Download STANDARD 1: Students solve equations and inequalities involving

ALGEBRA II WARMUP CST QUESTIONS
STANDARD 1: Students solve equations and inequalities involving absolute value
1. What is the complete solution to the equation: 3  6 x  1?
2. Which of the following are solutions to 3x  7  14 ?
3. Which of the following is equivalent to 2 x  9  3 ?
4. Which of the following is equivalent to 3x  5  19 ?
STANDARD 2: Students solve systems of linear equations and inequalities (in two or three variables) by
substitution, with graphs, or with matrices
1. For a wedding, Shereda bought several dozen roses and several dozen carnations. The roses cost $15 per dozen, and
the carnations cost $8 per dozen. Shereda bought a total of 17 dozen flowers and paid a total of $192. How many roses
did she buy?
2. A cashier at a restaurant made the chart below for popular lunch combinations. What is the individual price of
soup?
Soup + Salad = $4.25
Soup + Sandwich = $4.75
Salad + Sandwich = $5.50
3. To connect a VCR to a television set, you need a cable with special connectors at both ends. Suppose you buy a 6
foot cable for $15.50 and a 3 foot cable for $10.25. Assuming that the cost of a cable is the sum of the cost of the two
connectors and the cost of the cable itself, what would you expect to pay for a 4 foot cable?
4. A caterer is planning a party for 64 people. The customer has $150 to spend. A $39 pan of pasta feeds 14 people
and $12 sandwich tray feeds 6 people. How many pans of pasta and how many sandwich trays should the caterer
make?
5. What is the solution to the system of equations shown below?
2 x  y  3z  8

x  6 y  z  0
 6 x  3 y  9 z  24

ALGEBRA II WARMUP CST QUESTIONS
6. What is the solution of the following linear system?
2 x  5 y  3 z  10

3 x  y  4 z  8
5 x  2 y  7 z  12

7. What is the solution to the system of equations shown below?
2 x  4 y  3z  10

3x  y  6 z  15
5 x  2 y  z  25

8. What is the solution of the following linear system?
3x  2 y  3z  11

5 x  2 y  2 z  4
 x  y  z  7

STANDARD 3: Students are adept at operations on polynomials, including long division.
9. Divide: 2 x  7 2 x 4  21x3  35x 2  37 x  46
10. What is the quotient of (4 x3  11x 2  9 x  5)  ( x  4) ?
11. ( x 2  7 x  44)  ( x  4)
12. What is the result of dividing x3  9 x  5 by x  3 ?
13. Multiply: (3x 2  x  4)( 2 x  5) ?
14. Multiply: ( x  3)( x 2  4 x  9) ?
15. Multiply: ( x 2  x  3)( x 2  4 x  2) ?
ALGEBRA II WARMUP CST QUESTIONS
16. Factor: 8 x  1
3
17. (2 x 2  6 x  1)  2(4 x 2  3x  1)
18. What is the sum of 2 x 4  5 x3  8 x 2  x  10 and 8 x 4  4 x3  x 2  x  2 ?
19. (3x3  x  11)  (4 x3  x 2  x)
STANDARD 4: Students factor polynomials representing the difference of squares, perfect square trinomials,
and the sum and difference of two cubes.
1. Factor: 8a 3  c 3
2. Factor: 27 x 3  216
3. Factor: 2 x 3  54
4. Which of the following is the factorization of x 3  8 ?
A) ( x  2)( x 2  4 x  4)
B) ( x  2)( x 2  2 x  4)
C) ( x  2)( x 2  4 x  4)
D) ( x  2)( x 2  2 x  4)
5. The total area pf a rectangle is 4 x 4  9 y 2 . Which factors would represent the length times width?
6. Factor: 16 x 4  1
7. Factor: 32 x 6  2 x 2
ALGEBRA II WARMUP CST QUESTIONS
STANDARD 5: Students demonstrate knowledge of how real and complex numbers are related both
arithmetically and graphically. In particular, they can plot complex numbers as points in the plane.
1. If i  1 , what is the value of i 26 ?
2. i 31 
3. If i  1 , which point shows the location of 5  2i on the plane?
i
A) Point A
A
B) Point B
B
C) Point C
C

