Benchmark # 4 2014_ 2015
PLEASE Do NOT write on this test
Part 1: No Calculator
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1) Evaluate the following: 𝑠𝑖𝑛
(A) 0
(B)
11𝜋
4
√3
2
(C) −
√2
2
(D)
√2
2
(C) −
√2
2
(D) 1
(C) −
√2
2
(D) − 2
(E) −
(C) 2
(D) – 2
(E)
(C) 2
(D)
(E) −
√3
2
2) Evaluate the following: 𝑐𝑜𝑠3𝜋
(A) 0
(B)
√3
2
3) Evaluate the following: 𝑐𝑜𝑠 −
(A)
1
(B)
2
7𝜋
3
√3
2
4) Evaluate the following: 𝑠𝑒𝑐 −
5) Evaluate the following: 𝑐𝑜𝑡
2√3
3
5𝜋
3
6) Evaluate the following: 𝑡𝑎𝑛 −
√3
3
(E) −
2 √3
3
7𝜋
6
(B) −√2
(A) √2
√3
2
4
(B) −√2
(A) √2
1
𝜋
(B) −√2
(A) √2
(E) – 1
(C)
√3
3
(D)−
√3
3
(E) −
2 √3
3
√2
)
2
7) Find 𝐴𝑟𝑐𝑠𝑖𝑛 (
(A) −
𝜋
(B) −
6
𝜋
(C)
3
2𝜋
(D)
3
5𝜋
(E) answer not here
3
8) What are the solutions to: 2𝑠𝑖𝑛2 𝑥 + 3𝑠𝑖𝑛𝑥 + 1 = 0
(A)
7𝜋 11𝜋
6
,
6
(B)
𝜋 3𝜋 5𝜋
6
,
2
,
6
(C)
7𝜋 𝜋 11𝜋
6
, ,
2
6
(D)
7𝜋 3𝜋 11𝜋
6
,
2
,
6
(E) No solutions
9) What are the solutions to: 3𝑠𝑒𝑐 2 𝑥 − 4 = 0
(A)
𝜋
(B)
6
𝜋 5𝜋 7𝜋 11𝜋
6
,
6,
,
6
,
6
(C)
𝜋 2𝜋 4𝜋 5𝜋
3
,
3
,
3
,
3
(D)
10) Identify the following function as: (A) 𝑓(𝑥) = 𝑡𝑎𝑛−1 (𝑥)
𝜋 5𝜋
6
,
6
(E)
𝜋 5𝜋 7𝜋 11𝜋
6
,
6
,
6
,
6
(B) 𝑓(𝑥) = 𝑐𝑜𝑠 −1 (𝑥)
(C) 𝑓(𝑥) = 𝑠𝑖𝑛−1 (𝑥)
(D) 𝑓(𝑥) = tan(𝑥)
(E) 𝑛𝑜𝑛𝑒 𝑜𝑓 𝑡ℎ𝑒𝑠𝑒
11) What is the range of # 11 above?
𝜋
𝜋
(A) [− 2 , 2 ]
𝜋
𝜋
(B) (− 2 , 2 )
12) Identify the following function as:
(C) [0, 𝜋]
(D) (0, 2𝜋)
(E) none of these
(A) 𝑓(𝑥) = 𝑡𝑎𝑛−1 (𝑥)
(B) 𝑓(𝑥) = 𝑐𝑜𝑠 −1 (𝑥)
(C) 𝑓(𝑥) = 𝑠𝑖𝑛−1 (𝑥)
(D) 𝑓(𝑥) = tan(𝑥)
(E) 𝑛𝑜𝑛𝑒 𝑜𝑓 𝑡ℎ𝑒𝑠𝑒
13) What is the domain of # 13 above?
