美国高中数学测试题13-14MathPlacement (1)

Emma Willard School
Troy, New York 12180
MATHEMATICS PLACEMENT TEST
Purpose: The tests in this booklet are to help determine proper mathematics placement and minimize the need for course changes after the start of the academic year. It is important that the student work independently so that the test will give us a fair representation of her current knowledge and skills. The tests are for placement purposes only. They do not affect admissions decisions in any way. However, it is important to answer questions to the best of your ability in order for the mathematics department to place you properly.
Date student received test:_________________
Name: (please print)_______________________________ Date test taken:___________ Circle grade you are entering at Emma Willard School: 9 10 11 12 PG Phone number (with area code): _________________________________________________ E-mail address: _____________________________________________ (print legibly please) Name of most recent school attended and the city and state/country where it is located.
______________________________________________________________________________ What math course did you take this year, and what is your average grade at the time you are taking this test?
______________________________________________________________________________ Have you taken a full-year course called Geometry? Circle your answer. YES NO
If you were remaining in your current school district or at your current school, what would be the name of the course you would take next year?
______________________________________________________________________________ How do you view yourself as a mathematics student?
Please read the following carefully:
Directions to Parents: Please see that your daughter has a quiet place to complete the test in one sitting. The test has four levels. Each level is designed to be completed in less than an hour. Calculators may be used (EXCEPT on the Level One test), but texts and notes should not be used. We do not recommend extensive review prior to taking the test. It is meant to reflect your daughter’s accessible knowledge of her retained mathematics knowledge. It is not to your daughter’s advantage to obtain help on this test since proper placement is contingent on accurate assessment of her current knowledge of mathematics.
Directions to the Student: This booklet contains four tests, Level One, Level Two, Level Three, and Level Four. Do as much of all four parts of the tests as you can. The sooner you complete the tests and return them to the school, the sooner we can properly place you and start the process of creating your schedule for the fall. Print out the levels you wish to complete. Be sure to check that all diagrams and problems have printed correctly. Please mail in all parts that you have completed as soon as possible after you complete them. You may use a calculator except on Level One. It is important that you give these tests your serious consideration as they will be the main factor in determining the math course in which you are enrolled.Please show your work neatly next to the problem (including scratch work) as it is useful in our evaluation of your methods and skills. Do not use extra paper, and simplify all answers. Showing your work helps us see where your mistakes were and adds to proper assessment of your understanding and hence proper placement.
Calculator Use: It is important to note that all Emma Willard mathematics students will use the Texas Instruments TI-84 PLUS Graphing Calculator in all of our courses. These may be purchased at cost in our school store. While we teach students to use their calculators proficiently, we also stress the need to recognize problems that do not need a calculator and may require students to solve those problems without one. For this reason, on these placement tests we ask that you do as many problems as possible without the use of a calculator. On the Level One test, NO calculators are allowed.
Please complete as many questions as you can on all levels of the four tests. For example, if you are an entering freshman and have only completed an eighth grade math course, and are only capable of completing a few problems on the Level One part, THIS IS FINE. If you are an entering junior, and hope to have proper placement, complete as much as you can of all four tests.
The purpose of these tests is not to make a judgment about your mathematical ability. It is to assess how well you have been prepared for the sequence of courses
at Emma Willard. We strive to place new students in the course in which they will find the most appropriate challenge.
4/12
Revised
Level One Test
(total points = 108)
In all questions, SHOW YOUR WORK in the space under the question and place your final answer on the line provided to the right. Do NOT use a calculator on this test. 1. A suit that is regularly sold for $120 is now advertised on sale at 30% off. What is the sale price of the suit?
1. __________________
(3 pts) 2. A school’s ratio of boys to girls is 4:5. If there are 360 students, how many girls attend the school?
2. __________________
(3 pts) 3. Evaluate 2492x x −− when x = -1.
3. __________________ (3 pts)
In 4-8, simplify the expression.
