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CAT QUANT FORMULAS By 4 Time CAT 100%iler  Page Topic No Ratio And Proportion Formulas 1 Mixtures And Alligations Formulas 13 Profit And Loss, Discount Formulas...

CAT QUANT FORMULAS By 4 Time CAT 100%iler  Page Topic No Ratio And Proportion Formulas 1 Mixtures And Alligations Formulas 13 Profit And Loss, Discount Formulas 21 Interests (Simple Interest & Compound 28 Interest) Formulas Time, Speed, Distance &Work Formulas 36 Linear Equations Formulas 54 Quadratic Equations Formulas 62 Inequalities Formulas 71 Progressions and Series Formulas 77 Logarithms, Surds and Indices Formulas 89 Geometry Formulas 97 Set Theory And Venn Diagrams Formulas 136 Number System Formulas 145 Remainder Theorems Formulas 161 Permutations And Combinations Formulas 178 Bayes Theorem 185 (Conditional Probability) Formulas ALSO BUY FORMULAS BOOK BUY AT: CRACKU.IN/STORE ALSO BUY PREVIOUS PAPERS BOOK BUY AT: CRACKU.IN/STORE 1  CAT Ratio & Proportion Formulas Ratio and Proportions is one of the easiest concepts in CAT. Questions from this concept are mostly asked in conjunction with other concepts like similar triangles, mixtures and alligations. Hence fundamentals of this concept are important not just from a stand-alone perspective, but also to answer questions from other concepts A ratio can be represented as fraction a/b or using the notation a:b. In each of these representations ′a′ is called the antecedent and ′b′ is called the consequent. For a ratio to be defined, the quantities of the items should be of the same nature. We can not compare the length of the rod to the area of a square. Create Create Solved CAT Previous Papers PDF 2  However if these quantities are represented in numbers, i.e., length of a rod is ‘a’ cm and area of a square is ‘b’ sq.km, we can still define the ratio of these numbers as a:b The Ratio of the number a to the number b (b≠ 0) is 𝑎 also expressed as 𝑏 Example: As mentioned, a Ratio can be expressed or represented in a variety of ways. For instance, the 2 ratio of 2 to 3 can be expressed as 2:3 or 3 The order in which the terms of a ratio are written is important. For example, The ratio of the number of months having precisely 30 days to the number of 4 7 months with exactly 31 days, is , not 7 4 Take Free CAT Mock Tests 3 Properties of Ratios: It is not necessary for a ratio to be positive. When dealing with quantities of objects, however, the ratios will be positive. Only positive ratios will be considered in this notion. A ratio remains the same if both antecedent and consequent are multiplied or divided by the same non-zero number, i.e., 𝑎 𝑝𝑎 𝑞𝑎 𝑏 = 𝑝𝑏 = 𝑞𝑏 , p,q≠ 0 𝑎 𝑎/𝑝 𝑎/𝑞 𝑏 = 𝑏/𝑝 = 𝑏/𝑞 , p,q≠ 0 Create Create Solved CAT Previous Papers PDF 4  Two ratios in fraction notation can be compared in the same way that actual numbers can. 𝑎 𝑝 𝑏 = 𝑞 ⇔ 𝑎𝑞 = 𝑏𝑝 𝑎 𝑝 𝑏 > 𝑞 ⇔ 𝑎𝑞 > 𝑏𝑝 𝑎 𝑝 𝑏 < 𝑞 ⇔ 𝑎𝑞 < 𝑏𝑝 If antecedent > consequent, the ratio is said to be the ratio of greater inequality. If antecedent < consequent, the ratio is said to be the ratio of lesser inequality. Take Free CAT Mock Tests 5  If the antecedent = consequent, the ratio is said to be the ratio of equality. If a, b, x are positive, then 𝑎+𝑥 𝑎 If a > b, then < 𝑏+𝑥 𝑏 𝑎+𝑥 𝑎 If a < b, then > 𝑏+𝑥 𝑏 𝑎−𝑥 𝑎 If a > b, then > 𝑏−𝑥 𝑏 𝑎−𝑥 𝑎 If a < b, then < 𝑏−𝑥 𝑏 𝑎 𝑏 𝑐 𝑑 If 𝑝 = 𝑞 = 𝑟 = 𝑠 =... , then a:b:c:d:... = p:q:r:s:... Create Create Solved CAT Previous Papers PDF 6  𝑎 𝑐 If two ratios 𝑏 and 𝑑 are equal 𝑎 𝑐 𝑏 𝑑 𝑏 = 𝑑 ⟹ 𝑎 = 𝑐 (Invertendo) 𝑎 𝑐 𝑐 𝑏 𝑏 = 𝑑 ⟹ 𝑎 = 𝑑 (Alternendo) 𝑎 𝑐 𝑎+𝑏 𝑐+𝑑 𝑏 = 𝑑 ⟹ 𝑏 = 𝑑 (Componendo) 𝑎 𝑐 𝑎−𝑏 𝑐−𝑑 𝑏 = 𝑑 ⟹ 𝑏 = 𝑑 (Dividendo) 𝑎 𝑐 𝑎+𝑏 𝑏 = 𝑑 ⟹ 𝑎−𝑏 = (Componendo-Dividendo) 𝑎 𝑐 𝑝𝑎+𝑞𝑏 𝑝𝑐+𝑞𝑑 𝑏 = 𝑑 ⟹ 𝑟𝑎+𝑠𝑏 = 𝑟𝑐+𝑠𝑑 , for all real p, q, r, s such that pa+qb≠0 and rc+sd≠0 Take Free CAT Mock Tests 7  If a, b, c, d, e, f, p, q, r are constants and are not equal to zero 𝑎 𝑐 𝑒 ➔ = = = … then each of these ratios is 𝑏 𝑑 𝑓 𝑎+𝑐+𝑒... equal to 𝑏+𝑑+𝑓... 𝑎 𝑐 𝑒 ➔ = = =… then each of these ratios is 𝑏 𝑑 𝑓 𝑝𝑎+𝑞𝑐+𝑟𝑒... equal to 𝑝𝑏+𝑞𝑑+𝑟𝑓... 𝑎 𝑐 𝑒 ➔ = = =… then each of these ratios is 𝑏 𝑑 𝑓 1/𝑛 ( ) 𝑛 𝑛 𝑛 𝑝𝑎 +𝑞𝑐 +𝑟𝑒 +... equal to 𝑛 𝑛 𝑛 𝑝𝑏 +𝑞𝑑 +𝑟𝑓 +... 2 2 ➔ Duplicate Ratio of a : b is 𝑎 :𝑏 Create Create Solved CAT Previous Papers PDF 8 ➔ Sub-duplicate ratio of a : b is 𝑎 : 𝑏 3 3 ➔ Triplicate Ratio of a : b is 𝑎 : 𝑏 1/3 1/3 ➔ Sub-triplicate ratio of a : b is 𝑎 :𝑏 Proportions : A proportion is defined as an equalisation of ratios. As a result, if a:b = c:d is a ratio, the first and last terms are referred to as extremes, whereas the middle two phrases are referred to as means. When four terms a, b, c, and d are considered to be proportionate, a:b = c:d is the result. When three terms a, b, and c are considered to be proportionate, a:b = b:c is the result. Take Free CAT Mock Tests 9  A proportion is a statement that two ratios are equal; 2 8 for example 3 = 12 is a proportion. One way to solve a proportion involving an unknown is to cross multiply, obtaining a new equality. 