mathematics comprehensive exam 1 - ms. harris | … comprehensive exam 1 1. three cubes each of side...

MATHEMATICS Comprehensive Exam 1

1. Three cubes each of side 7 cm are joined end to end.

Find the surface area of the resulting rectangular prism.

a. 686 cm2 b. 752 cm2 c. 786 cm2 d. 838 cm2 e. 882 cm2

2. Find the midpoint of the line joining 4, 7, −8( ) and

−5, 3, − 4( ) .

a. −1, 10, −12( ) b. −0.5, 8, − 7.5( ) c. 9, 4, − 4( ) d. −3, 6, −8( ) e. −0.5, 5, − 6( )

3. A lady wants to but one cotton dress, one woolen and

one polyester dress from a textile shop. If there are 15 cotton varieties, 18 woolen and 12 polyester varieties, in how ways can she choose the three dresses?

a. 18 b. 216 c. 270 d. 2,772 e. 3,240

4. How long will it take a sum of money invested at 5%

per month simple interest to increase its value by 300%?

a. 2 years b. 30 months c. 45 months d. 5 years e. 66 months

5. In a right triangle with a 60° angle, the shorter leg has length 5 3 . What is the length of the longer leg?

a. 5

b. 5 3

c. 15

d. 10 3

3

e. 10 3

6. If < 1, − 3, − 7 > and < −x , − 9, 4 > are two perpendicular

movements in the x-y-z coordinate plane, find the value of x.

a. 0 b. –1 c. 1 d. –2 e. 2

7. Two poles of heights 6 m and 14 m stand on a flat

surface. If the distance between the feet of the poles is 15 m, find the distance between their tops.

a. 16 m b. 17 m c. 19 m d. 21 m e. 24 m

MATHEMATICS COMPREHENSIVE EXAM 1 PAGE 2 OF 6 DEMIDEC RESOURCES ©2011

8. Find the distance of the point 4, 1( ) from the line

3x − 4 y − 9 = 0 .

a. 0.2 b. 0.5 c. 1.2 d. 1.8 e. 2.5

9. Find the average rate of change of polynomial

y = x 3 − 4x + 7 over the interval −2, 4[ ] .

a. –4 b. –0.5 c. 2.7 d. 6.9 e. 8

10. If the line containing the points 3, y( ) and 2, 7( ) is

parallel to the line containing the points −1, 4( ) and

0, 6( ) , find the value of y.

a. –1 b. 3 c. 5 d. 9 e. 11

11. The number of diagonals in a regular polygon is 27.

Find the sum of interior angles of that polygon.

a. 900 b. 1060 c. 1260 d. 1620 e. 1800

12. In the following figure, find the value of arc angle x.

a. 31 b. 62 c. 92 d. 108 e. 123

13. The volume of a prism is equal to the volume of a

pyramid. If the base area of the prism is twice that of the pyramid, find the ratio of the height of the prism to that of the pyramid.

a. 2 : 3 b. 2 : 5 c. 1: 2 d. 1: 3 e. 1: 6

14. Find a relation between x and y such that the point

x , y( ) is equidistant from the points 7, 1( ) and 3, 5( ) .

a. x − y = 2 b. x = 2 y c. x + 2 y = 1 d. x + y = −1 e. x = y

46º

77º

xº


15. How many words (with or without meaning) of three distinct English letters are there?

a. 14,400 b. 15,000 c. 15,600 d. 16,900 e. 17,576

16. The perimeter of a regular octagon is 24 2 . Find the

ratio of its area to a similar octagon whose edge length is 4.

a. 6 2 :1 b. 9 : 8 c. 3 : 8 d. 7 :11 e. 4 : 5

17. A solid sphere of radius r is melted and cast into the

shape of a solid cone of height r. What is the radius of the base of the cone?

a. r

b. 32

r

c. 2r

d. 3r

e. 4r

18. Find the slope of y = x 2 + 3 x at the point

14

, 75

⎛⎝⎜

⎞⎠⎟ .

a. 12

b. 72

c. 158

d. 2516

e. 3316

19. Find n if n P4 : n−1P3 = 9 :1.

a. 5 b. 7 c. 9 d. 12 e. 13

20. ΔABC and ΔBDE are two equilateral triangles such

that D is the mid-point of BC. What is the ratio of the areas of triangles ABC and BDE?

a. 1:1 b. 1: 2 c. 2 :1 d. 1: 4 e. 4 :1

21. Find the number of coins 1.5 cm in diameter and 0.2

cm thick that need to be melted to form a right circular cylinder of height 10 cm and diameter 4.5 cm.

a. 250 b. 325 c. 395 d. 450 e. 515


22. A die is thrown 6 times. If getting an odd number is considered a success, what is the probability of at most 5 successes?

a. 1

64

b. 1112

c. 1316

d. 3132

e. 6364

23. A pack of tickets numbered 1 to 25 is shuffled and two

tickets are drawn. Find the probability that both the tickets drawn bear prime numbers.

a. 120

b.

