A plane area is bounded by the curve y2=4x and the line y=x.
1. Find the area bounded by the curves.
A. 3.33
B. 3.67
C. 2.67
D. 2.33
2. How far from the x-axis is the centroid of the curve?
A. 2.5
B. 3.3
C. 2
D. 3
3. Find the volume generated if the area is revolved about the x-axis.
A. 42.8
B. 25.8
C. 38.5
D. 33.5
Solution:
An equipment have a salvage value of P1 million at the end years. The straight-line depreciation charge is P2million.
1. What is the first cost of the machine?
A. P105 million
B. P101 million
C. P103 million
D. P110 million
2. What is its book value after 25 years?
A. P51 million
B. P55 million
C. P49 million
D. P45 million
3. At what year will its total depreciation be P30 million?
A. 14 years
B. 10 years
C. 18 years
D. 15 years
Solution:
A projectile is fired upward to the right from the top of a 50-m high building with an initial velocity of 200 m/s and an angle of 60° with the horizontal. A wind of 5 m/s is blowing horizontally to the left.
1. Determine the maximum height above the building that the projectile can reach.
A. 1247 m
B. 1087 m
C. 1632 m
D. 1529 m
2. Compute the time for which the projectile reaches its maximum height.
A. 17.7 sec.
B. 18.2 sec.
C. 12.5 sec.
D. 9.6 sec.
3. Determine the y-component of the velocity with which is hits the ground.
A. 196 m/s
B. 201 m/s
C. 185 m/s
D. 176 m/s
Solution:
The deflection angle from PC of two points A and B on a simple curve are 3°15’ and 8°15’, respectively. The chord distance between A and B is 40 m.
1. Find the length of curve from PC to A.
A. 32m.
B. 36m
C. 26m
D. 28m
2. Find the length of long chord from PC to B.
A. 62.3m
B. 65.9m
C. 71.4m
D. 76.8m
3. What is the radius of the curve?
A. 247.6m
B. 229.5m
C. 258.7m
D. 187.4m
Solution:
Determine the derivative of y with respect to x of the following equations:
1. x=y3/3
A. 1/y2
B. y2
C. y-0.5
D. y0.5
2. y=1/(1-u) where u = 1/(1+x)
A. x2
B. –x-2
C. x-2
D. –x2
3. x=y3/3.
A. (3x)3/2`
B. (3x)-3/2
C. (3x)2/3
D. (3x)-2/3
Solution:
Evaluate the following limits:
1. Limit of (x2-1)/(x-1) as x approaches 1.
A. 4
B. ¼
C. ½
D. 2
2. Limit of sin (1/x) as x approaches zero.
A. -0.99
B. -0.85
C. -0.75
D. -0.65
3. Limit of (x3-2x2+7)/(2x3+x-3) as x approaches infinity.
A. 2
B. 4
C. ½
D. ¼
Solution:
Given the circle x2 + y2 – 6x + 12y + 9 = 9.
1. What is the radius of the circle?
A. 7.21
B. 6.71
C. 6.28
D. 8.63
2. Where is the center of the circle?
A. (3,-6)
B. (-3,-6)
C. (-3,6)
D. (3,6)
3. Find the shortest distance from the line y = 2x + 10 to the center of the circle.
A. 6.87
B. 10.24
C. 8.49
D. 9.84
Solution:
Given an ellipse 16x2 + 25y2 + 128x – 150y + 38 = 0.
1. Determine the distance between the foci.
A. 4
B. 2.5
C. 3
D. 3.5
2. Determine the distance between the directrices.
A. 8.67
B. 9.67
C. 12.44
D. 8.33
3. Determine the center of the ellipse.
A. (-4,3)
B. (-4,-3)
C. (4,3)
D. (4,-3)
Solution:
Given the Matrix A and its inverse matrix B:
1. Determine the value of x.
A. -2
B. 5
C. 1
D. -7
2. Determine the value of y.
A. 1
B. 5
C. -7
D. -2
3. Determine the value of.z.
A. -2
B. -7
C. 5
D. 1
Solution:
A 75m x 75m borrow pit is to be divided into 9 square sections. The following are the elevations of the ground surface at the corners of each square section of the borrow pit.
| A=2.3m | B=3.1m | C=2.7m | D=2.8m |
| E=2.0m | F=1.8m | G=2.9m | H=3.6m |
| I=2.8m | J=2.5m | K=3.4m | L=4.2m |
| M=3.8m | N=4.2m | O=4.8m | P=5.5m |
Find the volume of earth to be excavated if the ground surface is to be leveled to elevation 5m below the elevation zero.
A. 48,658.5 m3
B. 46,896.5 m3
C. 45,562.5 m3
D. 42,635.5 m3
Solution:
The area bounded by the waterline of a reservoir and the contours at an interval of 2m is follows: A1 = 10,250 m2, A2 = 8,350 m2, A3 = 7,750 m2, A4 = 6,900 m2, and A5 = 5,250 m2, Calculate the volume of the reservoir by the prismoidal formula.
A. 61,333 m3
B. 60,867 m3
C. 61,500 m3
D. 71,200 m3
Solution:
Lines of levels are run form station A to station B over three different routes. The route length and difference in elevation between A and B are given in the following table.
| Route | Length (km) | Difference in Elevation (m) |
| 1 | 6.2 | 425.34 |
| 2 | 5.8 | 424.12 |
| 3 | 3.8 | 426.45 |
Determine the most probable value of the elevation of station B if station A is at elevation 187.235m.
