Download Practice Exam 3 - MDC Faculty Web Pages

Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the equation of the parabola described.
1) Focus at (-4, 0); directrix the line x = 4
A) y2 = -4x
B) y2 = 16x
C) x2 = -16y
2) Focus at (5, 0); vertex at (0, 0)
A) y2 = 20x
B) x2 = 5y
C) y2 = 5x
Find the vertex, focus, and directrix of the parabola. Graph the equation.
3) x2 = -16y
A) vertex: (0, 0)
focus: (4, 0)
directrix: x = -4
B) vertex: (0, 0)
focus: (-4, 0)
directrix: x = 4
1
D) y2 = -16x
D) x2 = 20y
1)
2)
3)
C) vertex: (0, 0)
focus: (0, -4)
directrix: y = 4
D) vertex: (0, 0)
focus: (0, 4)
directrix: y = -4
4) y2 = 12x
4)
A) vertex: (0, 0)
focus: (-3, 0)
directrix: x = 3
B) vertex: (0, 0)
focus: (3, 0)
directrix: x = -3
2
C) vertex: (0, 0)
focus: (0, 3)
directrix: y = -3
D) vertex: (0, 0)
focus: (0, 3)
directrix: y = -3
Find the equation of the parabola described.
5) Vertex at (1, 6); focus at (2, 6)
A) (y - 6)2 = 4(x - 1)
B) (x - 6)2 = 16(y - 6)
D) (x - 6)2 = -16(y - 6)
C) (y - 6)2 = -4(x - 1)
6) Vertex at (1, 4); focus at (1, 2)
A) (y - 4)2 = -4(x - 1)
B) (y - 4)2 = 4(x - 1)
D) (x - 1)2 = -8(y - 4)
C) (x - 1)2 = 8(y - 4)
Find the vertex, focus, and directrix of the parabola with the given equation.
7) (y - 3)2 = 8(x - 2)
A) vertex: (2, 3)
focus: (0, 3)
directrix: x = 4
C) vertex: (3, 2)
focus: (5, 2)
directrix: x = 1
B) vertex: (2, 3)
focus: (4, 3)
directrix: x = 0
D) vertex: (-2, -3)
focus: (0, -3)
directrix: x = -4
8) (y + 4)2 = -12(x - 1)
A) vertex: (1, -4)
focus: (-2, -4)
directrix: x = 4
C) vertex: (-1, 4)
focus: (-4, 4)
directrix: x = 2
B) vertex: (-4, 1)
focus: (-7, 1)
directrix: x = -1
D) vertex: (1, -4)
focus: (4, -4)
directrix: x = -2
Find the vertex, focus, and directrix of the parabola. Graph the equation.
3
5)
6)
7)
8)
9) (y + 2)2 = -8(x + 3)
9)
A) vertex: (2, 3)
focus: (1.75, 3)
directrix: x = 2.25
B) vertex: (3, 2)
focus: (3, 1.75)
directrix: y = 2.25
C) vertex: (-3, -2)
focus: (-3, -2.25)
directrix: y = -1.75
D) vertex: (-3, -2)
focus: (-3.25, -2)
directrix: x = -2.75
Solve the problem.
10) A searchlight is shaped like a paraboloid of revolution. If the light source is located 2 feet from the
base along the axis of symmetry and the opening is 8 feet across, how deep should the searchlight
be?
A) 8 ft
B) 4 ft
C) 2 ft
D) 0.3 ft
4
10)
11) A reflecting telescope contains a mirror shaped like a paraboloid of revolution. If the mirror is 16
inches across at its opening and is 2 feet deep, where will the light be concentrated?
A) 8 in. from the vertex
B) 0.7 in. from the vertex
C) 0.1 in. from the vertex
D) 0.3 in. from the vertex
Find the foci and vertices of the ellipse.
x2 y2
+
=1
12)
16
4
12)
A) foci at (0, - 2 3) and (0, 2 3)
vertices at (0, -4), (0, 4)
C) foci at (-4, 0) and (4, 0)
vertices at (-16, 0), (16, 0)
13)
11)
B) foci at (- 2 3, 0) and (2 3, 0)
vertices at (-4, 0), (4, 0)
D) foci at (0, -2) and (0, 2)
vertices at (0, -4), (0, 4)
x2 y2
+
=1
9
49
13)
B) foci at (0, - 2 10) and (0, 2 10)
vertices at (0, -7), (0, 7)
D) foci at (- 2 10, 0) and (2 10, 0)
vertices at (-7, 0), (7, 0)
A) foci at (0, 7) and (3, 0)
vertices at (0, 49), (9, 0)
C) foci at (0, -7) and (0, 7)
vertices at (0, -49), (0, 49)
5
Graph the ellipse and locate the foci.
x2 y2
+
=1
14)
9
4
14)
A) foci at (2 3, 0) and (-2 3, 0)
B) foci at (0,
5) and (0, - 5)
C) foci at ( 13, 0) and (- 13, 0)
D) foci at ( 5, 0) and (- 5, 0)
Find the center, foci, and vertices of the ellipse.
