Contents NUMBERS...3 1. Number System... 3 2. Sequence, Series and Progression... 8 ARITHMETIC... 12 3. Averages... 12 4. Ratio and Proportion... 16 5. Percentage... 20 6. Profit and Loss... 23 7. Interest... 26 8. Speed, Distance and Time... 29 9. Time and Work... 33 ALGEBRA... 36 10. Functions... 36 11. Quadratic and other equations... 40 12. Inequalities... 46 13. Logarithms... 48 GEOMETRY AND MENSURATION... 52 14. Geometry... 52 15. Mensuration... 59 16. Coordinate Geometry... 63 MODERN MATHS... 67 17. Permutations and Combinations... 67 18. Probability... 70 19. Set Theory... 73 2
NUMBERS Number System Number System covers various types of numbers viz. natural numbers, whole numbers, integers, rational and irrational numbers which constitutes the Real number system. Various Types of Numbers: Natural Numbers: All the positive numbers 1, 2, 3,.., etc. that are used in counting are called Natural numbers. Types of Natural Numbers based on divisibility: Prime Number: A natural number larger than unity is a prime number, if it does not have other divisors except for itself and unity. Examples for Prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, etc. Note: Unity, i.e. 1 is not prime number. Composite Numbers: Any number other than 1, which is not a prime number, is called a composite number. Examples for composite numbers are 4, 6, 8, 9, 10, 14, 15, etc. Whole Numbers: All counting numbers and 0, form the set of whole numbers. For example: 0, 1, 2, 3, etc. are whole numbers. Integers: All counting numbers, Zero and negative of counting numbers, form the set of integers. Therefore, -3, -2, -1, 0, 1, 2, 3,.. are all integers. Rational Numbers A number which can be expressed in the form p/q, where p and q are integers and q 0, is called a rational number. For example 2 can be written as 2/1, therefore 2 is also a rational number. Irrational Numbers Numbers which are not rational but can be represented by points on the number line called irrational numbers. Examples for Irrational numbers are,,, etc. Even and Odd numbers Numbers divisible by 2 are called even numbers whereas numbers that are not divisible by 2 are called odd numbers. For example 2, 4, 6, etc are even numbers and 3, 5, 7, etc are odd numbers. The following rules related to Even and Odd numbers are important: Factors and Co-primes: A number can be written in its prime factorization format. For example 72 =. The number of factors of a number N = The sum of factors of a number can be written as 3
The number of co-primes of a number can be written as The sum of co-primes of a number The number of ways of writing a number N as a product of two co-prime numbers = where n=the number of prime factors of a number. Product of all the factors of LCM or Least Common Multiple: For two numbers, HCF x LCM = product of the two. Factorization Method for LCM and HCF LCM: Here we can write all the given numbers in their prime factorization format. For example: Now take all primes number the given numbers and write their maximum powers. So LCM of 15, 18, 24 = HCF: Now, HCF of 12 and 18 = Tests of Divisibility of Numbers: Divisible by 2: if its unit digit of any of 0, 2, 4, 6, 8. Divisibility by 3: When the sum of its digits is divisible by 3. Divisibility by 9: When the sum of its digits is divisible by 9. Divisibility by 4: if the sum of its last two digits is divisible by 4. Divisibility by 8: If the number formed by hundred s ten s and unit s digit of the given number is divisible by 8. Divisibility by 10: When its unit digit is Zero. Divisibility by 5: When its unit digit is Zero or five. Divisibility by 11: if the difference between the sum of its digits at odd places and the sum of its digits at even places is either o0 or a number divisible by 11. Algebraic Formulae (a + b) 2 = a 2 + 2ab+ b 2 (a - b) 2 = a 2-2ab+ b 2 (a + b) 3 = a 3 + b 3 + 3ab(a + b) (a - b) 3 = a 3 - b 3-3ab(a - b) (a + b + c) 2 = a 2 + b2 + c 2 +2ab+2bc +2ca (a + b + c) 3 = a 3 + b 3 + c 3 + 3a 2 b + 3a 2 c + 3b 2 c + 3b 2 a + 3c 2 a + 3c 2 b + 6abc a 2 - b 2 = (a + b)(a b) 4