4. 3678x x −+−−
4. __________________ (3 pts)
5. 325()()x x x −−−
5. __________________ (3 pts)
6. 343762c c c +−
6. __________________
(3 pts)
7. a a a 462
⋅
7. __________________
(3 pts)
8. (3x )3
8. __________________
(3 pts)
In 9-12 solve the equation.
9. −+=−369d
9. _________________
(3 pts)
10. 25652x x +=−
10. _________________
(3 pts)
11. The formula 9532F C =+ converts Celsius temperature (C) to Fahrenheit (F). What is the Fahrenheit equivalent of 20o C?
11. _________________
(3 pts)
12. Solve for C in the formula 9
325
F C =+.
12. _________________
(3 pts)
13. Multiple Choice. To rent a truck for a day, a driver pays a $15 fee. She pays an additional 18 cents for each mile she drives. If the total cost in dollars is c and she drives d miles, then
A) d c =+15018. B) d c =+01815. C) c d =+15018. D) c d =+01815.
13. _________________
(3 pts)
14. Solve the equation for y . 236x y −=
14. _________________
(3 pts)
15. Is (1, 3) a solution of y x =−25? (Support your answer with work below.)
15. _________________
(3 pts)
16. Graph the line y x =−2
5
3 using slope & y-intercept. Do not
use a table of values. (1 square = 1 unit)
16. (3 pts)
x
a. Find its numerical slope.
17. a. __________________ (3 pts)
b. __________________ (3 pts)
18. A line passes through the points (4, -2) and (-9, 8). a. Find the slope of the line. Show work. b. Write an equation for this line.
18. a. __________________
(3 pts)
b. __________________ (3 pts)
19. Given the slope of a line is 5
3 and the point (-8, 2) is on the line. Write an equation for the line.
20. Solve the system:
6222=−=+y x y x
19. _________________ (3 pts)
20. _________________ (3 pts)
21. Multiply: ()()2135x x +−
21. _________________
(3 pts)
22. Simplify: ()x +82
22. _________________
(3 pts)
In 23-26, factor (using integers) the polynomial expression.
23. Factor: 51532ab a b −
23. _________________
(3 pts)
24. Factor: x 225−
24. _________________
(3 pts)
25. Factor: x x 2412+−
26. Factor: 12x 2−5x −2
25. _________________
(3 pts)
26. _________________
(3 pts)
27. Solve for x: ()()x x −+=530
27. _________________
(3 pts)
28. Solve the equation: 35
24t =
+
28. _________________
(3 pts)
in simplified radical form.
29. _________________
(3 pts)
30. Solve for x : 270x −=
30. _________________
(3 pts)
31. Solve for x : 24137x +=
31. _________________
(3 pts)
32. Simplify: 23()46()
32. _________________
(3 pts)
+
33. _________________
(3 pts)
34. Use the quadratic formula, x b b ac
a
=
−±−242 to solve the equation: 2602
x x −−=.
34. _________________
(3 pts)
Level Two Test
(total points = 100)
In all questions, SHOW YOUR WORK in the space under the question and place your final answer on the line provided to the right. You may use a scientific or graphing calculator.
1. Measure of angle x = ?
1. __________________
(2 pts) 2. In the figure, ||m n and 50.m x ∠=°
Find m y ∠.
2. _________________ (2 pts)
3. ?m y ∠=
3. _________________ (2 pts)
4. In the figure, 90m Q ∠=°and 40m SRT ∠=°. ?m P ∠=
4. _________________ (2 pts)
5. If is parallel to AB CD , then ?m x ∠=
5. _________________ (2 pts)
15°
x
55°
65°n
m x y 70°55°y
40°
T S R Q P
D
C B A 35°80°
x
6. In the plane figure, MN=NT=TV. Find the measure of TMN ∠.
6. _________________
(2 pts) 7. Which of the following reasons can always be used to prove two triangles congruent? Circle all that apply. No partial credit. (2 pts)
SSS AAA SAS SSA AAS ASA
8. Given the figure with DE BC , which of the following
proportions is not true? No partial credit.
