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ReFresher Take-Home Exam 1

Instruction: Select the correct answer for

each of the following questions. Mark only
one answer for each item by shading the box
corresponding to the letter of your choice on
the answer sheet provided.

component of the velocity of the stone if

the bird flew at an average speed of 12
m per second.
A. 14.484 m*
C. 16.745 m
B. 12.144 m
D. 18.192 m

Strictly no erasures allowed. Use pencil

no. 1 only.
NOTE: Whenever you come across a caret
(^) sign, it means exponentiation.

12. Find the equation of the normal to the

curve y=x2x2 at the point (1, 2).
A. y = -x-1*
C. y = 2x-1
B. y = -x+1
D. y = x+1

1.

Find the variance of the sample of

observations 2, 5, 7, 9, 12.
A. 14.91
C. 11.6*
B. 3.41
D. 5.27

2.

Suppose a die is loaded so that a 1

face lands uppermost 3 times as often
as any other face while all the other
faces occur equally often. What is the
probability of a 1 on a single toss?
A. 7/8
C. 3/8*
B. 1/8
D. 1/6

3.

4.

5.

6.

7.

A committee is composed of six

Democrats and five Republicans. Three
of the Democrats are men, and 3 of the
Republicans are men. If a man is
selected for chairman, what is the
probability that he is a Republican?
A. 2/3
C. 5/11*
B. 1/2
D. 3/5
A bucket, weighing 5 pounds when
empty, is loaded with 60 pounds of
sand, and then lifted (at constant speed)
10 feet. Sand leaks out of a hole in the
bucket at a uniform rate, and a third of
the sand is lost by the end of the lifting.
Find the work done in the lifting process.
A. 550 ft-lb*
C. 450 ft-lb
B. 625 ft-lb
D. 240 ft-lb
What is the period of the function
y=cos3x?
A. /2
C. 2/3*
B. 3/2
D. 2/
The region above the x-axis and under
the curve y = sinx, between x = 0 and x
= , is divided into two parts by the line
x = c. If the area of the left part is onethird the area of the right part, find c.
A. /2
C. 1.2
B. /3*
D. 4/
What is the imaginary part of the
complex number 2+j7?
A. 2
C. 7*
B. j
D. j7

13. If the average value of f(x) =x3 +bx- 2 on

[0,2] is 4,find b.
A. 3
C. 2
B. 5
D. 4*

14. Evaluate the Limit (z^10+1)/(z^6+1)

as z approaches i.
A. 3/5
C. 3/7
B. 2/10
D. 5/3*
15. Which of the following is an exact DE?
A. ( x^2 + 1 ) dx xydy = 0
B. 2xydx + ( 2 + x^2 ) dy = 0*
C. xdy + ( 3x 2y ) dx = 0
D. x^2 ydy ydx = 0
16. Write the differential equation of a family
of straight lines with slope and yintercept equal.
A. ydx + ( x 1 ) dy = 0
B. xdy ( y + 1 ) dx = 0
C. ydx ( x + 1 ) dy = 0*
D. ydy + ( x 1 ) dx = 0
17. Which of the following is not
transcendental function?
A. y=sinx
C. y=tanx
B. y=ex+1

D.

y x3 *

18. What is the inverse of the function

f(x)=ex?
A. x e
B. y=logx
y

C. y=lnx*
D. y=xe

19. The graph of the function f(x)=bx with

0<b<1
A. increases to the right ; y intercept -1
B. decreases to the right ; y intercept 1*
C. asymptotic to the left of the x-axis
D. is asymptotic to the y-axis
20. The region in the first quadrant bounded
by the curves y = x2 and y = x4.
A. 3/11
C. 2/15*
B. 11/2
D. 3/9

Determine the modulus and argument,

respectively, of the complex number Z
=2+j3.
A. 5.202, 65.23
C. 5.404, 56.34
B.3.606, 56.31*
D. 2,3

21. Determine the modulus and argument,

respectively, of the complex number Z
=2+j3.
A. 5.202, 65.23
C. 3.606,
56.31*
B. 5.404, 56.34
D. 2,3

Find the volume of a solid whose base

is enclosed by the circle x2 + y2 = 9 and
whose
cross-sections
taken
perpendicular to the x-axis are squares.
124
C. 144 *
164
D. 134

