Lecture Notes

Algebra Number System 1. The number 7 + 0i best describe as: a. Irrational number c. Imaginary b. Real number* d. Surd 2. Evaluate square root of negative 5 multiplied by square root of negative 14. a.  – square  – square root of 70* c. 70i b. Square root of 70 d.  – square  – square root of 70i 3. Compute the value of (1 + i)^7 Significant Figures 4. Which of the following statements is correct? a. 0.0010 has one significant figure b. 200 has two significant figures* c. 100.00 has five significant figures d. 1.000328 has three significant figures Least Common Multiple 5. What is the least common multiple of 15 and 18? 6. An Engineer goes to the gym every 12 days and a Nurse every 8 days. They are in t he gym exercising today. How many days will it be until they exercise together again? Greatest Common Factor 7. What is the greatest common factor of 70 and 112? 8. Two tankers contain 825 liters and 675 liters of diesel fuel respectively. Find the maximum capacity of a container which can measure the diesel fuel of both the tankers when used an exact number of times. System of Equations 9.

Solve the value of x:

 ⁄3   −⁄3 = 17⁄4

10. If 3^x = 9^y and 27^y=81^z, then find the value of x/y: 11. Resolve into partial fraction

+ −+

12. What is the remainder of the polynomial 4x3-8x2-9x+7 is divided by (2x – (2x – 3)?  3)? 13. Given f(x) = (x+3)(x-4)+4, if f(x) is divided by (x-k) the remainder is k. Solve the value of k. Logarithms 14. Solve the value of x: log of 81 base 3 is x. 15. If log 2 = x and log 3 = y, what is the value of log 1.2? Quadratic Equations 16. Find the value of x: square root of (20 – (20 – x)  x) = x. 17. Two reviewees attempt to solve a problem that reduces to a quadratic equation. One of the r eviewees made a mistake in the constant term and gave an answer of 8 and 2 for the roots. The other reviewee made a mistake in the coefficient of the first degree term and gave an answer of -9 and -1 for the roots. If you are to check their solutions, what would be the correct quadratic equation? Age Problem 18. Roberto is 25 years younger than his father. However his father will be twice h is age in 10 years. How old is Roberto? 19. The age of Diophantus, a Greek mathematician may ne calculated from the epitaph which reads as follows: “Diophantus passed 1/6 of his life in childhood, 1/12 in youth, 1/7 as a bachelor. Five years after his after his marriage was born a son who died 4 years before his fat her at half his father’s final age.” How o ld was Diophantus when he died? Mixture Problem 20. If an alloy containing 30% silver is mixed with a 55% silver alloy to get 800 pounds of 40% silver alloy, how much of the 30% silver alloy must be used? 21. The gasoline tank of a car contains 50 liters of gasoline and alcohol, the alcohol comprising 25%. How much of the mixture must be drawn off and replaced by alco hol so that the tank contain a mixt ure of which 50% is alcohol? Work Problem 22. Twelve men can finish the job in 16 days. Five men were working at the start and after 8 days 3 men were added. How many days will it take to finish the job? 23. Pedro can paint a fence 50% faster than Juan and 20% faster than Pilar, and together they can paint at a given fence in 4 hours. How long will it take Pedro to paint the same fence if he had to work alone? 24. One pipe can fill a tank in 5 hours and another pipe can fill the same tank in 4 hours. A drain pipe can empty the full content of the tank in 20 hours. With all three pipes open, how long will it t ake to fill the tank?

