Math Problems

ALGEBRA Problem No.1 The equation whose roots are the reciprocals of the roots of the equation, 2x2-3x-5=0 is Ans. 5x2+3x-2=0 Problem No.2 Find the sum of the roots of the equation 5x2-10x+2=0. Ans. X1+x2=2 Problem No.3 Find the value of k in the quadratic equation (2k+2)x2 + (44k)x + k-2=0 so that the roots are reciprocal of each other. Ans. -4 Problem No.4 From the equation 12x3-8x2+kx+18=0, find the value of k if one root is the negative of the other. Ans. -27 Problem No.5 If one root of 9x2-6x+k=0 exceeds the other by 2, find the value of k. Ans. -8 Problem No.6 If (x-3) is a factor of the polynomial x4-4x3-7x2+kx+24 what is the value of k? Ans. 22 Problem No.7 Find the sum and product of roots of the equation x3+2x223x-60=0 Ans. Sum -2, Product 60 Problem No.8 If 4x3+182+8x-4 is to be divided by 2x+3, compute the remainder. Ans. 11 Problem No.9 Find the 3rd term in the expansion of (x3+y)4. Ans. 6x6y2 Problem No.10 What is the sum of the coefficients in the expansion of (x+y-z)8? Ans.1 Problem No.11 Find the coefficient of (x+y)10 containing the term x7y3. Ans. 120 Problem No.12 When ax3+2x2-18x+7 is divided by (x+1) the remainder is 15. Find the value of a. Ans. 42 Problem No.13 Find the mean proportion of 4 and 36. Ans. 12

Problem No.14 The mean proportion between 12 and x is 6. Find x. Ans. 3 Problem No.15 Find the third proportional to 16 and 12. Ans. 9 Problem No.16 Three transformers are rated 5KVA, 10KVA and 25KVA respectively. The total cost of the transformer is

proportional to its KVA rating multiplied by the factor 1.0, 0.8 and 0.6 respectively, find the cost of the 10KVA rating. The total cost is 15000. Ans. 4286 Problem No.17 z varies directly as x and inversely as y2. If x=1 and y=2, then z=2. Find z when x=3, y=4. Ans. 1.5 Problem No.18 A clerk submitted the following reports. The average rate of production of radios is 1.5 unit for every 1.5 hours work by 1.5 workers. How many radios were produced in 1 month by 30 men working 200 hours during the month? Ans. 4000 units Problem No.19 Ten less than four times a certain number is 10. Determine the number. Ans. 5 Problem No.20 The sum of three consecutive integers is 24. Find the smallest integer Ans. 7 Problem No.21 The sum of the digits of a two digit number is 7. If the digits are reversed, the new number is 1 less than half the original number. Determine the original number. Ans. 52 Problem No.22 A number is divided into two parts such that when the greater part is divided by the smaller part the quotient is 3 and the remainder is 5. Find the smaller number if the sum of the two number is 37. Ans. 8 Problem No.23 The ratio of three numbers is 2:5:7. If 7 is subtracted from the second number, the resulting numbers form an arithmetic progression. Determine the smallest of the three numbers. Ans. 28 Problem No.24 One proposal in the agrarian reform program is to have a retention limit of 10 hectares. If a land owner was left with 10 hectares fewer than 40% of his land, after selling 6 hectares more than 70% of his land, what size of land did he initially owned? Ans. 40 hectares Problem No.25 The hypotenuse of a right triangle is 34cm. Find the lengths of the two legs if one leg is 14cm longer than the other leg. Ans. 16, 30 Problem No.26 The sum of two numbers is 21, and one number is twice the other. Find the numbers. Ans. 7, 14 Problem No.27 The product of 1/4 and 1/5 of a number is 500. What is the number? Ans. 100 Problem No.28

