M A T H E M A T I C S – C O M P R E H E N S I V E E X A M – P AR T 3 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. STRICTLY NO ERASURES ALLOWED. Use pencil No. 2 only. 1.
2.
By how much would the perimeter of a square be increased if its area is doubled? A. 41.42% B. 58.11% C. 73.21% D. 87.08%
3.
cos0 os0 cos1 os1 cos2 cos2 ... ... cos9 cos90 0 2 B. 1 −2 D. −1
Evaluate
14.
15.
B. D.
652.14 341.20
2 lnx
1
A. C. 5.
13.
Given the equation of the curve: y 0.01 0.01 1600 1600 x 2 . Find the area bounded by the curve and the x-axis. A. 472.41 C. 853.33
4.
12.
sin0 n 0 sin1 sin1 sin2 in2 ... ... sin9 sin90 0
Find the value of A. C.
REVI REVIEW EW – 2015
dx x 0.24 0.56
16.
B. D.
y2
A. C.
2y 5
A. C.
0.42 0.65
Determine the partial derivative with r espect to x of the function f (x (x, y) y) xy2 5y 6 . B.
y2 5
D.
2y
17.
18.
3
6.
7.
8.
What is the limit of
x2 18x 16 16 0
C. Evaluate
/6
0
A. C. 10.
11.
5 3x 3x 2 2x3
as x approaches +?
A. 3/5 B. −3/2 C. 3/2 D. −3/5 3 The volume of a pyramid whose base is a regular hexagon is 156 m . If the altitude of the pyramid is 5 m, find the length of one side of the base. A. 4m B. 6m C. 8m D. 10 m Two numbers have an arithmetic mean of 9 and a geometric mean of 4. These two numbers are the roots of the quadratic equation: A.
9.
3x 2x 4
2
x 18x 16 16 0
B.
x2 18x 16 16 0
D.
x2 18x 16 16 0
19.
20.
21.
27 23 765
B. D.
35 768
22.
Find the area bounded by the curve x2 6y and y2 6x . A. 18 sq. units B. 15 sq. units C. 12 sq. units D. 9 sq. units The height of a rectangular prism is 10 cm. Its length increases at the rate of 2 cm/s while its width decreases at the rate of 4 cm/s. When the length is 8 cm and the width is 6 cm, determine the rate at which the volume is changing. 3 3 A. +200 cm /s B. −200 cm /s 3 3 C. −440 cm /s D. +440 cm /s Find the area of the region in the first quadrant bounded by the curves y sin x , y cosx
Find dy/dx from the function y ln
B. D.
7x + 16 None of these
23.
B. D.
0.414 sq. units 0.667 sq. units
B.
x 1 . x 1
1 x2 1 2
1 x2 1
2 2 D. x2 1 x 1 If x is a positive integer except 1, then which of the following mathematical statements must be true of (x 1)(x)(x 1) ? C.
2 4
0.17 sq. units 0.05 sq. units
A. +6 B. +7.5 6 7.5 C. D. 3 The volume of a cone is 523.60 cm . The cone is inclined such that its axis is equal to 40 cm and the diameter of its circular base is 10 cm. How many degrees is the cone inclined? A. 60° B. 45° C. 30° D. 15°
A.
81
B. D.
B. D.
Find the slope of the curve 2y 9x 9x2 x3 6 0 at point (1, 7).