D
D) Point D
STANDARD 6: Students add, subtract, multiply, and divide complex numbers
1. Simplify: 16  2i    3  4i 2 
2. Simplify:
 3  6i   2  i  7  
3. Multiply:
3  7i  6  2i 
4. Multiply:
 4  3i 
5. Divide:
6  10i
2i
2
4
6. Divide:
ALGEBRA II WARMUP CST QUESTIONS
4  i
1  4i
STANDARD 7: Students add, subtract, multiply, divide, reduce, and evaluate rational expressions with
monomial and polynomial denominators and simplify complicated rational expressions, including those with
negative exponents in the denominator.
x3
6
 2

1.
x  5 x  3x  10
2.
2x 1
3
 2

x  8 x  16 x  16
3.
x2  x  3
3x


2
x  12 x  32 x  8
2
4. What is the difference
8x  3
7
 2
?
x  2 x  35 x  25
2
5. Which is the simplified form of
3a 2b3c 2
?
(a 1b 2c)3
6. Which is the simplified form of (4 x 2 y 5 )2 ?
7. Simplify
8. Simplify
9. What is
5 x3 y 9
:
20 x 2 y  2
2x2 y
:
6 xy1
20 x 4 8 x 3

?
27 y 2 15 y  5
ALGEBRA II WARMUP CST QUESTIONS
x 2  9 x  22
10. What is the quotient ( x  2) 
?
x 2  121
11.
4x  8
3x  6

x  3x  2 x  1
12.
12 xy 7 x 5 y 2

7 x4
4y
2
STANDARD 8: Students solve and graph quadratic equations by factoring, completing the square, or using the
quadratic formula. Students apply these techniques in solving word problems. They also solve quadratic
equations in the complex number system.
1. What are the solutions of x 2  12 x  61  0
2. Solve for x by using the quadratic formula: 3x 2  x  14
3. Solve for x by completing the square: x 2  2 x  10  0
4. There are two numbers with the following properties.
1. The second number is 3 more than the first number
2. The product of the two numbers is 9 more than their sum.
What are the numbers?
STANDARD 9: Students demonstrate and explain the effect that changing a coefficient has on the graph of a
quadratic function; that is students can determine how the graph of a parabola changes as a, b, and c vary in
2
the equation y  a  x  b   c
1. Which of the following most accurately describes the translation of the graph y   x  3  2 to the graph of
2
y   x  2  2 ?
2
A)
B)
C)
D)
up 4 and to the right 5
down 2 and 2 to the right
down 2 and 3 to the left
up 4 and to the left 2
ALGEBRA II WARMUP CST QUESTIONS
2
2
2. Which of the following sentences is true about the graphs of y  3  x  5   1 and y  3  x  5   1 ?
A)
B)
C)
D)
Their vertices are maximums
The graphs have the same shape with different vertices
The graphs have different shapes with different vertices
One graph has a vertex that is a maximum, while the other graph has a vertex that is a minimum.
STANDARD 10: Students graph quadratic functions and determine the maxima, minima, and zeros of the
function.
1. What are the zeros of y  x 2  13x  40 ?
2. Find the minimum point (vertex) on the graph of f  x   x2  4x  14
3. Find the maximum point (vertex) on the graph of f  x   3x2  12 x  1
4. Which is the graph of y  2  x  1  1 ?
A
2
B
C
STANDARD 11: (11.1) Students understand the inverse relationship between exponents and logarithms, and
use this relationship to solve problems involving logarithms and exponents.
1. What is the solution of the equation:
1
log 3 x  2
2
2. Write the exponential form: logb 37  2
3. If log10 x  2 , what is the value of x?
ALGEBRA II WARMUP CST QUESTIONS
1
4. Which equation is equivalent to log 3  x ?
9
3
1
B)    x
9
3
A)
1
 x3
9
C) 3x 
1
9
1
D) 39  x
(11.2) Students judge the validity of an argument according to whether the properties of real numbers,
exponents, and logarithms have been applied correctly at each step.
1. Which is the first incorrect step in simplifying log 4
Step 1: log 4
4
 log 4 4  log 4 64
64
Step 2:
 1 16
Step 3:
= -15
A) Step 1
4
?
64
B) Step 2
C) Step 3
D) Each step is correct
2. Jeremy, Michael, Shanan, and Brenda each worked the same math problem at the whiteboard. Each student’s work
is shown below. Their teacher said that while two of them had the correct answer, only one of them had arrived at the
correct conclusion using correct steps.
Jeremy’s Work
x3
x3 x 7  7
x
 x10 , x  0
Michael’s Work
x3
3 7
x x  7
x
 x 4 , x  0
Which is a completely correct solution?
A) Jeremy’s work
B) Shanan’s work
Shanan’s Work
x3
x3 x 7  7
x

