Entries Tagged 'MBA Study Material' ↓

A collection of questions that typically appear in the Common Admission Test (CAT) from the topic Number Theory.

These questions will guide you through your CAT and other MBA entrance exam preparation.

1. When 26854 and 27584 are divided by a certain two digit prime number, the remainder obtained is 47. Which of the following choices is a possible value of the divisor?

2. If both 112 and 33 are factors of the number a * 43 * 62 * 1311, then what is the smallest possible value of ‘a’?

3. What is the remainder when 9^1 + 9^2 + 9^3 + …. + 9^8 is divided by 6?

2. A number when divided by a divisor leaves a remainder of 24. When twice the original number is divided by the same divisor, the remainder is 11. What is the value of the divisor?

4. ‘a’ and ‘b’ are the lengths of the base and height of a right angled triangle whose hypotenuse is ‘h’. If the values of ‘a’ and ‘b’ are positive integers, which of the following cannot be a value of the square of the hypotenuse?

5. Let n be the number of different 5 digit numbers, divisible by 4 with the digits 1, 2, 3, 4, 5 and 6, no digit being repeated in the numbers. What is the value of n?

6. What is the reminder when 91 + 92 + 93 + …… + 99 is divided by 6?

7. For what value of ‘n’ will the remainder of 351^n and 352^n be the same when divided by 7?

8. How many keystrokes are needed to type numbers from 1 to 1000?

9. When 242 is divided by a certain divisor the remainder obtained is 8. When 698 is divided by the same divisor the remainder obtained is 9. However, when the sum of the two numbers 242 and 698 is divided by the divisor, the remainder obtained is 4. What is the value of the divisor?

10. Find the greatest number of five digits, which is exactly divisible by 7, 10, 15, 21 and 28.

11.Anita had to do a multiplication. Instead of taking 35 as one of the multipliers, she took 53. As a result, the product went up by 540. What is the new product?

12. Find the value of 1.1! + 2.2! + 3.3! + ……+n.n!

13. Let x, y and z be distinct integers. x and y are odd and positive, and z is even and positive. Which one of the following statements cannot be true?

14. When a number is divided by 36, it leaves a remainder of 19. What will be the remainder when the number is divided by 12?

15. A person starts multiplying consecutive positive integers from 20. How many numbers should he multiply before the will have result that will end with 3 zeroes?

16. How many different factors are there for the number 48, excluding 1 and 48?

17. How many zeros contained in 100!?
18. Two numbers when divided by a certain divisor leave remainders of 431 and 379 respectively. When the sum of these two numbers is divided by the same divisor, the remainder is 211. What is the divisor?

19. What number should be subtracted from x^3 + 4x^2 – 7x + 12 if it is to be perfectly divisible by x + 3?

20. What is the least number that should be multiplied to 100! to make it perfectly divisible by 350?

Questions along with respective explanations.

Q.1. Train A traveling at 60 km/hr leaves Mumbai for Delhi at 6 P.M. Train B traveling at 90 km/hr also leaves Mumbai for Delhi at 9 P.M. Train C leaves Delhi for Mumbai at 9 P.M. If all three trains meet at the same time between Mumbai and Delhi, what is the speed of Train C if the distance between Delhi and Mumbai is 1260 kms?

(1) 60 km/hr (2) 90 km/hr (3) 120 km/hr (4) 135 km/hr

Correct Answer - (3)

Solution:
All three trains meet at the same time between Delhi and Mumbai. Which means Train A and Train B are at the same point at that time. This will happen when Train B is overtaking Train A.

Train A starts 3 hours before Train B. Therefore, by the time Train B leaves Mumbai, Train A has covered 3 * 60 = 180 kms.

The relative speed between Train A and Train B = 90 - 60 = 30 kmph. Therefore, Train B will overtake Train A in = 6 hours from the time Train B leaves Mumbai. That is at 3 A.M, Train B will overtake Train A. The point between Mumbai and Delhi at which Train B overtakes Train A will be 6*90=540 kms from Mumbai.

Train C will also be at that point at 3 A.M while Train B is overtaking Train A. And Train C would have travelled 1260-540 = 720 kms in these 6 hours. Therefore, the speed of Train C = 120 km/hr.