12
If the rate of change of y 2x at x = 5 is four times the rate of change at x = a, what is the value of a? A. 1 C. 3 2 Differentiate: 7x + 16x A. 14x – 14x – 16 C. 14x + 16
0.22 sq. units 0.11 sq. units
and the y-axis A. 0.833 sq. units C. 1.100 sq. units
8
cos3A dA cos3A
363
How many ways can we arrange the letters of the word “HEAVEN” if the vowels are to be written together? A. 144 B. 72 C. 720 D. 360 Find the volume of the largest cylinder with circular base that can be inscribed in a cube of volume 27 cubic meters. 3 3 A. 12.12 m B. 21.21 m 3 3 C. 35.35 m D. 56.56 m A ladder 2 0 feet lon g is plac ed against a vertical wall. T he lower end of th e ladder b egins to slide away from the wall at the rate of 2 ft/s. How fast is the upper end of the ladder sliding down the wall when it is 12 feet above the base of the wall? 5/3 ft/s 7/3 ft/s A. B. C. D. 8/3 ft/s 10/3 ft/s Given trapezoid ABCD, AB is parallel to DC and AB
A. C.
It is always negative It is always divisible by 3
B. D.
It is always odd It is always divisible by 4 MDSD ☺
M A T H E M A T I C S – C O M P R E H E N S I V E E X A M – P AR T 3 24.
25.
26.
Given the curve x2 16 8y 8y . Compute the centroid of the area bounded by the given curve in the first quadrant. A. (1.8, 1.2) B. (1.2, 0.6) C. (1.5, 0.8) D. (1.6, 0.7) 3 The volume of a sphere is increasing at the rate of 9 m /s. When the volume of the sphere is 3 36 m , find the rate at which the surface area is increasing. 2 2 A. B. 3 m /s 4 m /s 2 2 C. D. 6 m /s 7 m /s sin x 1 cos x Simplify the trigonometric expression: 1 cos x sin x
27.
28.
29.
30.
33.
34.
35.
37.
38.
39.
40.
41.
42.
Given xy y2 1 , find dy dx .
C.
32.
36.
B.
A region is bounded by the curve y2 4x and the line x y . Find the distance of the centroid of the bounded area from the x-axis. A. 2.00 units B. 2.40 units C. 2.50 units D. 2.75 units The side of a regular dodecagon is 2 cm. Find the radius of the circumscribing circle. A. 3.035 cm B. 3.562 cm C. 3.732 cm D. 3.864 cm
A.
31.
sin2 x
cos cos2 x C. 1 D. 0 Three Shooters were practicing to hit a target. Their individual probabilities to hit a target are respectively: 1/6, 1/4 and 1/3. Each shoots once at the target. Given that only one hit the target, what is the probability that it was the first Shooter? A. 6/31 B. 5/36 C. 5/24 D. 31/72 A.
REVI REVIEW EW – 2015
y x 2y
y x 2y 2
x
B.
y 2x
D.
43.
x y 2x
2
The equation of an ellipse is 9x +16y – 144 – 144 = 0. Find the ratio of the lengths of minor axis to major axis. A. 0.75 B. 0.33 C. 0.51 D. 0.62 A balloon is rising vertically over a point A on the gr ound at the rate of 1 5 ft/sec. A point B on the ground is level with and 30 ft from A. When the balloon is 40 ft from A, at what rate is its distance from B changing? A. 10 ft/s B. 12 ft/s C. 13 ft/s D. 15 ft/s The sum of two angles is 1600 mils and their difference is 40 grads. Find the value of the bigger angle in degrees. A. 36° B. 48° C. 63° D. 74° Find the derivative of y 1 / (x (x 2) at x = 1. A. 1 B. −1 C. 0 D. The base radius of a cone is 32 m and its altitude is 54 m. Determine the altitude of a cylinder of the same volume whose base diameter is 48 m. A. 8m B. 16 m C. 24 m D. 32 m
44.
45.
46.
47.
48.