325

c.

225

d.

950

e.

775

24. The vertices of a quadrilateral are P 2, −1( ) , Q 3, 4( ) ,

R −2, 3( ) and S −3, − 2( ) . What type of quadrilateral is this?

a. kite b. trapezoid c. rectangle d. rhombus e. square

25. Find the value of z in the following figure if AB, PQ and RS are chords of the circle.

(Figure not drawn according to scale)

a. 1 b. 3 c. 6 d. 10 e. 11

4

7

2

1 5

z

Q

A

B P

R

S

M

N


26. In the figure below, two secants intersect outside the circle. The secant AB intersects with the chord MN inside the circle and bisects it. Find the length FC.


a. 5 b. 7 c. 8 d. 10 e. 13

27. The difference between compound and simple interests

earned annually on a principal amount x over 2 years at 5% per annum is $5. Find the value of x.

a. $1,000 b. $1,800 c. $2,000 d. $2,400 e. $2,500

28. A convex polygon has n vertices. Starting from a vertex A, a diagonal is drawn ending at vertex B, resulting in a pentagon. From B another diagonal is drawn to a vertex C resulting in a quadrilateral and a heptagon. Find the number of sides in the polygon.

a. 11 b. 12 c. 13 d. 14 e. 15

29. ABCD is a quadrilateral such that ∠D = 90 . A circle

with center O and radius r touches the sides AB, BC, CD and DA at P, Q, R and S respectively. If BC = 38 , CD = 25 and BP = 27 , find r.

a. 11 b. 14 c. 19 d. 25 e. 27

30. Find the equation of the line tangent to the curve

y = x 2 + 4x +1at the point where the x-coordinate is 3.

a. 3x − 2 y +8 = 0 b. 7x − 9 y + 2 = 0 c. 5x + 4 y − 6 = 0 d. 10x − y −8 = 0 e. x + 9 y − 3 = 0

A

B

O

E

M

N

4

2

2

3

C F


31. In the fig. below, AB is parallel to DC. Find the largest value x can take.

a. –2 b. 2 c. 7 d. 9 e. 14

32. Three numbers are chosen at random from 1 to 20.

Find the probability that they are consecutive.

a.

1380

b.

3190

c.

1342

d.

5364

e.

3176

33. A circle with diameter 16 has the largest possible cyclic regular hexagon inscribed inside it. Find the area of the hexagon.

a. 84 b. 96 3 c. 384 3 d. 176π e. 256π

34. Find the value of

limx→−1

x 3 +1x +1

⎛⎝⎜

⎞⎠⎟

.

a. 0 b. ∞ c. –3 d. 3 e. –5

35. If the different permutations of all the letters of the

word EXAMINATION are listed as in a dictionary, how many words are there in this list before the first word starting with E?

a. 226,800 b. 453,600 c. 907,200 d. 3,628,800 e. 9,979,200

A B

C D

O

3

x - 5

x - 3

3x - 19


1. We have 5 flags of different colors. How many different

signals can be generated using these flags if each signal requires the use of 2 flags, one below the other?

a. 10 b. 12 c. 16 d. 20 e. 25

2. A vertical stick 20 m long casts a shadow 10 m long on

the ground. At the same time, a tower casts a shadow 50 m long on the ground. What is the height of the tower?

a. 100 m b. 120 m c. 25 m d. 150 m e. 200 m

3. Find the number of diagonals in a 14 sided polygon.

a. 28 b. 54 c. 77 d. 91 e. 154

4. A man goes 10 m due west and then turns right and

walks 8 m. He then turns left and walks another 5 m in a straight line. How far is he from the starting point?

a. 89 m b. 10 m c. 109 m d. 12 m e. 17 m

5. Find the slope of tangent to the curve y = 3x 4 − 4x at x = 4 .

a. 688 b. 700 c. 752 d. 764 e. 786

6. The lateral surface area of a cylinder is 264 m2 and its

volume is 924 m3 . What is the ratio of its diameter to its height?

a. 5:3 b. 7:3 c. 7:6 d. 14:3 e. 14:9

7. How much time will it take for an amount of $750 to

yield $1,050 as interest at 2.5% of per month simple interest?

a. 4 years b. 4 years 8 months c. 4 years 11 months d. 5 years e. 5 years 4 months

8. Find the distance between the points

a cosθ + b sinθ , 0( ) and 0, a sinθ − b cosθ( ).

a. a2 + b2

b. a + b c. a

2 − b2 d. a2 + b2 e. a2 − b2


9. Find x if the line containing the points x , 4( ) and

3, −1( ) is perpendicular to the line containing the points

0, 7( ) and −4, 6( ) .