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Q.2 A train travels at an average speed of 90 km/hr without any stoppages. However, its average speed decrease to 60km/hr on account of stoppages. On an average, how many minutes per hour does the train stop?

(1) 12 minutes (2) 18 minutes (3) 24 minutes (4) 20 minutes

Correct Answer - (4)

Solution:
If it travelled at 90 km / hr, it would have crossed 90 kms in an hour. However, it covered only 60 kms due to stoppages.

The distance it covered decreased by 1/3 or it covered only 2/3rd of the distance that it can cover for which the traveling time would have been 2/3rd of an hour. The remaining 1/3rd of an hour was spent in stoppages. Therefore, the train stops on an average for 20 minutes every hour.
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Q.3 A man goes from city A to city B situated 60 kms apart by a boat. His onward journey was with the stream while the return journey was an upstream journey. It took him four and half hours to complete the round trip. If the speed of the stream is 10 km/hr, how long did it take him to complete the onward journey?
(1) 3 hours (2) 3.5 hours (3) 2.25 hours (4) 1.5 hours

Correct Answer - (4)

Solution:
The average speed for the round trip = km/hr
Let the speed during the onward journey be ‘D’ km/hr. Let the speed of the boat in still water be ‘B’ km/hr.
Therefore, D = B + S => D = B + 10 (As the speed of the stream is 10 km/hr).
Let the speed during the return journey be ‘U’ km/hr.
Therefore, U = B - S = B - 10

As the distance between A and B is the same as the distance between B and A, the average speed is given by the formula =
=> => 3B2 - 300 = 80B.
=> 3B2 - 80B - 300 = 0 => 3B2 - 90B + 10B - 300 = 0
=> 3B(B - 30)+10(B - 30) = 0
=> (B-30)(3B+10) = 0
=> B = 30 or B = -10/3

As speed is a positive quantity, B = 30.
Therefore, D = 30 + 10 = 40 km/hr and U = 30 - 10 = 20 km/hr.

His onward journey was done at a speed of 40 km/hr. The distance covered was 60 kms.
Therefore, the time taken for the onward journey = 1.5 hours

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Q. 4 The speed of a motor boat itself is 20 km/h and the rate of flow of the river is 4 km/h. Moving with the stream the boat went 120 km. What distance will the boat cover during the same time going against the stream?
(1) 80 km (2) 180 km (3) 60 km (4) 100 km

Correct Answer - (1)

Solution:
Let the distance to be covered by the boat when it is travelling against the stream be x.
The boat goes down the river at a speed of 20 + 4 = 24 km/h and up the river at a speed of 20 – 4 = 16 km/h.
Since the time taken is same 120/24 = x/16
Therefore, x = 80 km.

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Q. 5 A passenger train covers the distance between stations X and Y, 50 minutes faster than a goods train. Find this distance if the average speed of the passenger train is 60 kmph and that of goods train is 20 kmph.
(1) 20 kms (2) 25 kms (3) 45 kms (4) 40 kms
Correct Answer - (2)

Solution:
Let ‘d’ be the distance between the stations X and Y.
Time taken by the passenger train to cover the distance ‘d’ = d/60 hour
Time taken by the goods train to cover the distance ‘d’ = d/20 hour
Time difference between these two trains is given by 50 minutes or 50/60 hour

i.e., d/20 –d/60 = 50/60

d = 25kms.

How many number of times will the digit ‘7′ be written when listing the integers from 1 to 1000?

Sol:
7 does not occur in 1000. So we have to count the number of times it appears between 1 and 999. Any number between 1 and 999 can be expressed in the form of xyz where 0 < x, y, z < 9.

1. The numbers in which 7 occurs only once. e.g 7, 17, 78, 217, 743 etc. This means that 7 is one of the digits and the remaining two digits will be any of the other 9 digits (i.e 0 to 9 with the exception of 7). You have 1*9*9 = 81 such numbers. However, 7 could appear as the first or the second or the third digit. Therefore, there will be 3*81 = 243 numbers (1-digit, 2-digits and 3- digits) in which 7 will appear only once. In each of these numbers, 7 is written once. Therefore, 243 times.