The excess of the sum of the fourth and fifth parts over the difference of the half and third parts of a number is 119. Find the number. Ans. 420 Problem No.29 Mary was four times as old as Lea ten years ago. If she is now twice as old as Lea, how old is Mary? Ans. 30 Problem No.30 Ten years ago, I was three times as old as you are. In six more years I will be twice as old as you are. How old are you? Ans. 26 Problem No.31 A man is 25 years old and his son is 5. In how many yars will the father be three times as old as the son? Ans. 5 years Problem No.32 Ana is 5 years older than Beth. In 5 years, the product of their ages is 1.5 times the product of their present ages. How old is Beth now? Ans. 20 Problem No.33 Harry is 1/3 as old as Ron and eight years younger than Hermione. Find the sum of their ages if Harry is 8 years old. Ans. 48 Problem No.34 How many liters of a 25% acid solution must be added to 80 liters of a 40% acid solution to have a solution that is 30% acid? Ans. 160 liters Problem No.35 Determine how much water should be evaporated from 50kg of 30% salt solution to produce a 60% salt solution. All % by weight. Ans. 25kg Problem No.36 A container contains 40L of a wine which is 80% alcohol by volume. How many of the mixture should be removed and replaced by an equal volume of water so that the resulting solution will be 70% alcohol? Ans. 5L Problem No.37 How much gold and how much silver must be added to 100kg of an alloy containing 40% gold and 10% silver to produce an alloy containing 50% gold and 20% silver? Ans. 43.33 gold, 23.33 silver Problem No.38 Naruto has nickels, dimes, and quarters amounting to $1.85. if he has twice as many dimes as quarters, and the number of nickels is two less than twice the number of dimes, how many quarters does he have? Ans. 3 quarters Problem No.39 A ship propelled to move at 25 mi/hr in still water, travels 4.2 miles upstream in the same time that it can travel 5.8 miles downstream. Find the speed of the stream. Ans. 4 miles/hr Problem No.40

A yacht can travel 10 miles downstream in the same time as it goes 6 miles upstream. If the velocity of the river current is 3MPH, find the speed of the yacht in still water. Ans. 12 MPH Problem No.41 An airplane flying with the wind, took 2 hours to travel 1000km, and 2.5 hours in flying back, what was the wind velocity in kph? Ans. 50 kph Problem No.42 The velocity of an airplane in still air is 125 kph. The velocity of the wind due east is 25 kph. If the plane returns back to its base again after 4 hours, at what distance does the plane travel due east? Ans. 240 km Problem No.43 Find the harmonic mean between the numbers 3/8 and 4. Ans. 24/35 Problem No.44 Find the geometric mean between the terms -4 and -9. Ans. -6 Problem No.45 In a certain arithmetic progression the first, fourth and eight terms are themselves form a geometric progression. What is the common ration of the geometric progression? Ans. 4/3 Problem No.46 What is the sum of a geometric progression if there are 4 geometric means between 3 and 729? Ans. 1092 Problem No.47 Find the sum of the first five terms of the geometric progression if the third term is 144 and the sixth term is 486. Ans. 844 Problem No.48 If the sides of a right triangle are in arithmetic progression, then what is the ratio of its sides? Ans. 3:4:5 Problem No.49 An equipment cost $50,000 and depreciates 20% of the original cost during the first year, 16% during the second year, 12% during the third year, and so on, for five years. What is the value at the end of five years? Ans. $20,000 Problem No.50 Find the sum of the first 100 positive integers that is exactly divisible by 7. Ans. 35,350 Problem No.51 Find the missing term in the series: 1, 4, 9, _ , 25,..... Ans. 16 Problem No.52 A woman started a chain letter by writing to four friends and requesting each to copy the letter and send it to four other friends. If the chain was unbroken until the 5th set of letters was mailed, how much was spent for postage at $8.00 per letter? Ans. $10,912

Problem No.53 In a pile of logs, each layer contains one more log than the layer above and the top log contains just one log. If there are 105 logs in the pile, how many layers are there? Ans. 14 layers Problem No.54 The sum of the three numbers in an arithmetic progression is 33, if the sum of their squares is 461, find the numbers. Ans. 4, 11, 18 Problem No.55 A piece of paper is 0.05 inches thick. Each time the paper is folded in half, the thickness is doubled. If the paper was folded 12 times, how thick in feet the folded paper be? Ans. 17.07 ft. Problem No.56 A certain ball when dropped from a height rebounds 2/3 of the distance from which it last fell. Find the total distance travelled by the ball from the time it is dropped from a height of 60 meters until it strikes the ground the 5th time. Ans. 252.59 meters Problem No.57 If each bacterium in a culture divides into two bacteria every hour, how many bacteria will be present at the end of six hours if there are 4 bacteria at the start? Ans. 256 bacteria Problem No.58 If one third of the air in a tank is removed by each stroke of an air pump, what fractional part of the total air is removed in 6 strokes? Ans. 0.912 Problem No.59 Jose’s rate of doing work is three times as fast as bBong. On a given day Jose and Bong work together for four hours. Then Bong was called away and Jose finishes the rest of the job in two hours. How long would it take Bong to do the complete job alone? Ans. 22hours Problem No.60 Three men A, B, and C can do a piece of work in t hours working together. Working alone, A can do the work in six hours more, B in one hour more, and C in twice the time if all working together. How long would it take to finish the work if all working together? Ans. 40 minutes