a. 74

b. 174

c. − 3

2

d. − 5

8

e. 6

10. The radii of two cylinders are in the ratio 3:5 and their

heights are in the ratio 2:3. What is the ratio of their volumes?

a. 2:5 b. 3:5 c. 5:9 d. 6:25 e. 9:25

11. A book contains 100 pages. A page is chosen at random.

What is the chance that the sum of the digits on the page is equal to 9?

a. 0.05 b. 0.09 c. 0.1 d. 0.5 e. 0.9

12. How many different combinations of a captain, an assistant-captain and a goalkeeper are possible out of 18 players of which 3 are goalkeepers, assuming that a goalkeeper cannot be the captain or assistant-captain?

a. 630 b. 675 c. 918 d. 972 e. 4,896

13. If A −1, 3( ) , B 1, −1( ) and C 5, 1( ) are the vertices of

triangle ABC, find the length of the median through A.

a. 10 b. 13 c. 4 d. 21 e. 5

14. What is the ratio of the volume of a cube to that of the

largest sphere which will fit inside it?

a. 6 :π b. 3π : 4 c. 2π : 3 d. 3 : 4 e. 3 :π

15. The vertices of a quadrilateral are P −4, −1( ) ,

Q −2, − 4( ) , R 4, 0( ) and S 2, 3( ) . What type of quadrilateral is this?

a. rhombus b. square c. rectangle d. trapezoid e. kite


16. Three consecutive vertices of a parallelogram are

−2, −1( ) , 1, 0( ) and 4, 3( ) . Find the fourth vertex.

a. −3, 0( ) b. 1, 2( ) c. −1, 1( ) d. 0, 4( ) e. 2, 2( )

17. In how many ways can 4 girls and 3 boys be arranged

for a group photograph, if the girls are to sit on chairs in a row and the boys are to stand in a row behind them?

a. 12 b. 36 c. 48 d. 112 e. 144

18. Find the perimeter of a 30 − 60 − 90 triangle whose

hypotenuse is of length 18.

a. 9 3

b. 9 3+ 3( )

c. 18+ 9 3

d. 27 3

e. 27 + 3

19. Find the slope of the graph

y = − 1

2x 4 + 3x 2 − 7 at the

midpoint of −2, 8( ) and the origin.

a. 2 b. –3 c. –4 d. –7 e. 8

20. A field, in the shape of a regular hexagon with side length 120 m, is to be fenced in using aluminum wires along with creating partitions inside the field along the diagonals as shown below. What is the total cost of fencing and creating partitions at the rate of $3.50 per meter?

a. $720 b. $1,440 c. $2,520 d. $5,040 e. $10,800

21. How much interest is earned in 4 years on an investment of $3800 if it earns 7.5% APR compounded quarterly?

a. $1,315.23 b. $1,574.78 c. $4,093.12 d. $5,074.78 e. $5,115.23


22. Find the value of ∠ADC in the figure below.

a. 54 b. 72 c. 108 d. 126 e. 138

23. Find the distance of the point P 2, 1, −1( ) from the

plane x − 2 y + 4z = 9.

a. 3 2

b. 21

c. 13 21

21

d. 13

e. 13 2

24. Cone A with radius 1 and height 4 is similar to a cone B

with volume 64 2

3π . Determine the radius of the

cone B.

a. 2

b.

22

c. 4 2

3

d. 2 2

e. 2

25. If AB, AC and PQ are tangents in the figure below and

AB = 5 cm, find the perimeter of triangle APQ.

a. 5 cm b. 7.5 cm c. 10 cm d. 12.5 cm e. 15 cm

42o

66o

D

A

B

C

A

C

B

P Q X


26. Find the equation of the line that is parallel to

2x + 5 y − 7 = 0 and passes through the midpoint of the line segment joining the points 2, 7( ) and −4, 1( ) .

a. x − 5 y + 21= 0 b. x + 7 y −19 = 0 c. 2x + 7 y − 9 = 0 d. 2x + 5 y −18 = 0 e. 4x + 2 y − 7 = 0

27. Two chords AB and PQ intersect inside a circle as

shown below. If the chord AB is divided in the ratio 7:3, find the length of segment AO.


a. 2 b. 6 c. 14 d. 18 e. 22

28. A bag contains 4 black and 5 red balls. Six balls are

drawn at random from this bag. Determine the number of ways in which 3 black and 3 red balls can be drawn.

a. 20 b. 40 c. 72 d. 96 e. 132

29. In a hurdle race, a player has to jump over 10 hurdles.

The probability that he will jump over a hurdle is 56

.

What is the probability that he will knock down fewer than 2 hurdles?

a.

52

56

⎛⎝⎜

⎞⎠⎟

9

b.