25. The father can do a certain job in 9 days while his son can do the same job in 16 days. On a given day, they started working together, but after four days the son left and the father fi nished the job alone. How many more days did the father worked alone? Clock Problem 26. How many minutes after 7 o’clock will the hands of the clock be: together; opposite each other; and at right angles with each other? Motion Problem 27. Two turtles A and B start at the s ame time move towards each other at a di stance of 150 m. The rate of turtle A is 10 m/s while that of 20 m/s. A fly flies from one turtle to the other at the same time that the turtles start to move toward its each other. The rate of the fly is constant at 100 m/s. Determine the total distance travelled by the fly until the two turtle met. 28. Two jet planes traveling towards each other took the same time from two airports l ocated 4800 km apart. If they meet after 2 hours, determine the speed of the faster plane if it is flying at a speed of 160 kph faster than the other. Digit Problem 29. Given a 3-digit number. The sum of the digits is 14. The unit’s digit is half the ten’s digit. If the digits are reversed, the resulting number is 198 more than the origi nal number. Find the number. Arithmetic Progression 30. The 5th term of an AP is 17 and th e 3rd term is 10. Find the 8 th term. a. 32.5 c. 45.5 b. 27.5* d. 40.5 31. A family has 8 children, two of whom are t wins. The youngest is 3 years old and the oldest is 21 years old. Their ages are in arithmetic progression. There are three children younger than the twins. How old are the twins? a. 10 c. 14 b. 12* d. 15 Geometric Progression 32. If the 4th term of GP is 216 and the 6 th term is 1944, find the 8th term. a. 18692 c. 16486 b. 17496* d. 21664 33. If one-fourth of the air in a tank is removed by each stroke of an air pump, find the fract ional part of the air remaining after seven strokes of the pump. a. 0.75 c. 0.0123 b. 0.1336* d. 0.1645 Harmonic Progression 34. The 2nd and the 4th terms of a HP are 4/5 and t he -4. Find the 5th term. a. 1 c. -2 b. -1* d. 2 Variation Problem 35. The illumination receives from a light source varies inversely as the square of the distance from the source and directly as its candle power. At what distance from a 50-cp light would the illum ination be one-half that received at 20 ft from a 40-cp light? a. 10 sq. rt. Of 10* c. 10 sq. rt. Of 5 b. 5 sq. rt. Of 10 d. 3 sq. rt. Of 10 Diophantine Equation Problem 36. A man bought 20 chickens for P200. The cocks cost P 30 each, the hens P 15 each and the chicks at P 5 each. How many cocks did he buy? a. 1 c. 3 b. 2* d. 4 Venn Diagram Problem 37. A certain part can be defective because it has one or more out of three possible defects: insufficient tensile str ength, a burr or a diameter outside of tolerance limits. In a lot of 500 pieces: 19 have tensile strength, 17 have burr, 11 have unacceptable diameter, 12 have tensile strength and burr defects, 7 have tensile strength and diameter defects, 5 have burr and diameter defects, and 2 have three defect s. How many of the pieces have no defects? a. 450 c. 475* b. 500 d. 425

Trigonometry 1.

Which of the following is

10.

equivalent to 180 degrees?

2.

5.

shortest distance of tower C to

value of x.

the highway.

b.

100 grads

a.

0.350

a.

364 m

c.

π/2 rad

b.

0.250

b.

374 m

d.

all of these

c.

0.100

c.

384 m

d.

0.150

d.

394 m

What is the measure of 2.25 11.

At a point A south of a tower the

-835 degrees

angle of elevation of the top of

10 cm, and 14 cm. Determine the

-810 degrees

the tower is 50 degrees. At

radius of the inscribed and

c.

805 degrees

another point B, 200 m east of A,

circumscribing circle.

d.

810 degrees

the angle of elevation is 22

a.

2.45, 7.14

If the supplement of an angle θ is

degrees. Find the height of the

b.

2.45, 8.14

5/2 of its complement, find the

tower.

c.

3.43, 7.14

d.

5.43, 9.34

a.

85.9 m

a.

30°

b.

98.6 m

b.

50°

c.

85.3 m

c.

25°

d.

78.4 m

a.

0.281

d.

15°

An airplane having a speed of 120

b.

0.182

Find the angle in mils subtended

miles an hour in calm air is

c.

0.218

by a line 10 yards long at a

pointed in a direction 30°E of N.

d.

0.821

distance of 5000 yards.

A wind having a velocity of 15

12.

Solve for x in the equation: arctan(2x) + arctan(x) = π/4.

19.