Problem No.61 A boy in his bicycle intends to arrive of a certain time to a market that is 30km away from his school. After riding 10km, he rested half an hour and as a result he was obliged to ride the rest of the trip 2km/hr faster. What was his original speed? Ans. 8km/hr Problem No.62 A and B working together can finish a job in 5 days, B and C together can finish the same job in 4 days, and A and C in 2.5 days. In how many days can all of them do the job working together? Ans. 2.35 days

Problem No.63 A man and a boy can do in 15 days a piece of work which would be done by 7 men and 9 boys in 2 days. How long would it take one man to do it? Ans. 20 days Problem No.64 Seven carpenters and 5 masons earn a total of PhP2300 per day. At the same rate of pay 3 carpenters and 8 masons earn PhP2040. What are the wages per day of the carpenter and a mason? Ans. PhP200 Carpenter, PhP180 Mason Problem No.65 A pipe can fill a tank in 4 hours if the drain is open. If the pipe runs with the drain open for 1 hour and the pipe is then closed, the tank will be emptied in 40 minutes more. How long does it take the pipe to fill the tank if the drain is closed right at the start of filling? Ans. 1.6 hours Problem No.66 A tank can be filled in 48 minutes by two pipes running simultaneously. By the larger pipe it can be filled in 5 minutes less time than by the smaller. Find the time required for the larger pipe to fill it. Ans. 93.56 minutes Problem No.67 A would require 45 hours to varnish all the picture frame produced during a particular day. B who takes 2 minutes more to varnish each frame will require 60 hours. How many frames are there? Ans. 450 frames Problem No.68 Twenty eight men can finish the job in 60 days. At the start of the 16th day 5 men were laid off and after the 45th day 10 more men were hired. How many days were they delayed in finishing the job? Ans. 2.27 days Problem No.69 Pedro can paint a fence 50% faster than Juan and 20% faster than Pilar, and together they can paint a given fence in four hours. How long will it take Pedro to paint the same fence if he had to work alone? Ans. 10 hours Problem No.70 A laborer can finish a job in 4 days. Another laborer can finish the same job in 6 days. If both laborers plus a third laborer can finish the job in 2 days, how long will it take for the third laborer to finish the job alone? Ans. 12 days Problem No.71 It takes Butch twice as long as it takes Dan to do a certain piece of work. Working together they can do the work in six days. How long would it take Dan to finish the work alone? Ans. 9 days Problem No.72 The time required for two examinees to solve the same problem differs by two minutes together they can solve 32 problems in one hour. How long would it take for the slower problem solver to solve a problem? Ans. 5minutes Problem No.73

How many minutes after 2 o’clock will the hands of the clock be perpendicular for the first time? Ans. 27.273 minutes Problem No.74 What time after 2 o’clock will the hands of the clock extend in opposite directions for the first time? Ans. 2:43.64 Problem No.75 How soon after 7 o’clock will the hands of a clock be together? Ans. 38.18 minutes Problem No.76 How soon after 2 o’clock will the hands form a 60 degrees angle? Ans. 21.82minutes Problem No.77 How soon after 2 o’clock will the hands of the clock form a 40 degrees angle for the second time? Ans. 18.18 minutes