22. Evaluate
the
triple
integral
(x+y+z)dxdydz. The limit for x is from 0
to y, the limit for y is from 0 to z and the
limit for z is from 0 to 1.
A. 5/12*
C. 2/11
B. 7/20
D. 3/7

10. Find the equation of the tangent line

to the ellipse: 4x2+9y2=40, at the point
(1,2).
A. 2x+9y+20=0
C. 2x+9y20=0*
B. 9x-2y-20=0
D. 9x+2y+20=0

23. Find the volume of a

tetrahedron
formed by three mutually perpendicular
edges of lengths a,b,c.
A. a+b+c
C. a2b/c
B. abc/6*
D. 6(ab)/c

8.

9.

11. A stone is thrown in such a manner that

it would hit a bird at the top of a tree and
afterwards reach a height double that of
the tree. At the moment of throwing the
stone, the bird flies horizontally away.
Notwithstanding this, the stone hits the
bird.
Determine
the
horizontal

24. The limit of [1+(1/x)]x as x approaches

infinity is:
A.
C. e*
B. /e
D. e
2

25. If y x 3x x , find R at x=0.

A. 0.741
C. 0.471*
B. 0.147
D. 0.174

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Mathematics

26. Fruit flies are being bred in an enclosure

that can hold a maximum of 640 flies. If
the flies multiply exponentially, with a
growth constant K = 0.05 and where
time is measured in days, how long will
it take an initial population of 20 to fill
the enclosure?
A. 69 days*
C. 48 days
B. 39 days
D. 89 days
27. The curve represented with the
parametric equations x=1+t, y=1-t is:
A. line through (-1,1)
B. line through (1,1)*
C. parabola through (-1,-1)
D. point
28. What is the inverse of the function
f(x)=2x?
A.

y x2

C. y=lnx

B.

y log2 x

D. x=2y

29. The inverse of an exponential function is

A. exponential
C. logarithmic*
B. linear
D.
transcendental
30. A 5-lb monkey is attached to the end of
a 30-ft hanging rope that weighs 0.2
Ib/ft. The monkey climbs the rope to the
top. How much work has it done?
A. 550 ft-lb
C. 450 ft-lb
B. 625 ft-lb
D. 240 ft-lb*
31. The expansion of a gas in a cylinder
causes a piston to move so that the
volume of the enclosed gas increases
from 15 to 25 cubic inches. Assuming
the relationship between the pressure p
(lb/in2) and the volume v (in3) is pv 1.4
=60, find the work done.
A. 3.93 in-lb
C. 9.39 in-lb*
B. 2.59 in-lb
D. 3.99 in-lb
32. Find k so that A=(3,-2) and B=(1,k) are
parallel.
A. 2
C. -3/2
B. 2/3
D. -2/3*
33. Find the area inside one petal of the
curve r = sin2 and identify the curve
A. 1/2 ; 2 lemniscate
B. 9/2 ; cardioid
C. 16 : limacon
D. /8 ; four-leaved rose*
34. The curve r=e is known as:
A. Equiangular spiral* C. limacon
B. Hypocycloid
D. epicycloid
35. Find the length of r= 2 from =0 to
=5.
A. 10/3
C. 17/2
B. 19/3*
D. 21/5
36. Find the angle between the velocity
vectors v1=5i+2j+7k and v2 =4i+jk.
A. 55.20
C. 86.50
B. 66.40*
D. 42.35
5

n(3 2n)

37. Find the sum of the series n1

.
A. 115
C. 155*
B. 151
D. 551

38. Determine the mean value of the

2

function y 3x 4x 1 between x=-1

and x=2.
A. 4
C. 6*
B. 8
D. 2

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39. The
differential
equation
(9x^2+2y^2+2)dx + (4xy+12y^2) dy is
A. Homogeneous
B. Exact*
C. Bernoulli type
D. Linear
2

y x 3
40. Find the r.m.s value of
between x=1 and x=3.
A. 6.79
C. 7.69*
B. 9.76
D. 6.97
41.

