1. The numbers whose sum is 18, are in arithmetic progression. If the first term is multiplied by 2,
the second by 3, and the third by 6, the resulting numbers form a geometric progression. Find
the numbers.
2. A man bought 20 chickens for $20. The cocks cost $3 each, the hens are $1.5 each, and the
chicks at $0.5 each. How many did he buy?
3. A tank contains a mixture of acid and water. If 30 lit of acid are added the mixture will be 70%
acid. If 30lit of water are added the mixture will be 50% acid. Determine the percentage of acid
in the original mixture.
4. A vertical tower stands on horizontal ground. A surveyor positioned due east of the tower
measure the angle of elevation of the top as 48 degrees. he moves due south 30 m and
measures the angle of elevation of the top as 44 degrees. Determine the height of the tower.
5. If sin (u + v) = 0.766 and sin (u-v) = 0.176, find sinucoav.
6. What is the area in cm of the circle circumscribed about an equilateral triangle with a side of
10cm long?
7. If the edge of a cube is increased by 30%, by how much is the surface area increased?
8. Determine the volume of a right-angle truncated prism with the following dimensions: let the
corner of the triangular base be defined by A, B, and C. The length AB = 10ft, BC = 9ft and CA =
12ft. the sides at A, B and C are perpendicular to the triangular base and have the height of 8.6
ft, 7.1 ft, and 5.5 ft, respectively.
9. Find the angle formed by the lie lines 2x+y-8=0 and x=3y+4=0.
10. Find the area bounded by the curve 16x^2+9y^2=36y-108=0.
1. Which of the following is the exact differential equation of y’=2xy-x?
2. Which if the ff must be the value of c1 and c2 so that y(x)=c1e^2x+c2e^x+2sinx will satisfy y(0)=0
and y’(0)=1?
3. Solve y’=y^2x^3
4. Which of the ff is the general solution of y^2=cx?
5. A radium decomposes at a rate proportional to the amount at any instant. In 100 years, 100mg
of radium decomposes 96mg. How many mg will be left after 100 years?
6. According to the newton’s law of cooling, the rate at which a substance cools in air is directly
proportional to the difference between the temperature of the substances and that or air. if the
temperature of the air is 30 degrees and the substance cools from 100 degrees to 70 degrees in
15 minutes, how long will it take to cool 100 degrees to 50 degrees?
7. Rationalize (4+3i)/(2-1)
8. Given the matrix, solve for x and y
9. Which of the ff gives the magnitude of projection of 5i-3j-k on the vector 4i-15j+10k?
10. Given a square matrix below, what is the cofactor of element a32.
11. Simplify the expression i^29+i^21+1
12. What is the dot product of 1i+2j+3k and 4i-5J+5k?
Situation: The first branch of reverse curve has a radius of 200m. If the distance between the tangent
points is 110m assuming the reverse curves are in parallel.
1.
2.
3.
4.
What is the radius of the second curve?
Find the length of the first curve.
Find the total length of the curve.
A survey line was made measured on slopping ground and recorded as 117.84m. The difference
between the end was 5.88m. The tape was later found to be 30.66m when compared with a
standard of 30.48m. Calculate the horizontal length of line.
5. A line level was run from point 5 to 6, 8km apart. the average BS and also FS distance was 100m.
At every turning point, the rod settles by 3cm. Find the total error in the recorded elevation of
point 6.
Situation: A circular curve having an azimuth of back tangent equal to 205 degrees and the azimuth of
the forward tangent equal to 262 degrees. if the middle ordinate is 5.8m.
6.
7.
8.
9.
Compute the angle of intersection.
Compute the results of the curve.
Compute the length of the tangent.
The DMD of the line BC is equal to 426.20. the departure of the previous line AB is 117.56. What
is then the departure of the line BC.
10. A highway curve having a radius of 121.95m is banked so that there will no lateral pressure on
car’s wheel at a speed of 48kph. What is the angle of elevation of the embankment?
11. From station 0+040 with center height of 1.2m in fill, the ground line makes a uniform slope of
6.5% to station 0+100, whose center height in cut is 3.5m. Find the grade of the finished
roadway.
Situation: The data below shows the results of the flow of traffic at a certain part of highway by
observing the arrival times for four vehicles at two sections A and B that are 150m apart. The total
observation time at section A is 15 seconds.
VEHICLE
1
2
3
4
SECTION A
T
T+3
T+6
T+12
12. Determine the mean speed in kph.
13. determine the space mean speed.
14. Compute the flow at section A in vehicles per hour.
SECTION B
T+7.58
T+9.18
T+12.36
T+21.74
1. Water flows through an orifice at the vertical side of a large tank under a constant head of 2.4m.
how far horizontally from the vena contrata will jet strikes the ground 1.5m below the orifce?
2. A triangular channel with most efficient proportion discharges water at the rate of 1m/s.
Assuming n=0.018 and s=0.0021, calculate the normal depth of flow in meters.
