Wednesday, July 4, 2012

Aptitude Questions part 2

1: The largest copper producing country in the World is
1. Chile
2. Russia
3. South Africa
4. China
Ans: 1.
2: If the radius of a circle is diminished by 10%, then its area is diminished by:
1. 10%
2. 19%
3. 20%
4. 36%
Ans: 2.
3: A boat travels 20 kms upstream in 6 hrs and 18 kms downstream in 4 hrs.Find the speed of the boat in still water and the speed of the water current?
1. 1/2 kmph
2. 7/12 kmph
3. 5 kmph
4. none of these
Ans: 2.
4: At what time after 4.00 p.m. is the minutes hand of a clock exactly aligned with the hour hand?
1. 4:21:49.5
2. 4:27:49.5
3. 3:21:49.5
4. 4:21:44.5
Ans: 1.
5: A shop keeper sold a T.V set for Rs.17,940 with a discount of 8% and earned a profit of 19.6%.What would have been the percentage of profit earned if no discount was offered?
1. 24.8%
2. 25%
3. 26.4%
4. Cannot be determined
5. None of these
Ans: 5.
6: If (2x-y)=4 then (6x-3y)=?
1. 15
2. 12
3. 18
4. 10
Ans: 2.
7: A clock is set right at 8 a.m. The clock gains 10 minutes in 24 hours. What will be the true time when the clock indicates 1 p.m. on the following day?
1. 48 min. past 12
2. 38 min. past 12
3. 28 min. past 12
4. 25 min. past 12
Ans: 1.
8: What is the missing number in this series? 8 2 14 6 11 ? 14 6 18 12
1. 16
2. 9
3. 15
4. 6
Ans: 2.
9: Dinesh travelled 1200 km by air which formed 2/5 of his trip. One third of the whole trip, he travelled by car and the rest of the journey he performed by train. What was the distance travelled by train?
1. 600Km
2. 700Km
3. 800Km
4. 900Km
Ans: 3.
10: A train which travels at a uniform speed due to some mechanical fault after traveling for an hour goes at 3/5th of the original speed and reaches the destination 2 hrs late.If the fault had occurred after traveling another 50 miles the train would have reached 40 min earlier. What is distance between the two stations.
1. 300
2. 310
3. 320
4. 305
Ans: 1.
11: The average between a two digit number and the number obtained by interchanging the digits is 9. What is the difference between the two digits of the number?
1. 8
2. 2
3. 5
4. Cannot be determined
Ans: 4.
12: Pipe A can fill in 20 minutes and Pipe B in 30 mins and Pipe C can empty the same in 40 mins.If all of them work together, find the time taken to fill the tank
1. 17 1/7 mins
2. 20 mins
3. 8 mins
4. none of these
Ans: 1.
13: A person has 4 coins each of different denomination. What is the number of different sums of money the person can form (using one or more coins at a time)?
1. 16
2. 15
3. 12
4. 11
Ans: 2.
14: The simple interest on a certain sum of money for 3 years is 225 and the compound interest on the same sum at the same rate for 2 years is 153 then the principal invested is
1. 1500
2. 2250
3. 3000
4. 1875
Ans: 4.
15: A cow is tethered in the middle of a field with a 14 feet long rope. If the cow grazes 100 sq. ft. per day, then approximately what time will be taken by the cow to graze the whole field ?
1. 2 days
2. 6 days
3. 18 days
4. 24 days
5. None of these
Ans: 2.
16: 2 hours after a freight train leaves Delhi a passenger train leaves the same station travelling in the same direction at an average speed of 16 km/hr. After travelling 4 hrs the passenger train overtakes the freight train. The average speed of the freight train was?
1. 40
2. 30
3. 80
4. 60
Ans: 1.
17: The two colors seen at the extreme ends of the pH chart are:
1. Red and Blue
2. Red and Green
3. Green and Blue
4. Orange and Green
Ans: 1.
18: 8 15 24 35 48 63 _?

1. 70
2. 80
3. 75
4. 88
Ans: 2.
19: One of Mr. Horton, his wife, their son, and Mr. Horton’s mother is a doctor and another is a lawyer.
a) If the doctor is younger than the lawyer, then the doctor and the lawyer are not blood relatives.
b) If the doctor is a woman, then the doctor and the lawyer are blood relatives.
c) If the lawyer is a man, then the doctor is a man. Whose occupation you know?
1. Mr. Horton: he is the doctor
2. Mr. Horton’s son: she is the lawyer
3. Mr. Horton: he is the doctor
4. Mr. Horton’s mother: she is the doctor
Ans: 1.
20: In the given figure, PA and PB are tangents to the circle at A and B respectively and the chord BC is parallel to tangent PA. If AC = 6 cm, and length of the tangent AP is 9 cm, then what is the length of the chord BC?
1. 4 cm
2. 8 cm
3. 6 cm
4. 5 cm
Ans: 1.
21: Union Information and Broadcasting ministry recently gave an indication to change which of the following laws on a larger scale, as the existing provisions of the Act are inadequate to cater to the phenomenal growth of the print media in view of the liberalization of the government policies?
1. Press & Registration of Books Act, (PRB Act) 1867
2. The Delivery Of Books ‘And Newspapers’ (Public Libraries) Act, 1954
3. Indian Press (Emergency Powers ) Act 1931
4. none
Ans: 1.
22: 2 numbers differ by 5.If their product is 336,then the sum of the 2 numbers is:
1. 21
2. 51
3. 28
4. 37
Ans: 4.
23: Which number is the odd one out? 9678 4572 5261 3527 7768
1. 7768
2. 3527
3. 4572
4. 9678
5. 5261
Ans: 2.
24: Which one among the following has the largest shipyard in India
1. Kolkata
2. Kochi
3. Mumbai
4. Visakhapatnam
Ans: 2.
25: If x=y=2z and xyz=256 then what is the value of x?
1. 8
2. 3
3. 5
4. 6
Ans: 1.
26: A radio when sold at a certain price gives a gain of 20%. What will be the gain percent, if sold for thrice the price?
1. 280
2. 270
3. 290
4. 260
Ans: 4.
27: x% of y is y% of ?
1. x/y
2. 2y
3. x
4. can’t be determined
Ans: 3.
28: If the value of x lies between 0 & 1 which of the following is the largest?
1. x
2. x2
3. -x
4. 1/x
Ans: 4.
29: The tutor of Alexander the great was
1. Darius
2. Cyrus
3. Socrates
4. Aristotle
Ans: 4.
30: Thirty men take 20 days to complete a job working 9 hours a day. How many hour a day should 40 men work to complete the job?
1. 8 hrs
2. 71/2 hrs
3. 7 hrs
4. 9 hrs
Ans: 2.
31: Goitre caused by the deficiency of ………
1. Vitamin D
2. Iron
3. VItamin A
4. Iodine
Ans: 4.
32: Who invented Napier’s Bones
1. John Napier
2. William Oughtred
3. Charles Babbage
4. Napier Bone
Ans: 1.
33: The mass number of a nucleus is