2 𝑛 For example, to solve for n in the proportion 3 = 12 , cross multiply, obtaining 24=3n, then divide both sides by 3, to get n=8 Properties of proportions : If a:b = c:d is a proportion, then Product of extremes = product of means i.e., ad = bc Denominator addition/subtraction: a:a+b = c:c+d and a:a-b = c:c-d Create Create Solved CAT Previous Papers PDF 10  a, b, c, d,.... are in continued proportion means, a:b = b:c = c:d =.... 2 a:b = b:c then b is called mean proportional and 𝑏 = ac The third proportional of two numbers, a and b, is c, such that, a:b = b:c. ‘d’ is fourth proportional to numbers a, b, c if a:b = c:d Variations : If x varies directly to y, then x is said to be in directly proportional with y and is written as x ∝ y ➔ x = ky (where k is direct proportionality constant) ➔ x = ky + C (If x depends upon some other fixed constant C) Take Free CAT Mock Tests 11  If x varies inversely to y, then x is said to be in 1 inversely proportional with y and is written as 𝑥∝ 𝑦 1 ➔𝑥 = 𝑘 𝑦 (where k is indirect proportionality constant) 1 ➔𝑥 = 𝑘 +C 𝑦 (If x depends upon some other fixed constant C) If x ∝ y and y ∝ z then x ∝ z If x ∝ y and x ∝ z then x ∝ (y ± z) If a ∝ b and x ∝ y then ax ∝ by Free CAT Study Material CAT Previous papers (Download PDF) Join CAT Complete Course By IIM Alumni Create Create Solved CAT Previous Papers PDF 12  Our Results Students scored 99.9+ Percentile 45 in CAT 2023 Students scored 99.50+ Percentile 280 in CAT 2023 Students scored 99+ Percentile in 510 CAT 2023 Our Faculty Maruti Konduri Sayali Kale 4 Time CAT 100%iler CAT 99.97 %iler IIM Ahmedabad Alumnus IIM Ahmedabad Alumna +91-6303239042 cracku.in 14  CAT Mixtures And Alligations Formulas Mixtures and alligations are a common type of quantitative problem that may appear on the CAT. These problems involve mixing two or more substances to form a new mixture, and then finding the ratio or quantity of each substance in the mixture. Alligation is a specific method for solving mixture problems that involves representing the ingredients and the mixture as points on a line, and using the distance between these points to find the ratio of the ingredients in the mixture. There are many variations of mixture and alligation problems that may appear on the CAT, but they all Create Create Solved CAT Previous Papers PDF 15  involve some variation of this basic concept. To prepare for these types of problems, it is important to practise solving a variety of mixture and alligation problems, and to become familiar with the basic formulas and methods for solving them. Types of mixtures: Simple mixture: A simple mixture is formed by the mixture of two or more different substances. Example: Water and Wine mixture. Compound mixture: A Compound mixture is formed by the mixture of two or more simple mixtures. Example: one part of 'water and wine' mixture mixed with two parts of 'water and milk' mixture. Take Free CAT Mock Tests 16  If 𝑀1 and 𝑀2 are the values and 𝑄1 and 𝑄2 are the quantities of item 1 and item 2 respectively, and 𝑀𝐴 is the weighted average of the two items, then 𝑄1 𝑀2 − 𝑀𝐴 𝑄2 = 𝑀𝐴 − 𝑀1 Create Create Solved CAT Previous Papers PDF 17  Weighted average 𝑀𝐴 can be calculated by 𝑄1𝑀1+𝑄2𝑀𝐴 𝑀𝐴 = 𝑄1+𝑄2 The alligation rule can also be applied when cheaper substance is mixed with expensive substance 𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑜𝑓 𝑐ℎ𝑒𝑎𝑝𝑒𝑟 𝑃𝑟𝑖𝑐𝑒 𝑜𝑓 𝑐ℎ𝑒𝑎𝑝𝑒𝑟 − 𝑀𝑒𝑎𝑛 𝑝𝑟𝑖𝑐𝑒 𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑜𝑓 𝑑𝑒𝑎𝑟𝑒𝑟 = 𝑀𝑒𝑎𝑛 𝑃𝑟𝑖𝑐𝑒 − 𝑃𝑟𝑖𝑐𝑒 𝑜𝑓 𝑐ℎ𝑒𝑎𝑝𝑒𝑟 If two mixtures 𝑀1 and 𝑀2 , having substances 𝑆1 and 𝑆2 in the ratio a:b and p:q respectively are mixed, then in the final mixture, 𝑎 𝑝 𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑜𝑓 𝑆1 𝑀1 ⎡ 𝑎+𝑏 ⎤ +𝑀2 ⎡ 𝑝+𝑞 ⎤ ⎣ ⎦ ⎣ ⎦ 𝑄𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑜𝑓 𝑆2 = 𝑏 𝑞 𝑀1 ⎡ 𝑎+𝑏 ⎤ +𝑀2 ⎡ 𝑝+𝑞 ⎤ ⎣ ⎦ ⎣ ⎦ Take Free CAT Mock Tests 18  If there is a container with ‘a’ litres of liquid A and if ‘b’ litres are withdrawn and an equal amount of the mixture is replaced with another liquid B and if this operation is repeated ‘n’ times, then after the nth operation, Liquid A in the container 𝑎−𝑏 𝑛 =⎡ ⎤ × Initial quantity of A in the container ⎣ 𝑎 ⎦ 𝑎−𝑏 𝑛 𝐿𝑖𝑞𝑢𝑖𝑑 𝐴 𝑎𝑓𝑡𝑒𝑟 𝑛𝑡ℎ 𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑜𝑛 ⎡ ⎤ ⎣ 𝑎 ⎦ 𝐿𝑖𝑞𝑢𝑖𝑑 𝐵 𝑎𝑓𝑡𝑒𝑟 𝑛𝑡ℎ 𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑜𝑛 = 𝑎−𝑏 𝑛 1 −⎡ ⎤ ⎣ 𝑎 ⎦ Free CAT Study Material CAT Previous papers (Download PDF) Create Create Solved CAT Previous Papers PDF 19  Why Cracku is the Ultimate Preparation Platform for CAT? Our Offerings Daily schedule by 60 MBA Mocks experts (CAT+OMET) 