1
, x0
x4
Brenda’s Work
x3
3 7
xx  7
x
 x4 , x  0
C) Michael’s work
D) Brenda’s work
ALGEBRA II WARMUP CST QUESTIONS
STANDARD 12: Students know the laws of fractional exponents, understand exponential functions, and use
these functions in problems involving exponential growth and decay.
t
 1  300
1. A certain radioactive element decays over time according to the equation y  A   , where A = the number of
2
grams present initially and t = time in years. If 1000 grams were present initially, how many grams will remain after
900 years?
2. Bacteria in a culture are growing exponentially with time as shown in the table below.
Day Bacteria
0
100
1
200
2
400
Which of the following equations expresses the number of bacteria, y, present at any time, t?
A) y  100  2t
B) y  100  2t
C) y  2t
D) y   200  2t
3. If the equation y  2 x is graphed, which of the following values of x would produce a point closest to the x axis?
1
3
5
8
A)
B)
C)
D)
4
4
3
3
STANDARD 13: Students use the definition of logarithms to translate between logarithms in any base.
1. Which of the following is equivalent to log 2 7 ?
A) 7 2
B) 2 7
C) 7 log 2
D)
log 7
log 2
2. Identify the expression that is equivalent to log3 5
A)
log 5
log 3
B) log
5
3
C)
ln 3
ln 5
D) 5ln3
3. Use the change of base formula to identify the expression that is equivalent to log3 10
A)
ln 3
ln10
B) 10 log 3
C) ln
10
3
D)
1
log 3
STANDARD 14: Students understand and use the properties of logarithms to simplify logarithmic numeric
expressions and to identify their approximate values.
1. Evaluate log a 16 given that log a 2  0.4307
ALGEBRA II WARMUP CST QUESTIONS
2. Evaluate log a 18 given that log a 2  0.2789 and log a 3  0.4421
3. Evaluate log a 24 given that log a 2  0.4307 and log a 3  0.6826
4. What is the value of log 2
1
?
16
5. Evaluate: log16 4
STANDARD 15: Students determine whether a specific algebraic statement involving rational expressions,
radical expressions, or logarithmic or exponential functions is sometimes true, always true, or never true.
x 2  16
1. On a recent test, Jeremy wrote the equation
 x  4 . Which of the following statements is correct about the
x4
equation he wrote?
A)
B)
C)
D)
The equation is always true
The equation is always true, except when x = 4
The equation is never true
The equation is sometimes true when x = 4
2. If x is a real number, for what values of x is the equation
A)
B)
C)
D)
3x  9
 x  3 true?
3
all values of x
some values of x
no values of x
impossible to determine
n
3. Given the equation y  x where x > 0 and n < 0 which statement is valid for real values of y?
A) y > 0
B) y = 0
C) y < 0
D) y  0
STANDARD 16: Students demonstrate and explain how the geometry of the graph of a conic section (e.g. the
asymptotes, foci, eccentricity) depends on the coefficients of the quadratic equation representing it.
1. What is the focus of the parabola with the equation 2 x 2  120 y ?
2. What is the directrix of the parabola with the equation y 2  24 x ?
ALGEBRA II WARMUP CST QUESTIONS
3. Find the standard equation of the ellipse with vertices at  4,0  and foci at  3, 0 
A)
x2 y 2