1. Always less than its atomic number
2. Always more than its atomic number
3. Sometimes more than and sometimes equal to its atomic number
4. None of the above
Ans: 3.
34: A and B can do a piece of work in 45 days and 40 days respectively. They began to do the work together but A leaves after some days and then B completed the remaining work n 23 days. The number of days after which A left the work was
1. 9
2. 11
3. 12
4. 15
5. 16
Ans: 1.
35: Sam and Mala have a conversation. Sam says I am certainly not over 40 Mala Says I am 38 and you are at least 5 years older than me · Now Sam says you are at least 39 all the statements by the two are false. How old are they really?
1. Mala = 38 yrs, Sam =31 yrs.
2. Mala = 38 yrs, Sam = 41 yrs
3. Mala = 31 yrs, Sam = 41 yrs.
4. Mala = 45yrs, Sam = 41 yrs
Ans: 2.
36: What is the code name for Windows Vista?
1. Longhorn
2. Longhund
3. Stackspray
4. Pearl
Ans: 1.
37: On sports day, if 30 children were made to stand in a column, 16 columns could be formed. If 24 children were made to stand in a column, how many columns could be formed?
1. 20
2. 30
3. 40
4. 50
Ans: 1.
38: The probability that a man will be alive for 25 years is 3/5 and the probability that his wife will be alive for 25 years is 2/3. Find the probability that only the man will be alive for 25 years.
1. 2/5
2. 1/5
3. 3/5
4. 4/5
Ans: 2.
39: In a single throw of a dice, what is the probability of getting a number greater than 4?
1. 1/2
2. 2/3
3. 1/4
4. 1/3
Ans: 4.
40: If every alternative letter starting from B of the English alphabet is written in small letter, rest all are written in capital letters, how the month “September” be written. (1) SeptEMbEr (2) SEpTeMBEr (3) SeptembeR (4) SepteMber (5) None of the above
1. (1)
2. (2)
3. (3)
4. (5)
5. (4)
Ans: 4.
41: After allowing a discount of 11.11% ,a trader still makes a gain of 14.28 % .at how many precent above the cost price does he mark his goods?
1. 28.56%
2. 35%
3. 22.22%
4. None of these
Ans: 1.
42: Pipe A can fill in 20 minutes and Pipe B in 30 mins and Pipe C can empty the same in 40 mins.If all of them work together, find the time taken to fill the tank
1. 17 1/7 mins
2. 20 mins
3. none
4. 50 mins
Ans: 1.
43: There are 3 triplet brothers. They look identical. The oldest is John, he always tells the truth. The second is Jack, he always tells a lie. The third is Joe, he either tells the truth or a lie. Jimmie Dean went to visit them one day. He was wondering who was who. So he asked each person a question. He asked the one who was sitting on the left: “Who is the guy sitting in the middle?”. The answer was “He is John.” He asked the one who was sitting in the middle: “What is your name?”. The answer was “I am Joe.” He asked the one who was sitting on the right: “What is the guy sitting in the middle?”. The answer was “He is Jack.” Jimmie Dean got really confused. Basically, he asked 3 same questions, but he got 3 different answers. which is not true?
1. left most is joe
2. middle is jack
3. right is john
4. middle is john
Ans: 4.
44: A / B = C; C > D then
1. A is always greater than D
2. C is always greater than D
3. B is always less than D
4. none
Ans: 1.
45: Consider the following statements: 1. The Administrative Reforms Commission (ARC) had recommended that the Department of Personnel of a State should be put under the charge of the Chief Secretary of the State. 2. Chief Secretary of a State is not involved in any manner in the promotion of State Civil officers to the All-India Services. Which of the statements given above is/are correct?
1. Only 1
2. Only 2
3. Both 1 and 2
4. Neither 1 nor 2
Ans: 1.
46: The population of a town was 1,60,000 three years ago. If it increased by 3%, 2.5% and 5% respectively in the last three years, then the present population of the town is :
1. 1,77,000
2. 1,77,366
3. 1,77,461
4. 1,77,596
Ans: 2.
47: What is the population of India ?
1. 98 crores
2. More than 2 billion
3. More than 1 billion
4. Less than 96 crores
5. 96 crores
Ans: 3.
48: Some green are blue. No blue are white.
1. Some green are white
2. No white are green
3. No green are white
4. None of the above
Ans: 1.
49: What is the missing number in this series? 8 2 14 6 11 ? 14 6 18 12
1. 8
2. 6
3. 9
4. 11
Ans: 3.
50: Average age of students of an adult school is 40 years. 120 new students whose average age is 32 years joined the school. As a result the average age is decreased by 4 years. Find the number of students of the school after joining of the new students:
1. 1200
2. 120
3. 360
4. 240
Ans: 4.
51: On sports day,if 30 children were made to stand in a column,16 columns could be formed. If 24 children were made to stand in a column , how many columns could be formed?
1. 48
2. 20
3. 30
4. 16
5. 40
Ans: 2.
52: Which of the following numbers is divisible by 3? (i) 541326 (ii) 5967013
1. (ii) only
2. (i) only
3. (i) and (ii) both
4. (i) and (ii) none
Ans: 2.
53: A square is divided into 9 identical smaller squares. Six identical balls are to be placed in these smaller squares such that each of the three rows gets at least one ball (one ball in one square only). In how many different ways can this be done?
1. 81
2. 91
3. 41
4. 51
Ans: 1.
54: A man owns 2/3 of the market research beauro business and sells 3/4 of his shares for Rs.75000. What is the value of Business
1. 150000
2. 13000
3. 240000
4. 34000
Ans: 1.
55:    1,2,6,24,_?
1. 111
2. 151
3. 120
4. 125
Ans: 3.
56: The cost of 16 packets of salt,each weighing 900 grams is Rs.28.What will be the cost of 27 packets ,if each packet weighs 1Kg?
1. Rs.52.50
2. Rs.56
3. Rs.58.50
4. Rs.64.75
Ans: 1.
57: Ronald and Michelle have two children. The probability that the first child is a girl, is 50%. The probability that the second child is a girl, is also 50%. Ronald and Michelle tell you that they have a daughter. What is the probability that their other child is also a girl?
1. 1/2
2. 1/3
3. 1/4
4. 1/5
Ans: 2.
58: Find the value of (21/4-1)( 23/4 +21/2+21/4+1)
1. 1
2. 2
3. 3
Ans: 1.
59: The product of two fractions is 14/15 and their quotient is 35/24. the greater fraction is
1. 4/5
2. 7/6
3. 7/5
4. 7/4
Ans: 1.
60: 500 men are arranged in an array of 10 rows and 50 columns according to their heights. Tallest among each row of all are asked to fall out. And the shortest among them is A. Similarly after resuming that to their original podsitions that the shortest among each column are asked to fall out. And the tallest among them is B . Now who is taller among A and B ?
1. A
2. B
3. Both are of same height
Ans: 1.
61: Choose the pair of numbers which comes next 75 65 85 55 45 85 35
1. 25 15
2. 25 85
3. 35 25
4. 35 85
5. 25 75
Ans: 2.
62: A three digit number consists of 9,5 and one more number. When these digits are reversed and then subtracted from the original number the answer yielded will be consisting of the same digits arranged yet in a different order. What is the other digit?
1. 1
2. 2
3. 3
4. 4
Ans: 4.
63: ATP stands for:
1. Adenine triphosphate
2. Adenosine triphosphate
3. Adenosine Diphosphate
4. Adenosine tetraphosphate
Ans: 2.
64: Veselin Tapolev who became the World Champion recently, is associated with which of the following games/sports ?
1. Chess
2. Golf
3. Snooker
4. Badminton
5. None of these
Ans: 1.
65: A piece of cloth cost Rs 35. if the length of the piece would have been 4m longer and each meter cost Re 1 less , the cost would have remained unchanged. how long is the piece?
1. 10
2. 11
3. 12
Ans: 1.