The

graph

r a bcos is:

of

the

equation

A. Spiral of Archimedes C. Cardioid*

B. Circle
D. Lemniscate
42. Find the area enclosed by the curve

r 1 cos and the radius vectors at

=0 and =/2.
A. 2.178*
B. 1.287

C. 2.718
D. 1.817

43. Which of the following functions is

exponential?
A.

esinx

C. x10
x

1
2

B. sinx2
D.
*
44. Find the area enclosed by the curve y =
4 cos3x, the x-axis and ordinates x = 0
and x = /6
A. 1.333 sq. units*
C. 1.677 sq.
units
B. 2.111 sq. units
D. 3.211 sq.
units
45. Evaluate

3i 2
i 0

A.
B.

120,942
60

C. 90*
D. 180,945

46. A long piece of tin 60 feet wide is to be

made into a trough by bending up two
sides. Find the inclination of each side if
the carrying capacity is a maximum?
A. 30 deg
C. 60 deg*
B. 45 deg
D. 90 deg
47. If f(x) = x squared and g(x) = 2x, what is
the value of f(g(-3)) g(f(-3))?
A. 54
C. -18
B. 18*
D. -54
48. Find the term involving x squared in the
expansion of (x^3 + k/x)^10.
A. 210 k sq. x sq.
C. 92 k sq. x sq.
B. 32 k sq. x sq.
D. 120 k^7 x
sq.*
49. A student is to answer 8 out 10
questions on an exam. Find the number
of choices.
A. 45*
C. 40
B. 50
D. 60
50. The volume of a square pyramid is 384
cu.cm. Its altitude is 8 cm. How long is
an edge of the base?
A. 11
C. 13
B. 12*
D. 14
51. Find the equations of the lines of slope
-3/4 that form with the coordinate axes a
triangle of area 24 sq. units.
A. 3x+4y=+24*
C. 3x+4y=+15
B. 3x+4y=+35
D. 3x+4y=+8
52. Studies have shown a particular
television commercial is understood by
a 25% of pre school pupils and 80% of
grade school pupils. If a television

ReFresher Take-Home Exam 1

advertising agency randomly selected
one preschooler and one gradeschooler, what is the probability neither
child would understand the commercial,
assuming the children reactions are
independent?
A. 0.15*
C. 0.50
B. 0.25
D. 0.12

53. Evaluate lim ( z squared + 1 )/( z raised

to the 6 + 1 ) as z approaches to i.
A. 2( 1 + i )/2
C. 1/3*
B. ( - 4/3 ) 4i
D. 12 + 6i
54. A post office can accept for mailing only
if the sum of its length and its girth ( the
circumference of its cross section ) is at
most 100 in. What is the maximum
volume of
a rectangular box with
square cross section that can be
mailed?
A. 5432.32 cu.in.
C. 2592.25
cu.in.
B. 1845.29 cu.in.
D. 9259.26
cu.in.*
55. Find the second derivative of the
function y = 5x cubed + 2x + 1.
A. 2x
C. 30x*
B. x
D. 24x
56. If

log x 4 and logz y 5

logxy is equal to
A. 20
B. 4/5

, the value of

C. 5/4*
D. 1/20

57. Find x if versed sin x = 0.423.

A. 54.76*
C. 53.3
B. 43.8
D. 48.93
58. Find the average value of 1/x on [3, 5].
A. 0.51
C. 0.255*
B. 0.551
D. 0.125
59. Two circles have radius of 4 cm. and 12
cm., respectively. If the distance
between their centers is 30 cm.,
compute the length of the common
internal tangent to the two circles.
A. 27.32
C. 57.73
B. 25.37*
D. 24.06
60. The diameters of the two circles that are
tangent internally are 18 and 8
respectively. What is the length of the
tangent segment from the center of the
larger circle to the smaller circle?
A. 2
C. 3*
B. 4
D. 5
61. Find the volume of the solid of
revolution formed by revolving the
region bounded by y = x 3 + 3, x = 1, x =
0 and y = 0 about x = 1.
A. 9.20
C. 9.59
B. 9.74 *
D. 9.4
62. A cylindrical glass 5 inches in diameter
and 8 inches in altitude is filled with ice
tea. It is then tilted to a position enough
for the ice tea level to coincide with the
center of the base. How much ice tea is
drained from the glass?
A. 123.75 in3
C. 134.65 in3
B. 154.32 in3
D. 120.34 in3
63. Identify the graph of the equation x2 +y2
+2x - 2y +2=0.
A. circle
C. empty set
B. point*
D. hyperbola
64. Find the acute angles between the two
planes 2x-y+z=8 and x+y+2z-11=0.