𝜋
𝜋
(A) [− 2 , 2 ]
𝜋
𝜋
(B) (− 2 , 2 )
(C) [0, 𝜋]
14) Constructed response. See additional sheet for details
(D) (0, 2𝜋)
(E) all reals
BLANK UNIT CIRCLE
14) Constructed Response: Graph and find all characteristics of the following
𝟏
𝒇(𝒙) = − 𝒔𝒊𝒏 ( 𝒙 − 𝝅) + 𝟑
𝟐
Amplitude
Frequency
Period
Critical Values
Phase Shift
Vertical Shift
15) Which matrix equals 2 [
2 −7
0 −5
]+ [
]
−4 3
−3 6
4 −4
A) [
]
−2 18
B) [
−24 60
D) [
]
−6 30
E) [
−1 0
16) What is the product of [−2 1
3 2
4
1
3 ] and [0
−1
5
19 −2
A) [ 13 2 ]
−2 −3
19 −2
B) [ 13 −6]
−2 −3
19
D) [13
2
21 −2
E) [ 13
8]
−2 −3
−2
2]
−3
4
−24
]
−14
18
C) [
6
−22
]
−12 20
42 −104
]
−18
76
−2
1]
−1
19 −2
C) [ 13 2 ]
−2 −7
Part 2: With a calculator
PLEASE do NOT write on the test
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A = √𝑠(𝑠
− 𝑎)(𝑠 − 𝑏)(𝑠 − 𝑐) 𝑤ℎ𝑒𝑟𝑒 𝑠 =
𝑎+𝑏+𝑐
1
A = 𝑎𝑏𝑠𝑖𝑛𝐶
2
2
𝑎
𝑎2 = 𝑏 2 + 𝑐 2 − 2𝑏𝑐𝑐𝑜𝑠(𝐴)
𝑠𝑖𝑛𝐴
𝝅𝒙
Use the following equations for # 16 - # 18 𝒇(𝒙) = −𝟐𝐜𝐨 𝐬 ( 𝟐 −
𝟐𝝅
𝟑
=
𝑏
𝑠𝑖𝑛𝐵
=
𝑐
𝑠𝑖𝑛𝐶
) + 𝟐𝟏
16) The period of the function is:
(A) 4𝜋
(B) 4
(C)
𝜋
(D) 𝜋
2
(E) None of these
17) The phase shift of the function is:
(A) right 4
(B) left
4
3
(C) left
4𝜋
3
(D) right
4
3
(E) None of these
18) The vertical shift of the function is:
(A) up 21
(B) down 2
(C) up 2
(D) down 21
(E) None of these
19) Identify the function that accompanies the following graph:
(A) 𝑓(𝑥) = 4 sin(𝑥) − 1
(B) 𝑓(𝑥) = 4 cos(𝑥) − 1
(D) 𝑓(𝑥) = −4 cos(𝑥) + 1
(E) None of these
(C) 𝑓(𝑥) = −4sin(𝑥) − 1
20) Find the standard form of the equation of the specified ellipse: Vertices (0, ±5); Foci (±7, 0)
𝑥2
(A) 25 +
𝑦2
=7
74
𝑥2
(B) 74 +
𝑦2
25
=7
𝑥2
(C) 74 +
𝑦2
𝑥2
=1
25
(D) 25 +
𝑦2
74
=1
(E) None of these
21) Find the standard from of the equation of the specified ellipse: Foci (3, 3), (- 1, 3); Major axis length of 6
(A)
(C)
(𝑥−1)2
5
(𝑥−1)2
9
+
+
(𝑦−3)2
9
(𝑦−3)2
5
=1
(B)
=1
(D)
(𝑥−3)2
5
(𝑥−3)2
9
+
+
(𝑦−1)2
9
(𝑦−1)2
5
=1
=1
(E) None of these
22) What type of conic is the following: 9𝑦 2 − 𝑥 2 + 2𝑥 + 54𝑦 + 62 = 0
(A) parabola
(B) ellipse
(C) circle
(D) hyperbola
(E) None of these
23) What type of conic is the following: 9𝑥 2 + 4𝑦 2 − 54𝑥 + 40𝑦 + 37 = 0
(A) parabola
(B) ellipse
(C) circle
24) Identify the foci from the following ellipse:
(A) (±2, 0)
(B) (0, ±2)
𝑥2
5
+
𝑦2
9
(D) hyperbola
(E) None of these
(D) (0, ±√13)
(E) None of these
=1
(C) (±√13, 0)
25) The center and vertices of the following hyperbola:
(𝑥−1)2
4
−
(𝑦+2)2
=1
1