66: In a journey of 15 miles two third distance was travelled with 40 mph and remaining with 60 mph.How muvh time the journey takes
1. 40 min
2. 30 min
3. 120 min
4. 20 min
Ans: 4.
67: Solid cube of 6 * 6 * 6. This cube is cut into to 216 small cubes. (1 * 1 * 1).the big cube is painted in all its faces. Then how many of cubes are painted at least 2 sides.
1. 56
2. 45
3. 23
4. 28
Ans: 1.
68: Find the average of first 40 natural numbers.
1. 40
2. 35
3. 30.6
4. 20.5
5. None of these
Ans: 4.
69: 1, 5, 14, 30, ?, 91
1. 45
2. 55
3. 60
4. 70
5. None of these
Ans: 2.
70: There is a shortage of tubelights, bulbs and fans in a village – Gurgaon. It is found that
a) All houses do not have either tubelight or bulb or fan.
b) Exactly 19% of houses do not have just one of these.
c) Atleast 67% of houses do not have tubelights.
d) Atleast 83% of houses do not have bulbs.
e) Atleast 73% of houses do not have fans.
1. 42 %
2. 46 %
3. 50 %
4. 54 %
5. 57 %
Ans: 1.
71: If 9 engines consume 24 metric tonnes of coal, when each is working 8 hours a day; how much coal will be required for 8 engines, each running 13 hours a day, it being given that 3 engines of the former type consume as much as 4 engines of latter type.
1. 22 metric tonnes.
2. 27 metric tonnes.
3. 26 metric tonnes.
4. 25 metric tonnes.
Ans: 3.
72: To 15 lts of water containing 20% alcohol, we add 5 lts of pure water. What is % alcohol.
1. 20%
2. 34%
3. 15%
4. 14%
Ans: 3.
73: In page preview mode:
1. You can see all pages of your document
2. You can only see the page you are currently working
3. Satyam BPO Services
4. You can only see pages that do not contain graphics
Ans: 4.
74: A house wife saved Rs. 2.50 in buying an item on sale .If she spent Rs.25 for the item ,approximately how much percent she saved in the transaction ?
1. 8%
2. 9%
3. 10%
4. 11%
Ans: 2.
75: I have trouble _____.
1. to remember my password
2. to remembering my password
3. remember my password
4. remembering my password
Ans: 4.
76: Superheroes Liza and Tamar leave the same camp and run in opposite directions. Liza runs 1 mile per second (mps) and Tamar runs 2 mps. How far apart are they in miles after 1 hour?
1. 10800 mile
2. 19008 mile
3. 12300 mile
4. 14000 mile
Ans: 1.
77: A = 5, B = 0, C = 2, D = 10, E = 2. What is then AB + EE – (ED)powerB + (AC)powerE = ?
1. 113
2. 103
3. 93
4. 111
Ans: 2.
78: A man can row upstream at 8 kmph and downstream at 13 kmph.The speed of the stream is?
1. 2.5 kmph
2. 4.2 kmph
3. 5 kmph
4. 10.5 kmph
Ans: 1.
79: Find what is the next letter. Please try to find. O,T,T,F,F,S,S,E,N,_ What is that letter?
1. B
2. S
3. Q
4. T
5. O
Ans: 4.
80: There are 3 societies A, B, C. A lent cars to B and C as many as they had Already. After some time B gave as many tractors to A and C as many as they have. After sometime c did the same thing. At the end of this transaction each one of them had 24. Find the cars each originally had.
1. A had 21 cars, B had 39 cars & C had 12 cars
2. A had 39 cars, B had 39 cars & C had 12 cars
3. A had 39 cars, B had 21 cars & C had 19 cars
4. A had 39 cars, B had 21 cars & C had 12 cars
Ans: 4.
81: A papaya tree was planted 2 years ago. It increases at the rate of 20% every year. If at present, the height of the tree is 540 cm, what was it when the tree was planted?
1. 432 cm
2. 324 cm
3. 375 cm
4. 400 cm
Ans: 3.
82: A boy has Rs 2. He wins or loses Re 1 at a time If he wins he gets Re 1 and if he loses the game he loses Re 1. He can loose only 5 times. He is out of the game if he earns Rs 5. Find the number of ways in which this is possible?
1. 14
2. 23
3. 16
4. 12
5. 10
Ans: 3.
83: Five racing drivers, Alan, Bob, Chris, Don, and Eugene, enter into a contest that consists of 6 races. The results of all six races are listed below: Bob always finishes ahead of Chris. Alan finishes either first or last. Eugene finishes either first or last. There are no ties in any race. Every driver finishes each race. In each race, two points are awarded for a fifth place finish, four points for fourth, six points for third, eight points for second, and ten points for first. If Frank enters the third race and finishes behind Chris and Don, which of the following must be true of that race?
1. Eugene finishes first.
2. Alan finishes sixth.
3. Don finishes second.
4. Frank finishes fifth.
5. Chris finishes third.
Ans: 4.
84: A is twice as good a workman as B and together they finish a piece of work in 18 days.In how many days will A alone finish the work?
1. 27
2. 26
3. 25
4. 24
Ans: 1.
85: Daal is now being sold at Rs. 20 a kg. During last month its rate was Rs. 16 per kg. By how much percent should a family reduce its consumption so as to keep the expenditure fixed?
1. 20 %
2. 40 %
3. 3%
4. 2%
Ans: 1.
86: The sum of 5 successive odd numbers is 1075. What is the largest of these numbers?
1. 215
2. 223
3. 219
4. 217
Ans: 3.
87: A man sells two buffaloes for Rs. 7,820 each. On one he gains 15% and on the other, he loses 15%. His total gain or loss in the transaction is
1. 2.5% gain
2. 2.25% loss
3. 2% loss
4. 5% loss
5. None of these
Ans: 2.
88: One ship goes along the stream direction 28 km and in opposite direction 13 km in 5 hrs for each direction.What is the velocity of stream?
1. 1.5 kmph
2. 2.5 kmph
3. 1.8 kmph
4. 2 kmph
Ans: 1.
89: Which one of the words given below is different from others?
1. Orange
2. Grape
3. Apricot
4. Raspberry
5. Mango
Ans: 3.
90: Complete the series: 5, 20, 24, 6, 2, 8, ?
1. 12
2. 32
3. 34
4. 36
Ans: 1.
91: A can have a piece of work done in 8 days, B can work three times faster than the A, C can work five times faster than A. How many days will they take to do the work together
1. 3 days
2. 8/9 days
3. 4 days
4. None of the above
Ans: 2.
92: 7 Pink, 5 Black, 11 Yellow balls are there. Minimum no. atleast to get one black and yellow ball
1. 17
2. 13
3. 15
4. 19
Ans: 1.
93: (1/10)18 – (1/10)20 = ?
1. 99/1020
2. 99/10
3. 0.9
4. none of these
Ans: 1.
94: Three friends divided some bullets equally. After all of them shot 4 bullets the total number of bullets remaining is equal to the bullets each had after division. Find the original number divided?
1. 18
2. 20
3. 54
4. 8
Ans: 1.
95: A sum of Rs. 427 is to be divided among A, B and C in such a way that 3 times A’s share, 4 times B’s share and 7 times C’s share are all equal. The share of C is
1. Rs.84
2. Rs.76
3. Rs.98
4. RS.34
Ans: 1.
96: There are 20 poles with a constant distance between each pole. A car takes 24 second to reach the 12th pole.How much will it take to reach the last pole.
1. 41.45 seconds
2. 40.45 seconds
3. 42.45 seconds
4. 41.00 seconds
Ans: 1.
97: An emergency vehicle travels 10 miles at a speed of 50 miles per hour. How fast must the vehicle travel on the return trip if the round-trip travel time is to be 20 minutes?
1. 72 miles per hour
2. 75 miles per hour
3. 65 miles per hour
4. 78 miles per hour
Ans: 2.
98: 12% of 580 + ? = 94
1. 24.4
2. 34.4
3. 54.4
4. 65.4
Ans: 1.
99: There is a certain relation between two given words on one side of : : and one word is given on another side of : : while another word is to be found from the given alternatives, having the same relation with this word as the given pair has. Select the best alternative. Horse : Jockey : : Car : ?
1. Mechanic
2. Chauffeur
3. Steering
4. Brake
Ans: 2.
100: Which of the following numbers should be added to 11158 to make it exactly divisible by 77?
1. 9
2. 8
3. 7
4. 5
Ans: 3.
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Mathematical Formulae