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Mathematics

A. 30
B. 60*

C. 45
D. 40

65. Find the equation of the perpendicular

bisector of the segment joining the
points (2,6) and (-4,3).
A. 2x-4y+5=0
C. 4x+2y-5=0*
B. 2x+4y-5=0
D. 5x-2y+4-0
66. The curve represented by x=2cos,
y=3sin is:
A. helix
C. ellipse*
B. circle
D. spiral

67. If the volume of a cube is increased

by 30%, by how much is the surface
area increased?
A. 42.22%
C. 69% *
B. 19.11%
D. 33%
68. A conical tank, 10meters deep and 8
meters across at the top, is filled with
water to a depth of 5 meters. The tank is
emptied by pumping the water over the
top edge. How much work is done in the
process?
A. 5283 kJ
C. 4580 kJ
B. 1625 kJ
D. 1283 kJ*
69. It is desired to give a spherical tank of
diameter 20 feet ( 240 inches ) a coat of
paint 0.01 inch thick. Estimate how
many gallons of paint will be required.
A. 7.83*
C. 6.84
B. 8.37
D. 8.64
70. A 5-foot girl is walking toward a 20-foot
lamppost at the rate of 6 feet per
second. How fast is the tip of her
shadow (cast by the lamp) moving?
A. 6 ft/s
C. 8 ft/s*
B. 5 ft/s
D. 4 ft/s
71. Find the arclength of the spiral r=e
from =0 to =In2.
A. 3
C. 2*
B. 22
D. 33
72. Find the area inside one loop of the r 2 =
cos2 and identify the curve
A. 1/2 ; lemniscate*C. 9/2 ; cardioid
B. 16 : limacon
D. /8 ; 4-leaved
rose
73. Find the locus of a point the sum of the
squares of whose distances from (2,3)
and (-1,-2) is 34.
A. x2+y2-x-y-8=0* C. x2+y2-5x-y+8=0
B. x2+y2+2x-3y-5=0 D. x2+y2-4x-3y8=0
74. If the point (2, k) lies on the line with
slope m = 3 passing through the point
(1, 6), find k.
A. 6
C. 9*
B. 8
D. 4
75. Find the height of the right circular
cylinder of maximum volume V that can
be inscribed in a sphere of radius R.
A. 3R/2
C. 3 (R/2)
B. 2R/3*
D. 2/3R
76. Find an equation of a line through (4,-2)
and at a perpendicular distance of 2
units from the origin.
A. 4x+2y-10=0
C. 4x+3y-10=0*
B. 3x-4y+10=0
D. 5x-4y+12=0
77. A boat is being pulled into a dock by a
rope that passes through a ring on the
bow of the boat. The dock is 8 feet
higher than the bow ring. How fast is the
boat approaching the dock when the
length of rope between the dock and the
boat is 10 feet, if the rope is being
pulled in at the rate of 3 feet per
second?

