Problem 74 Problem 85

Problem 74 If 3x3- 4x2y + 5xy2 + 6y3 is divided by ( x2 - 2xy + 3y2 ), the remainder is: Ans. 0 Problem 75 The arithmetic mean of 6 numbers is 17. If two numbers are added to the progression, the new set of number will have an arithmetic mean of 19. What are the two numbers if their difference is 4? Ans. 23, 27 Problem 76 Log 6 845 =? Ans. 3.761 Problem 77 What is the sum of the coefficients of the expansion of ( 2x - 1 )20? Ans. 0 Problem 78 Find k so that 4x2 + kx + 1 = 0 will only have one real solution. Ans. 4 Problem 79 Pedro can paint a fence 50% faster than Juan and 20% faster than Pilar and together they can paint a given fence in 4 hours. How long will it take Pedro to paint the same fence if he had to work alone? Ans. 10 hrs. Problem 80 A and B can do a piece of work in 42 days, B and C in 31 days, and A and C in 20 days. Working together, how many days can all of them finish the work? Ans. 18.9 Problem 81 At what time after 12:00 noon will the hour hand and the minute hand of a clock first form an angle of 1200? Ans. 12:21.818 Problem 82 A boat takes 2/3 as much time to travel downstream from C to D, as to return, If the rate of the river’s current is 8kph, what is the speed of the boat in still water? Ans.40 Problem 83 How many terms of the sequence -9, -6, -3,… must be taken so that the sum is 66? Ans.11 Problem 84 How many 4 digit numbers can be formed without repeating any digit, from the following digits 1, 2, 3, 4 and 6. Ans. 120

Problem 85 A bag contains 3 white and 5 black balls. If two balls are drawn in succession without replacement, what is the probability that both balls are black? Ans. 5/14 Problem 86 Find the value of x in the equation csc x + cot x = 3. Ans.  /5 Problem 87 In a triangle, find the side c if angle C = 100 0, side b = 20, and side a = 15. Ans.27 Problem 88 A man finds the angle of elevation of the top of a tower to be 30 degrees. He walks 85 m. nearer the tower and finds its angle of elevation to be 60 degrees. What is the height of the tower? Ans.73.61 meters Problem 89 The frustum of a regular triangular pyramid has equilateral triangles for its bases. The lower and upper base edges are 9m and 3m, respectively. If the volume is 118.2 cu.m., how far apart are the base? Ans. 7m Problem 90 A right circular cone with an altitude of 9m is divided into two segments, one is a smaller circular cone having the same vertex with an altitude of 6m. find the ratio of the volume of the two cones. Ans. 8:27 Problem 91 The volume of the sphere is 36  cu.m. The surface area of this sphere in sq. m. is: Ans. 36  Problem 92 The midpoint of the line segment between P 1( x, y ) and P 2 (-2, 4 ) is Pm ( 2, -1 ). Find the coordinate of P1. Ans. ( 6, -6 ) Problem 93 Find the distance from the point ( 5, -3 ) to the line 7x - 4y -28 = 0. Ans. 2.36 Problem 94 What is the equation of the line that passes through ( 4, 0 ) and is parallel to the line x -y - 2 = 0? Ans. y - x + 4 = 0 Problem 95 The raduis of the circle 2x2 + 2y2 -3x + 4y -1 = 0 is: Ans. 33 / 4

Problem 96 Find the area ( in sq. units ) bounded by the parabolas x 2 2y = 0 and x2 + 2y - 8 = 0. Ans. 10.7 Problem 97 An ellipse has the equation 16x2 + 9y2 + 32x - 128 = 0. Its eccentricity is: Ans. 0.66 Problem 98 The semi-major axis of an ellipse is 4 and its semi-minor axis is 3. The distance from the center to the directrix is: Ans. 6.047 Problem 99 If y = cos x , what is dy/dx? Ans. -sin x Problem 100

(x  1)3 x Find the derivative of Ans.

