High School – Mathematics – Exam 2
2012-13 Governor’s Cup Practice Questions
Student Instructions:
 Write your complete ID code on your answer sheet.
 For each question, choose the BEST answer. If you change your answer, erase well.
1.
This part time mathematician is credited with discovering geometry independently of Descartes. A lawyer
by profession, he dabbled in math and it is generally accepted that he was not interested in publishing his
work as he would make casual remarks in the margins of books instead of writing formal proofs. Who was
this French lawyer who left the following note in the margin regarding the proof of xn + yn = zn that has no
integer solutions for n > 2?
"I have discovered a truly marvelous proof of this, which however the margin is not large enough to contain."
A.
B.
C.
D.
2.
Blaise Pascal
Simeon-Denis Poisson
Pierre de Fermat
Lazare Carnot
Which of the following numbers has the greatest value when converted to a decimal number?
A.
22A11
B.
11346
C.
DE20
D.
1B312
Questions 3 and 4 refer to the figure pictured to the right.
3.
Calculate the area of the shaded region inside the rectangle shown to the
right.
A.
170 m2
B.
160 m2
C.
150 m2
D.
140 m2
4.
What is the perimeter of the shaded region in the figure pictured to the
right?
A.
39 + √29 + √109 m
B.
39 + 2√29 + √109 m
C.
42 + 2√29 + √109 m
D.
39 + √29 + √34 m
5.
Find the oblique asymptote for the following rational function.
𝑓(𝑥) =
A.
B.
C.
D.
𝑥2 − 𝑥 − 6
𝑥+1
y=x+2
y=x–2
y=x+1
y=x–1
Questions 6 and 7 refer to the graph pictured below.
2012-13 Governor’s Cup Practice Questions – Mathematics – Exam 2 - Page 1
6.
The graph shown below can be modeled by the following quadratic regression equation.
y = 0.0129x2 – 1.21x + 28.67
What is the absolute difference between the predicted value of fatal crashes per 100 million miles driven
obtained from the regression equation and the
actual value shown on the graph for the drivers
of age 45? (Round the regression equation
computation to the nearest tenth)
A.
1.7
B.
1.5
C.
0.3
D.
0.1
7.
Which of the following statements best
describes the difference between the actual
value and the predicted value obtained in
question 6?
A.
The difference between the predicted
value and the actual value is very small, 0.1, which is acceptable for a regression equation with a
correlation coefficient near 0.9.
B.
Obtaining a difference of 0.3 between the predicted and the actual value is within ± 15%, an
acceptable range for quadratic regression equations with R2 = 0.799.
C.
The visual representation, though it appears to be a parabola, may be flawed because the scale on
the x-axis does not have equally spaced divisions which resulted in a regression equation that
differs from the graph and in turn yielded a much lower prediction for the age 45 drivers.
D.
With a very low correlation coefficient, 0.33, the large difference in the two values can be expected.
8.
Y is directly proportional to the square of x and inversely proportional to the square root of z. Determine k,
the constant of proportionality if x = 5 and z = 16 when Y = 10.
A.
1.6
B.
0.625
C.
16
D.
6.25
9.
Use the Venn diagram to the right to answer how many elements
belong to set A but not set B?
A.
7
B.
13
C.
8
D.
12
10.
Match the function to the description.
x.
f(x) = 3x4 – 2x2 + 5
y.
f(x) = 2x5 – 7x3 + 4x
z.
f(x) = x3 + x2
A.
B.
C.
D.
1.
2.
3.
Function is odd
Function is neither odd nor even
Function is even
x – 1, y – 3, z – 2
x – 1, y – 2, z – 3
x – 3, y – 2, z – 1
x – 3, y – 1, z – 2
2012-13 Governor’s Cup – Mathematics – Practice Exam 2- Page 2
11.
𝑓
Let 𝑓(𝑥) = √9 − 𝑥 2 and 𝑔(𝑥) = √𝑥. Find the domain of 𝑔 (𝑥).
A.
B.
C.
D.
[0, 3]
(0, 3)
[0, 3)
(0, 3]
12.
Find the volume for the cone to the nearest hundred that has a radius of
21meters and a height of 54 meters, as shown to the right.
A.
74, 900 m3
B.
74, 800 m3
C.
24, 900 m3
D.
24, 800 m3
13.
Solve the system of equations for x and y. What is the value of x + y?
x – 3y = -58
3x – y = -70
A.
B.
C.
D.
6
-6
-32
32
14.
The mnemonic device, “A Smart Trig Class”, is used to remember which trigonometric functions are
positive in each quadrant. Which of the following trigonometric functions is positive in quadrant III?
A.
secant
B.
cotangent
C.
cosecant
D.
no positive trigonometric functions in the quadrant III
15.
To the nearest hundredth radian approximate all angles, Ө, in the interval [0, 2π) that satisfy sin Ө = 0.4195.
A.
24.80, 155.22
B.
0.43, 0.86
C.
0.86, 2.71
D.
0.43, 2.71
16.