3. A rectangular channel 5.5m wide by 1.3m deep is lined with a smooth stone, well laid and has a
hydraulic slope of 0.002. use n=0.013. What is the capacity of the channel?
4. Water flows in a triangular V-notch channel with vertex 60 degrees, at a velocity of 1.2m/s. Find
the normal depth of flow id the channel is laid on a slope of 0.0017. use n=0.014.
Situation: A meter diameter cast iron pipe is 845m long and has a head loss of 1.11m.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
Find the discharge capacity of water for the pipe by Hazen-William’s formula. C=130.
Find the discharge capacity of water for the pipe by Manning’s formula. n=0.013.
Find the discharge capacity of water for the pipe by Darcy Weisbach formula. f=0.014.
A block of wood floats in water with 20cm projecting above water surface. The same block when
placed in oil (s=0.80) floats with 15cm projecting above the oil surface. Find the specific gravity
of the block.
A body having a specific gravity of 0.70 floats on a liquid with specific gravity of 0.80. The
volume of the body above the liquid surface is what percent of its total volume?
A trapezoidal dam 4m on top, 18m at the bottom and 24m high has a water 20m deep acting on
its vertical side. What is the factor of safety against overturning? Assume weight of masonry to
be 2400kg/m.
A stone weighs 105 lb in air. When submerged in water it weighs 67lb. Find the specific gravity
of the stone.
An open cylinder vessel 1m in diameter contains water to a depth of 3.56m. If it is rotated at a
speed of 80 rpm about its vertical axis, determine the least depth of the vessel so that no water
will be spilled out.
A piece of wood floats in water with 5cm projecting above the water surface. When placed in
glycerins (s=1.35), the wood floats 7.5cm projecting above the glycerin surface. Determine the
height of the wood.
Determine the absolute pressure at a depth of 10m below the free surface of oil 9s=0.75) if the
barometric reading is 752mm Hg.
Situation: A tank 12m high is to be filled with oil having a unit weight of 9.40 kN/m as shown. neglect
the weight of the tank (unit weight of sand=18.88 kN/cum, unit weight of clay = 17.31 kN/cum, LI =
0.645, PL = 20%, e = 1.27, w = 40%).
1. Which of the ff gives the compression index of the clay?
2. Which if the ff most nearly gives the settlement under the center of the tank?
3. Which of the ff most nearly gives the minimum depth in the ground to which the tank must be
placed to minimize settlement?
4. A constant head permeability laboratory test is conducted on a certain soil. the diameter of the
sample is 5.50cm, the length is 18cm, and the constant head is 45cm. the mass of water
collected in 50sec is 86 grams. Determine the coefficient of permeability in cm/min.
5. During a falling head permeability test, the head falls from 80cm to 65cm in 83 seconds. the
specimen has a diameter of 5cm and a length of 10cm. the diameter of the standpipe used was
0.8 cm and the temperature during the test was 90 degrees Celsius.
Situation: The soil under a dam has four layers with diff coefficients of permeability.
LAYER
1
2
3
4
DEPTH
4m
8m
12m
3m
COEF. OF PERMEABILITY
5cm/hr
3cm/hr
2cm/hr
1cm/hr
6. Which if the ff most nearly gives the equivalent vertical coefficient of permeability?
7. Which of the ff nearly gives the transmissibility of the soil when the water table is at the ground
surface?
8. If the void ratio is 0.60 and the hydraulic gradient is 0.0018, which of the ff most nearly gives the
interstitial velocity of water moving through the soil?
Situation: The figure shows a layer of soil in a tube 100mm x 100mm in cross section. Water supplied to
maintain a constant head difference of H across the sample. hydraulic conductivities of each soil are
tabulated below.
SOIL
A
B
C
D
E
K (m/s)
0.005
0.0025
0.00625
0.004
0.002
9. Which of the ff gives the equivalent coefficient of permeability of the system in the direction of
flow?
10. If H=1.20m, which of the ff nearly gives the rate of flow?
11. if H=1.50m, which of the ff most nearly gives the rate of flow?
Situation: A 300mm diameter test well penetrates 27m below the static water table. Water is pumped at
a rate of 69 L/sec for 24 hrs. The water levels at two observation wells 35m and 98m from the test well
were lowered by 1.10m and 0.56m, respectively.
12. Which of the ff most nearly gives the rate of the flow in MLD?
13. Which of the ff most nearly gives the coefficient of permeability of the aquifer?
14. Which of the ff most nearly gives the transmissivity of the aquifer?
Situation: A flow net around a single row of sheet piles in a permeable soil is shown below. K=8x10^2mm/sec.
15. Which of the ff most nearly gives the rise above the ground surface when a piezometer is placed
at point A.
16. Which of the ff mostly gives the rate of seepage through the flow channel per m width?
17.