100 Concept 1000+ Videos Notes Weekly Live classes Doubt-solving by By IIMA Alumni experts 18,000+ Solved Access till Questions Jan 10th, 2025 Ready to ace CAT 2024? Enroll Now +91-6303239042 cracku.in 21  CAT Profit And Loss, Discount Formulas Profit, Loss and Discount is a very important topic for CAT and a significant number of questions are asked from this topic every year. The number of concepts in these topics is limited and most of the problems can be solved by applying the formulae directly. This document covers various formulas, tips and shortcuts of Profit, Loss and Discount topics. Create Create Solved CAT Previous Papers PDF 22 Profit and Loss Cost Price: The amount paid to purchase an article or the cost of manufacturing an article is called Cost Price (C.P) Selling Price: The price at which a product is sold is called Selling price (S.P) Marked Price: ➔ The price at which an article is marked is called Marked price (M.P) ➔ If S.P>C.P, then Profit or Gain, P = S.P – S.P ➔ If C.P>S.P, then Loss, L = C.P – S.P ➔ % Profit or Gain percentage or Profit Take Free CAT Mock Tests 23  𝑃𝑟𝑜𝑓𝑖𝑡 Percentage = 𝐶.𝑃 × 100 𝐿𝑜𝑠𝑠 %Loss = 𝐶.𝑃 × 100 ➔ Discount = M.P – S.P (If no discount is given, then M.P = S.P) 𝐷𝑖𝑠𝑐𝑜𝑢𝑛𝑡 ➔ %Discount = 𝑀.𝑃 × 100 ➔ Total increase in price due to two subsequent increases of X% and Y% is (𝑋 +𝑌+ 𝑋𝑌 100 )% ➔ If two items are sold at same price, each at Rs. x, one at a profit of P% and other at a loss of P% then 2 𝑃 there will be overall loss of 100 % Create Create Solved CAT Previous Papers PDF 24  2 2𝑃 𝑥 ➔ The absolute value of loss = 2 2 100 −𝑃 ➔ If C.P of two items is the same, and by selling each item he earned p% profit on one article and p% loss on another, then there will be no loss or gain. ➔ If a trader professes to sell at C.P but uses false weight, then 𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 Gain% = 𝑇𝑟𝑢𝑒 𝑊𝑒𝑖𝑔ℎ𝑡 × 100% Difference represents the difference in claimed weight and true weight ; claimed weight > true weight ➔ S.P =( 100 + 𝑃𝑟𝑜𝑓𝑖𝑡% 100 ) C.P (if S.P > C.P) S.P =( ) 100 + 𝐿𝑜𝑠𝑠% ➔ C.P (if S.P < C.P) 100 Take Free CAT Mock Tests 25  ➔ C.P =( 100 × 𝑆.𝑃 100 + 𝑃𝑟𝑜𝑓𝑖𝑡% ) C.P (if S.P > C.P) C.P =( )C.P (if S.P > C.P) 100 × 𝑆.𝑃 ➔ 100 + 𝑃𝑟𝑜𝑓𝑖𝑡% 𝑦 ➔ Buy x get y free, then the %discount = 𝑥+𝑦 × 100 (here x+y articles are sold at C.P of x articles.) ➔ When there are two successive discounts of a% and b% are given then the, Resultant discount = (𝑎+𝑏− 𝑎×𝑏 100 ) ➔ If C.P of x article is equal to the selling price of y articles then the, 𝑦−𝑥 Resultant profit % or loss % = 𝑦 × 100 Create Create Solved CAT Previous Papers PDF 26 Our Courses Exclusive Course Complete CAT Study Room Comprehensive Course + For CAT 2024 For CAT 2024 Daily Target Combo EMI Available Scholarship Available CAT Daily Target CAT Daily Target 20 CAT Mock + 3 Tests every weekday + CAT Mock Test with video solution 45 Sectional Tests Get upto 50% discount on the Complete Cracku package Apply for Scholarship +91-6303239042 cracku.in 28  CAT Simple Interest & Compound Interest Formulas Simple Interest (S.I) and Compound Interest (C.I) is one of the easiest topics in the CAT quant section. Every year, a significant number of questions appear from each of these sections and students should aim to get most questions right from these topics. The number of concepts that are tested from these topics is limited and most of the problems can be solved by applying the formulae directly. Many students commit silly mistakes in this topic due to complacency, which should be avoided. Create Create Solved CAT Previous Papers PDF 29  In Simple Interest, the principal and the interest calculated for a specific year or time interval remains constant. In Compound Interest, the interest earned over the period is added over to the existing principal after every compounding period and thus, the principal and the interest change after every compounding period. For the same principal, positive rate of interest and time period (>1 year), the compound interest on the loan is always greater than the simple interest. Simple Interest The sum of principal and the interest is called Amount. Amount (A) = Principal (P) + Interest (I) Take Free CAT Mock Tests 30  The Simple Interest (I) occurred over a time period (T) for R% (rate of interest per annum), 𝑃𝑇𝑅 I= 100 Compound Interest The amount to be paid, if money is borrowed at Compound Interest for N number of years, 𝑅 𝑁 A=P (1 + 100 ) The Interest occurred, I = A – P 𝑅 𝑁 I=P (1 + 100 ) -P If the interest is compounded half yearly, then 𝑅/2 2𝑁 Amount, A = P (1 + 100 ) Create Create Solved CAT Previous Papers PDF 31  If the interest is compounded quarterly, then 𝑅/4 4𝑁 Amount, A = P (1 + 100 ) If interest Rate is 𝑅1% for first year, 𝑅2% for 𝑟𝑑 second year and 𝑅3% for 3 year