3(x  1)2 (x  1)3  x x2

Problem 101 A statue 3 m high is standing on a base of 4 m high. If an observer’s eye is 1.5 m above the ground, how far should he stand from the base in order that the angle subtended by the statue is a maximum? Ans. 3.71 m Problem 102 A balloon is rising vertically over a point A on the ground at the rate of 15 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? Ans. 12 ft/sec. Problem 103 Evaluate the integral of e x2 1 2x dx. Ans. e x2 1 + C Problem 104 Integrate the square root of ( 1 - cos x ) dx. x Ans. -2 2 cos +C 2 Problem 105 2 2y Evaluate ( x2 + y2 ) dx dy. 1 0 Ans. 35/2

 

Problem 106 Evaluate the integral of

xdx (x  1)8

if it has an upper limit of 1

and a lower limit of 0. Ans. 0.022 Problem 107 Given is the area in the first quadrant bounded by x 2 = 8y, the line x = 4 and the x-axis. What is the volume generated by revolving this area about the y-axis? Ans. 50.26 Problem 108 A plane is headed due east with air speed of 240 kph. If a wind of 40 kph is blowing from the north, find the ground speed of the plane. Ans.243 kph Problem 109 A 50 kilogram block of wood rest on the top of the smooth plane whose length is 3 m. and whose altitude is 0.8 m. How long will it take for the block to slide to the bottom of the plane when released? Ans. 1.51 seconds Problem 110 A bank charges 12% simple interest on a P300.00 loan. How much will be repaid if the load is paid back in one lump sum after three years? Ans. P408.00 Problem 111 A student has money given by his grandfather in the amount of 20,000.00. How much money in the form of interest will he get if the money is put in a bank that offers 8% rate compounded annually, at the end of 7 years? Ans. P34,276.48 Problem 112 A man expects to receive P20,000 in 10 years. How much is that money worth now considering interest at 6% compounded quarterly? Ans. P11,025.25 Problem 113 What is the present worth of P27,000.00 due in 6 years if money is worth 13% and is compounded semi-annually? Ans. P12,681.00

GEODETIC ENGINEERING Mathematics

Problem 1 Simplify the expression: lim x 4

x 2  16 x4

Ans. 8 Problem 2

1  cos x x 0 x2

Evaluate the lim Ans. 0.5 Problem 3

Evaluate the limit of

x2  4 as x approaches infinity. x4

Ans. infinity Problem 4 What is the first derivative of y = 5x Problem 5 2 What is the first derivative of y = 2e x . Problem 6 What is the first derivative of y = 3x ex. Problem 7 What is the first derivative of y = 2 Sec (x2 + 1). Ans. 4x Sec (x2 + 1) tan (x2 + 1) Problem 8 What is the first derivative of y = ln Cos x. Ans. –tan x Problem 9 Find the slope of the tangent to the curve y = x 4 – 2x2 + 8 through point (2,16). Problem 10 What is the slope of the curve x2 + y2 -6x + 10y + 5 = 0 at point (1,0). Problem 11 Find the angle that the line 2y – 9x -18 = 0 makes with the x-axis. Ans. 77.470 Problem 12 Find the radius of curvature at (4,4) of the curve y2 – 4x = 0. Ans. 22.36

Problem 13 If y = x3 – x, what is the maximum value of y? Ans. 0.385 Problem 14 Divide 120 into two parts so that the product of one and the square of the other is maximum. Find the numbers. Ans. 80 and 40 Problem 15 The sum of the two numbers is S. What is the maximum value of the sum of their cubes? S3 Ans. 4 Problem 16 A rectangular lot having an area of 5000 m 2 is to be fenced on three sides. Find the least amount of fencing needed. Ans. 200 m Problem 17 A box with open top and square base is to be constructed using a minimum amount of material. If the volume of the box formed is 32 cu. m, what should be the height of the box in meters? Ans. 2 m Problem 18 What is the height of the biggest box with open top and square base that can be made out of a 48 sq. ft card board? Ans. 2 m Problem 19 The volume of the closed cylindrical tank is 11.3 cu.m. If the total surface area is a maximum, what is its base radius? Ans. 1.216 m Problem 20 A box is to be constructed from a piece of zinc 20 inches square by cutting equal squares from each corner and turning up the zinc to form the sides. What is the volume of the largest box that can be constructed? Ans. 592.59 cu. in Problem 21 A farmer has 340 m of fencing with which he wishes to fence in his two separate lots, one square and the other a rectangle twice as long as its wide. Find the dimensions of each lot so that the sum of the areas shall be maximum.