56

⎛⎝⎜

⎞⎠⎟

8 16

⎛⎝⎜

⎞⎠⎟

2

c.

52

16

⎛⎝⎜

⎞⎠⎟

10

d.

16

⎛⎝⎜

⎞⎠⎟

7

+ 56

⎛⎝⎜

⎞⎠⎟

3

e.

56

⎛⎝⎜

⎞⎠⎟

18

30. A prism has a regular hexagonal base with each side of

the hexagon equal to 6 cm. If height of prism is 5 cm, find the total surface area of the prism.

a. 54 3 cm2 b. 108 3 cm2 c. 54 3 + 210 cm2 d. 108 3 +180 cm2 e. 144 3 + 210 cm2

14

6

A

B

Q

O

P


31. The letters of the word MALENKOV are placed in a line at random. Find the chance that the three vowels are together.

a.

156

b. 328

c. 18

d. 542

e. 1

18

32. The sum of the interior angles of a polygon is 1440 . If

each side has length 8, find its area.

a. 183.4 b. 492.8 c. 764.9 d. 985.6 e. 1,040.5

33. Find the average rate of change of the polynomial

y = 2x − 7x 2 +18 over the interval 3, 5[ ] .

a. –2 b. –9 c. –18 d. –27 e. –54

34. What is the value of x in the figure below?

a. 116

b.

125

c. 5

d. 173

e. 7


limx→1

x 2 −1+ x −1x 2 −1

.

a. 0

b. ∞

c. 2

d.

12

e. 2+ 2

2

6

5

10

2

4

x

A B O

E

C

F

D


1. Find the value of ∠BOC in the figure below if AB and

AC are tangents to the circle.

a. 21 b. 69 c. 90 d. 111 e. 159

2. There are 14 doors in a hall. In how many ways can a

person enter the hall through a door and leave it by a different door?

a. 144 b. 168 c. 182 d. 196 e. 210

3. A 24 m long wire is attached to the top of a pole of height 18 m and has a stake attached to the other end. How far from the base of the pole should the stake be driven so that the wire is taut?

a. 6 m b. 10 2 m c. 6 7 m d. 21 m e. 30m

4. If ΔABC and ΔDEF are similar such that 2AB = DE

and BC = 8 , then what is the length of EF?

a. 16 b. 12 c. 8 d. 4 e. 2

5. What is the chance that a non-leap year selected at

random contains 53 Sundays?

a.

25

b.

17

c.

27

d.

153

e.

1365

A

C

B

O

69o


6. What is the sum of all the exterior angles of a nonagon?

a. 360 b. 720 c. 1260 d. 1620 e. 1980

7. Water in a 3 m wide and 1.2 m deep canal is flowing

with velocity of 10 km/hr. How much water flows through it in 2 hours?

a. 36,000 m3 b. 54,000 m3 c. 72,000 m3 d. 144,000 m3 e. 288,000 m3


limx→ a

x + ax + a

.

a. 1

b.

aa

c. a

d. 2

e. 2 a

9. Find the slope of the graph y2 = 3x 2 + x at the point

−2, 10( ) .

a. −11 10

20

b. –11

c. − 25 6

12

d. 10

e.

107

10. What is the area of a right isosceles triangle whose

hypotenuse has length 7 6 ?

a. 73.5 sq. units b. 112.5 sq. units c. 147 sq. units d. 197.5 sq. units e. 294 sq. units

11. A five year cash certificate with a maturity value of $140

is purchased for $50. Find the annual rate of simple interest.

a. 7% b. 14% c. 18% d. 27% e. 36%


12. If ΔABC and ΔDEF are similar triangles such that

ABDE

= BCEF

= CAFD

= 25

, then find area ΔABC( ) : area

ΔDEF( ) .

a. 2:5 b. 5:2 c. 4:25 d. 25:4 e. 125:8

13. If the distance between the points 4, p( ) and 1, 0( ) is 5,

then find p given that 4, p( ) lies in fourth quadrant.

a. –5 b. –3 c. –1 d. –4 e. –7

14. Find the area of a rhombus whose vertices are A 8, 4( ) ,

B 0, − 2( ) , C −10, − 2( ) and D −2, 4( ) .

a. 20 b. 36 c. 48 d. 60 e. 82

15. An ancient pyramid in Egypt is found to have a square

base with edge length equal to 150 meters. Assuming that it is 80% filled with sand, find the amount of sand in the pyramid if its height is equal to 80 meters.

a. 480,000 m2 b. 600,000 m2 c. 1,440,000 m2 d. 1,800,000 m2 e. 1,920,000 m2

16. Find the value of x in the following figure.

a. 4 b. 5 c. 6 d. 7 e. 8

17. Three dice are each rolled once. Find the chance of

getting a sum of 5.

a.