(x - 3)2 (y + 1)2
+
=1
15)
36
16
15)
A) center at (3, -1)
foci at (- 2 5, -1), (2 5, -1)
vertices at (6, -1), (-6, -1)
C) center at (-1, 3)
foci at (-1 + 2 5, 3), (-1 - 2 5, 3)
vertices at (-3, -1), (9, -1)
B) center at (3, -1)
foci at (3 + 2 5, 3), (3 - 2 5, 3)
vertices at (6, -1), (-6, -1)
D) center at (3, -1)
foci at (3 + 2 5, -1), (3 - 2 5, -1)
vertices at (-3, -1), (9, -1)
6
16) 4x2 + 5y2 - 56x + 30y + 221 = 0
(x - 7)2 (y + 3)2
+
=1
A)
5
4
16)
center: (-7, 3); foci: (-6, 3), (-8, 3); vertices: (-9.2, 3), (-4.8, 3)
(x - 7)2 (y + 3)2
+
=1
B)
4
5
center: (-7, 3); foci: (-6, 3), (-8, 3); vertices: (-9.2, 3), (-4.8, 3)
(x - 7)2 (y + 3)2
+
=1
C)
5
4
center: (7, -3); foci: (8, -3), (6, -3); vertices: (9.2, -3), (4.8, -3)
(x - 7)2 (y + 3)2
+
=1
D)
4
5
center: (7, -3); foci: (8, -3), (6, -3); vertices: (9.2, -3), (4.8, -3)
7
Graph the equation.
(x + 1)2 (y + 2)2
+
=1
17)
16
9
17)
A)
B)
C)
D)
Solve the problem.
18) An arch in the form of a semiellipse is 52 ft wide at the base and has a height of 20 ft. How wide is
the arch at a height of 12 ft above the base?
A) 41.6 ft
B) 35.5 ft
C) 20.8 ft
D) 17.7 ft
Find an equation for the hyperbola described.
19) Vertices at (0, ±6); asymptote the line y =
A)
y2
x2
=1
100
9
B)
3
x
10
18)
19)
y2
x2
=1
36 400
C)
8
y2
x2
=1
36 100
D)
y2
x2
=1
400 36
Find the center, transverse axis, vertices, and foci of the hyperbola.
x2
y2
=1
20)
121 144
A) center at (0, 0)
transverse axis is y-axis
vertices at (0, -11) and (0, 11)
foci at (- 265, 0) and ( 265, 0)
C) center at (0, 0)
transverse axis is x-axis
vertices at (-11, 0) and (11, 0)
foci at (-12, 0) and (12, 0)
20)
B) center at (0, 0)
transverse axis is x-axis
vertices at (-11, 0) and (11, 0)
foci at (- 265, 0) and ( 265, 0)
D) center at (0, 0)
transverse axis is x-axis
vertices at (-12, 0) and (12, 0)
foci at (- 265, 0) and ( 265, 0)
Find an equation for the hyperbola described. Graph the equation.
21) Center at (0, 0); focus at (2 13, 0); vertex at (6, 0)
A)
B)
x2 y2
=1
16 36
y2 x2
=1
36 16
9
21)
C)
D)
y2 x2
=1
16 36
x2 y2
=1
36 16
Find an equation for the hyperbola described.
1
9
6
22) Vertices ( , -3) and (- , -3); asymptote the line y + 3 = (x + 2)
2
2
5
A)
4(x + 2)2 (y + 3)2
=1
25
9
B)
(x + 2)2 4(y + 3)2
=1
9
25
C)
4(x - 2)2 (y - 3)2
=1
25
9
D)
(y + 3)2 4(x + 2)2
=1
9
25
10
22)
Find the center, transverse axis, vertices, foci, and asymptotes of the hyperbola.