8. _________________ (2 pts) A. 456x = B. 4512y = C. 456x = D. 496
x x =+
9. Which one of the following sets of points is not collinear?
A. B & D
B. D, A & H
C. D, B & G
D. G, C &B
9. __________________
(2 pts) 10. Refer to the diagram in #9. The intersection of plane P and plane Q is:
10. _________________
(2 pts) A. line KC B. line AC C. line GC D. line PQ E. point C
N
35°V T
M
11. Given ABC Δ such that 50m A ∠=°and 64m B ∠=°. The longest side of the triangle is
A. AB
B. AC
C. BC
D. all sides are equal
E. not enough information is given
11. _________________
(2 pts)
12. Two similar triangles have areas in the ratio 4:9. The ratio of a pair of corresponding sides is
12. _________________
(2 pts) A. 4:9 B. 16:81 C. 9:4 D. 2:3
13. ABCD is a parallelogram. Solve for x.
13. _________________
(2 pts)
14. If the side of an equilateral triangle measures 6, then what is the measure of an altitude of the triangle?
14. _________________
(3 pts)
15. What is the radius of circle O if PQ=12.
15. _________________
(3 pts)
16. Find the sum of the interior angles in a hexagon.
16. _________________
(3 pts)
17. A 6 foot ladder leans against a wall. Its top touches a point on the wall 4 feet above the floor. How many feet is the bottom of the ladder from the base of the wall? Draw a diagram and put your answer in simplest radical form. 17. _________________
(3 pts)
18. A rhombus has diagonals of 20 and 16. What is the perimeter of the rhombus? Draw a diagram and put your answer in simplest radical form. 18. _________________
(3 pts)
19. In right triangle ABC, AC=12. What is the perimeter of the triangle? 19. _________________
(3 pts)
20. Find the area and perimeter of the rectangle below.
20.
Area = ______________
(2
pts) Perimeter = __________
(2 pts)
B
21. Write the following proof either in a two-column format or written in paragraph form. (6 pts)
Given: DA bisects BDC ∠
BD=DC Prove: AB=AC
22. What is the area of ADB Δin square units?
22. _________________ (3 pts) 23. What is the area of parallelogram ABCD in square units?
23. _________________
(3 pts)
B
C
24. What is the area of a circle whose circumference is 12π.
24. _________________
(2 pts)
25. Square ABCD is inscribed in circle O. OA=4. What is area of the shaded region in square units?
25. _________________
(3 pts)
26. In the circle below, what is the degree measure of arc AB?
26. _________________
(3 pts)
27. The edge of a cube is 2 cm. Find the total surface area of the cube. Include units of measurements in your answer.
27. _________________
(3 pts)
28. The diameter and height of a cone both measures 4 cm. What is the volume of the cone?
28. _________________
(3 pts)
A
29. Find the distance between the points (-2, 3) and (1, -4). 29. _________________
(3 pts)
30. Write an equation of the line perpendicular to
1
5
2
y x
=+going
through the point (3, 4). 30. _________________
(3 pts)
31. Given the line segment with endpoints (1, -4) and (3, 2), determine the coordinates (x, y) of the midpoint. 31. _________________
(3 pts)
32. List the letters of the answer(s) that give (s) you enough information to determine that the quadrilateral is a parallelogram. No partial credit.
A. Both pairs of opposite sides are parallel.
B. Two pairs of consecutive sides are congruent.
C. Diagonals are congruent.
D. Consecutive angles supplementary. 32. _________________
(3 pts)
33. Name the vector from A(3, 5) to B(7, 0). 33. _________________
(2 pts)
In 34-39, define the geometric term in your own words, a diagram alone is NOT sufficient.
(2 pts each)
34. Complementary Angles
35. Isosceles Right Triangle
36. Alternate Interior Angles
37. Altitude of a Triangle
38. Rhombus
39. Median of a Triangle
Level Three Test
(total points = 101)
In all questions, SHOW YOUR WORK in the space under the question and place your final answer on the line provided to the right. You may use a scientific or graphing calculator.