5216

b. 16

c. 19

d. 1

36

e. 124

A

B

C

D

O

x

x+1

3

10


18. The vertices of a quadrilateral are A 4, 3( ) , B 6, 4( ) ,

C 5, 6( ) and D 3, 5( ) . What type of quadrilateral is this?

a. Kite b. Trapezoid c. Rectangle d. Rhombus e. Square

19. A cylindrical vessel of radius 4 cm contains water. A

solid sphere of radius 3 cm is lowered into the water until it is completely immersed. Find the rise of water level in the vessel.

a. 9 cm

b. 92

cm

c. 94

cm

d. 72

cm

e. 74

cm

20. What is the slope of the line A on which a

perpendicular from the origin intersects at the point

−2, 9( ) ?

a. − 2

9

b. 29

c. − 9

2

d. 92

e. –9

21. Which of the following is perpendicular to the direction < 21, 9, − 3 > ?

a. < 7, 3, 4 > b. < 5, − 5, 3 > c. < 4, − 4, 2 > d. < 2, 7, 1 > e. < 1, − 3, − 2 >

22. How many three digit numbers are there having no

digit as 5?

a. 648 b. 729 c. 861 d. 900 e. 999

23. Find the point on the curve y = x − 2( )2 at which the

tangent is parallel to the chord joining the points 2, 0( )and 4, 4( ) .

a. 3, 1( ) b. 3, − 4( ) c. 0, 4( ) d. −2, 1( ) e. 3, 2( )

24. How many different arrangements can be made from

the letters of the word ‘ENGINEERING’?

a. 14,641 b. 69,300 c. 277,200 d. 3,326,400 e. 39,916,800


25. The barrel of a fountain pen, cylindrical in shape, is 7 cm long and 5 mm in diameter. A full barrel of ink in the pen will be used up on writing 330 words on an average. How many words would use up a bottle containing 200 cm3 of ink?

a. 9,610 b. 24,024 c. 48,035 d. 132,078 e. 240,115

26. If the mid-point of the line joining 3, 4( ) and k, 7( ) is

x , y( ) and 2x + 2 y +1= 0 , find the value of k.

a. –15 b. –8 c. 3 d. 7 e. 12

27. Find the distance of the point 0, −1( ) from the line

joining the points 1, 3( ) and −2, 6( ) .

a. 3 2

b. 5 2

2

c.

95

d. 17

e. 53


y = −x 2 + 2 between the points where it intersects the positive x-axis and the positive y-axis.

a. 0 b. − 2

c.

22

d. –2 e. 2− 2

29. Find the length AC in the figure below.

a. 14.75 b. 15.82 c. 16.63 d. 17.59 e. 18.02

8

7 19 A

B

Q

C

P


30. If the average of three interior angles of an octagon is

107 , what is the average of the other five interior angles?

a. 151.8 b. 174.6 c. 205.6 d. 223.8 e. 253.4

31. The compound interest on a certain sum of money for

2 years is $52 whereas simple interest on same sum of money for 2 years is $50. Find the interest rate, if it is compounded annually.

a. 18% b. 12.5% c. 10% d. 8% e. 5%

32. A regular hexagon, with side length equal to 5, has a

circle inscribed inside it touching all the sides of the hexagon. Find the area of the inscribed circle.

a. 5 3π

2

b. 105π

4

c. 324π

11

d. 26π

e. 75π

2

33. If the cost of painting a square is $84, find the cost of painting a regular decagon given that they have sides of equal length and the painting cost per square meter is constant.

a. $89.2 b. $168.5 c. $357.4 d. $646.8 e. $983.1

34. Gary takes a step forward with a probability of 0.3 and

backwards with a probability of 0.7. Find the probability that at the end of 9 steps he is one step away from his starting point.

a.

92

⎛⎝⎜

⎞⎠⎟

0.3( )7 0.7( )2

b.

94

⎛⎝⎜

⎞⎠⎟

0.3( )4 0.7( )5

c.

94

⎛⎝⎜

⎞⎠⎟

0.3( )6 0.7( )3

d.

95

⎛⎝⎜

⎞⎠⎟

0.3( )4 0.7( )4

e.