Numbers
-----------
1. A number is divisible by 2, if its unit’s place digit is 0, 2, 4, or 8

2. A number is divisible by 3, if the sum of its digits is divisible by 3

3. A number is divisible by 4, if the number formed by its last two digits is divisible by 4

4. A number is divisible by 8, if the number formed by its last three digits is divisible by 8

5. A number is divisible by 9, if the sum of its digits is divisible by 9

6. A number is divisible by 11, if, starting from the RHS,
(Sum of its digits at the odd place) – (Sum of its digits at even place) is equal to 0 or 11x

7. Product of two numbers = Their H. C. F. × Their L. C. M.

Simple Interest
-----------------
1. Let Principle = P, Rate = R% per annum and Time = T years. Then,

a. S.I. = ( P × R × T ) / 100

b. P = ( 100 × S.I. ) / ( R × T ),

c. R = ( 100 × S.I. ) / ( P × T ),

d. T = ( 100 × S.I. ) / ( P × R ).

Compound Interest
---------------------
1. Let Principle = P, Rate = R% per annum and Time = T years. Then,

I. When interest is compounded Annually,
Amount = P (1 + R/100)N

II. When interest is compounded Half-yearly:
Amount = P (1 + R/2/100)2N

III. When interest is compounded Quarterly:
Amount = P (1 + R/4/100)4N

2. When interest is compounded Annually, but the time is in fraction, say 3⅞ years.
Then, Amount = P (1 + R/100)3 × (1 + ⅞R/100)

3. When Rates are different for different years, say R1%, R2%, R3% for 1st, 2nd, and 3rd year
respectively,
Then, Amount = P (1 + R1/100) (1 + R2/100) (1 + R3/100)

4. Present worth of Rs. x due n years hence is given by:
Present Worth = x / (1 + R/100)n

Logarithms
-------------
1. Logarithm: If a is a positive real number, other than 1 and am = x, then we write m = loga x
and say that the value of log x to the base a is m.

2. Properties of Logarithms:

a. loga (xy) = loga x + loga y

b. loga (x/y) = loga x - loga y

c. logx x = 1 (i.e. Log of any number to its own base is 1)

d. loga 1 = 0 (i.e. Log of 1 to any base is 0)

e. loga (xp) = p loga x

f. loga x = 1 / logx a

g. loga x = logb x / logb a
= log x / log a (Change of base rule)

h. When base is not mentioned, it is taken as 10

i. Logarithms to the base 10 are known as common logarithms

Algebra
-----------
1. (a + b)2 = a2 + 2ab + b2

2. (a - b)2 = a2 - 2ab + b2

3. (a + b)2 - (a - b)2 = 4ab

4. (a + b)2 + (a - b)2 = 2(a2 + b2)

5. (a2 – b2) = (a + b)(a - b)

6. (a3 + b3) = (a + b)(a2 - ab + b2)

7. (a3 – b3) = (a - b)(a2 + ab + b2)

8. Results on Division:
Dividend = Quotient × Divisor + Remainder

9. An Arithmetic Progression (A. P.) with first term ‘a’ and Common Difference ‘d’ is given
by:
[a], [(a + d)], [(a + 2d)], … … …, [a + (n - 1)d]
nth term, Tn = a + (n - 1)d
Sum of first ‘n’ terms, Sn
= n/2 (First Term + Last Term)

10. A Geometric Progression (G. P.) with first term ‘a’ and Common Ratio ‘r’ is given by:
a, ar, ar2, ar3, … … …, arn-1
nth term, Tn = arn-1
Sum of first ‘n’ terms Sn = [a(1 - rn)] / [1 - r]

11. (1 + 2 + 3 + … … … + n) = [n(n + 1)] / 2

12. (12 + 22 + 32 + … … … + n2) = [n(n + 1)(2n + 1)] / 6

13. (13 + 23 + 33 + … … … + n3) = [n2(n + 1)2] / 4

Percentage
------------------
1. To express x% as a fraction, we have x% = x / 100