(x - 2)2 (y - 3)2
=1
23)
9
4
A) center at (3, 2)
transverse axis is parallel to x-axis
vertices at (0, 2) and (6, 2)
foci at (3 - 13, 2) and (3 + 13, 2)
2
2
asymptotes of y - 2 = - (x - 3) and y - 2 = (x - 3)
3
3
B) center at (2, 3)
transverse axis is parallel to x-axis
vertices at (0, 3) and (4, 3)
foci at (2 - 13, 3) and (2 + 13, 3)
3
3
asymptotes of y - 3 = - (x - 2) and y - 3 = (x - 2)
2
2
C) center at (2, 3)
transverse axis is parallel to x-axis
vertices at (-1, 3) and (5, 3)
foci at (2 - 13, 3) and (2 + 13, 3)
2
2
asymptotes of y - 3 = - (x - 2) and y - 3 = (x - 2)
3
3
D) center at (2, 3)
transverse axis is parallel to y-axis
vertices at (2, 0) and (2, 6)
foci at (2, 3 - 13) and (2, 3 + 13)
3
3
asymptotes of y + 3 = - (x + 2) and y + 3 = (x + 2)
2
2
11
23)
Graph the hyperbola.
(x - 2)2 (y + 2)2
=1
24)
4
9
24)
A)
B)
C)
D)
Write the augmented matrix of the given system of equations.
25)
5x +7z = 46
-2y +8z = 58
6x +7y +8z = 73
A)
B)
C)
5 0 7 46
5 0 7
5 7 0 46
0 -2 8 58
0 -2 8
-2 8 0 58
6 7 8 73
6 7 8
6 7 8 73
12
25)
D)
5 0 6 46
0 -2 7 58
7 8 8 73
26)
2x +5y +9z = 80
8x +4y +5z = 68
-2x +4y +4z = 44
A)
2 8 -2 80
5 4 4 68
9 5 4 44
26)
B)
C)
2 5 9 80
8 4 5 68
-2 4 4 44
2 5 9
8 4 5
-2 4 4
D)
80 9 5 2
68 5 4 8
44 4 4 -2
Solve the system of equations using matrices (row operations). If the system has no solution, say that it is inconsistent.
27)
27)
2x + 2y = 2
3x + 3y = 3
A) x = 4, y = -3; (4, -3)
B) x = -3, y = 4; (-3, 4)
C) x = -3, y = -4; (-3, -4)
D) inconsistent
28)
28)
2x + 4y - z = 20
x + 3y + 8z = 66
7x + y + z = 32
A) x = 3, y = 6, z = 5; (3, 6, 5)
C) x = -3, y = 5, z = 6; (-3, 5, 6)
B) x = 3, y = 5, z = 6; (3, 5, 6)
D) inconsistent
Solve using elimination.
29)
x2 + y2 = 41
29)
x2 - y2 = 9
A) x = 5, y = 4; x = 4, y = 5; x = -5, y = -4; x = -4, y = -5
or (5, 4), (4, -5), (-5, -4), (-4, -5)
B) x = -5, y = -4; x = -4, y = -5 or (-5, -4), (-4, -5)
C) x = 5, y = -4; x = 5, y = 4 or (5, -4), (5, 4)
D) x = 5, y = 4; x = -5, y = 4; x = 5, y = -4; x = -5, y = -4
or (5, 4), (-5, 4), (5, -4), (-5, -4)
30)
30)
3x2 - 3y2 = 36
4x2 + 5y2 = 84
A) x = 4, y = -2; x = 4, y = 2 or (4, -2), (4, 2)
B) x = 4, y = 2; x = 2, y = 4; x = -4, y = -2; x = -2, y = -4
or (4, 2), (2, 4), (-4, -2), (-2, -4)
C) x = 4, y = 2; x = -4, y = 2 ; x = 4, y = -2; x = -4, y = -2
or (4, 2), (-4, 2), (4, -2), (-4, -2)
D) x = -4, y = -2; x = -2, y = -4 or (-4, -2), (-2, -4)
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Solve the problem.
31) A hall 130 feet in length was designed as a whispering gallery. If the ceiling is 25 feet high
at the center, how far from the center are the foci located?
13
31)
Answer Key
Testname: MAC1140_PE4
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
12)
13)
14)
15)
16)
17)
18)
19)
20)
21)
22)
23)
24)
25)
26)
27)
28)
29)
30)
31)
D
A
C
B
A
D
B
A
D
C
B
B
B
D
D
C
C
A
B
B
D
A
C
A
A
B
B
B
D
C
60 ft
14