A light is placed on the ground 30 f eet from a building. A man 6 f eet tall walks from the light towards the building at the rate of 5 feet per second. Find the rate at which the length of his shadow on the wall is changing when he is 15 feet from the building. A. 4 ft/sec B. 6 ft/sec C. 8 ft/sec D. 10 ft/sec There are 12 persons attending a party. In how many ways can 7 among them be seated on a round table of 7 seats? A. 570420 B. 570240 C. 570042 D. 570024 Find the area bounded by the curve 16x 2 9y 2 36y 10 108 0 . A. 18.85 square units B. 37.70 square units C. 23.56 square units D. 47.21 square units Compute the volume generated by revolving about the y-axis the area in the first quadrant 2 bounded by the curve x = 10y, the line x = 4 and the x-axis. A. 40.21 B. 41.21 C. 42.21 D. 43.21 Find the rate of change of the circumference of a circle with respect to the area, when the area is equal to 4. A. 0.80 B. 0.75 C. 0.50 D. 0.35 Find the area of a cyclic quadrilateral whose sides are 4 cm, 5 cm, 8 cm and 11 cm. A. 60.25 sq. cm B. 50.25 sq. cm C. 40.25 sq. cm D. 48.65 sq. cm A line with a slop e of −2 forms an angle of 45° with another line. Find the slope of the other line. A. 2 B. 2.5 C. 3 D. 3.5 b The linear equation that has the solution x is a bx a 0 ab x 0 A. B. ax b 0 ab x 0 C. D. A box contains 24 red balls, 27 green balls, and 30 blue balls. Three balls are drawn in succession without replacement; find the probability that all three balls drawn are red? A. 0.02372 B. 0.02704 C. 0.02601 D. 0.02195 The altitude and diameter of a cone are equal. Find the distance in terms of the altitude from the base through which a cutting plane may be passed that will divide the cone into two equal volumes. A. 0.2063H B. 0.3145H C. 0.6855H D. 0.7937H Given is the area bounded by the curve y sqrt sqrt of (x) (x) , the line x = 1, the line x = 3, and the x-axis. If it is revolved about the x-axis, determine the volume of the solid formed. A. B. 2 C. 3 D. 4 3 If the volume of a sphere is 36 cm . Find its surface area. 2 2 A. 25.81 cm B. 48.28 cm 2 2 C. 52.81 cm D. 79.09 cm A curve has an equation of y cos x . If the area bounded by the curve from x = 0 to x =
/2
is revolved about the x-axis, determine the volume of solid generated. A. 2.47 cu. units B. 3.29 cu. units C. 4.93 cu. units D. 9.87 cu. units MDSD ☺
M A T H E M A T I C S – C O M P R E H E N S I V E E X A M – P AR T 3 49.
50.
51.
52.
53.
54.
55.
56.
57.
58.
59.
A particle moves according to the equation s t4 3t3 3t2 t 3 where s is in meters and t in seconds. Determine the time when the velocity is increasing. A. 0.25 < t < 1.00 B. 0.50 < t < 1.00 C. t < 0.50 or t > 1.00 D. t < 0.25 or t > 1.00 Find the equation of the hyperbola which has the line 2x – 2x – 3y 3y = 0 as one of its asymptotes. 2 2 2 2 A. 2x – 3y – 3y = 6 B. 3y – 2x – 2x = 6 2 2 2 2 C. 4x – 9y – 9y = 36 D. 9y – 4x – 4x = 36 The area enclosed by the ellipse 4x2 9y2 36 is revolved about the line x 3 . What is the volume generated? A. 370.3 cu. units B. 365.1 cu. units C. 360.1 cu. units D. 355.3 cu. units
REVI REVIEW EW – 2015 62.
velocity at t 3 . A. 44 C. 64 63.
64.