2. To express a / b as a percent, we have a / b = (a / b × 100) %

3. If ‘A’ is R% more than ‘B’, then ‘B’ is less than ‘A’ by
OR
If the price of a commodity increases by R%, then the reduction in consumption, not
to increase the expenditure is
{100R / [100 + R] } %

4. If ‘A’ is R% less than ‘B’, then ‘B’ is more than ‘A’ by
OR
If the price of a commodity decreases by R%, then the increase in consumption, not to
increase the expenditure is
{100R / [100 - R] } %

5. If the population of a town is ‘P’ in a year, then its population after ‘N’ years is
P (1 + R/100)N

6. If the population of a town is ‘P’ in a year, then its population ‘N’ years ago is
P / [(1 + R/100)N]

Profit & Loss
---------------

7. If the value of a machine is ‘P’ in a year, then its value after ‘N’ years at a depreciation of
‘R’ p.c.p.a is
P (1 - R/100)N

8. If the value of a machine is ‘P’ in a year, then its value ‘N’ years ago at a depreciation of
‘R’ p.c.p.a is
P / [(1 - R/100)N]

9. Selling Price = [(100 + Gain%) × Cost Price] / 100
= [(100 - Loss%) × Cost Price] / 100

Ratio & Proportion
-----------------------
1. The equality of two ratios is called a proportion. If a : b = c : d, we write a : b :: c : d and
we say that a, b, c, d are in proportion.
In a proportion, the first and fourth terms are known as extremes, while the second and
third are known as means.

2. Product of extremes = Product of means

3. Mean proportion between a and b is=(ab)^(1/2)

4. The compounded ratio of the ratios (a : b), (c : d), (e : f) is (ace : bdf)

5. a2 : b2 is a duplicate ratio of a : b

6. a^(1/2).b^(1/2) : is a sub-duplicate ration of a : b

7. a3 : b3 is a triplicate ratio of a : b

8. a1/3 : b1/3 is a sub-triplicate ratio of a : b

9. If a / b = c / d, then, (a + b) / b = (c + d) / d, which is called the componendo.

10. If a / b = c / d, then, (a - b) / b = (c - d) / d, which is called the dividendo.

11. If a / b = c / d, then, (a + b) / (a - b) = (c + d) / (c - d), which is called the componendo &
dividendo.

12. Variation: We say that x is directly proportional to y if x = ky for some constant k and we
write, x α y.

13. Also, we say that x is inversely proportional to y if x = k / y for some constant k and we
write x α 1 / y.

Area
------
1. Rectangle:

a. Area of a rectangle = (length × breadth)

b. Perimeter of a rectangle = 2 (length + breadth)

2. Square:

a. Area of square = (side)2

b. Area of a square = ½ (diagonal)2

3. Area of 4 walls of a room
= 2 (length + breadth) × height

4. Triangle:

a. Area of a triangle = ½ × base × height

b. Area of a triangle = , where
s = ½ (a + b + c), and a, b, c are the sides of the triangle

c. Area of an equilateral triangle = / 4 × (side)2

d. Radius of incircle of an equilateral triangle of side a = a / 2

e. Radius of circumcircle of an equilateral triangle of side a = a /

5. Parallelogram/Rhombus/Trapezium:

a. Area of a parallelogram = Base × Height

b. Area of a rhombus = ½ × (Product of diagonals)

c. The halves of diagonals and a side of a rhombus form a right angled triangle with
side as the hypotenuse.

d. Area of trapezium = ½ × (sum of parallel sides) × (distance between them)

6. Circle/Arc/Sector, where R is the radius of the circle:

a. Area of a circle = πR2

b. Circumference of a circle = 2πR

c. Length of an arc = Ө/360 × 2πR

d. Area of a sector = ½ (arc × R)
= Ө/360 × πR2

Volume & Surface Area
-------------------------
1. Cuboid:
Let length = l, breadth = b & height = h units Then,

a. Volume = (l × b × h) cu units

b. Surface Area = 2 (lb + bh + hl) sq. units

c. Diagonal = units

2. Cube:
Let each edge of a cube be of length a. Then,

a. Volume = a3 cu units

b. Surface Area = 6a2 sq. units

c. Diagonal = ( l^2 + b^2 +h^2)^1/2 units

3. Cylinder:
Let radius of base = r & height (or length) = h. Then,

a. Volume = (πr2h) cu. units

b. Curved Surface Area = (2πrh) sq. units
c. Total Surface Area = 2πr(r + h) sq. units

4. Cone:
Let radius of base = r & height = h. Then,

a. Slant height, l =(h^2 + R^2)^1/2 units

b. Volume = (⅓ πr2h) cu. units

c. Curved Surface Area = (πrl) sq. units

d. Total Surface Area = πr(r + l) sq. units

5. Sphere:
Let the radius of the sphere be r. Then,

a. Volume = (4/3 πr3) cu. units

b. Surface Area = (4πr2) sq. units

6. Hemi-sphere:
Let the radius of the sphere be r. Then,

a. Volume = (2/3 πr3) cu. units

b. Curved Surface Area = (2πr2) sq. units

c. Total Surface Area = (3πr2) sq. units

Stocks & Shares
-----------------
1. Brokerage: The broker’s charge is called brokerage.

2. When stock is purchased, brokerage is added to the cost price.

3. When the stock is sold, brokerage is subtracted from the selling price.

4. The selling price of a Rs. 100 stock is said to be:

a. at par, if S.P. is Rs. 100 exactly;

b. above par (or at premium), if S.P. is more than Rs. 100;

c. below par (or at discount), if S.P. is less than Rs. 100.

5. By ‘a Rs. 800, 9% stock at 95’, we mean a stock whose face value is Rs. 800, annual
interest is 9% of the face value and the market price of a Rs. 100 stock is Rs. 95.

True Discount
----------------
1. Suppose a man has to pay Rs. 156 after 4 years and the rate of interest is 14% per
annum. Clearly, Rs. 100 at 14% will amount to Rs. 156 in 4 years. So, the payment of
Rs. 100 now will clear off the debt of Rs. 156 due 4 years hence. We say that:
Sum due = Rs. 156 due 4 years hence;
Present Worth (P.W.) = Rs. 100;
True Discount (T.D.) = Rs. (156 - 100)
= (Sum due) – (P.W.)

2. T.D. = Interest on P.W.

3. Amount = (P.W.) + (T.D.)

4. Interest is reckoned on R.W. and true discount is reckoned on the amount

5. Let rate = R% per annum & time = T years. Then,

a. P.W. = (100 × Amount) / (100 + [R × T])
= (100 × T.D.) / (R × T)

b. T.D. = (P.W.) × R × T / 100
= ([Amount] × R × T) / (100 + [R × T])

c. Sum = ([S.I.] × [T.D.]) / ([S.I.] – [T.D.])

d. (S.I.) – (T.D.) = S.I. on T.D.

e. When the sum is put at compound interest, then
P.W. = Amount / (1 + R/100)T

Banker’s Discount
---------------------
1. Banker’s Discount (B.D.) is the S.I. on the face value for the period from the date on
which the bill was discounted and the legally due date.