GEODETIC ENGINEERING Mathematics

Problem 22 A wall 3 m high is 2.44 m away from a building. What is the length of the shortest ladder that can reach the building with one end resting on the ground outside the wall? Ans. 7.68 m Problem 23 A 3 m statue stands on top of a 4 m pedestal whose base is on a level ground. How far should a man stand from the base such that the angle subtended by the statue at the eyes of the man would be maximum. The man’s eyes are 4.92 ft above the ground. Ans. 3.7 m Problem 24 Water flows at the rate of 2000 cc per sec into a vertical cylindrical tank 120 cm in diameter and 6 m high. How fast is the water level rising in cm per sec. Ans. 0.177 cm/s Problem 25 How fast is the surface area of a sphere changing when the radius is 25 cm, if the radius is changing at the rate of one mm per sec. Ans. 62.8 cm2/sec Problem 26 A lady 1.65 m tall walks towards a lamp post 2.65 m high at a speed of 1.5 m/s. How fast does her shadow shorten in m/s. How fast does the end of her shadow move with respect to the lamp post. Ans. 2.74 m/s Problem 27 A 15 feet ladder leans against a vertical wall. If the lower end starts sliding horizontally at the rate of 2 ft/s, how fast is the upper end going down when it is 9 feet above the ground. Ans. 2.67 ft/s Problem 28 As a bicycle passes under a skywalk at a speed of 1.2m/s, a man directly above it walks on the skywalk at a speed of one meter per sec. The skywalk is at right angle to the road, 3 sec later, the man and the bicycle will be separating at the rate of 1.57 m/s. Find the height of the skywalk. Ans. 6.80 m Problem 29 A body is projected vertically upward such that the height from the ground in feet, is h = 50t – 16.1t 2 where t is in seconds. After how many seconds will the velocity be 12 feet/sec.

Problem 30 Evaluate the integral of

2dx from x = 0 to x = 1. x  2x  3 2

Problem 31

2dx

Evaluate the integral of

9  2x2

from x = 0 to x = 1.

Problem 32 Evaluate the integral of

2dx from x = 1 to x = 2. 3  2 x2

Problem 33 Evaluate the integral of

4 x dx from x = 0 to x = 3. ex

Problem 34 3 Evaluate the integral of e x x 2 dx from x = 0 to x = 1. Problem 35 Evaluate the integral of e3x dx from x = 0 to x = 3. Problem 36 Evaluate the integral of 23 x dx if the lower limit is 0 and the upper limit is 3. Problem 37

5 xdx if the lower limit is 1 and 5  x2

Evaluate the integral of the upper limit is 2.

Problem 38 If x = t2 and y = t3, evaluate the integral of xy dx from x = 1 to x = 4. Problem 39 If x  5cos  and y  2sin  , evaluate the integral of xydx from x = 0 to x = 5. Problem 40 Evaluate the integral of sin 2 d if the lower limit is 0 and the upper limit is 2 . Problem 41 Evaluate the integral of x cos x dx if the lower limit is 0 and the upper limit is

 2

.

Problem 42 Evaluate the integral of 3cos8 x sin 2 x dx if the lower limit is 0 and the upper limit is

 2

.

GEODETIC ENGINEERING Mathematics

Problem 43 Determine the area bounded by the curve x 2 = 5y and the line x = 3y. Ans. 0.154

Problem 53 Find the centroid of the area in the first quadrant bounded by the curve y2 = 4ax and the latus rectum. Ans. (3/5 of a, 0)

Problem 44 Determine the area enclosed by the curve x 2 = 9y and y2 = 9x. Ans. 27

Problem 54 A line having an equation y = 2x intersects the curve y2 = 8x. Find the centroid of the area. Ans. (0.8, 2.0)

Problem 45

Problem 55 An engineer promised to pay P36,000 pesos at the end of 90 days. He was offered a 10% discount if he pays in 30 days. Find the rate of interest. Ans. 66.6 %

What is the area bounded by the curves x = 4 y=4

y and

x Ans. 85.33

Problem 46 What is the area bounded by the parabola x2 = 3y – 3 and x2 = y. Ans. 1.63 Problem 47 Find the area bounded by y = (11 – x ) 1/2 , the lines 3x = 2 and x = 10 and the x – axis. Ans. 21.487 Problem 48 The area bounded by the curve y2 = 12x and the line x = 3 is revolved about the line x = 3. What is the volume generated? Ans. 180.96 Problem 49 The area is in the first quadrant bounded by x 2 = 8y, the line x = 4 and the x – axis. What is the volume generated by revolving this area about the y – axis? Ans. 50.26 Problem 50 Find the volume generated by revolving the area enclosed by the curve x2 + y2 = 25 about the line y – 8 = 0. Ans. 3948 Problem 51 The area is in the first quadrant bounded by x 2 = 8y, the line y – 2 = 0 and the y – axis. What is the volume generated by revolving this area about the line y – 2 = 0? Ans. 26.81 Problem 52 Find the centroid of the area bounded by y = 4 – x 2 in the first quadrant. Ans. (0.75, 1.6)