1
7
9
B)
4. Find the vertices of the hyperbola:
x2 y 2

1
16 9
 x  2
16
2
C)
 y  1

9
x2 y 2

1
16 7
D)
x2 y 2

1
16 9
2
1
STANDARD 17: Given a quadratic equation of the form ax 2  by 2  cx  dy  e  0 , students can use the method
for completing the square to put the equation into standard form and can recognize whether the graph of the
equation is a circle, ellipse, parabola, or hyperbola. Students can then graph the equation.
1. What conic does the equation x 2  5 x  10 y  11  0 represent?
2. Identify the graph of 4 x2  4 y 2  6 x  16 y  11  0
A) Circle
B) Ellipse
C) Hyperbola
D) Parabola
3. Match the graph with the correct equation.
A) x 2  16 y 2  4 x  4 y  8  0
B) x 2  16 y 2  4 x  4 y  67  0
3.
C) x  16 y  4 x  64 y  52  0
D) x 2  4 y 2  4 x  16 y  4  0
2
2
4. Match the graph with the correct equation.
x2 y 2
A)

1
16 4
B)
x2 y 2

1
4 16
y 2 x2
 1
C)
16 4
D)
y 2 x2
 1
4 16
4.
ALGEBRA II WARMUP CST QUESTIONS
STANDARD 18: Students use fundamental counting principles to compute combinations and permutations.
1. Natalie wants to create several different 8-character screen names. She wants to use arrangements of the first three
letters of her first name (nat), followed by arrangements of the 5 digits in her zip code (92562). How many different
screen names can she create in this way?
2. Andrew needs to create a 7-character password. He wants to use the arrangements of the first 3 letters of his first
name (and) followed by arrangements of the letters in his middle name (john). How many different password can he
create in this way?
3. Allison went to a buffet restaurant for lunch. She had to choose from four main courses, three side dishes, three
drinks, and two desserts. How many different combinations could she have created?
4. Brooks was going to buy a car. The model he wanted came in three different exterior colors, with two different
interiors (cloth and leather), and three different stereo options. How many ways could Brooks configure the car?
STANDARD 19: Students use combinations and permutations to compute probabilities
1. Jack and Jill are among 12 students who have applied for a trip to Washington D.C. Two students from the group
will be selected at random for the trip. What is the probability that Jack and Jill will be the two students?
2. Alex’s math teacher told the class that if he rolled three dice and they all turned up one, he would let them skip their
homework for a month. What is the probability of this actually happening?
3. Suppose you have the following five books in your backpack: Chemistry, Biology, Calculus, Physics, and
Psychology. Without looking, you pick a book. You replace the book and repeat this process. What is the probability
that the first time you pick Physics or Chemistry, and the second time you picked Physics?
STANDARD 20: Students know the binomial theorem and use it to expand binomial expressions that are
raised to positive integer powers
3
1.  3 y  4  
2.
 4 y  1
3.
 3 y  1
4

5

ALGEBRA II WARMUP CST QUESTIONS
STANDARD 22: Students find the general term and the sums of arithmetic series and of both finite and infinite
geometric series.
1. What is the sum of the infinite geometric series
1 1 1 1
    ...
2 4 8 16
2. Find the ninth term of the arithmetic sequence with a1  4 and d  10 . Assume that the series begins with 1.
9 27 81
3. Find the common ratio of the geometric series: 4,3,  , ,  ,...
4 16 16
STANDARD 24: Students solve problems involving functional concepts, such as composition, defining the
inverse function and performing arithmetic operations on functions.
1. Which expression represents f  g  x   if f  x   x2 1 and g  x   x  3 ?
2. Given f  x   x2 and g  x   x  5 , find g  f  x   .
3. If f  x   x  4 and g  x   3x , find f  g  2   .
STANDARD 25: Students use properties from number systems to justify steps in combining and simplifying
functions.
2
1. If f  x   x2  2x  1 and g  x   3  x  1 , find f  x   g  x  ?
2. Given f  x   6 and g  x   2 x2 1 , find f  x   g  x  ?
3. Given f  x   2 x  4 and g  x   1  3x , find f  x   g  x  ?
STANDARD P.S. 2: Students know the definition of conditional probability and use it to solve for probabilities
in the finite sample spaces.
1. A box contains 4 large red marbles, 8 large yellow marbles, 2 small red marbles, and 6 small yellow marbles. If a
marble is drawn at random, what is the probability that it is red given that it is one of the small marbles?
2. A jar contains 2 red, 3 blue, and 4 green marbles. Niki draws one marble from the jar and then Tom draws a
marble. What is the probability that Niki will draw a green marble and Tom will draw a blue marble?
3. What is the probability that a family will have four children, all of them girls?