2. Banker’s Gain (B.G.) = (B.D.) – (T.D.) for the unexpired time

3. When the date of the bill is not given, grace days are not to be added

4. B.D. = S.I. on bill for unexpired time

5. B.G. = (B.D.) – (T.D.)
= S.I. on T.D.
= (T.D.)2 / P.W.

6. T.D. =(P.W x B.G)^1/2

7. B.D. = (Amount × Rate × Time) / 100

8. T.D. = (Amount × Rate × Time) / (100 + [Rate × Time])

9. Amount = (B.D. × T.D.) / (B.D. – T.D.)

10. T.D. = (B.G. × 100) / (Rate × Time)

Partnership
--------------
1. If a number of partners have invested in a business and it has a profit, then
Share Of Partner = (Total_Profit × Part_Share / Total_Share)
Chain Rule

2. The cost of articles is directly proportional to the number of articles.

3. The work done is directly proportional to the number of men working at it.

4. The time (number of days) required to complete a job is inversely proportional to the
number of hours per day allocated to the job.

5. Time taken to cover a distance is inversely proportional to the speed of the car.
Time & Work

6. If A can do a piece of work in n days, then A’s 1 day’s work = 1/n.

7. If A’s 1 day’s work = 1/n, then A can finish the work in n days.

8. If A is thrice as good a workman as B, then:
Ratio of work done by A and B = 3 : 1,
Ratio of times taken by A & B to finish a work = 1 : 3

Pipes & Cisterns
-----------------
1. If a pipe can fill a tank in ‘x’ hours and another pipe can empty the full tank in ‘y’ hours
(where y > x), then on opening both the pipes, the net part of the tank filled in 1 hour is
(1/x – 1/y)

Time And Distance
----------------------
1. Suppose a man covers a distance at ‘x’ kmph and an equal distance at ‘y’ kmph, then
average speed during his whole journey is
[2xy / (x + y)] kmph

Trains
--------
1. Lengths of trains are ‘x’ km and ‘y’ km, moving at ‘u’ kmph and ‘v’ kmph (where, u > v) in
the same direction, then the time taken y the over-taker train to cross the slower train is
[(x + y) / (u - v)] hrs

2. Time taken to cross each other is
[(x + y) / (u + v)] hrs

3. If two trains start at the same time from two points A and B towards each other and after
crossing they take a and b hours in reaching B and A respectively.
Then, A’s speed : B’s speed = ( b^(1/2) : a^(1/2) ).

4. x kmph = (x × 5/18) m/sec.

5. y metres/sec = (y × 18/5) km/hr.

Boats & Streams
-------------------
1. If the speed of a boat in still water is u km/hr and the speed of the stream is v hm/hr,
then:
Speed downstream = (u + v) km/hr.
Speed upstream = (u - v) km/hr.

2. If the speed downstream is a km/hr and the speed upstream is b km/hr, then:
Speed in still water = ½ (a + b) km/hr.
Rate of stream = ½ (a - b) km/hr.

Alligation or Mixture
----------------------
1. Alligation: It is the rule that enables us to find the ratio in which two or more ingredients at
the given price must be mixed to produce a mixture at a given price.