Problem 56 A man is required to pay P57,500 at the end of 15 days or P60,000 at the end of 60 days. Determine the rate of interest. Ans. 34.78% Problem 57 What is the principal amount if the amount of interest at the end of 21/2 year is P4500 for a simple interest of 6% per annum? Ans. P30,000.00 Problem 58 How long must a P40,000 note bearing 4% simple interest run to amount to P41,350.00 Ans. 0.844 yrs Problem 59 A bank charges 12% simple interest on a P300 loan. How much will be repaid in lump sum after 3 years? Ans. P408.00 Problem 60 Compute the interest for an amount of P200,000.00 for a period of 8 years. 1. If it was made at a simple interest rate of 16%. 2. If it was made at 16% compounded annually. 3. If it was made at 16% compounded quarterly. 4. If it was made at 16% compounded semiannually. 5. If it was made at 16% compounded monthly. 6. If it was made at 16% compounded bi-monthly. 7. If it was made at 16% compounded continuously. Problem 61 How long would it take your money to double itself. 1. If it is invested at 6% simple interest 2. If it is invested at 6% compounded semi-quarterly. 3. If it is invested at 6% compounded semi-annually. 4. If it is invested at 6% compounded annually.

GEODETIC ENGINEERING Mathematics

Problem 62 An engineer construct a house and sells it on monthly installments of P25,000.00 and it will be fully paid in 7 yrs. If the rate of interest is 12% compounded monthly, what is the equivalent cash price of the house in thousands of pesos? Ans. P1,416,211.00

Problem 69 The following terms of payment for an annuity are as follows: Periodic payment = 20,000.00 Payment Interval = 1 month Interest rate = 18% compounded monthly Terms = 15 years

Problem 63 A man paid 10% down payment of P200,000 for a house and lot and agreed to pay the balance on monthly installments for 5 years at an interest rate of 15% compounded monthly. What is the monthly installment? Ans. P42,821.87

1. What is the present worth of all the payments if it is paid at the end of each month? Ans. P1,241,911.25 2. What is the difference between the sums of an annuity due and an ordinary annuity on these payments? Ans. P271,687.35 3. What is the difference between the present values of an annuity due and an ordinary annuity on these payments? Ans. P18,628.67

Problem 64 A Geodetic Engineer wants to provide his newly born son a lump sum of one million pesos when he starts college at the end of 17 yrs. To achieve this, he deposits at the end of every month a certain amount in a fund that pays an interest rate of 12% compounded monthly. How much is the monthly deposit? Ans. P1,512.16 Problem 65 A service car whose price was P540,000 was bought with a down payment of P162,000 and monthly installment of P10,874.29 for 5 years. What was the rate of interest if compounded monthly? Ans. 24% Problem 66 A debt of P100,000 with interest of 7% compounded annually will be retired at the end of 10 years through the accumulation of a sinking fund invested at 6% compounded semi-annually. How much must be deposited in the sinking fund at the end of every six months? Ans. P7,320.89 Problem 67 An engineer is entitled to received P25,000 at the beginning of each year for 18 years. What is the present value of this annuity at the time he is suppose to receive the first payment if the rate of interest is 4% compounded annually? Ans. P329,142.00 Problem 68 An endowment of P10,000 will start now and continues every 3 months interval for 8 years. If money is worth 5% compounded quarterly, find the sum of the annuity at the end of 8th year. Ans. P395,386

Problem 70 A boy is entitled to 10 yearly endowments of P30,000 each starting at the end of the eleventh year from now. If the rate of interest is 8% compounded annually, what is the value of these endowment now? Ans. P93,241.98 Problem 71 A man invested P100,000 every end of the year for 10 yrs, then waited for another 10 yrs for his money to grow. If his investment earned 8% compounded annually, what would be the sum of his investments at the end of the 20th year? Ans. P3,127,540 Problem 72 An engineer bought an equipment for P500,000. Other expenses including installation amounted to P30,000. At the end of its estimated useful life of 10 years, the salvage value will be 10% of the first cost. Using Straight Line Method of Depreciation, what is the book value after 5 yrs? Ans. P291,500.00 Problem 73 A machine costing P1,800,000 has a useful life of 8 yrs. If the total depreciation at the end of 4th year is P800,000, what is the salvage value of the machine? Use Straight Line Method. Ans. P200,000.00