96

⎛⎝⎜

⎞⎠⎟

0.3( )6 0.7( )3

35. How many three letter words (with or without

meaning) containing exactly one vowel can be made using the first 10 letters of the alphabet where repetition is not allowed?

a. 42 b. 126 c. 252 d. 378 e. 756


1. If ΔABC and ΔDEF are similar triangles such that

∠A = 47 and ∠E = 83 , then find ∠C .

a. 47 b. 50 c. 55 d. 83 e. 60

2. What is the distance between two parallel tangents of a

circle of radius 6 cm?

a. 3 cm b. 3 2 cm c. 6 cm d. 6 2 cm e. 12 cm

3. Find the sum of the interior angles of a 29-sided

polygon.

a. 4500 b. 4680 c. 4860 d. 5040 e. 5220

4. There are five types of hats in a showroom. In how

many ways can two people purchase one hat each so that the hats do not match?

a. 25 b. 21 c. 20 d. 16 e. 12

5. A fair die is rolled again and again until it shows a 6. How many items are there in the sample space?

a. 1 b. 6 c. 216 d. 46,656 e. an infinite number

6. The volume of an octagonal prism is 176 cm3 . Find the

area of the octagonal base if height of prism is 11 cm.

a. 8 cm2 b. 16 cm2 c. 24 cm2 d. 32 cm2 e. 40 cm2

7. If the area of a right isosceles triangle is 49, find the

length of its hypotenuse.

a. 7 b. 7 2 c. 14 d. 14 2 e. 21

8. ΔABC is such that AB = 3 cm , BC = 2 cm and

CA = 2.5 cm . If ΔDEF is similar to ΔABC and EF = 4 cm , then find the perimeter of ΔDEF .

a. 7.5 cm b. 9.5 cm c. 15 cm d. 22.5 cm e. 24.5 cm


9. In how many ways can a Hockey Team of eleven players be selected out of 14 players, if one of these players has already been chosen as captain?

a. 286 b. 364 c. 1,092 d. 1,716 e. 2,184

10. Determine the most specific classification for the

quadrilateral with vertices in the x-y coordinate plane at

P 5, 5( ) , Q 6, − 2( ) , R −7, − 2( ) and S −4, 5( ) .

a. rectangle b. square c. rhombus d. trapezoid e. kite


limz→1

z13 −1

z16 −1

.

a. 0

b. ∞

c. 12

d. − 1

2

e. 2

12. Which of the following direction is perpendicular to the plane 3x + 2 y − 7z = 8 ?

a. < 1, 2, 1 > b. < 7, − 3, 1 > c. < −7, 3, −1 > d. < 3, 2, − 7 > e. < −3, − 2, 7 >

13. In the figure given below, find the value of arc angle

DE.

a. 8

b. 17 c. 22.5 d. 37 e. 41


y = 2x( )3 − 7 over the interval − p, p[ ] .

a. p b. 2 p c. 4 p2 d. 8p2 e. 16 p3

A B

E

C

D 37

o

82o


15. Find the area of the isosceles trapezoid shown below in which AB CD .

a. 25 3 b. 45 2 c. 75 d. 75 3 e. 110 2

16. What is the volume of the greatest sphere that can be

cut out from a cylindrical log of wood of base radius 1 and height 5?

a. 16π

b. 43π

c. 256π

d. 12512

π

e. 500

3π

17. Simple interest on a certain sum is

9

16of the sum. Find

the rate percent and time if both are equal.

a. 5 b. 7.5 c. 10 d. 12.5 e. 15

18. A speedboat leaves an island and sails due east with a speed of 100 km/hr. At the same, another speedboat leaves the same island and sails due south at a speed of 120 km/hr. How far apart will be the two boats after 1.5 hours?

a. 132.5 km b. 157 km c. 182.6 km d. 220 km e. 234.3 km

19. If the points A −5, 7( ) , B 3, 1( ) , C x , 3( ) and D 1, y( )

form a parallelogram, find x − y .

a. –4 b. 0 c. 3 d. 7 e. 13

20. Find the perimeter of the triangle formed by the points

0, 0( ) , 1, 0( ) and 0, 1( ) .

a. 1± 2 b. 2 +1 c. 3 d. 2+ 2 e. 2 2

10 10

10

20 A B

C D


21. How many spherical bullets, approximately, each of 5 cm in diameter can be cast from melting a rectangular block of metal 1.1 m ×1 m × 0.5 m ? Round off the answer to the nearest integer.

Use π = 3.14( )

a. 15,817 b. 8,408 c. 4,203 d. 2,101 e. 1,047

22. A box contains 20 balls out of which p are red. The

probability of getting a red ball is

p20

. If 10 more red

balls are added, the probability of getting a red ball doubles. Find the value of p.

a. 1 b. 2 c. 3 d. 4 e. 5

23. How many numbers greater than 20000 can be formed

by using the digits 0, 1, 2, 3 and 4, no digit being repeated in any number?

a. 12 b. 36 c. 54 d. 72 e. 120

24. If f 7( )happens to equal the number of diagonals in a

polygon where f x( ) = x 2 + 9x −8 , then find the number of sides of the polygon.

a. 13 b. 15 c. 16 d. 19 e. 21

25. At what point of the curve y = x 2 does the tangent make an angle of 45with the x-axis, given that the tangent has a positive slope?

a. − 1

4, 116

⎛⎝⎜

⎞⎠⎟

b.

1, − 34

⎛⎝⎜

⎞⎠⎟

c.