If 77° + 0.40x = arctan (cot 0.25x),

2.5 mils

miles an hour is blowing from the

b.

2 mils

northwest. Find the speed of the

a.

10°

c.

4 mils

airplane relative to the ground.

b.

30°

d.

1 mil

solve for x.

a.

223 mph

c.

20°

If the sine of angle A is given as K,

b.

175 mph

d.

40°

what would be the tangent of

c.

117 mph

d.

124 mph

a.

hK/o

b.

13.

20.

The corresponding sides of the two similar triangles are in the

Solve angle A of an oblique

ratio 3:2. What is the ratio of

aK/h

triangle ABC, if a = 25, b = 16, and

their areas?

c.

ha/K

C = 94.1°.

d.

hK/a

a.

A man, who can row 6 kph in still b. c.

rate and direction of his motion. a.

7.5 km/hr @ 53.13°

b.

8.5 km/hr @ 54.14°

c.

4.5 km/hr @ 53.13°

d.

6.5 km/hr @ 54.14°

a.

9:4

52 degrees and 40

b.

3:4

minutes

c.

6:4

50 degrees and 30

d.

9:5

minutes

rate of 4.5 kph. Find the actual

d.

21.

Solve the other side c of the

54 degrees and 30

spherical triangle whose given

minutes

parts are: a = 72° 27' , b = 61° 49'

49 degrees and 32

and C = 90°. a.

9° 5'

A pole cast a shadow 15 m long

b.

8° 11'

when the angle of elevation of

c.

7° 11'

In which quadrant will angle A

the sun is 61°. If the pole is

d.

terminates if sec A is positive and

leaned 15° from the vertical

csc A is negative?

minutes

directly towards the sun,

isosceles spherical triangle with

determine the length of the pole.

given A = B = 64° 38' and side b =

14.

22.

Solve side

11° 5' c

based on the given

a.

I

b.

II

a.

23.45 m

81° 14'.

c.

III

b.

15.67 m

a.

12° 51'

d.

IV

c.

54.23 m

b.

13° 51'

If sin A = 3/5 and A is in the

d.

34.56 m

c.

14° 51'

d.

15° 51'

second quadrant while cosine B =

9.

18.

a.

current of a river flowing at the

8.

The sides of a triangle are 8 cm,

b.

water, heads directly across the

7.

17.

a.

angle A?

6.

and sin 2A = 3.939x, find the

3200 mils

value of θ?

4.

is 26° N of E. Approximate the

a.

revolutions counter-clockwise?

3.

If sin A = 2.571x, cos A = 3.06x,

15.

Three circles of radii 3, 4 and 5

7/25 and B is in the first

inches, respectively are tangent

quadrant, find sin (A+B).

23.

Find the difference in time and

to each other externally. Find the

distance between Manila (14° 36'

a.

0.936

largest angle of a triangle formed

N, 121° 05' E) and Tokyo (35° 39'

b.

-0.60

by joining the centers.

N, 139° 45' E).

c.

0.4

a.

72.6 degrees

d.

-0.82

b.

75.1 degrees

Which is identically equal to (sec

c.

73.4 degrees

x + tan x) ?

d.

73.3 degrees

a. b. c. d.

 e−ta csc x - 1

 −ta

csc x + 1

16.

Points A and B, 1000 m apart are

a.

nautical miles b.

the bearing of a tower C is 32° W of N and from B, the bearing of C

54 mins, 1613 nautical miles

c.

plotted on straight highway running East and West. From A,

74 mins, 1613

64 mins, 1520 nautical miles

d.

84 mins, 1520 nautical miles

Differential Calculus 1. 2. 3.

Differentiate: Differentiate: Differentiate:

4. 5. 6.

3 at x = 2 Differentiate: x Find the second derivative of y with respect to w of the function y = (2w^2 – 4)(3w^2 + 4). Find the partial derivative of 2x^2y + xy^2 with r espect to x. − Evaluate:

7. 8.

y = (x + 1) / (x + 2) y = sq. rt. (2 – 3x^2) y = cos (2x-3)

√ 1  

lim  −  →  −+5 Evaluate: lim   →∞  −  lim →0

(csc2)

9.