3
Find the second derivative of y x x 1 when x = 1. A. 36 B. 40 C. 48 D. 56 How many numbers less than 7000, with no repeated digits can be formed by use of the digits 0, 3, 5, 7, 4, 2? A. 131 B. 371 C. 240 D. 265 Given one large coin with 4.8 cm in diameter and many small coins with 1.5 cm in diameter. Determine the number of small coins that maybe arrange tangentially around the large coin without overlapping. A. 15 B. 13 C. 11 D. 10 Find the volume of the solid generated by rotating the circle x2 y2 16 about the line y 14 14 0 . A. 6632.37 cu. units B. 4421.58 cu. units C. 3316.19 cu. units D. 2210.79 cu. units A flagstaff s tanding on top of a tower 80 ft high subtends an angl e of arctan(1/9) at a point 100 ft from the foot of the tower. Find the height of the flagstaff. A. 18 ft B. 20 ft C. 24 ft D. 30 ft From the equation y ax ax3 bx2 cx d , the critical point is located at the origin and the inflection point is at (2, 4). Find th e value of “a”. A. 1/4 B. −1/4 C. 3/2 D. −3/2 A solid material is in the form of a rectangular parallelepiped 4 ft 6 ft 8 ft. The solid is cut completely to form cubes 1 ft 1 ft 1 ft. How many cubes will there be? A. 156 B. 188 C. 192 D. 208 (5) 2 Find y of the function y = sin x. A. 2cos2x B. −8cos2x C. −4sin2x D. 32sinxcosx
60.
Find the slope of the line tangent to the curve y e5x at x = 0.
61.
A. 2 B. 4 C. 5 D. 7 The axis of a hyperbola that passes through its foci is known as: A. conjugate axis B. transverse axis C. minor axis D. major axis
The distance a body travels is a function of time and is given by x t 16t 8t2 . Find its
65.
66.
67.
69.
70.
54 74
A region is bounded by the cur ve y2 4x and the line x y . Find the volume generated if the bounded area is rotated about the x-axis. A. 35.19 cu. units B. 33.51 cu. units C. 52.36 cu. units D. 63.36 cu. units If the side of an equilateral triangle decreases at the constant rate of 2 inches per minute, find the time rate of change of the area when the side is 6 inches. 2 2 A. 4.56 in /min B. 5.64 in /min 2 2 C. 7.69 in /min D. 10.39 in /min If the altitude to the hypotenuse of a right triangle is 8, determine the lengths of the segments of the hypotenuse formed by the altitude. A. 6 and 10 B. 5 and 13 C. 4 and 16 D. 3 and 22 In the expression a1/n , there is no real root if A. n is even and a is positive B. C. n is odd and a is negative D.
n is even and a is negative n is odd and a is positive
For the formula: R E C , find the maximum error in R if C = 20 with possible error of 0.1 and E = 120 with possible error of 0.05. A. 0.0275 C. 0.0523
68.
B. D.
B. D.
0.0325 0.0752
Find the minimum value for th e slope of the tangent to th e curve of f x x5 x3 2x . A. m = −1 B. m = −1/2 C. m = −2 D. m = −2/3 A c ardboard 20 cm 20 cm is to be formed into a box by cutting four equal squares and folding the edges. Find the volume of the largest box formed. 3 3 A. 559.29 cm B. 592.59 cm 3 C. 529.95 D. 525.99 cm A test consists of 5 questions and to pass the test, a student has to answer at least 4 questions correctly. Each question has three possible answers of which only one is correct. If a student guesses on each question, what is the probability that the student will pass the test? A. 0.0453 B. 0.0321 C. 0.0589 D. 0.0114
MDSD ☺
M A T H E M A T I C S – C O M P R E H E N S I V E E X A M – P AR T 3
REVI REVIEW EW – 2015
ANSWERS:
1
A
26
D
51
D
2
B
27
A
52
A
3
C
28
A
53
B
4
A
29
D
54
B
5
A
30
C
55
B
6
B
31
A
56
B
7
B
32
B
57
B
8
D
33
C
58
C
9
B
34
B
59
D
10
C
35
D
60
C
11
C
36
A
61
B
12
B
37
B
62
C
13
B
38
B
63
B
14
C
39
A
64
D
15
B
40
C
65
C
16
C
41
C
66
B
17
B
42
C
67
A
18
C
43
C
68
C
19
C
44
A
69
B
20
B
45
A
70
A
21
B
46
D
22
A
47
C
23
C
48
A
24
C
49
C
25
C
50
C
MDSD ☺