then the Amount, A=P (1 + ) ( 1 + ) ( 1 + ) 𝑅1 100 𝑅2 100 𝑅3 100 If a difference between C.I and S.I for certain sum at same rate of interest is given, th 2 𝑃𝑟𝑖𝑛𝑐𝑖𝑝𝑎𝑙 (𝑃) = (𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝐶𝐼 𝑎𝑛𝑑 𝑆𝐼) * (100/𝑅) When interest is compounded annually but time is in fraction, let a b c then the Amount, ( ) 𝑏 𝑎 ( ) 𝑐 𝑅 𝑅 A=P 1+ 100 1+ 100 Take Free CAT Mock Tests 32  Installments and Present Worth: If R is the rate of interest per annum, then the present worth of Rs. ‘K’ due N years hence is represented as 𝐾 Present worth = 𝑅 𝑁 (1+ ) 100 If an amount of ‘P’ is borrowed for ‘n’ years at ‘r’% per annum, and ‘x’ is the installment that is paid at the end of each year starting from the 𝑛 first year, then 𝑥= 𝑃 ( )( 𝑟 100 ) 𝑟 1+ 100 𝑛 (1+ ) −1 100 𝑟 Create Create Solved CAT Previous Papers PDF 33  Try our courses absolutely FREE! Approach to Reading Logical Reasoning: Puzzles: Comprehension Arrangements Einstein Puzzle Subject: CAT VARC Subject: CAT LRDI Subject: CAT VARC Data Interpretation Arithmetic: Arithmetic: Basics Ratio & Proportion Time & Work Subject: CAT LRDI Subject: CAT QA Subject: CAT QA Free CAT Mock in XAT : Free MBA Previous Year Latest Pattern Decision Making Papers Subject: CAT Mock Subject: XAT DM Subject: CAT Papers +91-6303239042 cracku.in 36  CAT Time, Speed, Distance & Work Formulas Time, Distance and Work is the most important topic for CAT Quant Section & all competitive exams. The questions from this topic vary from easy to difficult. This formula sheet covers the most importance tips that helps you to answer the questions in a easy, fast and accurate way 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 = 𝑆𝑝𝑒𝑒𝑑 × 𝑇𝑖𝑚𝑒 𝑆𝑝𝑒𝑒𝑑 = 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑇𝑖𝑚𝑒 | 𝑇𝑖𝑚𝑒 = 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑆𝑝𝑒𝑒𝑑 Create Create Solved CAT Previous Papers PDF 37  While converting the speed in m/s to km/hr, multiply it by ( ) ⇒ 1 m/s = 3.6 km/h 18 5 While converting km/hr into m/sec, we multiply by ( ) 5 18 If the ratio of the speeds of A and B is a : b, then ➔ The ratio of the times taken to cover the same distance is 1/a : 1/b or b : a. ➔ The ratio of distance travelled in equal time intervals is a : b , 𝑇𝑜𝑡𝑎𝑙 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑇𝑟𝑎𝑣𝑒𝑙𝑙𝑒𝑑 Average Speed = 𝑇𝑜𝑡𝑎𝑙 𝑇𝑖𝑚𝑒 𝑇𝑎𝑘𝑒𝑛 ➔ If a part of a journey is travelled at speed 𝑆1 𝑘𝑚/ℎ𝑟 in 𝑇 ℎ𝑜𝑢𝑟𝑠 and remaining part at speed 𝑆2 𝑘𝑚/ℎ𝑟 1 in 𝑇 ℎ𝑜𝑢𝑟𝑠 then, 2 Take Free CAT Mock Tests 38  Total distance travelled = 𝑆 𝑇 + 𝑆 𝑇 𝑘𝑚 1 1 2 2 𝑆1𝑇1+𝑆2𝑇2 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑆𝑝𝑒𝑒𝑑 = 𝑇1+𝑇2 𝑘𝑚/ℎ𝑟 If 𝐷1 𝑘𝑚 is travelled at speed 𝑆1 𝑘𝑚/ℎ𝑟, and 𝐷2 𝑘𝑚 is travelled at speed 𝑆 𝑘𝑚/ℎ𝑟 then, 2 𝐷1+𝐷2 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑆𝑝𝑒𝑒𝑑 = 𝐷1 𝐷2 𝑘𝑚/ℎ𝑟 𝑆1 + 𝑆2 In a journey travelled with different speeds, if the distance covered in each stage is constant, the average speed is the harmonic mean of the different speeds. Suppose a man covers a certain distance at x km/hr and a equal distance at y km/hr Create Create Solved CAT Previous Papers PDF 39  Then the average speed during the whole journey is 2𝑥𝑦 𝑥+𝑦 𝑘𝑚/ℎ𝑟 In a journey travelled with different speeds, if the time travelled in each stage is constant, the average speed is the arithmetic mean of the different speeds. If a man travelled for a certain distance at x km/hr and for equal amount of time at the speed of y km/hr then the average speed during the whole journey is 𝑥+𝑦 2 𝑘𝑚/ℎ𝑟 Constant Distance Let the distance travelled in each part of the journey be 𝑑1, 𝑑2 & 𝑑3 and so on till 𝑑𝑛 and the speeds in each part be 𝑠1, 𝑠2, 𝑠3 and so on till 𝑠𝑛 Take Free CAT Mock Tests 40 If 𝑑1 = 𝑑2 = 𝑑3 =......... = 𝑑𝑛 = 𝑑, then the average speed is the harmonic mean of the speeds 𝑠1, 𝑠2, 𝑠3 and so on till 𝑠. 𝑛 Constant Time Let the distance travelled in each part of the journey be 𝑑1, 𝑑2 𝑎𝑛𝑑 𝑑3 and so on till 𝑑𝑛 and the speeds in each part be 𝑡 , 𝑡2, 𝑡3 and so on till 𝑡𝑛. 1 If 𝑡1 = 𝑡2 = 𝑡3 =......... = 𝑡𝑛 = 𝑡, then the average speed is the harmonic mean of the speeds 𝑠1, 𝑠2, 𝑠3 and so on till 𝑠. 𝑛 Create Create Solved CAT Previous Papers PDF 41  Clocks ➔ Calculating the angle/position of the hands 0 Speed of hour hand = 0. 5 per minute 0 Speed of minute hand = 6 per minute 0 Relative speed of two hands = 5. 