Evaluate:

10. 11. 12. 13.

Find the maximum value of y given: y = x^3 – 9x^2 + 15x – 3 Find the point of inflection of the curve y = x^3 – 6x^2 +12 Given the curve y^2 = 5x – 1 at point (1, -2), find the equation of tangent and normal to the curve. The sum of two positive numbers is 21. The product of one of the numbers by the square of the other is to be maximum, what are the numbers? What is the area of the largest rectangle that can be inscribed in a semi-circle of radius 10cm? Find the largest area of a rectangle that can be inscribed in the ellipse 4x^2 + 9y^2 = 36 Two posts, one 8 m and the other 12 m high ar e 15 m apart. If the posts are supported by a cable running from t he top of the first post to a stake on the ground and then back to the top of the second post, find the distance from the lower post to the stake to use mi nimum amount of wire? A kite, at a height of 60 ft is moving horizontally at a rate of 5 ft/s away from the boy who flies it. How fast is the cord being released when 100 ft are out? A spherical snowball is melting in such a way that its surface area decreases at the r ate of 1 sq. in per min. How fast the radius decreasing when r = 3 in.? A point moves on the parabola y = x^2 – 9x such that Vx = 2. Find the velocity and acceleration at (9, 0). The height (in feet) at any time ( in seconds) of a projectile thrown vertic ally is: h(t) = -16t^2 + 256t. What is the projectiles average velocity for the fi rst 5 seconds of travel? How fast is the projectile travelling 10 seconds after it is thrown? If the radius of the sphere is measured as 5 in with a possible error of 0.02 inch, find the approximate error an d the percentage error in the computed value of the volume? A rectangular parallelepiped is measured with 5 cm le ngth, 3 cm width and 2 cm thickness. If there were errors in measurement of 0.01 cm, 0.002 cm and 0.001 cm respectively, then what is the percentage error in the c omputed volume? Find the approximate radius of curvature of the f unction y = x^2 – 3x + 1 at the point (1, -1).

14. 15. 16.

17. 18. 19. 20.

21. 22.

23.

Integral Calculus 1.

2.

Evaluate: a. b. c. d. Evaluate: a. b. c. d.

3.

4.

5.

6.

7.

8.

9.

Evaluate:

∫sin  

½ x – ¼ sin2x + C ¼ x – ¼ sin2x + C ½ x – ½ sin2x + C ½ x – ¼ sin4x + C

10.

  ∫ (+)(+) 0.24 0.40 0.30 0.19

∫0 ∫0  3 

11.

d. 25.32 The area bounded by the curve y 2 = 12x and the line x = 3 is revolved about the line x = 3. What is the volume generated? a. 190 b. 181 c. 188 d. 184 What is the surface area generated by revolving the

√ 2



a. 1 b. 2 c. 4 d. 5 What is the approximate area under the curve y = 1/x between y = 2 and y = 10? a. 0.48 b. 1.6 c. 2.1 d. 3.0 Find the area between the curve y = cosh x and the xaxis from x = 0 to x = 1. a. 1.333 sq. units b. 1.667 sq. units c. 1.125 sq. units d. 1.175 sq. units Find the area bounded by the parabola x 2 + y + 5 = 6x and the x-axis. a. 32/3 b. 16/3 c. 31/2 d. 21/6 What is the area within the curve r 2 = 16 cos θ? a. 30 b. 36 c. 34 d. 32 Determine the length of the curve x = 2 (2t + 3) 3/2 , y = 3 (3 +t)2 from t = -1 to t = 3. a. 101.5 b. 103.7 c. 107.3 d. 109.2 Find the volume of the solid generated when the area

√ 

bounded by the curve y =  , the y-axis, and the line y = 2 is rotated about the y-axis. a. 20.11 b. 21.32 c. 23.53

12.

13. 14.

15. 16.

17.