12

, 14

⎛⎝⎜

⎞⎠⎟

d. − 1

2, 14

⎛⎝⎜

⎞⎠⎟

e.

3, − 25

⎛⎝⎜

⎞⎠⎟

26. In the figure below, OC is radius of circle and has

length 7 cm. AB is a chord of length 10 cm and is perpendicular to the radius. Find the length of segment CD.

a. 17 +1cm b. 24 cm c. 5+ 3 7 cm d. 6 7 cm e. 7 − 24 cm

A B

O

C

D


27. The sum of the slopes of a line and it’s perpendicular is

5524

. What is the slope of the line?

a.

25

b. 52

c. 38

d. 83

e. 58

28. At which of the following points is the slope of the

tangent to the curve y = x 3 equal to the y-coordinate of the point?

a. 0, 1( ) b. 1, 1( ) c. 2, 8( ) d. 3, 27( ) e. 5, 125( )

29. From a paper sheet of area 708 m2 , a regular nonagon

with side length 8 m is cut and the remaining paper is thrown away. Find the amount of paper wasted in the process.

a. 288 m2 b. 312 m2 c. 352 m2 d. 396 m2 e. 424 m2

30. A sum of money, x, becomes eight fold in 15 years when interest is compounded annually. What was the amount value after 5 years?

a. 2x

b. 83

x

c. 3x

d. 113

x

e. 4x

31. Find the distance of the point −1, 1( ) from the line

12 x + 6( ) = 5 y − 2( ) .

a. 1.8 b. 3 c. 5 d. 7.25 e. 9

32. A right angled triangle whose sides are 3 cm, 4 cm and

5 cm is revolved about the side of length 4 cm. Find the total surface area of the resulting figure.

a. 15π b. 21π c. 24π d. 25π e. 36π


33. In the figure below a circle is shown with center O. Two secants intersect each other outside the circle at A. Find the perimeter of the circle given that secant AC passes through the center of the circle.

a. 5.5π b. 11π c. 18π d. 25.6π e. 30.25π

34. On a multiple choice examination with three possible

answers for each of the five questions, what is the probability that a candidate would get four or more correct answers just by guessing?

a. 14

b. 781

c.

1577

d.

3128

e. 11243

35. In how many ways can the Venn-diagram shown below be shaded?

a. 8 b. 32 c. 128 d. 256 e. 512


1. If radius OB of the circle shown below is 8 and length

of tangent AB is 15, find the length of AO.

a. 7 b. 12 c. 17 d. 161 e. 305

2. How many 3 digit alphanumeric (combination of

letters and numbers) passwords are possible with each character distinct?

a. 17,576 b. 24,336 c. 33,696 d. 42,840 e. 46,656

3. The slope of a line is

49

. If the line is rotated clockwise

270 , then find the new slope.

a. 49

b. − 4

9

c. 98

d. 94

e. − 9

4

4. If the area of a 30 − 60 − 90 triangle is 49 3

8, find

the length of hypotenuse.

a. 3.5

b. 7 2

2

c. 7

d. 11

e. 14 2


limx→−2

1x+ 1

2x + 2

.

a. − 1

4

b. − 1

2

c. − 3

2

d. − 3

4

e. −2

6. The volume of two spheres are in the ratio 125:8. Find

the ratio of their surface areas.

a. 5:2 b. 25:4 c. 25:8 d. 125:8 e. 125:16


7. Find the slope of normal to the curve y = 4x 3 − 2x 5 at the point 2, − 32( ) .

a. − 1

24

b.

1112

c. –36

d. –112

e. 75

8. A certain sum of money, x, doubles itself in 10 years at

a certain rate of simple interest. Starting from the initial investment and the same rate of interest, how much time would it take for the money to quadruple?

a. 20 years b. 25 years c. 30 years d. 35 years e. 40 years

9. In the following figure find the value of ∠BOC .

a. 55 b. 68 c. 84 d. 112 e. 136

10. One end of a diameter of a circle is 7, −1( ) and the coordinates of the center are 2, 1( ) . Find the coordinates of the other end of the diameter.

a. 4.5, 0( ) b. 5, − 2( ) c. 2.5, −1( ) d. 0, 3( ) e. −3, 3( )

11. Find the equation of the tangent to the curve y = x 3 at

2,8( ) .

a. y = 12x −16 b. y = 8x − 24 c. y = 8−16x d. y = x − 24 e. y = 12x − 32

12. What is the sum of the interior angles of a 15-sided

polygon?

a. 2340 b. 2700 c. 3060 d. 4360 e. 5400

13. Which of the following is perpendicular to the direction

< −1,8,−3 > ?

a. <1,−8,3 > b. < 2,−5,1> c. < 4,−1,−4 > d. < −1,0,3 > e. < 0,3,−5 >


14. How many different response sheets are possible for a multiple choice test with 10 questions, where each questions has 5 choices?

a. 5 10!( )

b. 53

c. 55

d. 510

e. 510

10!