5 per minute The angle (in degrees) between hour hand and minute hand at time H:M can be represented 11 0 as: θ = || 2 𝑀 − 30𝐻|| ➔ In a 12 hour period: The hour hand and the minute hand meet 11 times 0 A 180 angle is formed between the two hands 11 times 0 A 90 angle is formed between the two hands 22 times Take Free CAT Mock Tests 42 ➔ In a well functioning clock, both the hands meet 720 after every Mins. 11 ➔ It is because the relative speed of the minute hand 11 with respect to the hour hand = degrees per 2 minute. Erroneous Clocks ➔ An erroneous clock is a clock which loses or gains time at a constant rate. ➔ In case of an erroneous clock losing/gaining ‘x’ sec per minute, It will lose/gain ‘x’ minutes per hour. Create Create Solved CAT Previous Papers PDF 43  It will show the correct time after every ‘720/x’ hours. The clock will show the same time again after ‘y’ hours where: 720 y= 60+𝑥 if the clock gains time. 720 y= 60−𝑥 if the clock loses time.’ If the clock is set right at ‘Q’ AM/PM. Then the time ‘T’ shown by the clock after ‘h’ hours pass on a correct clock would be: T = 𝑄 + (ℎ × 𝑥) if the clock gains time. T = 𝑄 − (ℎ × 𝑥) if the clock loses time.’ Take Free CAT Mock Tests 44  If the clock is set right at ‘Q’ AM/PM. If ‘h’ hours pass on the erroneous clock, then the actual time ‘T’ shown by T=Q+y; y(in hours) 60ℎ = 60+𝑥 𝑖𝑓 𝑡ℎ𝑒 𝑐𝑙𝑜𝑐𝑘 𝑔𝑎𝑖𝑛𝑠 𝑡𝑖𝑚𝑒 60ℎ = 60−𝑥 𝑖𝑓 𝑡ℎ𝑒 𝑐𝑙𝑜𝑐𝑘 𝑙𝑜𝑠𝑒𝑠 𝑡𝑖𝑚𝑒. Circular Tracks If two people are running on a circular track with speeds in ratio a:b where a and b are co-prime, then ➔ They will meet at 𝑎 + 𝑏 distinct points if they are running in opposite directions. Create Create Solved CAT Previous Papers PDF 45 ➔ They will meet at |𝑎 − 𝑏| distinct points if they are running in same direction If two people are running on a circular track having perimeter ‘l’, with speeds ‘m’ and ‘n’, 𝐼 ➔ The time for their first meeting = (𝑚 +𝑛) (when they are running in opposite directions) 𝐼 ➔ The time for their first meeting = |(𝑚 −𝑛)| (when they are running in the same direction) If a person P starts from A and heads towards B and another person Q starts from B and heads towards A and they meet after a time 't' then, t= (𝑥 × 𝑦) where x = time taken (after meeting) by P to reach B and y = time taken (after meeting) by Q to reach A. Take Free CAT Mock Tests 46  A and B started at a time towards each other. After crossing each other, they took 𝑇 ℎ𝑟𝑠, 𝑇2ℎ𝑟𝑠 1 respectively to reach their destinations. If they travel at constant speed 𝑆1 and 𝑆2 respectively all over the 𝑆1 𝑇2 journey, then 𝑆2 = 𝑇1 Trains ⇒ Two trains of length 𝐿1 and 𝐿2 travelling at speed 𝑆1 and 𝑆2 cross each other in a time 𝐿1 + 𝐿2 =𝑆 + 𝑆2 (If they are going in opposite directions) 1 𝐿1 + 𝐿2 = (If they are going in the same directions) |𝑆1 − 𝑆2| Create Create Solved CAT Previous Papers PDF 47  Time & Work ⇒ If X can do a work in ‘n’ days, the fraction of work 1 X does in a day is 𝑛 ⇒If X can do work in ‘x’ days, and Y can do work in ‘y’ days, then the number of days taken by both of them 𝑥×𝑦 together is 𝑥+𝑦 ⇒ If 𝑀1 men work for 𝐻1 ℎ𝑜𝑢𝑟𝑠 per day and worked for 𝐷1days and completed 𝑊1 work, and if 𝑀2 men work for 𝐻2 hours per day and worked for 𝐷2 days and completed 𝑊2 work, then 𝑀1𝐻1𝐷1 𝑀2𝐻2𝐷2 𝑊1 = 𝑊2 Take Free CAT Mock Tests 48  Boats & Streams ⇒ If the speed of water is 'W' and speed of a boat in still water is ‘B’ ⟹ Speed of the boat (downstream) is B+W ⟹ Speed of the boat (upstream) is B-W The direction along the stream is called downstream. And, the direction against the stream is called upstream. ⇒ If the speed of the boat downstream is x km/hr and the speed of the boat upstream is y km/hr, then 𝑥+𝑦 ⟹ Speed of the boat in still water = km/hr 2 𝑥−𝑦 ⟹ Rate of stream = 2 𝑘𝑚/ℎ𝑟 Create Create Solved CAT Previous Papers PDF 49 ⟹ While converting the speed in m/s to km/hr, multiply it by ( ) ⇒ 1 m/s = 3.6 km/h 18 5 ⟹ While converting km/hr into m/sec, we multiply by ( ) 5 18 Pipes & Cisterns ⇒ Inlet Pipe : A pipe which is used to fill the tank is known as Inlet Pipe. ⇒ Outlet Pipe : A pipe which can empty the tank is known as outlet pipe. If a pipe can fill a tank in ‘x’ hours then the part 1 filled per hour = 𝑥 Take Free CAT Mock Tests 50  If a pipe can empty a tank in ‘y’ hours, then the 1 part emptied per hour = 𝑦 If a pipe A can fill a tank in ‘x’ hours and pipe can empty a tank in ‘y’ hours, if they are both active at the same time, then 1 1 The part filled per hour = 𝑥 − 𝑦 (𝐼𝑓 𝑦 > 𝑥) 1 1 The part emptied per hour = 𝑦 − 𝑥 (𝐼𝑓 𝑥 > 𝑦) Some Tips and Tricks Some of the questions may consume a lot of time. While solving, write down the equations without any eros once you fully understand the given problem. Create Create Solved CAT Previous Papers PDF 51  The few extra seconds can help you avoid silly mistakes. Check if the units of distance, speed and time match up. If you see yourself adding a unit of distance like m to a unit of speed m/s, you would realise you have possibly missed a term. Choose to apply the concept of relative speed wherever possible since it can greatly reduce the complexity of the problem. In time and work, while working with equations, ensure that you convert all terms to consistent units like man-hours. Free CAT Study Material CAT Previous papers (Download