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A. 6 ft/s
B. 5 ft/s*

C. 8 ft/s
D. 4 ft/s

78. The lower edge of a mural, 12 ft high, is

6 ft above an observers eye. Under the
assumption that the most favorable view
is obtained when the angle subtended
by the mural at the eye is a maximum,
at what distance from the wall should
the observer stand?
A. 8.48 ft
C. 15.5 ft
B. 10.4 ft*
D. 12.7 ft
79. Sand falling from a chute forms a
conical pile whose altitude is always
equal to 43 the radius of the base. How
fast is the volume increasing when the
radius of the base is 3 ft and is
increasing at the rate of 3 in/min?
A. 3 ft3/min*
C. 2 ft3/min
3
B. 2/3 ft /min
D. /3 ft3/min
80. Under certain conditions, cane sugar in
water is converted into dextrose at a
rate proportional to the amount that is
unconverted at any time. If, of 75 grams
at time t = 0, 8 grams are converted
during the first 30 min, find the amount
converted in 1 1/2 hours.
A. 16.2 g
C. 21.6 g*
B. 26.1 g
D. 12.6 g
81. The solid has a base, which is a circle of
radius
r.
Each
cross
section
perpendicular to a fixed diameter of the
circle is an isosceles triangle with
altitude equal to one-half of its base.
Find the volume of the solid.
A. (2/3)r
C. (5/4)r
B. (4/3)r*
D. (3/5)r
82. Box 1 contains 30 red and 70 white
balls, box 2 contains 50 red and 50
white balls, and box 3 contains 75 red
and 25 white balls. The three boxes are
all emptied into a large box, and a ball is
selected at random. If the selected ball
is red, what is the probability that it
came from box 1?
A. 0.193*
C. 0.122
B. 0.256
D. 0.109
83. If the probability that a basketball player
sinks the basket at 3-point range is 2/5,
determine the probability of shooting 5
out of 8 attempts.
A. 12.4%*
C. 25.4%
B. 10.4%
D. 16.8%
84. A man is dealt a poker hand (5 cards)
from an ordinary deck of playing cards.
In how many ways can he be dealt with
a straight flush?
A. 36
C. 27
B. 40*
D. 45
85. How many answer patterns are there in
a standard multiple choice exam with 10
items?

A. 2,405,657
B. 1,048,576*

C. 1,230,872
D. 2,251,200

86. A student is to answer 8 out of 10

questions on an exam. How many
choices does he have if he must answer
at least 4 of the first 5questions?
A. 40
C. 30
B. 35*
D. 45
87. Find

the

0 x 360

values
,

of x,
such

where
that

A.
B.

6
7

2 sin x 5 cos x 1 0
A.
B.
88. If

120 or 240*
115 or 320
f(x)=3x and
?

C. 100 or 240
D. 136 or 310
g(x)=7x-1,

what

is

f og 4
A.
B.

51
81

C. 22
D. 19

89. The sides of a triangle are 8 cm, 10 cm,

and 14 cm. Determine the radius of the
inscribed and circumscribing circle.
A. 2.45, 7.14*
C. 3.45, 8.14
B. 3.45, 8.14
D. 2.54,4.71
lim

2x 2 5x 6

3
2
90. Evaluate x x 6x x 2 .

A. - 1
C. 1
B.
D. 0 *
91. 160 joules of work is required to stretch
a spring 0.80 m from its natural length.
Find the work done required to stretch
the spring an additional 0.4 m.
A. 205 joules
C. 210 joules
B. 215 joules
D. 200 joules *
92. Find the center of mass for a system
with m1 = 5 at (2,3), m2 = 8 at (-3, -1), m3
= 4 at (-1, 6) and m4 = 3 at (1,-2).

A.

3 5
,
4 4

B.

3 5
,
4 4

3 5
,
C. 4 4
3 5
,
D. 4 4 *

93. Find the centroid of the region bounded

by f(x) = 3x 2 and g(x) = x2.

A.
B.

3 12
,
2 5 *
3 12
,
2 5

3 12
,
C. 2 5
3 6
,
D. 2 5

94. Find the value of k such that the

quadratic equation 6(x2 + 2x) = k(2x 1)
+ 5x2 has the double root x = 3.

C. 8
D. 9 *
tan

95. If

3
2
2
,

5
12 and

evaluate sin .

A.

5
12

B.

5
12

.
2

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Mathematics

ReFresher Take-Home Exam 1

96. If

sin

5
C. 13 *
5
13
D.
2
2 ,

tan 1 ,

3
2
and 2
, find the exact value of
sin( ) .
A. 0
B. 2

C. 1 *
D. 1

eix eix
2i
97. Evaluate
.
A. sin x*
B. cos x

C. tan x
D. cot x

98. Convert r 6 tan sec into rectangularcoordinate equation.

2
A. y 6x

2
C. x 6y *

2
B. y 6x

2
D. x 6y

99. Express the following complex number

2
2

6 cos
isin
3
3

in rectangular form.
.
A. 3 3i 3

C. 3 3i 3 *

B. 3 3i 2

D. 3 3i 3

2
3
100. Solve for x: (ln x) 2ln x 5
A. e
C. e3
B. e5
D. e and e5 *

IMPORTANT!
Answer sheets must be submitted next
meeting together with your solution.
NO ANSWER SHEET, NO HANDOUTS
NEXT MEETING.
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