(A) (1, - 2) and vertices (3, - 2), (-1, - 2)
(B) (- 2, 1) and vertices (3, - 2), (- 1, - 2)
(C) (1, - 2) and vertices (- 2, 3), (- 2, - 1)
(D) (2, - 1) and vertices (- 2, 3), (- 2, - 1)
(E) None of these
26) Find the equations of the asymptotes of the hyperbola: −25𝑥 2 + 4𝑦 2 − 150𝑥 − 8𝑦 − 321 = 0
2
5
(A) Asymptotes: y = ± 5 𝑥
2
(C) Asymptotes: y = 5 𝑥 +
(B) Asymptotes: y = ± 2 𝑥
11
5
2
,𝑦 = −5𝑥 −
1
5
5
(D) Asymptotes: y = 2 𝑥 +
17
2
5
,𝑦 = −2𝑥 −
13
2
(E) None of these
27) Identify the equations that represent hyperbolas.
(i) 4𝑥 2 + 6𝑦 2 + 3𝑥 + 4𝑦 − 7 = 0
(ii) 4𝑥 2 − 6𝑦 2 + 2𝑥 + 7𝑦 − 2 = 0
(iii) 8𝑥 2 + 10𝑦 2 + 4𝑥 + 8𝑦 + 19 = 0
(A) iii only
(B) i and iii only
(C) i only
(D) ii only
(E) None of these
28) Find the area of the triangle given a = 12.32, b = 8.46, and c = 15.05
(A) 52.11
(B) 2715.21
(C) 12.31
(D) 151.56
(E) None of these
29) If < A = 10°, < C = 135°, and c = 45, find a from the oblique triangle.
(A) 16.28
(B) .06
(C) 11.05
(D) 183.24
(E) None of these
(D) 12
(E) None of these
30) If a triangle has < B = 120°, a = 4, and c = 6, find its area.
(A) 6
(B) 10.39
(C) 20.78
31) Find the remaining side of a triangle if < A = 115°, b = 15 cm, and c = 10 cm.
(A) 19.71
(B) 13.75
(C) 53.11
(D) 21.26
(E) None of these
32) How many solutions exist given the following information: < A = 58°, a = 4.5 and b = 5
(A) 0
(B) 1
(C) 2
(D) 3
(E) None of these
33) How many solutions exist given the following information: < A = 60°, a = 4 and b = 14
(A) 0
(B) 1
(C) 2
(D) 3
(E) None of these
34) Find the area of a triangular lot containing side lengths that measure 24 yards and 18 yards and form an
angle of 80°.
(A) 212.72
(B) 37.52
(C) 425.44
(D) 75.02
(E) None of these
Constructed Response
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35) Graph the equation:
𝒚𝟐 − 𝒙𝟐 − 𝟒𝒚 − 𝟐𝒙 − 𝟏𝟑 = 𝟏
(Be sure to plot the center, plot the foci, AND the vertices and co-vertices. Then draw the conic)
Center: ______________________________
Vertices ___________________
Co-vertices _________________
Foci: _________________________________
To approximate the length of a marsh, a surveyor walks 425 meters from point A to point B. Then the
surveyor turns 65° and walks 300 meters to point C. (see figure).
36) Approximate the length of AC of the marsh (round answer to the nearest hundredth) ____________
37) Approximate the area of ∆𝐴𝐵𝐶 (round answer to the nearest hundredth) ______________