2. The numbers in which 7 will appear twice. e.g 772 or 377 or 747 or 77 In these numbers, one of the digits is not 7 and it can be any of the 9 digits ( 0 to 9 with the exception of 7). There will be 9 such numbers. However, this digit which is not 7 can appear in the first or second or the third place. So there are 3 * 9 = 27 such numbers. In each of these 27 numbers, the digit 7 is written twice. Therefore, 7 is written 54 times.

3. The number in which 7 appears thrice - 777 - 1 number. 7 is written thrice in it. Therefore, the total number of times the digit 7 is written between 1 and 999 is 243 + 54 + 3 = 300
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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?

Ans : 4

Sol. Let the digit unknown be n.

The given number is then 900+50+n=950+n.
When reversed the new number is 100n+50+9=59+100n.
Subtracting these two numbers we get 891-99n.
The digit can be arranged in 3 ways or 6 ways.
We have already investigated 2 of these ways.
We can now try one of the remaining 4 ways. One of these is n 95

100n+90+5=891-99n
or 199n =796
so, n=4
the unknown digit is 4.
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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?

Ans. 20

Sol:

Total number of children=30*16=480
Number of columns of 24 children each =480/24=20.
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The accuracy is highest in such questions, and the time spent solving them can be quite less if one has done a good amount of practice. The questions that are asked in logical reasoning require one to find out whether the given conclusion is valid or not. For example:

1. All drunken drivers meet with an accident.
Salman is a drunken driver.
Conclusion – Salman will meet with an accident
This one is pretty straightforward. Since Salman belongs to a group, all members of which meet with an accident, he would meet with an accident.

2. Most males are intelligent.
Mr. X is a male
Conclusion – Mr. X may or may not be intelligent
Here there are two possibilities. Either Mr. X is a part of the group of intelligent males or he is not. Hence, it follows that he may or may not be intelligent.

In the exam the questions that are asked have two statements followed by two conclusions and you have to determine which conclusion is correct. Obviously the questions are not so easy as the above examples and may require one to draw venn diagrams to solve the questions. E.g.

1. Statements –
some students are smart
all students are hardworking

Conclusions – (i) some hardworking students are smart
(ii) some smart students are hardworking
Solution – Both the conclusions are correct, as some students are definitely both smart and hardworking.

2. Statement–
tennis players get married only to models

Conclusions – (i) Madhu is married to a tennis player
(ii) Madhu is not married to a tennis player
Solution – Here either Madhu can be married to a tennis player or to someone who is not a tennis player. So either conclusion (i) or (ii) follow but not both.

3. Statements –
some roses are red
some red are black.

Conclusions – (i) no black is a rose.
(ii) no rose is a black
Solution – Here both the statements are possibilities. It may be that some roses are black or no roses are black. Hence neither of the conclusions follows.

You need to adopt different approaches for different passages.
These are as follows:

Read Passage –> Answer Questions:

This is the traditional way or reading the passage first and then approaching the questions.

Answer Questions -> Read Passage –> Answer Questions:

This approach most commonly used is to have a look at the questions first and then read the passage, underlining part of the passage where an answer to the questions read may be found. After reading the passage, one can go to the questions and if need be just revisit the underlined part. However, while one is reading the questions, one must not waste a lot of time reading any inference-based questions and just keep a look out for names, events and terms that can be found in the passage.

Answer Questions -> Read Passage -> Answer Questions -> Read Passage:

In this approach, one never ends up reading the passage. A question is taken one at a time and its answer is searched for in the passage. Obviously not many inference-based questions can be answered in this way, so these questions will have to be left. But if it were a fact-based passage, these types of questions would be minimal. Also one can take advantage that usually the answers to the questions would lie sequentially in the passage.

You could also follow this approach. Around one-third of the passage is read, which gives one a fair idea of the structure of the passage. Then questions are looked at and all those questions whose answers lie in the part of passage read are attempted. Then for the remaining passage the same approach is followed. This also helps in saving time searching for the answers as the amount of passage unread is less and also one understands the flow of the passage. A few inferential questions can also be answered using this approach.

The above approaches coupled with a proper selection of the passages to be attempted should help one comfortably clear the cut-offs of the Reading Comprehension section.