PERMUTATION, PROBABILITY

COMBINATION,

and

Problem No.78 How many permutations can be made out of the letters in the word RONNJ taken 3 at a time? Ans. 60 Problem No.79 How many permutations can be made out of the letters in the word DANIELLE? Ans. 10,080 Problem No.80 How many ways can you draw 2 jacks and 2 aces from a deck of 52 cards? Ans. 36 ways Problem No.81 How many ways can you go to a date with one or more of your 7 girlfriends? Ans. 128 ways Problem No.82 How many 4-digit even numbers can be formed from the digits 0,1,2,3,4,5,6,7,8, and 9 if each digit is to be used only once in each number? Ans. 3240 numbers Problem No.83 A club has 25 members, 4 of whom are ChE’s. In how many ways can a committee of 3 be formed so as to include at least one ChE? Ans. 970 ways Problem No.84 A guy has 8 flowers of different variety. In how many ways can he select at least 2 or more flowers to form a bouquet? Ans. 247 ways Problem No.85

How many numbers between 3000 and 5000 can be formed from the digits 0,1,2,3,4,5, and 6 if repetition is not allowed? Ans. 240 numbers Problem No.86 In how many ways can a group of 6 people be seated on a row of 6 seats if a certain 2 refuse to sit next to each other? Ans. 480 ways Problem No.87 How many different 8-digit numbers can be formed from the digits 2,2,2,5,5,7,7, and 7. Ans. 560 ways Problem No.88 In how many ways can 10 different magazines be divided among A, B, and C so that A gets 5 magazines, B 3 magazines, and C 2 magazines? Ans. 2520 ways Problem No.89 How many are less than 330 in a group of 3 digit numbers formed from the digits 0,1,2,3,4, and 5 where no digit may be repeated in a given number? Ans. 52 numbers Problem No.90 Four different coloured flags can be hung in a row to make code signals. How many signals can be made if a signal consists of the display of one or more flags? Ans. 64 signals Problem No.91 A semiconductor company will hire 7 men and 4 women. In how many ways can the company choose from 9 men and 6 women who qualified for the position? Ans. 540 ways Problem No.92 There are 14 teams in a tournament. Each team is to play with each other only once. What is the minimum number of days can they all play without any team playing more than one game in any day? Ans. 13 days Problem No.93 The lotto uses numbers 1-42. A winning combination consists of 6 different numbers in any order. What are your chances of winning it? Ans. 1/5,245,786 Problem No.94 There are 50 tickets in a lottery in which there is a first and second prize. What is the probability of a man drawing a prize if he owns 5 tickets? Ans. 1/5 Problem No.95 Two balls are drawn one at a time from a basket containing 4 black and 5 white balls. If the first ball is returned before the second ball is drawn, find the probability that both balls are black. Ans. 0.198 Problem No.96 A number between 1 and 10,000 is randomly selected. What is the probability that it will be divisible by 4 and 5? Ans. 0.05 Problem No.97 A bag contains 4 white and 3 black balls. Another bag contains 3 white and 5 black balls. If one ball is drawn from

each bag, determine the probability that the balls drawn will be 1 white and 1 black. Ans. 29/56

Problem No.98 In a single throw of a pair of dice, what is the probability of having totals of 7 or 8? Ans. 11/36 Problem No.99 Find the probability of getting exactly 12 out of 30 questions in a true or false questions. Ans. 0.08 Problem No.100 A coin is biased so that the head is twice as likely to occur as tail. If it is thrown three times, find the probability of having at least 2 heads. Ans. 20/27 Problem No.101 A ChE class of 40 students took examination in ChE Thermodynamics and Unit Operations. If 30 passed in ChE Thermodynamics, 36 passed in unit operations and 2 failed in both subjects, how many students passed in both subjects? Ans. 28 students Problem No.102 In a restaurant, 100 tables were set. Of these tables, 15 had dish A only, 20 had dish B only, 15 had dish C only, 10 had dishes A and B only, 15 had dishes B and C only, 15 had dishes C and A only, and 10 had all these dishes. In how many tables can dishes A or C be found? Ans. 80 tables Problem No.103 In a certain party, each one drinks coke or beer or whiskey or all. The count of drinks was 400 coke, 500 beer and 300 whiskey in total. If 100 drinks coke and beer, 200 drinks beer and whiskey, and one who drinks whiskey do not drink coke, how many are in the party? Ans. 900 Problem No.104 A survey of 500 television viewers produced the following results: 285 watch football games 195 watch hockey games 115 watch basketball games 45 watch football and basketball 70 watch football and hockey 50 watch hockey and basketball 50 do not watch any How many watch Hockey games only? Ans. 95 Television Viewers Trigonometry and Plane Geometry Problem No.105 If sec(2x-3)=[(1/sin(5x-9)], determine the value of x in degrees. Ans. 14.57 Problem No.106