a 3 b 3 = (a b) (a 2 + ab + b 2 ) a 3 + b 3 = (a + b) (a 2 - ab + b 2 ) (a + b) 2 + (a - b) 2 = 4ab (a + b) 2 - (a - b) 2 = 2(a 2 + b 2 ) If a + b +c =0, then a 3 + b 3 + c 3 = 3abc Laws of indices a a a m n m n ( a ) 0 1 1 2 m n mn m n m n 1 a a a a a a 1 m 1 and a a a p 1 q p q p 1 q q 1 q a a and a a a ( a ) a p a ( a ) ( a) q q p q p m Binomial theorem Expansion is, n C 0 n C 1 n C 2 n C r n C n-1 n C n Middle term of Binomial theorem: If n is an odd number in the binomial theorem, then the middle terms are If n is an even number in the binomial theorem, then the middle term is n C 0 + n C 1 + n C 2 +.+ n C n = n C 1 + n C 3 + n C 5 +. = n C 0 + n C 2 + n C 4 +.= Some basic values using Binomial theorem: term. terms. Previous year questions: Number System is an important topic from MBA Entrance exam perspective as the quantitative aptitude section covers around a quarter of the total number of questions from this chapter. Here are some previous year questions which came in CAT Examination: 5
1. How many integers, greater than 999 but not greater than 4000, can be formed with the digits 0, 1, 2, 3 and 4, if repetition of digits is allowed? (CAT 2008 Exam) 1) 499 2) 500 3) 375 4) 376 5) 501 Answer: 4) Explanation: The minimum number that can be formed is 1000 and the maximum number that can be formed is 4000. As 4000 is the only number in which the first digit is 4. First, let us calculate the numbers less than 4000 and then we will add 1 to it. First digit can be 1, 2 or 3. Remaining 3 digits can be any of the 5 digits. Total numbers that can be formed, which are less than 4000 = 3 5 5 5 = 375 Total numbers that satisfy the given condition 375 + 1 = 376 2. A confused bank teller transposed the rupees and paise when he cashed a cheque for Shailaja, giving her rupees instead of paise and paise instead of rupees. After buying a toffee for 50 paise, Shailaja noticed that she was left with exactly three times as much as the amount on the cheque. Which of the following is a valid statement about the cheque amount? 1) Over Rupees 13 but less than Rupees 14 2) Over Rupees 7 but less than Rupees 8 3) Over Rupees 22 but less than Rupees 23 4) Over Rupees 18 but less than Rupees 19 5) Over Rupees 4 but less than Rupees 5 Answer: 4) Explanation: (CAT 2007 Exam) Suppose the cheque for Shailaja is of Rs. X and Y paise As per the question: 299 X 50 97 Now the value of Y should be an integer. Checking by options only for, Y is an integer and the value of. 3. A group of 630 children is arranged in rows for a group photograph session. Each row contains three fewer children than the row in front of it. What number of rows is not possible? (CAT 2006 Exam) 1) 3 2) 4 3) 5 4) 6 5) 7 Answer: 4) Explanation: Let there be n rows and a students in the first row. Number of students in the second row = a + 3 Number of students in the third row = a + 6 and so on. The number of students in each row forms an arithmetic progression with common difference = 3. 6
The total number of students = The sum of all terms in the arithmetic progression = = 630 Now consider options. n=3 a = 207 n=4 = 630 = 630 a = 153 Similarly, when n = 5, a= 120. As n= 6,, n = 7, is an integer, only n = 6 is not possible. 7
Sequence, Series and Progression Sequence: Sequences can be defined as a logically ordered list of elements related to each other through some relationship. Series: A series is defined as a sum of a sequence of terms. Some basic types of sequences: Sum or Difference Type of Sequence In this type of sequence, the next number is determined by adding or subtracting some quantity to the previous term. For example, find the 7th term of the sequence 1, 2, 4, 7, 11, 16.. Solution: 1 st term is 1; 2nd term is 1 + 1 = 2; 3 rd term is 2 + 2 = 4; 4 th term is 4 + 3 = 7; 5 th term is 7 + 4 = 11; 6 th term is 11 + 5 = 16 This sequence can be expressed as next term = previous term + position of the previous term. Therefore, the 7 th term of this sequence would be 16 + 6 = 22 Cumulative Sequence Consider the sequence 1, 2, 3, 5, 8, 13, 21.. Analyzing the above sequence would reveal that after the second term, the next term is the sum of the previous two terms. Hence, the sequence is broken up in the following manner. 