Selecting the correct passages:

The number of passages asked will be minimum five. If 3 are attempted it is a fair attempt. Passages should be selected based on the following.
1. The Comfort level with the topic of the passage
2. Degree of Difficulty level of the language used
3. Trying Understanding the structure of the passage
4. Knowing Length of the passage and number of the questions asked

Management:

1. Any Edition of Harvard Business Review
2. Case Studies of Businessworld Magazine
3. What They Don’t Teach You At Harvard Business School - Mark H. McCormack

Physics/Science:

1. Brief History of Time - Stephen Hawking
2. Any work of Isaac Asimov
3. The Tao of Physics - Fritjof Capra

Philosophy:

1. Zen And The Art of Motor Cycle Maintenance
2. The Fountainhead - Ayn Rand

Abstract Topic:

Futureshock, Power Shift, Third Wave - Alvin Toffler

Literary Works:

1. God of Small Things - Arundhati Roy
2. Midnight’s Children - Salman Rushdie

Business:

Made In Japan - Akio Morito

Religion:

Any book written by Swami Vivekanand

Sports:

Idols - Sunil Gavaskar

General:

Uncommon Wisdom - Fritjof Capra

Law:

Any of John Grisham’s works

Medical:

Doctors - Erich Segal

Anthropology/Sociology:

1. Manwatching
2. The Naked Ape - Desmond Morris

Note: The above list is just an indicative list.

Lessons in Imperative Sentences-
Study the following base form of the verbs and see how they are used in the imperative sentences. Imperative sentences are the sentences, which express commands, advice, and suggestions. The second person ‘you’ is always addressed in the imperative sentences.

In the following sentences, ‘you’ is the subject, though understood: -

1. Believe in God.
2. Worship Ma Sarswatiji, Ma Durgaji, Ma Luxmiji.
3. Don’t be superstitious.
4. Don’t deceive anyone.
5. Always speak the truth.
6. Vishwamitra said to Rama, “Send Ram and Lakshman with me. Don’t suspect my intention.”
7. Get up early in the morning. Have a long walk. Do exercises and yoga to keep yourself fit.
8. Never lose your patience. Be brave and courageous.
9. Don’t pluck flowers, as they enhance the beauty of nature.
10. Drink eight glasses of water daily, as it is very conducive to health.
11. Never tease the poor and the weak.
12. Do justice to get justice from God.
13. Don’t advise anyone unless you are asked for.
14. Work at the behest of your conscience.
15. The officer said to the servant, “Bring a glass of water. Always keep the office neat and clean.”
16. Be considerate to others and never think negatively.
17. The principal told the students, “Discipline yourself at every step of your life. Attend your classes regularly and punctually. Don’t come late. Pay attention to what your teachers teach you.”
18. Be honest, as honesty always pays; sooner or later.
19. Try to understand the motive of your opponents before you take any step against them.
20. Mother said to her son, “Mend your ways, if you want me to lead a happy life.”

Pay Attention: In the above-mentioned sentences, the verbs in their base form are “believe, worship, be, deceive, speak, send, get up, lose, do, pluck, drink, tease, advise, work, bring, keep, think, discipline, attend, come, pay, try, mend”.

Lessons in Modal Auxiliaries-

Modal Auxiliaries:

The modal verbs are - shall, will, can, may, must, should, would, could, might, ought to, need, dare.

Pay Attention: Most of the modal verbs are always followed by the first form (bare infinitive) of verbs in Active Voice, and be + 3rd form of the verb in Passive Voice.

A.. ‘Will’ and its past form ‘Would’

It (will) is used with second and third persons (you, thou, he, she, they, all the nouns).

in Future Tense -

1. Time will decide what will happen in coming days.
2. What will you decide in my absence?
3. The best students of our college will be awarded by the manager.
4. The problems regarding the condition of the poor will be discussed later on.
5. Who will feed you if you don’t work?
6. Nothing will be done if we await the right time.
7. I asked my friend when he would go to Mumbai.

Pay Attention: Conjunctions showing time and condition are not followed by ’shall’ or ‘will. Such conjunction are - if, till, provided, in case, supposing, unless, until, till, as long as, so long as, as if, as though, before, when, as soon as, since, while, after. The past form of ‘will’ or’ shall’ (would, should) is also not used.