15. Find the length of the longest rod than can be

transported in a cuboid with dimensions 7 m × 4 m ×11 m .

a. 13.2 m b. 15.3 m c. 16.4 m d. 17.9 m e. 18.1 m

16. A bag contains 5 white and 7 red balls. Two balls are

drawn. What is the probability that one is white and the other red?

a. 4

33

b. 722

c. 5

33

d. 3566

e. 13

132

17. Two regular pentagonal prisms are similar to each other. The smaller of these has a base length 3 and the

larger has base length of 72

. Find the ratio of their

surface areas.

a. 6:7 b. 49:4 c. 49:9 d. 36:49 e. 216:343

18. In question 17 above, find the ratio of the volumes of

the two prisms.

a. 6:7 b. 49:4 c. 49:9 d. 36:49 e. 216:343

19. If three points 0, 0( ) ,

3, 3( ) and 3, λ( ) form an

equilateral triangle, then what is the value of λ ?

a. − 3

b. –3

c. 3

d. − 3

3

e.

33

20. How many numbers are there between 100 and 1000 in

which all the digits are distinct?

a. 144 b. 364 c. 504 d. 648 e. 720


21. Find the volume of the largest cone that can be cut out of a cube of side length 9 cm.

a. 243π

4 cm3

b. 266π

9 cm3

c. 465π

4 cm3

d. 548π

3 cm3

e. 729π

2 cm3

22. The vertices of a quadrilateral are A 6, 3( ) , B 1, 7( ) ,

C −3, 2( ) and D 2, − 2( ) . What type of quadrilateral is this?

a. Kite b. Trapezoid c. Rectangle d. Rhombus e. Square

23. The letters of the word MUMMY are placed at random

in a row. What is the probability that the letters on the extreme ends are both M’s?

a. 5

36

b. 120

c.

415

d. 3

10

e.

115

24. Find the ratio of the area of a regular heptagon to that of a regular decagon given that they have sides of same

length. Use tan 450

7⎛⎝⎜

⎞⎠⎟

= 2.08 .

a.

1077

b.

2655

c. 2548

d. 1342

e. 3791

25. Each side of a rhombus is 10 cm. If one of its diagonals

is 16 cm find the length of the other diagonal.

a. 8 cm b. 10 cm c. 12 cm d. 14 cm e. 16 cm

26. A tent is to be made out in the form of a hemisphere

along with a base made of the same cloth. Find the amount of cloth needed to make such a tent whose diameter of base is 42 m, given that 3 m2 of cloth is wasted in the process.

a. 4,160.22 m2 b. 4,157.22 m2 c. 4,154.22 m2 d. 16,632.22 m2 e. 16,635.22 m2


27. Find the average rate of change of the function

y = sin2 x over the interval

π3

, 3π4

⎡⎣⎢

⎤⎦⎥

.

a.

12π

b. 2π3

c.

35π

d. 3π4

e.

75π

28. Find the distance of the point P 8, 7, − 2( ) from the

plane 5x − 3 y + 4z + 9 = 0

a. 2 2

b.

3 55

c. 5 2

2

d. 5 7

e. 7 3

29. If the population of an area grew from 1 million to 4

million in 20 years, what is the average rate of population growth per decade?

a. 50% b. 100% c. 200% d. 300% e. 400%

30. Determine the number of 5 card combinations out of a deck of 52 cards if each selection of 5 cards has exactly one king.

a. 99,820 b. 194,580 c. 318,540 d. 583,264 e. 778,320

31. If a represents the number of diagonals of a hexagon

and b represents the number of diagonals in a heptagon, then find b

2 − a2 .

a. 81 b. 115 c. 126 d. 158 e. 196

32. Find the value of x in the following figure.

a. 1 b. 2 c. 3 d. 4 e. 5


33. If in the figure shown below ∠PAE = 90 , then find the length of EP.

a. 4 10 b. 13.5 c. 5 7 d. 18.6 e. 9 2

34. It is known that 10% of certain articles manufactured

are defective; what is the probability that in a random sample of 12 such articles, 9 are defective?

a.

310

⎛⎝⎜

⎞⎠⎟

9

b. 18339

108

c. 160381011

d.

3649

11010

⎛⎝⎜

⎞⎠⎟

e. 454796

1012

35. If a cone is cut into two parts by a horizontal plane passing through the midpoint of its axis, find the ratio of the volumes of the resulting cone and the original cone.

a. 1:1 b. 1:2 c. 1:4 d. 1:6 e. 1:8

mathematics comprehensive exam 1 - ms. harris | … comprehensive exam 1 1. three cubes each of side...

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