.

	A. n + 4!
	B. 4!
	C. (n + 3)!
	D. 1

If the given sequence is a geometric sequence, find the common ratio.

, , , ,

	A.
	B. 30
	C.
	D. 4

Solve the problem. Round to the nearest dollar if needed. Looking ahead to retirement, you sign up for automatic savings in a fixed-income 401K plan that pays 5% per year compounded annually. You plan to invest $3500 at the end of each year for the next 15 years. How much will your account have in it at the end of 15 years?

	A. $77,295
	B. $75,525
	C. $76,823
	D. $73,982

Find the term indicated in the expansion.

(x – 3y)¹¹; 8th term

	A. -721,710x7y4
	B. -721,710x4y7
	C. 240,570x7y4
	D. 240,570x4y8

Find the probability. Two 6-sided dice are rolled. What is the probability that the sum of the two numbers on the dice will be greater than 10?

	A. 1/12
	B. 5/18
	C. 3
	D. 1/18

Does the problem involve permutations or combinations? Do not solve. A club elects a president, vice-president, and secretary-treasurer. How many sets of officers are possible if there are 15 members and any member can be elected to each position? No person can hold more than one office.

	A. permutations
	B. combinations

Find the slope of the tangent line to the graph of f at the given point.

f(x) = at ( 36, 6)

	A.
	B. 12
	C. 3
	D.

Use properties of limits to find the indicated limit. It may be necessary to rewrite an expression before limit properties can be applied.

	A. 16
	B. does not exist
	C. -16
	D. 0

Use properties of limits to find the indicated limit. It may be necessary to rewrite an expression before limit properties can be applied.

(2x² + 2x + 3)²

	A. -9
	B. 9
	C. does not exist
	D. 1

Complete the table for the function and find the indicated limit.

	A. -0.0300, -0.0200, -0.0100, 0.0100, 0.0200, 0.0300 limit = -1
	B. -0.0300, -0.0200, -0.0100, 0.0100, 0.0200, 0.0300 limit = 0
	C. -0.0300, -0.0200, -0.0100, 0.0100, 0.0200, 0.0300 limit = 0.1
	D. -0.0300, -0.0200, -0.0100, 0.0100, 0.0200, 0.0300 limit = 1

Use the definition of continuity to determine whether f is continuous at a.

f(x) = 5x⁴ – 9x³+ x – 7a = 7

	A. Not continuous
	B. Continuous

Find the slope of the tangent line to the graph of f at the given point.

f(x) = x²+ 5x at (4, 36)

	A. 13
	B. 21
	C. 9
	D. 3

Use the definition of continuity to determine whether f is continuous at a.

f(x) = a = 4

	A. Not continuous
	B. Continuous

Graph the function. Then use your graph to find the indicated limit. f(x) = 7e^x , f(x)

	A. 0
	B. 7
	C. 1
	D. -7

The graph of a function is given. Use the graph to find the indicated limit and function value, or state that the limit or function value does not exist.

a. f(x)
b. f(1)

	A. a.  f(x) = 1 b. f(1) = 0
	B. a. f(x) does not exist b. f(1) = 2
	C. a. f(x) = 2 b. f(1) = 2
	D. a. f(x) = 2 b. f(1) = 1

Choose the table which contains the best values of x for finding the requested limit of the given function.

	A.
	B.
	C.
	D.

Choose the table which contains the best values of x for finding the requested limit of the given function.

(x²+ 8x – 2)

	A.
	B.
	C.
	D.

Determine for what numbers, if any, the given function is discontinuous.

f(x) =

	A. 5
	B. None
	C. 0
	D. -5, 5

Complete the table for the function and find the indicated limit.

	A. -1.22843; -1.20298; -1.20030; -1.19970; -1.19699; -1.16858 limit = -1.20
	B. -2.18529; -2.10895; -2.10090; -2.09910; -2.09096; -2.00574 limit = -2.10
	C. -4.09476; -4.00995; -4.00100; -3.99900; -3.98995; -3.89526 limit = -4.0
	D. 4.09476; 4.00995; 4.00100; 3.99900; 3.98995; 3.89526 limit = 4.0

The function f(x) = x³describes the volume of a cube, f(x), in cubic inches, whose length, width, and height each measure x inches. If x is changing, find the average rate of change of the volume with respect to x as x changes from 1 inches to 1.1 inches.

	A. 2.33 cubic inches per inch
	B. -3.31 cubic inches per inch
	C. 23.31 cubic inches per inch
	D. 3.31 cubic inches per inch

The graph of a function is given. Use the graph to find the indicated limit and function value, or state that the limit or function value does not exist.

a. f(x)
b. f(3)

	A. a. f(x) = 3 b. f(3) = 5
	B. a. f(x) = 5 b. f(3) = 5
	C. a. f(x) = 4 b. f(3) does not exist
	D. a. f(x) does not exist b. f(3) = 5

Use the definition of continuity to determine whether f is continuous at a.

f(x) = 
a = -5

	A. Not continuous
	B. Continuous

Use the graph and the viewing rectangle shown below the graph to find the indicated limit.

( x² – 2)

[-6, 6, 1] by [-6, 6, 1]

	A. (x2 – 2) = -6
	B. (x2 – 2) = 2
	C. (x2 – 2) = -2
	D. (x2 – 2) = 6

Use properties of limits to find the indicated limit. It may be necessary to rewrite an expression before limit properties can be applied.

5

	A. -5
	B. 0
	C. 5
	D. 2

Find the derivative of f at x. That is, find f ‘(x). f(x) = 7x + 8; x = 5

	A. 40
	B. 8
	C. 35
	D. 7

Graph the function. Then use your graph to find the indicated limit.

f(x) = , f(x)

	A. 6
	B. -2
	C. -6
	D. 2