1. Write an equation of a line perpendicular to 523x y +=and
containing the point (4, -1).
1. __________________ (3 pts)
2. State the radius and center of the circle with the equation: 22(3)16x y +−=
2.
Radius:______________
(1 pts)
Center:______________
(2 pts)
3. State the domain & range of the relation graphed below.
3.
Domain:_____________
(1 pts)
Range:_______________
(1 pts)
4. Is the relation in #3 a function? Explain why or why not?
4. __________________
(3 pts)
5. Identify whether the function 2()3g x x =− is an even function, odd function or neither. 5. __________________
(3 pts)
(2, -1) x y
6. Given the point (1, -3) is on the graph ()y f x =, what point is on the graph of.
a. 3()y f x =
b. (3)3y f x =++
c. 2()3y f x =−+
6.
a. __________________
(3 pts)
b. __________________
(3 pts)
c. __________________
(3 pts)
7. Let 2()1f x x =+and ()23g x x =−. Find (())g f x and simplify.
7. __________________
(3 pts)
8. If ()32f x x =−, then 1()f x −=_?_
8. __________________
(3 pts)
9. Find the EXACT (no decimals) x -intercept(s), y -intercept, and vertex of the parabola 22810y x x =−−algebraically. Show your work below. 9.
x -int:________________
(2 pts)
y -int:________________
(1 pts)
vertex:_______________
(3 pts)
10. Simplify over the set of complex numbers.
b. 2(12)i −
10.
a. __________________
(3 pts)
b. __________________
(3 pts)
11. Simplify: 0235
43
()x y z x yz
−−. Write your answer with positive exponents.
11. _________________
(3 pts)
12. Simplify: 23
6427−⎛⎞
⎜⎟
⎝⎠
. Write your answer as a simplified
fraction.
12. _________________
(3 pts)
13. Graph the function 1
()f x x
=
. Be certain to plot at least three points and include any asymptotes as dashed lines.
(3 pts) (1 square =1 unit)
14. Write 239=in logarithmic form.
14. _________________
(3 pts)
x
15. Evaluate 2log 8.
15. _________________
(3 pts)
16. Solve 450x =. Round to the nearest thousandth.
16. _________________
(3 pts)
17. Write as a single logarithm with base 6:1662log 9log 5+.
17. _________________
(3 pts)
18. Graph. 2()log f x x =. Be certain to plot at least three points and include any asymptotes as dashed lines.
(3 pts)(1 square =1 unit)
19. Write an equation for a polynomial function that has the roots 0, -5 and 1/3. There is no need to simplify your answer.
19. _________________
(3 pts)
20. What is the quotient when 321110x x x +−+is divided by 2x −? 20. _________________
(3 pts)
x
21. Calculate θto the nearest tenth of a degree.
21. _________________
(3 pts)
22. Give the exact value (no decimal) of sin 240°.
22. _________________
(3 pts)
23. Solve 1
cos 2
x =− for x in degrees where 0
360x °≤<°.
23. _________________
(3 pts)
22. State the amplitude and period of the graph of 3cos(2)y x =.
22.
Amp. =______________
(1 pts)
Period=______________
(2 pts)
23. Graph at least one full period of ()2tan f x x =. Be certain to lines.
23. (3 pts)
125θ
24. Graph at least one full period of ()sin f x x =−. Be certain to plot at least three points and include any asymptotes as dashed lines.
24. (3 pts)
25. Give the exact solution (no decimals) of sin 0x =for x in radians 02x π≤<.
25. _________________
(3 pts)
26. Find the exact radian value for 1sin (−.
26. _________________
(3 pts)
27. Give the exact solution (no decimals) of cos x =for x in radians 02x π≤<.
27. _________________
(3 pts)
28. Give the exact solution(s) (in radians) of 1tan (1)x −−=.
28. _________________
(3 pts)
Level Four Test
(total points = 75)
In all questions, SHOW YOUR WORK in the space under the question and place your final answer on the line provided to the right. You may use a scientific or graphing calculator.