1. Will you take your bath before you leave for your office?
2. As long as atom-bombs continue to be made, threats of war will remain.
3. The students will continue their agitation till the Government takes its decision back.
4. We shall hold a meeting after the students have gone home.
5. As soon as we discard ‘laziness’, we shall find that we shall be able to do a lot.

1. PIN code started in India in the year….

a. 1962
b. 1964
c. 1968
d. 1972

2. You would associate the establishment of the Banaras Hindu University (BHU) with.
a. Subramaniam Bharti
b. Pt. Madan Mohan Malviya
c. Jawaharlal Nehru
d. Gobind Vallabh Pant

3. Who presides the Lok Sabha?
a. Prime Minister
b. President
c. Vice President
d. Speaker

4. Who discovered Aspirin?
a. Alexander Fleming
b. Felix Hoffmann
c. Robert Koch
d. Joseph Lister

5. Which of the following is NOT one of the sacred texts of Buddhism called Tripitakas?
a. Suttapitaka
b. Vinayapitaka
c. Dharmapitaka
d. Gyanapitaka

6. Which of the following Indian states has the lowest sex ratio {number of females per thousand males}?
a. West Bengal
b. Haryana
c. Punjab
d. Uttar Pradesh

7. Which of the following is the oldest printed book of the world?
a. The Gutenberg Bible
b. Hiraka Sutra
c. Les Hommes de bonne Volante
d. Old King Cole

8. This personality was the first Indian to receive the Magsaysay Award in the year 1958. Can you name him from the given options?
a. CD Deshmukh
b. Mother Teresa
c. Acharya Binoba Bhave
d. Jaya Prakash Narayan

9. Who is known as the father of Geometry?
a. Pythagoras
b. Euclid
c. Euler
d. None of these

10. Rovers cup is associated with …
a. Hockey
b. Football
c. Cricket
d. Golf

11. This Indian state is the least densely populated state. Name it from the given options.
a. Sikkim
b. Arunachal Pradesh
c. Assam
d. Manipur

12. What is the approximate age of the earth?
a. 3.8 billion years
b. 4.2 billion years
c. 4.6 billion years
d. 4.8 billion years

13. Which Indian state was the first one to implement the Panchayati Raj in the whole state?
a. Uttar Pradesh
b. Bihar
c. Rajasthan
d. Gujarat

14. This personality is NOT the winner of Nobel Prize.
a. Kofi Annanb
b. Mother Teresac
c. VS Naipaul
d. Richard Nixon

15. The largest state of India (in terms of area) is…
a. Uttar Pradesh
b. Rajasthan
c. Madhya Pradesh
d. Maharashtra

16. The line of equator does no pass through…
a. Kenya
b. Somalia
c. Uganda
d. Egypt

17. National Institute of Oceanography is based in …
a. Mumbai
b. Chennai
c. Panaji
d. Kochi

18. Poverty and Un-British Rule in India is a famous work of …
a. Dada Bhai Naoroji
b. Mahatma Gandhi
c. AO Hume
d. BG Tilak

19. The first ancient Olympic games took place in the year
a. 476 BC
b. 576 BC
c. 676 BC
d. 776 BC

20. When one crosses the International Date Line from west to east, then the date is said to be set back by…
a. one hour
b. twelve hours
c. one day
d. None of these

21. Equinoxes occur when
a. the days are longer than the nights
b. the nights are longer than the days
c. the nights and days are equal in length
d. the difference between the length of nights and days is the highest

22. Which of the following is NOT a permanent member of United Nation’s Security Council?
a. Russia
b. China
c. France
d. Germany

23. You would associate the theory of ‘Laissez Faire’ with …
a. JK Galbraith
b. JM Keynes
c. Adam Smith
d. David Ricardo

24. Wealth of Nations is authored by….
a. Adam Smith
b. JK Galbraith
c. JM Keynes
d. None of these

25. This personality is considered, as the political guru of Mahatma Gandhi. Can you identify him from the given options?
a. Bal Gangadhar Tilak
b. Gopal Krishna Gokhale
c. Dadabhai Naoroji
d. Mahadev Gobind Ranande

ANSWERS:

1. d. 1972
2. b. Pt. Madan Mohan Malviya
3. d. Speaker
4. b. Felix Hoffman
5. d. Gyanapitaka
6. b. Haryana
7. b. Hiraka Sutra
8. c. Acharya Binoba Bhave
9. b. Euclid
10. b. Football
11. b. Arunachal Pradesh
12. c. 4.6 billion years
13. c. Rajasthan
14. d. Richard Nixon
15. b. Rajasthan
16. d. Egypt
17. c. Panaji
18. a. Dada Bhai Naoroji
19. d. 776 BC
20. c. one day
21. c. the nights and days are equal in length
22. d. Germany
23. c. Adam Smith
24. a. Adam Smith
25. b. Gopal Krishna Gokhale

1. Etihad Airways is the name of the national airlines of…

a. Afghanistan
b. Iraq
c. Kuwait
d. United Arab Emirates

2. Who was the chairman of the twelfth finance commission of India?
a. MS Ahluwalia
b. C Rangarajan
c. Rakesh Mohan
d. Bimal Jalan

3. General Insurance Corporation of India (GIC) is chaired by…
a. N Rangachari
b. RK Joshi
c. MK Garg
d. Ashok Lahiri

4. This site/search engine was in the news in January 2005 as it had plans to offer the famous libraries of the world online. Can you identify the name of the engine in question from the given options?
a. Yahoo. com
b. Google.com
c. Altavista. com
d. AOL

5. Aramco is one of the world’s biggest….
a. Cement firm
b. Steel firm
c. IT firm
d. Oil firm

6. Mukand Steel is the key company owned by…
a. Birlas
b. Bajajs
c. Tatas
d. Ruias

7. ‘Cottons’ is the name of garment collection by….
a. Madura Garments
b. Century Textiles
c. Indigo Nation
d. Raymond

8. Servo is an oil brand from….
a. HPCL
b.IOC
c. BPCL
d. None of these

9. Progeon is the name of the BPO arm of this Indian software giant. Identify it from the given operations.
a. Wipro Technologies
b. Infosys Technologies
c. HCL Technologies
d. Polaris Softwares

10. According to the Union Budget 2005-2006, which of the following represents the outlay for the defence sector?
a. Rs. 66,000 cr
b. Rs. 77,000 cr
c. Rs. 83,000 cr
d. Rs. 90,000 cr

11. ‘Mahila card’ is an international debit card meant exclusively for women. This card belongs to the stable of …
a. ICICI Bank’
b. UTI Bank
c. Induslnd Bank
d. Canara Bank

12. Which of the following represents the limit of foreign direct investment (FDI) in the construction sector?
a. 49 per cent
b. 66 per cent
c. 74 per cent
d. 100 per cent

13. Who is the chairman of IDBI?
a. M. Damodaran
b. VP Shetty
c. PP Vora
d. KV Kamath

14. You would associate Corp Kisan Card with …
a. Canara Bank
b. Corporation Bank
c. ICICI Bank
d. HDFC Bank

15. Itz Cash, a pre-paid card, belongs to the stable of…
a. Tatas
b. Essel group
c. Essar group
d. None of these

16. Who among the following is the CMD of National Thermal Power Corporation (NTPC)?
a. VP Sinha
b. b. C P Jain
c. AK Purwar
d. PS Shenoy

17. ‘Aapka Fayda’ is the punch line that you would associate with….
a. Sahara Samay
b. NDTV India
c. CNBC TV 18
d. Zee Business

18. ‘Data One’ is the name of the broadband service of ….
a. Reliance Infocomm
b. BSNL
c. MTNL
d. Tata Indicom

19. Food-World, Giant, Health 8′ Glow and Music World are all retail outlets of …
a. Pantaloon
b. RPG
c. Shopper’s Stop
d. HLL

20. Sarthak Behuria is the chairman-cum-managing director of…
a. ONGC
b. lac
c. BPCL
d.HPCL

Answer Key:

1. d. United Arab Emirates
2. b. C. Rangarajan
3. b. RK Joshi
4. b Google.com
5. d. Oil firm
6. b. Bajajs
7. b. Century Textiles
8. b. 10C
9. b. Infosys Technologies
10. c. Rs. 83.000 cr
11. c. Induslnd Bank
12. d. 100 per cent
13. b. VP Shelly
14. b. Corporation Bank
15. b. Essel Group
16. b. CP Jain
17. d. Zee business
18. b. BSNL
19. b. RPG
20. b. 10C