Find the amplitude, period, and phase shift for y = cos (2x – π) + 2.
A.
amplitude = 2, period = π, phase shift = 𝜋⁄2
B.
amplitude = 1, period = 𝜋⁄2, phase shift = 𝜋⁄2
C.
amplitude = 1, period = π, phase shift = 𝜋⁄2
D.
amplitude = 1, period = 𝜋⁄2, phase shift = π
17.
Pictured to the right is a square based pyramid with slant height s and altitude
h. What is the lateral area of the faces expressed in terms of s and Ө, the
angle formed by h and s?
A.
S.A. = 4s2 sin Ө
B.
S.A. = 4s2 cos Ө
C.
S.A. = 4s sin Ө
D.
S.A. = 4s cos Ө
2012-13 Governor’s Cup – Mathematics – Practice Exam 2- Page 3
18.
A ping pong paddle comes with 6 plastic balls in a box that is a rectangular prism. The box is just big
enough to hold the 6 balls. The radius of a ball is 21 millimeters. What is the volume of the air in the box
surrounding the balls?
A.
222,264 – 37,044π mm3
B.
222,264 – 74,088π mm3
C.
444,528 – 74,088π mm3
D.
444,528 – 37,044π mm3
19.
How many distinct outcomes are possible using the
combination of a spinner with four equally likely
outcomes, a fair coin, and another spinner with 3
equally likely outcomes?
A.
9
B.
24
C.
36
D.
48
20.
Match the quadrant with the given conditions.
x.
cos Ө > 0 and sin Ө < 0
y.
csc Ө > 0 and cot Ө < 0
z.
sec Ө < 0 and tan Ө > 0
A.
x – 3, y – 1, z – 2
B.
x – 1, y – 2, z – 3
C.
x – 2, y – 3, z – 1
D.
x – 3, y – 2, z – 1
1.
2.
3.
Quadrant II
Quadrant III
Quadrant IV
21.
A math quiz consists of 8 true-or-false questions. If a student guesses on all eight questions what is the
probability that the student will guess 7 correct answers and 1 incorrect answer?
A.
0.015625
B.
0.031250
C.
0.062500
D.
0.390625
22.
Solve the following equation for x.
A.
− 4⁄99
B.
− 44⁄99
44⁄
C.
99
4⁄
D.
99
23.
Sequence A = {(n-1)(n-2)(n-3)} for all positive integers. What is the sum of the first five terms?
A.
0
B.
6
C.
30
D.
90
24.
Simplify (-2 + 2i)10 to the form a + bi. What is the sum of a and b?
A.
1,024
B.
-2,048
C.
-17,408
D.
-32,768
2-100x = (0.5)x-4
2012-13 Governor’s Cup – Mathematics – Practice Exam 2- Page 4
(y + 1)2 = -12(x + 2)
25.
Find the vertex and focus of the following parabola.
A.
vertex (-2, -1), focus (5, -1)
B.
vertex (-2, -1), focus (-5, -1)
C.
vertex (2, 1), focus (5, -1)
D.
vertex (2, 1), focus (-5, -1)
26.
Find the zeroes of f(x)
A.
0, 1
B.
0, -2
C.
0, 2
D.
1, 2
27.
Around 11:00 a.m. a 260 feet tall building casts a shadow that measures
70 feet from the base of the building as shown in the drawing to the right.
What is the length of the dotted line shown in the drawing?
A.
50√29 feet
B.
270 feet
C.
25√29 feet
D.
330 feet
28.
A parking lot is in the shape of a right triangle. The shorter leg measures 160 meters. The hypotenuse is 50
meters longer than the length of the longer leg. What is the length of the longer leg of the triangle?
A.
281 meters
B.
261 meters
C.
251 meters
D.
231 meters
29.
The only requirement for map coloring is that two regions
sharing a common boundary have different colors. If two
regions meet at only one point, they do not share a common
boundary. What is the minumum number of colors needed to
color the map shown to the right?
A.
8
B.
5
C.
4
D.
3
30.
Worldwide there are approximately 132 million births per year. Express this quantity in births per second
rounded to the nearest tenth.
A.
0.4 births per second
B.
252.0 births per second
C.
100.8 births per second
D.
4.2 births per second
31.
What is the number of terms in the finite arithmetic sequence with first term 5, last term 11, and sum of
224?
A.
27
B.
28
C.
29
D.
30
= -x2e-x + 2xe-x.
2012-13 Governor’s Cup – Mathematics – Practice Exam 2- Page 5
32.
What is the first derivative of 𝑦
A.
𝑦 = −5𝑥 −6 +
B.
𝑦 = 5𝑥 −6 +
C.
𝑦 = −5𝑥 −6 −
D.
33.
𝑦 = 5𝑥 −6 −
= 𝑥 −5 −
1
1
𝑥
?
𝑥2
1
𝑥2
1
𝑥2
1
𝑥2
Find the area between the following curve and the x-axis over the interval [0, 5].
y = -x2 + 9
A.
B.
C.
D.