2. Mean Price: The cost price of a quantity of the mixture is called the mean price.


Numerical solution of nonlinear equations
------------------------------------------
We want to develop numerical methods for solving nonlinear
equations of the form
f(x) = 0. (1)
Examples:
(i) f(x) = x2 − A = 0, A > 0 (⇒ x = ±√A).
(ii) f(x) = x3 − 3x2 + 2 = 0.
(iii) f(x) = 1
x − A = 0 (⇒ x = 1
A).
(iv) f(x) = sin x − 1
x = 0.
Numerical solution of nonlinear equations
Idea: Approximate Eqn. (1) by an iterative process (recursion)
xn+1 = F(xn), x1 is given, (2)
such that
lim
n→∞
xn = X, with f(X) = 0.
Easy to implement (1) on a computer.
Goal: find a systematic way for obtaining iterative process of the
form (2) for solving equations of the form (1).
Numerical solution of nonlinear equations
Consider Example (i), f(x) = x2 − A = 0, A > 0.
x2 − A = 0 ⇒ 2x2 − x2 = A
⇒ 2x2 = x2 + A ⇒ x =
1
2 x +
A
x .
Use the following iterative scheme:
xn+1 =
1
2 xn +
A
xn . (3)
Choose x1 (initial guess), use (3) to calculate approximately the
square root of A. Let A = 25 ⇒ X = ±5.
Numerical solution of nonlinear equations
Choose x1 = 1:
x1 = 1 ⇒ x2 = 13.0 ⇒ x3 = 7.4615
⇒ x4 = 5.406 ⇒ x5 = 5.0152 . . .
i.e. the iterative process converges to X = 5. Take now x1 = −1.
x1 = −1 ⇒ x2 = −13.0 ⇒ x3 = −7.4615
⇒ x4 = −5.406 ⇒ x5 = −5.0152 . . .
i.e. the iterative process converges to X = −5. Choose x1 = 8:
x1 = 8 ⇒ x2 = 5.5625 ⇒ x3 = 5.0284
⇒ x4 = 5.0001 . . .
The iteration process converges faster to the root of the equation.
Numerical solution of nonlinear equations
When we have an iterative procedure of the form (2) which we use
to calculate the roots of (1) we want to address the following issues.
1. Does the iterative process converge, i.e. is there a limit of xn as
n → ∞?
2. Does the iterative process converge to the right value (i.e. one
of the roots of the equation F(x) = 0)?
3. How does its accuracy increase with n (i.e. is it practical)?
4. How does the choice of x1 affect the convergence of the method
and the value to which it converges (in the case when the
equation has more than one roots)?
A iterative process of the form (2) for solving an equation of the
form (1) is called a fixed point method.
Numerical solution of nonlinear equations
Consider example (ii) x3 − 3x2 + 2 = 0:
x3 − 3x2 + 2 = 0 ⇒ x3 = 3x2 − 2
⇒ x = 3 −
2
x2 .
• Use the iterative process
xn+1 = 3 −
2
x2
n
.
• If this method converges, to which of the 3 roots of the
equation does it converge?
Numerical solution of nonlinear equations
If an iterative process is to be convergent, then we have
lim
n→∞xn+1 − xn = 0,
i.e. the distance between two subsequent elements of the sequence
{xn}∞
n=1 generated by (2) becomes smaller as n increases.
We expect that the starting value x1 affects the eventual
convergence of the iterative process to a limit. It might be that the
iterative process that we have chosen converges only for certain
values of x1.
If we sketch the curves y = x, y = F(x), their intersection satisfies
x = F(x). Start with x1. The iterates generated by (2) should give
x = X.
Numerical solution of nonlinear equations
Calculation of √A.
• Consider the iterative process
xn+1 =
1
2 xn +
A
xn ,
• which approximates √A.
• Assume 0 < x2 =" 1">
1
2 √A +
A
√A
= √A ⇒ x2 > √A.
Numerical solution of nonlinear equations
Now:
x3 =
1
2 x2 +
A
x2
< n =" 2,"> √A. Then xn+1 < n =" 1," a =" 1" a =" 1" a =" 1" 1 =" xn(2" x =" 0" 1 =" Ax" 1 =" 2" x =" x(2" xn =" 1" x1 =" 0" x1 =" 2" xn =" 0," n =" 1,">
2
A ⇒ lim
n→∞
xn = −∞.
Numerical solution of nonlinear equations
Summary
• An iterative process can converge or diverge, depending on the
initial choice x1.
• The root to which the iterative process converges depends on
the initial choice x1.
• The convergence can be fast (quadratic).
How to choose the iterative process for solving f(x) = 0.
A given equation f(x) = 0 may often be written in several different
ways as x = F(x).
Example: f(x) = x2 − 6x + 2 = 0 with roots X1,2 = 3 ± √7. It
can be written as x = F(x) in many different ways:
1. x = x
2
−2
2x−6 ⇒ the iterative process is xn+1 = x
2
−2
2xn−6 :
x2−6x+2 = 0 ⇒ 2x2−6x+2 = x2 ⇒ 2x2−6x = x2−2 ⇒ x =
x2 − 2
2x − 6
.
2. x = 6 − 2
x ⇒ the iterative process is xn+1 = 6 − 2
xn
:
x2 − 6x + 2 = 0 ⇒ x2 = 6x − 2 ⇒ x = 6 −
2
x
.
3. x = 1
6x2 + 1
3 ⇒ the iterative process is xn+1 = 1
6x2
n + 1
3 :
6x = x2 + 2 ⇒ x =
1
6
x2 +
1
3
.
How to choose the iterative process for solving f(x) = 0.
The resulting different iterative processes xn+1 = F(xn) have, in
general, different convergence properties. We want to choose the
iterative process that has the best convergence properties.
This means that not only does it converge to the roots of the
equation f(x) = 0, but also that it converges sufficiently fast. We
want to develop criteria that enable us to choose the best iterative
process for solving a nonlinear equation of the form f(x) = 0.
How to choose the iterative process for solving f(x) = 0.
Suppose that xn+1 = F(xn) with limn→∞ xn = X and
f(X) = 0 ⇒ X = F(X). Let us write
X = xn + ǫn, n = 1, 2, . . .
i.e.
X = approximation + error.
We want to calculate ǫn+1 as a function of ǫn. Assume that ǫn is
small so that we can use the Taylor series expansion:
xn+1 = F(xn) ≈ F(X) − ǫnF′(X) +
1
2
ǫ2
nF′′(X).
But xn+1 = X + ǫn+1 ⇒
ǫn+1 = X − xn+1 ≈ X − F(X) + ǫnF′(X) −
1
2
ǫ2
nF′′(X)
= ǫnF′(X) −
1
2
ǫ2
nF′′(X).
⇒ The size of ǫn+1, relative to the size of ǫn, depends on F′(X).
How to choose the iterative process for solving f(x) = 0.
Indeed:
ǫn+1
ǫn
= F′(X) + O(ǫn). (4)
DEFINITION 1 Assume that F′(X) 6= 0. Then the iterative
process xn+1 = F(xn) is called a first order process.
The convergence of a first order process is guranteed when
F′(X) <> 1. The case F′(X) = 1 requires further study.
How to choose the iterative process for solving f(x) = 0.
We can also define second order processes:
DEFINITION 2 Assume that F′(X) = 0 and that F′′(X) 6= 0.
Then the iterative process xn+1 = F(xn) is called a second order
process.
For a second order process we have that
ǫn+1 = −
1
2
ǫ2
nF′′(X) + O(ǫ3
n) ⇒ ǫn+1 = Cǫ2
n
and the convergence is much faster (quadratic as opposed to
linear). It is to our advantage to use a second order iterative
process to solve the equation f(x) = 0.
How to choose the iterative process for solving f(x) = 0.
Examples
1. The square root formula F(x) = 1