What is the maximum value of 3-2cosѳ? Ans. 5 Problem No.107 Find the value of x in the equation (square root of 5)^(2cox)=5. Ans. 0 Problem No.108 Simplify the equation sin2 x(1+cot2x). Ans. sin2 x(1+cot2x)=1 Problem No.109 If sinxcosx + sin2x =1, what are the values of x in degrees? Ans. 20.9, 69.1 Problem No.110 If sin3x = cos6y then: Ans. x+2y=30 Problem No.111 Evaluate arccot[2cos(arcsin0.5)] Ans. 30˚ Problem No.112 The trigonometric expression (2tanx)/(1-tan2x) is equal to? Ans. tan2x Problem No.113 Solve for x in the equation: Arctan(x+1) + Arctan(x-1) = Arctan12 Ans. 1.34 Problem No.114 (1+secx)/(tanx+sinx) is equal to? Ans. cscx Problem No.115 A clock has a dial face of 12 in. radius. The minute hand is 9 in. while the hour hand is 6 in. The plane of rotation of the hour hand is 2in. above the plane of rotation of the minute hand. Find the distance between the tips of the minute hand and the hour hand at 5:40 a.m.. Ans. 9.17 inches Problem No.106 Two towers are 60m apart from each other. From the top of the shorter tower, the angle of elevation of the taller tower is 40˚. How high is the taller tower if the height of the smaller tower is 40m? Ans. 90.35m Problem No.117 Considering the earth to be sphere of radius 6400km, find the radius of the 60th parallel of latitude. Ans. 3200 km Problem No.118 From a point on level ground, the angle of elevation of the top and bottom of the abs-cbn tower situated on the top of the hill are measured as 48˚ and 40˚, respectively. Find the height of the hill if the height of the tower is 116 feet. Ans. 358.49 feet Problem No.119 A ladder, with its foot in the street, makes an angle of 30˚ with the street when its top rest on a building on one side of the street and makes an angle of 40˚ with the street when its top rest on the other side of the street. If the ladder is 50 feet long, how wide is the street? Ans. 81.6 feet Problem No.120

A wall is 15 ft. high and 10 ft. from a building, find the length of the shortest ladder which will just touch the top of the wall and reach a window 20.5 ft. above the ground. Ans. 42.54 feet Problem No.121 A pole tilts toward the sun at an angle 10˚ from the vertical and casts a shadow 9 meters long. If the angle of elevation from the tip of the shadow to the top of the pole is 43˚, how tall is the pole? Ans.10.2m Problem No.122 From a helicopter flying at 30,000 feet, the angles of depression of two cities are 28˚ and 55˚. How far apart are the two cities? Ans. 35,415.57 feet Problem No.123 Two angles are adjacent and form an angle of 120˚. If the larger angle is 20˚ less than three times the smallest angle, find the larger angle. Ans. 85˚ Problem No.124 A pine tree is broken over by the wind forms a right triangle with the ground. If the broken part makes an angle of 50˚ with the ground and the top of the tree is now 20 ft. from its base, how tall was the pine tree? Ans. 55 feet Problem No.125 A ball 5 feet in diameter rolls up an incline of 18˚20’. What is the height of the center of the ball above the base of the incline when the ball has rolled up 5 feet up the incline? Ans. 4 feet Problem No.126 A vertical pole consists of two parts, each one half of the whole pole. At a point in the horizontal plane which passes through the foot of the pole and 36m from it, the upper part of the pole subtends an angle whose tangent is 1/3. How high is the pole? Ans. 36 Problem No.127 If the sides of the triangle are 2x+3, x2+3x+3, and x2+2x, find the largest angle. Ans. 120˚ Problem No.128 The angle of elevation of the top of the tower from point A is 23˚30’. From another point B, the angle of elevation of the top of the tower is 55˚30’. The points A and B are 217.45 apart and on the same horizontal plane as the foot of the tower. The horizontal angle subtended by A and B at the foot of the tower is 90˚. Find the height of the tower. Ans. 89.5m Problem No.129 The sum of the sides of a triangle is equal to 100 cm. if the angles of the triangle are in the continued proportions of 1:2:4, compute the shortest side. Ans. 19.8 Problem No.130 The sides of a triangle which contains an area of 2400 cm2 are in continued proportions of 3:5:7. Find the smallest side of the triangle. Ans. 57.67