1 st term is 1; 2 nd term is 2; 3 rd term is 1 + 2 = 3; 4 th term is 3 + 2 = 5; 5 th term is 5 + 3 = 8; 6 th term is 8 + 5 = 13; 7 th term is 13 + 8 = 21 As can be seen from above, the next term is the sum of the previous two terms. Hence, 8 th term is 21 + 13 = 34 In these types of sequences the pattern is formed with the help of its previous terms. Arithmetic Progression (AP) A succession of numbers is said to be in Arithmetic Progression if the difference between any term and the previous term is constant throughout. In other words, the difference between any of the two consecutive terms should be the same. If the first term of an AP is a and the common difference is d, then the progression could take either of the forms: Or For example: 3, 7, 11, 15 where a = 3 and d = 4 8, 2, 4, 10 where a = 8 and d = 6 Thus for any arithmetic progression beginning with a and having a common difference d, its nth term is determined by For an Arithmetic Progression, the sum of the first n terms is given by 8
Note: When three terms are in Arithmetic Progression, the middle term is the arithmetic mean of the other two. It is always convenient to take three terms in an AP as a d, a and a + d. Similarly four terms in AP could be taken as a 3d, a d, a + d and a + 3d. Geometric Progression (GP) The terms in a progression are said to be in Geometric Progression when they increase or decrease by a constant multiplying factor. The constant factor is called the common ratio and it is found by dividing any term to the preceding term. If the first term is a and the common ratio is r, and then the progression takes the form: The nth term of the geometric progression is given by For a geometric progression, the sum of the first n terms is given by For r = 1, the sum of n terms is indeterminate. Harmonic Progression (HP) The progression a 1, a 2, a 3 is called an HP if,,...is an HP. If a, b, c are in HP, then b is the harmonic mean between a and c. In this case, Previous year questions: Sequence, Series and Progression are interrelated concepts and they are also one of the most frequently occurring topics in Quantitative Aptitude section of CAT Examination. Here are few examples: 1. The number of common terms in the two sequences 17, 21, 25,..., 417 and 16, 21, 26, 466 is (CAT 2008 Exam) 1) 78 2) 19 3) 20 4) 77 5) 22 Answer: 3) Explanation: The first sequence can be written as 17, 17 + 4, 17 + 8,.. 417 and second sequence can be written as 16, 16 + 5, 16+10,..., 466 The common difference for the first sequence is 4 and that for the second sequence is 5 and both the sequences have 21 as the first common term. 9
Common terms are 21, 21 + L, 21 + 2L, [Here, L = LCM of 4 and S = 20] Common terms are 21, 21 + 20, 21 + 40, The common terms have a common difference of 20 and first term as 21. 417-21 = 396 and 396/20 = 19.8, 19 terms are common, other than 21. The total number of terms which are common to both the sequences = 19 + 1 = 20 2. Find the sum: (CAT 2008 Exam) 1) 2) 3) 4) 5) Answer: 1) Explanation: Consider only 1 st term, Now consider first two terms, Similarly, 3. Consider the set S = {1, 2, 3,..., 1000). How many arithmetic progressions can be formed from the elements of S that start with 1 and end with 1000 and have at least 3 elements? (CAT 2006 Exam) 1) 3 2) 4 3) 6 4) 7 5) 8 Answer: 4) Explanation: Let there be terms in the arithmetic progression having 1 as the first term and 1000 as the last. Let be the common difference. Then,.. (i) Factors of 999 are 1, 3, 9, 27, 37, 111, 333 and 999 Substituting in equation (i) 10
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