5⁄
9
98⁄
3
10⁄
3
10⁄
9
34.
Find tan (A - B) given that cos A = − 5⁄13, with A in quadrant II, and sin B = 15⁄17 with B in quadrant II.
−171⁄
A.
220
−3
B.
⁄20
3⁄
C.
20
−21
D.
⁄220
35.
For matrix A below find the cofactor A32.
−9
2
−12 6
Matrix A = |
2
18
3
3
A.
B.
C.
D.
9 4
0 1
|
4 −3
0 5
-399
-33
112
-567
log4 (5 + x) = 3
36.
Solve the following equation for x.
A.
-2
B.
29
C.
59
D.
76
37.
What is the length of the vertical transverse axis of the hyperbola with equation 25 − 49 = 1?
A.
5
B.
7
C.
10
D.
14
𝑦2
2012-13 Governor’s Cup – Mathematics – Practice Exam 2- Page 6
𝑥2
38.
Which of the following is an equation for an ellipse that has a horizontal major axis of length 8 and a minor
axis of length 5?
A.
B.
C.
D.
4𝑥 2
16
𝑥2
+
25
4𝑥 2
25
𝑥2
16
+
25
𝑦2
=1
=1
+
+
𝑦2
4
𝑦2
4
4𝑦 2
25
=1
=1
39.
What is the absolute difference between the geometric mean and the arithmetic mean of the numbers 12 and
48?
A.
6
B.
4
C.
2
D.
0
40.
What is the limit of the following function as x approaches 1?
𝑥 3 − 𝑥 2 + 3𝑥 − 3
𝑓(𝑥) = 4
𝑥 − 𝑥3 + 𝑥 − 1
A.
0
B.
1
C.
2
D.
3
41.
Write the vector w in the form of a + bi given that the magnitude of vector w = 5 and the angle created by
the vector in quadrant I and the x-axis is 45o?
A.
𝒘=
B.
𝒘=
C.
𝒘=
D.
𝒘=
5√2
𝑖
2
5√3
2
−
5√2
𝑗
2
5
𝑖+ 𝑗
2
5√3
𝑖+
𝑗
2
2
5√2
5√2
𝑖+
𝑗
2
2
5
42.
Which of the following polar equations produces a cardioid?
A.
r = 2 + 3 cos (Ө)
B.
r = 2 + 2 cos (Ө)
C.
r = 8 + 2 cos (Ө)
D.
r = 3 + 2 sin (Ө)
43.
Let f(x) = 5x – 7 and g(x) = 3x2 – x + 2. Find g(f(3).
A.
73
B.
186
C.
172
D.
65
2012-13 Governor’s Cup – Mathematics – Practice Exam 2- Page 7
44.
What is the remainder when f(x) = 3x5 – 4x3 + x + 5 is divided by p(x) = x3 – 2x + 7?
A.
-21x2 + 5x – 9
B.
21x2 – 5x + 9
C.
10x2 – 3x + 7
D.
-10x2 + 3x – 7
45.
Simplify the complex fraction
A.
B.
C.
D.
46.
13
17
16
17
13
17
16
17
−
+
+
−
16
17
13
17
16
17
13
17
7−𝑖
3−5𝑖
to the form a + bi.
𝑖
𝑖
𝑖
𝑖
Express 2 sin 5Ө cos 3Ө as a sum of trigonometric functions.
A.
cos 8Ө + sin 2Ө
B.
sin 8Ө + sin 2Ө
C.
sin 8Ө + cos 2Ө
D.
𝑠𝑖𝑛 8Ө + 𝑠𝑖𝑛 2Ө
2
47.
Calculate the obtuse angle between the vectors ⟨4, −7⟩ and ⟨−6, 1⟩ to the nearest tenth of a degree.
A.
50.8
B.
69.7
C.
110.3
D.
129.2
48.
Calculate the integral ∫
ln(2) 𝑥
𝑒 𝑑𝑥
ln(4)
A.
B.
C.
D.
6
2
-2
4
49.
Let v = i - 4j - 3k and w = -6i + 2j – k. Find the cross product, v x w.
A.
10i + 19j – 22k
B.
14i + 19j + 12k
C.
-14i – 9j + 12k
D.
-10i – 19j + 22k
50.
Consider the set of positive integers less than 10. What is the number of proper subsets for this set of
numbers?
A.
255
B.
256
C.
511
D.
512
2012-13 Governor’s Cup – Mathematics – Practice Exam 2- Page 8
Mathematics - High School –Exam 2
2012-13 Governor’s Cup Practice Questions
1. C
26. C
2. D
27. A
3. A
28. D
4. B
29. C
5. B
30. D
6. A
31. B
7. C
32. A
8. A
33. C
9. B
34. D
10. D
35. D
11. D
36. C
12. C
37. C
13. B
38. D
14. B
39. A
15. D
40. C
16. C
41. D
17. A
42. B
18. C
43. B
19. B
44. A
20. A
45. C
21. B
46. B
22. A
47. D
23. C
48. C
24. D
49. A
25. B
50. C