Download MATH 2412 TINAL EXAMINATION. ANSWER ALL QUESTIONS

MATH 2412 TINAL EXAMINATION.
ANSWER ALL QUESTIONS. TIME 1.5HRS.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Fill in the blank using the word product, sum, quotient, or difference.
α + β
α-β
1) The formula sin α + sin β = 2 sin cos can be used to change a 2
2
the of two sines into
1)
of a sine and a cosine expression.
B) difference, product
D) sum, product
A) difference, quotient
C) sum, quotient
Solve the problem.
2) The population of a town is increasing by 500 inhabitants each year. If its population at the
beginning of 1990 was 24,390, what was its population at the beginning of 1998?
A) 389,960 inhabitants
B) 28,390 inhabitants
C) 27,890 inhabitants
D) 194,980 inhabitants
3) A company models its yearly expenses in millions of dollars using the equation
f(t) = 0.03t3 - 0.6t2 + 1.15t + 2.5 where t = 0 represents 1989. The companyʹs account manager
2)
3)
decides to adjust the model so that t = 0 corresponds to 1999 rather than 1989. To do this, she
obtains g(t) = f(t + 10). Use the Binomial Theorem to express g(t) in descending powers of t.
A) 0.03t3 - 1.5t2 + 22.15t + 0.65
B) 0.03t3 + 0.3t2 - 1.85t - 16
C) 0.03t3 + 0.3t2 + 22.15t + 8.3
D) 0.03t3 - 1.5t2 - 4.85t + 20.3
4) If a projectile is fired at an angle θ and initial velocity v, then the horizontal distance traveled by
1
the projectile is given by D = v 2 sin θ cos θ. Express D as a function of 2θ.
16
1
A) D = v2 sin 2θ
8
C) D = 1 2
v sin 2θ
16
Find the product AB, if possible.
5)
2
A = -6 2 8 , B = -6
8 -8 5
-9
A)
-6 2 8
8 -8 5
2 -6 -9
C) AB is not defined.
B) D = 1 2
v cos 2θ
16
D) D = 1 2
v sin 2θ
32
4)
5)
B) -96 19
D)
-96
19
1
Solve the system by the method of your choice. Identify systems with no solution and systems with infinitely many
solutions, using set notation to express their solution sets.
6)
6) y = 10 - 2x
8x + 4y = 40
A) {(5, 0)}
B) {(0, 10)}
D) ∅
C) {(x, y) 2x + y = 10}
Find the area of the triangle having the given measurements. Round to the nearest square unit.
7) C = 125°, a = 3 yards, b = 8 yards
A) 20 square yards
B) 7 square yards
C) 10 square yards
D) 39 square yards
7)
Two sides and an angle (SSA) of a triangle are given. Determine whether the given measurements produce one triangle,
two triangles, or no triangle at all. Solve each triangle that results. Round lengths to the nearest tenth and angle
measures to the nearest degree.
8)
8) B = 50°, b = 2, a = 20
A) no triangle
B) A = 48°, C = 81°, c = 22
C) A = 49°, C = 82°, c = 23.5
D) A = 46°, C = 83°, c = 19
Use DeMoivreʹs Theorem to find the indicated power of the complex number. Write the answer in rectangular form.
9) 3(cos 15° + i sin 15°) 4
9)
A)
81 81 3
+ i
2
2
B)
81 3 81
+ i
2
2
C) 81i
D)
81 81
+ i
2
2
Find the exact value of the expression, if possible. Do not use a calculator.
π
10) cos-1 cos - 4
A) - π
4
B)
3π
4
C)
π
4
10)
D)
5π
4
Rewrite the equation in a rotated xʹyʹ-system without an xʹyʹ term. Express the equation involving xʹ and yʹ in the
standard form of a conic section.
11) 31x2 + 10 3xy + 21y2 -144 = 0
11)
A)
xʹ2 yʹ2
+ = 1
9
4
B)
xʹ2 yʹ2
+ = 1
4
9
C) xʹ2 = -4 2yʹ
D) yʹ2 = -4 2xʹ
Use Cramerʹs rule to determine if the system is inconsistent system or contains dependent equations.
12) 9x + 9y = -27
18x + 18y = -29
A) system contains dependent equations
B) system is inconsistent
12)
Express the repeating decimal as a fraction in lowest terms.
13) 0.6
2
A)
3
13)
3
B)
5
3
C)
50
2
20
D)
3
Find the radian measure of the central angle of a circle of radius r that intercepts an arc of length s.
14) r = 2 inches, s = 14 inches
1
A) -7 radians
B) radians
C) 7°
D) 7 radians
7
Solve the equation on the interval [0, 2 π).
15) tan x + sec x = 1
π
A) 0
B)
4
15)
C)
5π
4
D) no solution
Complete the identity.
16) tan x(cot x - cos x) = ?