PDF) Take Free CAT Mock Tests 52 Success Stories Deveswar Mandava CAT 2023 - 99.92%ile The extensive question bank based on difficulty level helped me practice a lot based on the areas where I was struggling and what level of questions were those. Priyadarshini Das CAT 2023 - 99.91%ile Cracku helped me to maintain regularity with their daily tests. The materials and sectional tests helped to brush up my concepts. The mock tests are also at par with the level of questions in the actual exam and solving and analysing these mock tests helped to improve my score. Rishab Ram CAT 2023 - 99.89%ile Cracku had some of the best test material and the dash cat test series was the most accurate to the actual cat and prepared me in the best way, and not to mention the daily target question quality was really very good and it helped me keep up on a daily basis. Trisha Awari CAT 2023 - 99.33%ile I am very grateful for Sayali ma'am and Maruti sir's course videos and livestreams; they helped me strengthen my concepts and build confidence. Dashcats, daily targets and the extensive study material ensure that I was prepared for whatever the exam threw at me. Read more success stories here +91-6303239042 cracku.in 54  Linear Equations Linear equations is one of the foundation topics in the Quant section on the CAT. Hence, concepts from this topic are useful in solving questions from a range of different topics. A linear equation is an equation which gives a straight line when plotted on a graph. Linear equations can be of one variable or two variable or three variable. Generally, the number of equations needed to solve the given problem is equal to the number of variables Let a, b, c and d are constants and x, y and z are variables. A general form of single variable linear equation is a𝑥 + b = 0. A general form of two variable linear equations is a𝑥+b𝑦 = c. Create Create Solved CAT Previous Papers PDF 55  A general form of three variable linear equations is a𝑥+b𝑦+c𝑧 = d. Equations with two variables: ➔ Consider two equations a𝑥+b𝑦 = c and m𝑥+n𝑦 = p. Each of these equations represent two lines on the x-y coordinate plane. The solution of these equations is the point of intersection. 𝑎 𝑏 𝑐 ➔ If 𝑚 = 𝑛 ≠ 𝑝 then the slope of the two equations is equal and so they are parallel to each other. Hence, no point of intersection occurs. Therefore no solution. 𝑎 𝑏 ➔ If 𝑚 ≠ 𝑛 then the slope is different and so they intersect each other at a single point. Hence, it has a single solution. Take Free CAT Mock Tests 56  𝑎 𝑏 𝑐 ➔ If 𝑚 = 𝑛 = 𝑝 then the two lines are the same and they have infinite points common to each other. So, infinite solutions occur. General Procedure to solve linear equations: ➔ Aggregate the constant terms and variable terms ➔ For equations with more than one variable, eliminate variables by substituting equations in their place. ➔ Hence, for two equations with two variables x and y, express y in terms of x and substitute this in the other equation. ➔ For Example: let x+y = 14 and x+4y = 26 then x = 14-y (from equation 1) substituting this in equation 2, we get 14-y+4y = 26. Hence, y = 4 and x = 10. Create Create Solved CAT Previous Papers PDF 57 ➔ For equations of the form 𝑎𝑥 + 𝑏𝑦 = 𝑐 and 𝑚𝑥 + 𝑛𝑦 = 𝑝, find the LCM of b and n. Multiply each equation with a constant to make the y term coefficient equal to the LCM. Then subtract equation 2 from equation 1. ➔ Example: Let 2x+3y = 13 and 3x+4y = 18 are the given equations (1) and (2). LCM of 3 and 4 is 12. Multiplying (1) by 4 and (2) by 3, we get 8x+12y = 52 and 9x+12y = 54. (2) - (1) gives x=2, y=3 ➔ If the system of equations has n variables with n-1 equations then the solution is indeterminate. ➔ If system of equations has n variables with n-1 equations with some additional conditions Take Free CAT Mock Tests 58  (for eg. the variables are integers), then the solution may be determinate. ➔ If a system of equations has n variables with n-1 equations then some combination of variables may be determinable. ➔ For example, if 𝑎𝑥 + 𝑏𝑦 + 𝑐𝑧 = 𝑑 and 𝑚𝑥 + 𝑛𝑦 + 𝑝𝑧 = 𝑞, if a, b, c are in Arithmetic progression and m, n and p are in AP then the sum x+y+z is determinable. ➔ Equations with three variables: Let the equations be 𝑎1𝑥 + 𝑏1𝑦 + 𝑐1𝑧 = 𝑑1, 𝑎2𝑥 + 𝑏2𝑦 + 𝑐2𝑧 = 𝑑2 and 𝑎3𝑥 + 𝑏3𝑦 + 𝑐3𝑧 = 𝑑3. ➔ Here we define the following matrices. Create Create Solved CAT Previous Papers PDF 59 ➔ If Determinant of D ≠ 0, then the equations have a unique solution. ➔ If Determinant of D = 0, and at least one but not all of the determinants Dx, Dy or Dz is zero, then no solution exists. ➔ If Determinant of 𝐷 = 0, and all the three of the determinants Dx, Dy and Dz are zero, then there are infinitely many solutions. ➔ Determinant can be calculated by D = 𝑎 (𝑏 𝑐 − 𝑐 𝑏 ) − 𝑏 (𝑎 𝑐 − 𝑐 𝑎 ) + 𝑐 (𝑎 𝑏 − 𝑏 𝑎 ) 1 2 3 2 3 1 2 3 2 3 1 2 3 2 3 Free CAT Study Material Take Free CAT Mock Tests 60  Our Results Students scored 99.9+ Percentile 45 in CAT 2023 Students scored 99.50+ Percentile 280 in CAT 2023 Students scored 99+ Percentile in 510 CAT 2023 Our Faculty Maruti Konduri Sayali Kale 4 Time CAT 100%iler CAT 99.97 %iler IIM Ahmedabad Alumnus IIM Ahmedabad Alumna +91-6303239042 cracku.in 62  Quadratic Equations Quadratic Equations is also an important topic For CAT Exam. The theory involved in this topic is very simple and students should be comfortable with some basic formulas and concepts. The techniques like option elimination, value assumption can help to solve questions from this topic quickly. This pdf covers all the important formulas and concepts related to Quadratic Equations. General Quadratic equation will be in the form of 2 𝑎𝑥 + 𝑏𝑥 + 𝑐 = 0 Create Create Solved CAT Previous Papers PDF 63  The values of ‘x’ satisfying the equation are called roots of the equation. 2 −𝑏 ± 𝑏 −4𝑎𝑐 The value of roots, p and q = 2𝑎 The above formula is known as the Shreedhara Acharya's Formula, after the ancient Indian Mathematician who derived it. −𝑏 Sum of the roots = p+q = 𝑎 𝑐 Product of the roots = p × q = 𝑎 If 'c' and 'a' are equal then the roots are reciprocal to each other. Take Free CAT Mock Tests 64  If b = 0, then the roots are equal and are opposite in sign. 2 ➔ Let D denote the discriminant, D = 𝑏 − 4𝑎𝑐. Depending on the sign and value of D, nature of the roots would be as follows: D < 0 and |D| is not a perfect square: Roots will be in the form of p+iq and p-iq where p and q are the real and imaginary parts of the complex roots. p is rational and q is irrational. D < 0 and |D| is a perfect square: Roots will be in the form of p+iq and p-iq where p and q are both rational. Create Create Solved CAT Previous Papers PDF 65  −𝑏 D=0 ⇒ Roots are real and equal. X = 2𝑎 D > 0 and D is not a perfect square: Roots are conjugate surds D > 0 and D is a perfect square: Roots are real, rational and unequal ➔ Signs of the roots: Let P be product of roots and S be their sum P > 0, S > 0 : Both roots are positive P > 0, S < 0 : Both roots are negative P < 0, S > 0 : Numerical smaller root is negative and the other root is positive P < 0, S < 0 : Numerical larger root is negative and the other root is positive Take Free CAT Mock Tests 66  2 Minimum and maximum values of 𝑎𝑥 + 𝑏𝑥 + 𝑐 = 0 2 4𝑎𝑐−𝑏 If a > 0: minimum value = and occurs at 4𝑎 −𝑏 𝑥= 2𝑎 2 4𝑎𝑐−𝑏 If a < 0: maximum value = and occurs at 4𝑎 −𝑏 𝑥= 2𝑎 𝑛 𝑛−1 ➔ If 𝐴𝑛𝑋 + 𝐴𝑛−1𝑋 +.... + 𝐴1𝑋 + 𝐴0, then −𝐴𝑛−1 Sum of the roots = 𝐴𝑛 𝐴𝑛−2 Sum of roots taken two at a time = 𝐴 𝑛 Create Create Solved CAT Previous Papers PDF 67  −𝐴𝑛−3 Sum of roots taken three at a time = and so on 𝐴𝑛 𝑛 [(−1) 𝐴0] Product of the roots = 𝐴𝑛 ➔ Finding a quadratic equation: If roots are given: 2 (𝑥 − 𝑎)(𝑥 − 𝑏) = 0 ⇒ 𝑥 − (𝑎 + 𝑏)𝑥 + 𝑎𝑏 = 0 If sum s and product p of roots are given: 2 𝑥 − 𝑠𝑥 + 𝑝 = 0 If roots are reciprocals of roots of equation 2 𝑎𝑥 + 𝑏𝑥 + 𝑐 = 0, then equation is 2 𝑐𝑥 + 𝑏𝑥 + 𝑎 = 0 Take Free CAT Mock Tests 68  If roots are k more than roots of 2 𝑎𝑥 + 𝑏𝑥 + 𝑐 = 0 then equation is 2 𝑎(𝑥 − 𝑘) + 𝑏(𝑥 − 𝑘) + 𝑐 = 0 2 If roots are k times roots of 𝑎𝑥 + 𝑏𝑥 + 𝑐 = 0 2 then equation is 𝑎(𝑥/𝑘) + 𝑏(𝑥/𝑘) + 𝑐 = 0 Descartes Rules: A polynomial equation with n sign changes can have a maximum of n positive roots. To find the maximum possible number of negative roots, find the number of positive roots of f(-x). An equation where the highest power is odd must have at least one real root. Free CAT Study Material Create Create Solved CAT Previous Papers PDF 69  Why Cracku is the Ultimate Preparation Platform for CAT? Our Offerings Daily schedule by 60 MBA Mocks experts (CAT+OMET) 100 Concept 1000+ Videos Notes Weekly Live classes Doubt-solving by By IIMA Alumni experts 18,000+ Solved Access till Questions Jan 10th, 2025 Ready to ace CAT 2024? Enroll Now +91-6303239042 cracku.in 71  CAT Inequalities Formulas The topic Inequalities is one of the few sections in the quantitative part which can throw up tricky questions. The questions are often asked in conjunction with other sections like ratio and proportion, progressions etc. The theory involved in Inequalities is limited and therefore, students should be comfortable with learning the basics, which involves operations such as addition, multiplication and changing of signs of the inequalities. The scope for making an error is high in this section as a minor mistake in calculation (like forgetting the sign) can lead to a completely different answer. Create Create Solved CAT Previous Papers PDF 72  The modulus of x, |x| equals the maximum of x and –x is –|x| ≤ x ≤ |x| For any two real numbers 'a' and 'b’, ➔ a > b => -a < -b ➔ |a| + |b| ≥ |a + b| ➔ |a| - |b| ≤ |a - b| ➔ |a. b| = |a| |b| ➔ |a| > |b| ⇒ a > b (if both are +ve) ⇒ a < b (if both are -ve) For any three real numbers X, Y and Z; if X > Y then X+Z > Y+Z If X > Y and 1. Z is positive, then XZ > YZ Take Free CAT Mock Tests 73  2. Z is negative, then XZ < YZ 1 1 3. If X and Y are of the same sign, 𝑋 < 𝑌 1 1 4. If X and Y are of different signs, 𝑋 > 𝑌 1 For any positive real number, 𝑥+ 𝑥 ≥2 For any real number x >1, 1 𝑥 2 < ⎡1 + ⎤ < 2. 