Problem No.131 In triangle ABC, angle A is 80˚ and point D is inside the triangle. If BD and CD are bisectors of angle B and C, solve for the angle BDC. Ans. 130˚ Problem No.132 An airplane can fly at an air speed of 300mph. if there is a wind blowing towards the east at 50mph, what should be the planes compass reading in order for its course to be 30˚. What will be the plane’s groundspeed if it flies at this course? Ans. 21.7˚, 321.86mph Problem No.133 Find the area of a hexagon inscribed in a circle with area of 33.16 cm2 Ans. 27.4 Problem No.134 A circle having a diameter of 8cm is inscribed in a sector of a circle whose central angle is 80˚. Find the area of the sector. Ans.72.92 cm2 Problem No.135 A circle with radius 6 has half its area removed by cutting off a border of uniform width. Find the width of the border. Ans. 1.76 Problem No.137 Two equilateral triangles, each with sides 12cm, overlap each other to form a hexagram. Determine the overlapping area in cm2. Ans. 41.57 Problem No.138 A regular pentagon has sides of 20cm. an inner pentagon with sides of 10 cm is inside and concentric to the larger pentagon. Determine the area inside the larger pentagon but outside the of the smaller pentagon. Ans. 516.2 Problem No.139 In triangle ABC, angle C=70˚, A=45˚, and side AB=40m. What is the length of the median drawn from the vertex A to side BC? Ans. 36.3 Problem No.140 What is the area of a circle circumscribing an equilateral triangle with sides 10 cm long? Ans. 104.72 Problem No.141 A non-square rectangle is inscribed in a square so that each vertex of the rectangle is at the trisection point of the different sides of the square. Find the ratio of the area of the rectangle to the area of the square. Ans. 9/4 Problem No.142 Two triangles have equal bases. The altitude of one triangle is 3 units more than its base while the altitude of the other is 3 units less than its base. Find the altitudes if the areas of the triangles differ by 21 square units. Ans. 4,10 Problem No.143 A piece of wire is shaped to enclose a square whose area is 169 cm2. It is then reshaped to enclose a rectangle whose length is 15 cm. what is the area of the rectangle?

Ans. 165

Problem No.144 A trapezoid has an area of 360m2 and altitude of 20 m. its two bases in meters have ratio of 4:5. The bases are? Ans. 16,20 Problem No.145 The diagonals of a parallelogram are 18 cm and 30 cm respectively. One side of parallelogram is 12cm. find the area of the parallelogram. Ans. 216 Problem No.146 A quadrilateral have sides equal to 12 cm, 20 cm, 8 cm, and 17 cm respectively. If the sum of the two opposite angles is 225˚, find the area of the parallelogram. Ans. 168.18 Problem No.147 The sides of a cyclic quadrilateral are 3cm, 3cm, 4cm, and 4cm. find the radius of the circle that can be inscribed in it. Ans. 1.71 Problem No.148 How many diagonals can be drawn from a dodecagon? Ans. 54 Problem No.149 Find the area of a regular polygon whose side is 25m and apothem is 17.2m. Ans. 1075m2 Problem No.150 Find the area of a pentagon which is circumscribing a circle having an area of 420.6cm2 Ans. 486.29 cm2 Problem No.151 In a circle with diameter of 10 meters, a pentagram touching its circumference is inscribed. What is the area of the part not covered by the pentagram? Ans. 50.47 m2 Problem No.152 A square of area equal to 72cm2 is inscribed in a circle. The circle is inscribed in a hexagon. Find the area of the hexagon. Ans. 124.71 Analytic Geometry Problem No.154 In a circle with diameter of 10 meters, a pentagram touching its circumference is inscribed. What is the area of the part not covered by the pentagram? Ans. 50.47 m2 Problem No.155 A square of area equal to 72cm2 is inscribed in a circle. The circle is inscribed in a hexagon. Find the area of the hexagon. Ans. 124.71