2 􀀀x + A
x .
F′(x) =
1
2 1 −
A
x2 ⇒ F′(√A) =
1
2 1 −
A
A = 0
⇒ We have to calculate the second derivative
at the root X = √A.
F′′(x) =
A
x3 ⇒ F′′(√A) =
1
√A
.
⇒ Second Order Process.
⇒ The convergence is quadratic.
How to choose the iterative process for solving f(x) = 0.
Examples
2. The reciprocal formula F(x) = x (2 − Ax).
F′(x) = (2 − Ax) + x(−A) = 2 − 2Ax
⇒ F′ 1
A = 2 − 2A 1
A = 0
⇒ We have to calculate the second derivative
at the root X =
1
A
.
F′′(x) = −2A ⇒ F′′ 1
A = −2A.
⇒ Second Order Process.
⇒ The convergence is quadratic.
How to choose the iterative process for solving f(x) = 0.
Examples
3. The quadratic equation f(x) = x2 − 6x + 2 = 0. The two roots
of this equation are R1 = 3 + √7, R2 = 3 − √7. Consider the
following 4 different iterative processes.
(a) xn+1 = 6 − 2
xn
.
(b) xn+1 = 1
6x2
n + 1
3 .
(c) xn+1 = √6xn − 2.
(d) xn+1 = x2
−2
2xn−6 .
How to choose the iterative process for solving f(x) = 0.
The performance of the the four different iterative processes is
summarized in the following table.
Iterative process R1 attainable R2 attainable order of process
xn+1 = 6 − 2
xn
. NO YES 1
xn+1 = 1
6x2
n + 1
3 . YES NO 1
xn+1 = √6xn − 2. NO YES 1
xn+1 = x2
−6
2xn−6 . YES YES 2
⇒ The fourth iterative process performs better: both roots are
attainable and it converges faster.
Q: Is there a systematic method for finding the best method for a
given nonlinear equation f(x) = 0? A: Yes, the
Newton–Raphson Method.
The Newton–Raphson Method
Consider the equation f(x) = 0 and let X be the root of this
equation: f(X) = 0. Let x1 be our initial approximation and write
X = x1 + ǫ1.
Use the Taylor series expansion to obtain
0 = f(X) = f(x1 + ǫ1) = f(x1) + ǫ1f′(x1) + . . .
Assuming that f′(x1) 6= 0, we solve this equation for ǫ1 to obtain
ǫ1 ≈ −
f(x1)
f′(x1)
.
Hence, a better approximation to X than x1 would appear to be
x2 = x1 −
f(x1)
f′(x1)
.
The Newton–Raphson Method
We can repeat this procedure to obtain the iterative process
xn+1 = xn −
f(xn)
f′(xn)
=: F(xn). (6)
This is the Newton–Raphson iterative process. The
Newton–Raphson process converges to a particular root of f(x) = 0
if x1 is suitably chosen, assuming that all the roots are attainable.
The process is usually second order convergent if X is a
simple root:
The Newton–Raphson Method
F(x) = x −
f(x)
f′(x) ⇒
F′(x) = 1 −
(f′(x))2 − f(x)f′′(x)
(f′(x))2 =
(f′(x))2 − (f′(x))2 + f(x)f′′(x)
(f′(x))2
=
f(x)f′′(x)
(f′(x))2 ⇒
F′(X) =
f(X)f′′(X)
(f′(X))2 = 0, since f(X) = 0.
If X is not a simple root then the Newton–Raphson process is
usually first order, but it still converges.
The Newton–Raphson Method: Examples
1. The square root formula:
f(x) = x2 − A = 0 ⇒ f′(x) = 2x ⇒
xn+1 = xn −
f(xn)
f′(xn)
= xn −
x2
n − A
2xn
= xn −
xn
2
+
A
2xn
=
1
2 xn +
A
xn .
The Newton–Raphson Method: Examples
2. The reciprocal formula:
f(x) =
1
x − A = 0 ⇒ f′(x) = −
1
x2 ⇒
xn+1 = xn −
f(xn)
f′(xn)
= xn −
1
xn − A
− 1
x2
n
= xn + x2
n 1
xn − A
= xn + xn − Ax2
n
= xn(2 − Axn).
The Newton–Raphson Method: Examples
3. A quadratic equation:
f(x) = x2 − 6x + 2 = 0 ⇒ f′(x) = 2x − 6 ⇒
xn+1 = xn −
f(xn)
f′(xn)
= xn −
x2
n − 6xn + 2
2xn − 6
=
xn(2xn − 6) − x2
n + 6xn − 2
2xn − 6
=
2x2
n − 6xn − x2
n + 6xn − 2
2xn − 6
=
x2
n − 2
2xn − 6
.
The Newton–Raphson process converges to R1 or R2 depending
on the initial guess:
• x1 < r1 =" 3"> 3 ⇒ xn → R2, R2 = 3 + √7.
• x1 = 3 ⇒ x2 = +∞, the method diverges.
The Newton–Raphson Method: Examples
4. The equation sin x − 1
x = 0.
f(x) = sin x −
1
x
= 0 ⇒ f′(x) = cos x +
1
x2 ⇒
xn+1 = xn −
f(xn)
f′(xn)
= xn −
sin xn − 1
xn
cos xn + 1
x2
n
= xn − x2
n
sin xn − 1
xn
x2
n cos xn + 1
= xn −
x2
n sin xn − xn
x2
n cos xn + 1
.
The equation sin x − 1/x = 0 has infinitely many solutions. Let
us find the first positive solution R = 1.11416. Start with
x1 = 0.5.
x1 = 0.5 ⇒ x2 = 0.8117 ⇒ x3 = 1.0413
⇒ x4 = 1.1095 ⇒ x5 = 1.1141.
The Newton–Raphson Method: Examples
0.5 1 1.5 2 2.5 3 3.5 4
−1
−0.5
0
0.5
1
1.5
2
sin(x)
1/x
0.5 1 1.5 2 2.5 3 3.5 4
−1.5
−1
−0.5
0
0.5
a. f1(x) = sin(x), f2(x) = 1/x. b. f(x) = sin(x) − 1/x.
The Newton–Raphson Method: Examples
0 5 10 15 20 25 30
−1
−0.5
0
0.5
1
1.5
2
sin(x)
1/x
f1(x) = sin(x), f2(x) = 1/x.
The Newton–Raphson Method: Examples
We want to show that a solution to the equation
f(x) = sin x − 1
x = 0 exists in the interval (
4 ,
2 ). We have that
f′(x) = cos x +
1
x2 > 0 for x ∈ hπ
4
,
π
2 i.
Consequently, the function f(x) is strictly increasing in the interval
[
4 ,
2 ]. Furthermore,
f(π/4) = sin(π/4) −
1
π/4
=
√2
2 −
4
π
< 2 =" 1"> 0.
Since f(x) is continuous and strictly increasing, there exists an
R ∈ (
4 ,
2 ) such that f(R) = 0. This shows that there exists a root
of the equation f(x) = 0 in the interval (
4 ,
2 ).
The Newton–Raphson Method: Examples
We want to show that the equation
f(θ) = sin(θ) − θ cos(θ) −
1
2
π = 0
in the interval (
2 , 2
3 ).
f′(θ) = cos θ − cos θ + θ sin θ = θ sin θ > 0 for θ ∈ π
2
,
2π
3 .
Consequently, the function f(x) is strictly increasing in the interval
[
2 , 3
2 ]. Furthermore,
f(
π
2
) = sin(
π
2
) −
π
2
cos(
π
2
) −
π
2
= 1 −
π
2
< 3 =" 0.3424"> 0.
Since f(x) is continuous and strictly increasing, there exists an
R ∈ (
2 , 2
3 ) such that f(R) = 0. This shows that there exists a
root of the equation f(x) = 0 in the interval (
2 , 2
3 ).
The Newton–Raphson Method: Examples
Now we find the Newton–Raphson process for this equation. We
have that
f′(θ) = θ sin(θ).
The Newton–Raphson process is
θn+1 = θn −
sin θn − θn cos θn −
2
θn sin θn
.
The root of the equation f(θ) = sin(θ) − θ cos(θ) − 1
2π = 0 in the
interval (
2 , 2
3 ) is R = 1.9057. We calculate it using the
Newton-Raphson process with x1 =
2 :
x1 =
π
2 ⇒ x2 = 1.9342 ⇒ x3 = 1.9058 ⇒ x4 = 1.9057.
The Newton–Raphson Method: Examples
1.5 1.6 1.7 1.8 1.9 2 2.1 2.2
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0 2 4 6 8 10
−8
−6
−4
−2
0
2
4
6
8
a. θ ∈
2 , 2
3 . b. θ ∈ [0, 10].
f(θ) = sin(θ) − θ cos(θ) − 1
2π.
Numerical solution of nonlinear equations
Final Remarks
• One can also define higher order processes:
F′(X) = 0, F′′(X) = 0, . . . F(n)(X) 6= 0.
• This is an nth order process.
• There are many other methods for solving nonlinear equations
other than the Newton–Raphson method: Bisection method,
secant method etc.
• The Newton–Raphson methods works also for equations on the
complex plane:
f(z) = 0, z ∈ C.