18.

parabola y = x 2 from x = 0 to x =  about the y-axis? a. 12.74 b. 14.98 c. 11.65 d. 13.61 Find the volume formed by revolving the triangle whose vertices are (1, 1), (2, 4) and (3, 1) about the line 2x – 5y = 10. a. 52 b. 63 c. 60 d. 56 Find the centroid of a semi-circular arc of radius r. Find the coordinates of the centroid of the solid generated by revolving the area bounded by y = 4x  – x2 and y = 0 about the x = 0. Find the moment of inertia of a circle of 5 cm radius with respect to its tangent. Determine the moment of inertia about the x-axis of the area bounded by the curve x 2 = 4y, the line x = -4, and the x-axis. a. 12.19 b. 10.52 c. 13.22 d. 11.67 A spring having a modulus of 8.8 N/mm has a natural length of 250 mm. Work equal to 90, 000 N-mm is exerted in pulling it from a length of 300 mm to a length L mm. Find the final length L. a. 503.22 mm b. 401.51 mm c. 403.32 mm d. 510.21 mm A circular water main 4 m in diameter is closed by a bulkhead whose center is 40 m below the surface of the water in the reservoir. Find the force on the bulkhead. a. 4032 kN b. 4931 kN c. 4760 kN d. 4321 kN

Mechanics Statics 1.

A force of 100 kg acting horizontally to the right is to b e combined with one of 50 kg acting upward and to the right at an angle of 15 deg. with the verti cal. Solve for the resultant force R and the angle between R and the 100 kg force.

2.

The coordinates of two points in space are A ( 1,2,3) and B (4, -1, 2). Determine the angle between r adius vector A and B.

3.

A block weighing 500 kN rest on a ramp inclined at 25 deg. with the horizontal. Determine the force tending to move block down the ramp.

4.

Determine the magnitude of the r esultant force of the following coplanar fo rces: 15 N, 30 deg.; 55 N, 80 deg.; 90 N, 210 deg.; 130 N, 260 deg.

5.

A load of 100 lb is hung from the middle of the rope which is stretc hed between two rigid walls 30 ft apart. Due to the load the rope sags 4 feet in the middle. Determine the tension in the rope.

6.

A simply supported beam is five meters i n length. It carries a uniformly distributed load including its own weight of 300 N/m and concentrated load of 100 N, 2 meters from the left end. Find the reaction A at the left end.

7.

A 500-kg block is resting on a 30 deg. inclined plane with µ = 0.3. Find t he required force P acting horizontally t hat will start the block up the plane.

Dynamics 1.

A train changes its speed uniformly from 60 mi/hr to 30 mi/hr in a distance o f 1500 ft. What is its acceleration?

2.

A car travels with an initial velocity of 36 kph. If it is decelerating at a rate of 3 m/s 2, how far in meters, does it travel before stopping?

3.

A stone is thrown vertically upward at the rat e of 20 m/s. Find the highest point reached by the stone.

4.

A balloon, rising vertically with a speed of 16 ft/s, releases sandbag at an instant when the balloon i s 64 ft high. What is the position of the sand bag i n feet after its release 1/4 second l ater?

5.

A particle's position (in inches) along the x-axis after t seconds of t ravel is given by the equation x= 24t2 - t3 + 10. What is the particle's average velocity during t he first 3 seconds of travel? Where is the particle and how fast it is moving after 3 seconds?

6.

A baseball is thrown from a horizontal plane with an initial velocity of 100 m/s at an angle of 30 degrees above the horizontal. How long after will the ball attains its original vertical level?

7.

A projectile with a muzzle velocity of 550 m/s is fired from a gun on top of a cliff 460 m above sea level at a certain angle with respect to the horizontal. If the projectile hits the ocean surface 49.2 seconds after being fired, determine the horizontal range of the projectile.

8.

An automobile weighing 1455 kg traverses a 45-degree cruve at a constant speed of 48 kph. Assuming no banking on the curve, find the force acting on the tires necessary to maintain motion along the curve.