A) 1 - sin x
14)
16)
B) - sec2 x
C) 1
D) 0
The rectangular coordinates of a point are given. Find polar coordinates of the point. Express θ in radians.
17) (3 3, 3)
π
π
π
π
B) 3, C) 6, D) 6, A) 3, 6
3
6
3
17)
Solve the right triangle shown in the figure. Round lengths to one decimal place and express angles to the nearest tenth
of a degree.
18)
18) a = 3.7 cm, b = 1.7 cm
A) A = 24.7°, B = 65.3°, c = 4.1 cm
C) A = 60.8°, B = 29.2°, c = 4.1 cm
B) A = 27.4°, B = 62.6°, c = 5.4 cm
D) A = 65.3°, B = 24.7°, c = 4.1 cm
Find a positive angle less than 360° or 2π that is coterminal with the given angle.
19) 446°
A) 86°
B) 223°
C) 76°
Solve the system by the addition method.
20) 2x - 7y = 11
7x - 4y = 18
B) {(-2, -1)}
A) {(2, 1)}
19)
D) 266°
20)
C) {(-2, 1)}
D) {(2, -1)}
21) x2 + y2 = 113
x2 - y2 = -15
21)
B) {(7, 8), (-7, 8), (7, -8), (-7, -8)}
D) {(7, 8), (8, 7), (-7, -8), (-8, -7)}
A) {(-7, -8), (-8, -7)}
C) {(7, -8), (7, 8)}
3
Evaluate the trigonometric function at the quadrantal angle, or state that the expression is undefined.
3π
22) cot 2
A) undefined
B) -1
C) 0
Find the exact value by using a sum or difference identity.
23) sin 255°
2( 3 - 1)
2( 3 + 1)
B)
A) 4
4
22)
D) 1
23)
C) -
2( 3 - 1)
4
D)
2( 3 + 1)
4
Use a right triangle to write the expression as an algebraic expression. Assume that x is positive and in the domain of the
given inverse trigonometric function.
24)
24) cos(tan-1 x)
A) x x2 + 1
B)
x2 + 1
x2 + 1
C)
x2 - 1
x2 - 1
D)
x x2 + 1
x2 + 1
Show that the equation is not an identity by finding a value of x for which both sides are defined but not equal.
25)
25) sin 2x + sin 4x = sin 5x
π
C) 0
D) π
A) 2π
B)
2
Write the expression as the cosine of an angle, knowing that the expression is the right side of the formula for cos (α - β)
with particular values for α and β.
26) cos (175°) cos (55°) + sin (175°) sin (55°)
26)
A) cos (190°)
B) cos (220°)
C) cos (120°)
D) cos (210°)
Find the quotient z1
z2
of the complex numbers. Leave answer in polar form.
27)
27) z 1 = 4i
z 2 = -6 + 6i
A)
7π
7π
2
cos + i sin 4
4
3
B)
2
7π
7π
cos - i sin 4
4
3
C)
2
π
π
cos + i sin 4
4
3
D)
2
π
π
cos - i sin 4
4
3
Find the standard form of the equation of the hyperbola satisfying the given conditions.
2
28) Endpoints of transverse axis: (0, -4), (0, 4); asymptote: y = x
5
A)
y2 x2
- = 1
16 25
B)
y2 x2
- = 1
25
4
C)
y2
x2
- = 1
16 100
Find the angle between the given vectors. Round to the nearest tenth of a degree.
29) u = 2j, v = 7i - 2j
A) 74.1°
B) 123.3°
C) -15.9°
4
D)
28)
y2
x2
- = 1
100 16
29)
D) 105.9°
30) u = -5i + 3j, v = 4i + 6j
A) 92.7°
30)
B) 102.7°
C) 46.4°
D) 36.4°
Find the sum of the infinite geometric series, if it exists.
1 1 2
31) - + - . . .
6 3 3
A) - 64
3
B)
31)
128
3
C)
1
18
D) does not exist
Use the vertex and the direction in which the parabola opens to determine the relationʹs domain and range.
32) x = -(y - 1)2 + 6
A) Domain: (-∞, ∞)
Range: (-∞, ∞)
C) Domain: (-∞, ∞)
Range: (-∞, 6]
Evaluate the determinant.
33)
200
392
797
A) 162
32)
B) Domain: (-∞, 6)
Range: (-∞, ∞)
D) Domain: (-∞, 6]
Range: (-∞, ∞)
33)
B) 90
C) 95
5
D) -90
Answer Key
Testname: M2412FINAL
1) D
2) B
3) B
4) D
5) D
6) C
7) C
8) A
9) A
10) C
11) B
12) B
13) A
14) D
15) A
16) A
17) D
18) D
19) A
20) D
21) B
22) C
23) A
24) B
25) B
26) C
27) A
28) C
29) D
30) A
31) D
32) D
33) B
6