A. 613.685
B. 612.386
C. 613.125
D. 612.712
Solution:
A vertical summit curves have tangent grades of 2.2% and -1.8%. The required slopping sight distance 100m. Find the length of the curve.
A. 105.82m
B. 94.15m
C. 112.41m
D. 87.36m
Solution:
The deflection angle from PC of two points A and B on a simple curve are 3°15’ and 8°15’, respectively. The chord distance between A and B is 40m. Find the length of curve from PC to A.
A. 32m
B. 36m
C. 26m
D. 28m
Solution:
What is the probable error of the mean of the following measurements? d1 = 141.50, d2 = 141.6, d3 = 141.7
A. ± 0.03498
B. ± 0.03489
C. ± 0.03984
D. ± 0.03894
Solution:
A 50-kN automobile passes through a circular curve. If the impact factor is 0.3, find the force that is pulls the automobile away from the center of the curve.
A. 30kN
B. 5kN
C. 45kN
D. 15kN
Solution:
A 45cm x 45cm square plate ABCD of uniform thickness is supported by three vertical strings. The first string is located at B, the second is located 15cm from A alongside AD, and the third is at the point midway of side CD. What percent of the load is carried by the second support?
A. 60%
B. 40%
C. 30%
D. 20%
Solution:
A block is resting on a plane surface. What angle can the plane be inclined when the block start to slide if the coefficient of friction between the block and surface is 0.28?
A. 15.64°
B. 19.32°
C. 12.54°
D. 17.84°
Solution:
Determine the annual cost of a structure that requires P15M to build with a salvage value of P2M after 12 years, if interest rate is 6%.
A. P1,856,324
B. P2,025,547
C. P1,670,601
D. P1,532,025
Solution:
An equipment costing P60,000 have book value of P4,000 after 8years. Find the constant percentage of the declining book value.
A. 22.36%
B. 28.72%
C. 21.87%
D. 25.36%
Solution:
Fifty thousand pesos is deposited annually at an interest rate of 6% per annum compounded continuously. How many years will it accumulate to P1,000,000?
A. 16.74
B. 10.57
C. 13.42
D. 12.54
Solution:
How many years will a deposit triple itself at an interest rate of 7% per annum compounded continuously?
A. 13.6
B. 11.1
C. 16.2
D. 17.5
Solution:
Find the moment of inertia with respect to the X-axis of the area bounded by the parabola x2 = 8y, the line y = 2 and the Y-axis.
A. 9.143
B. 10.365
C. 8.652
D. 11.214
Solution:
The velocity of a particle is defined by the equation υ = 8t2 + 4t + 1, where υ is in m/min and t is in minutes. Find the distance after 2 minutes.
A. 28.3m
B. 25.7m
C. 36.7m
D. 31.3m
Solution:
The sides of a triangle sides are 9.61m, 4.25m, and 8.62m. Find the width of the largest rectangle that can be inscribed in the triangle with one side along the 9.61m side.
A. 4.81m
B. 1.91m
C. 3.51m
D. 2.71m
Solution:
Water flows at the rate of 10ft3/min into a vertical tank. The water surface in the tank is rising at the rate 4 in/min. Find the radius of the tank.
A. 3.9ft
B. 4.2ft
C. 3.5ft
D. 3.1ft
Solution:
A circle with radius of 5 has its center at (4, 8). Find the equation of the centers of family of circles tangent to the given circles tangent to the given circle and the X-axis.
A. y2 – 4x – 26y + 55 = 0
B. x2 – 8x – 26y + 55 = 0
C. x2 – 9x – 24y + 55 = 0
D. x2 + 8x – 20y + 64 = 0
Solution:
Find the radius of a sphere having the following equation: x2 + y2 + z2 – 4x + 6y – 10z + 13 = 0
A. 4
B. 5
C. 3
D. 6
Solution:
Find the length of the latus rectum of the parabola x2 – 6x = 12y + 51.
A. 12
B. 3
C. 6
D. 16
Solution:
The sides of a spherical triangle ABC are a = 35.25°, b = 63.58°, and c = 85.47°. Find the measure of angle B.
A. 123.4°
B. 68.4°
C. 28.9°
D. 48.6°
Solution:
The upper base of a frustum of a pyramid is 2.5m by 4m and the lower base is 5m by 8m. Find its volume if the distance between the bases is 6m.
A. 140 cm3
B. 160 cm3
C. 130 cm3
D. 120 cm3
Solution:
A rectangle whose width is 4 units and height of 2 units has its center at coordinate (10, 12). The 4-unit side is parallel to the Y-axis. Find the volume generated by revolving the area about the Y-axis.
A. 536.45 cu.units
B. 502.66 cu.units
C. 587.12 cu.units
D. 457.65 cu.units
Solution:
The base of a truncated prism is a rectangle with length twice its width. The corner edges have heights of 12m, 12m, 16m, and 16m respectively. If the volume of the prism is 8,200 cu.m., find the length of its base.
A. 17.11
B. 34.23
C. 28.62
D. 14.31
Solution:
Find the area of a regular octagon circumscribing a circle of area 126 cm2.
A. 142 cm2
B. 168 cm222
C. 156 cm2
D. 133 cm2
Solution:
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