1. Solve for x (no decimals):ln 4x =.
1. __________________ (2 pts)
2. To the nearest tenth, solve for x in the triangle shown below.
2. __________________
(3 pts) 3. Use the Law of Cosines to solve for the missing side. x
5
8
60°
3. __________________
(3 pts)
4. Find the sum of the infinite geometric series: 111
......2832+++
4. __________________
(2 pts)
5.Calculate the 145th term in the arithmetic sequence: 6, 10, 14,…
5. __________________
(2 pts)
12x 30°
100°
6. Find 4
353
lim 7n n n
→∞−
6. __________________ ( 3 pts)
7. Find 5435432
6871
lim 39106n n n n n n n n
→∞−+−+−−+
7.__________________ (3pts)
8. A nearby ice cream shop has 31 flavors of ice cream to choose from. In how many ways can you choose a bowl of three flavors?
8. __________________
(3 pts)
9. Write an equation for the ellipse graphed below. (1 square =1 unit)
9. __________________
(3 pts)
10. Solve showing steps: 5557log log 2log 80x +=. Express answer as a decimal rounded to the hundredths. 10.__________________ (3 pts)
x
11. Women’s heights are normally distributed with a mean of 64 inches and a standard deviation of 2.5 inches. Please use the standard normal distribution table on pages 7 and 8 to answer the following questions.
a. Find the women’s height at the 80th percentile.
b. What is the standard value (z-score) of a women’s height of 61 inches?
c. What is the probability that a women has a height of 67 inches or more? 11.
a.__________________
(3 pts)
b.__________________
(3 pts)
c.__________________
(3 pts)
12. If the junior class has 80 members, then how many different ways can the class choose four members to be president, vice president, secretary, and treasurer? 12.__________________
(3 pts)
13. The weather forecaster says that there is a 10% chance of rain for each day.
a.What is the probability that it rains at least one of the next
three days?
b.What is the probability that it will rain exactly 2 days of
the next 14 days? Round to thousandths. 13.
a.___________________
(3 pts)
b.___________________
(3pts)
14. Given vectors [1,3]u =−and [5,2]v =−, make an accurate sketch on the grid of 2u v −. Include the resultant vector in your sketch. (1 square = 1 unit)
(3 pts)
15. Prove this trigonometric identity. Show all steps.
cos cos 2tan 1sin 1sin θθθθθ−=−+−
(3 pts)
16. Given the identity:
2222cos 2cos sin 12sin 2cos 1x x x x x =−=−=−.
Find the EXACT radian solutions for cos 23sin 2x x =−+ where 02x π≤<. 16. _________________
(3 pts)
17. A box with a square base and no top has volume 8 cubic meters. The material for the base cost $8 per square meter, and the material for the sides cost $6 per square meter. Express the cost, C, of the materials to make the box as a function of the width, w, of the bases. 17. _________________
(3 pts)
18a. Write an equation for a hyperbola with these points: Center at(0,2)
One vertex at (0,3)
−
One focus at (0,9)
b. Write the asymptote equations for the hyperbola. 18a._________________
(3 pts)
b. __________________
(3 pts)
19. 9 students are chosen at random for the prom committee from a group that includes 5 sophomores, 12 juniors, and 10 seniors. What is the probability that the committee will have 1 sophomore, 5 juniors, and 3 seniors on it? 19._________________
(3 pts)
20. Does the graph of 25y xy −= have y axis − symmetry?
Justify your answer algebraically showing work.
20.__________________ (3 pts)
21. An airplane flying northeast with a speed of 500 miles per
hour encounters a 100 mile per hour wind blowing to the west.
a. What is the new speed of the plane rounded to the
hundredths place?
b. To the nearest whole degree, How many degrees will the wind blow the plane off its northeast course?
21. a.___________________ (3 pts)
b.___________________ (3pts)。