8 ⎣ 𝑥⎦ As x increases, the function tends to an irrational number called 'e' which is approx. equal to 2.718 If |x| ≤ k then the value of x lies between –k and k, or –k ≤ x ≤ k If |x| ≥ k then x ≥ k or x ≤ -k Create Create Solved CAT Previous Papers PDF 74  2 If 𝑎𝑥 + 𝑏𝑥 + 𝑐 < 0 then (x-m)(x-n) < 0, and if n > m, then m < x < n 2 If 𝑎𝑥 + 𝑏𝑥 + 𝑐 > 0 then (x-m)(x-n) > 0 and if m < n, then x < m and x > n 2 If 𝑎𝑥 + 𝑏𝑥 + 𝑐 > 0 but m = n, then the value of x exists for all values, except x is equal to m, i.e., x < m and x > m but x ≠ m Free CAT Study Material CAT Previous papers (Download PDF) Join CAT Complete Course By IIM Alumni Take Free CAT Mock Tests 75 Our Courses Exclusive Course Complete CAT Study Room Comprehensive Course + For CAT 2024 For CAT 2024 Daily Target Combo EMI Available Scholarship Available CAT Daily Target CAT Daily Target 20 CAT Mock + 3 Tests every weekday + CAT Mock Test with video solution 45 Sectional Tests Get upto 50% discount on the Complete Cracku package Apply for Scholarship +91-6303239042 cracku.in 77  Progressions & Series Progressions and Series is one of the important topics for CAT and a significant number of questions appear in the examination from this section every year. Some of the questions from this section can be very tough and time consuming while the others can be very easy. The trick to ace this section is to quickly figure out whether a question is solvable or not and not waste time on very difficult questions. Some of the questions in this section can be answered by ruling out wrong choices among the options available. This method will both save time and improve accuracy. There are many shortcuts which will be of vital importance in answering this section. Create Create Solved CAT Previous Papers PDF 78  There are 3 standard types of progressions ➔ Arithmetic Progression ➔ Geometric Progression ➔ Harmonic Progression Arithmetic progression (A.P): If the sum or difference between any two consecutive terms is constant then the terms are said to be in A.P (Example: 2,5,8,11 or a, a+d, a+2d, a+3d…) If ‘a’ is the first term and ‘d’ is the common difference then the general ‘n’ term is 𝑇𝑛 = 𝑎 + (𝑛 − 1)𝑑 Sum of first ‘n’ terms in 𝑛 A.P = 2 [2𝑎 + (𝑛 − 1)𝑑] Take Free CAT Mock Tests 79  Number of terms in 𝐿𝑎𝑠𝑡 𝑡𝑒𝑟𝑚−𝐹𝑖𝑟𝑠𝑡 𝑡𝑒𝑟𝑚 A. P = 𝐶𝑜𝑚𝑚𝑜𝑛 𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 +1 Sum of all terms of an 𝑛 A. P= 2 [𝐹𝑖𝑟𝑠𝑡 𝑡𝑒𝑟𝑚 + 𝐿𝑎𝑠𝑡 𝑇𝑒𝑟𝑚] Properties of A.P: If a, b , c, d,…. are in A.P and ‘k’ is a constant then a-k, b-k, c-k,… will also be in A.P ak, bk, ck,…will also be in A.P 𝑎 𝑏 𝑐 , , will also be in A.P 𝑘 𝑘 𝑘 Create Create Solved CAT Previous Papers PDF 80 Geometric progression (G.P): If in a succession of numbers the ratio of any term and the previous term is constant then that number is said to be in Geometric Progression. 2 3 Ex :1, 3, 9, 27 or a, ar, 𝑎𝑟 ,𝑎𝑟 𝑛−1 The general expression of an G.P, 𝑇𝑛 = 𝑎𝑟 (where ‘a’ is the first terms and ‘r’ is the common ratio) Sum of ‘n’ terms in G.P, 𝑛 𝑛 𝑎(1−𝑟 ) 𝑎(𝑟 −1) Sn= 1−𝑟 (if r1) Properties of G.P: If a, b , c, d,…. are in G.P and ‘k’ is a constant then Take Free CAT Mock Tests 81  ➔ ak, bk, ck,…will also be in G.P ➔ a/k, b/k, c/k will also be in G.P Sum of term of infinite series in G.P, 𝑎 𝑆∞ = 1−𝑟 (− 1 < 𝑟 < 1) Harmonic progression (H.P): If a, b, c, d,.…..are unequal numbers then they are 1 1 1 said to be in H.P if 𝑎 , 𝑏 , 𝑐 ,……are in A.P The ‘n’ term in H.P is 1/(nth term in A.P) Properties of H.P : If a, b, c, d,…are in H.P, then a+d > b+c ad > bc Create Create Solved CAT Previous Papers PDF 82 Arithmetic Geometric Series: A series will be in arithmetic geometric series if each of its terms is formed by the product of the corresponding terms of an A.P and G.P. The general form of A.G.P series is a,(𝑎 + 2𝑑)𝑟, 2 (𝑎 + 2𝑑)𝑟 ,.... Sum of ‘n’ terms of A.G.P series 𝑛−1 𝑎 (1−𝑟 ) [𝑎+(𝑛−1)𝑑] 𝑠𝑛 = 1−𝑟 + 𝑟𝑑 1−𝑟 + 𝑟𝑛 (1−𝑟) (𝑟 ≠ 1) 𝑛 𝑠𝑛 = 2 [2𝑎 + (𝑛 − 𝑑)] Sum of infinite terms of A.G.P series 𝑎 𝑑𝑟 𝑠∞ = 1−𝑟 + 2 (|𝑟| < 1) (1−𝑟) Take Free CAT Mock Tests 83 Standard Series: 𝑛(𝑛+1) The sum of first ‘n’ natural numbers = 2 The sum of squares of first ‘n’ natural numbers 𝑛(𝑛+1)(𝑛+2) = 6 The sum of cubes of first ‘n’ natural numbers 𝑛(𝑛+1) 2 =[ ] 2 2 The sum of first ‘n’ odd natural numbers = 𝑛 The sum of first ‘n’ even natural numbers = n(n+1) In any series, if the sum of first n terms is given by 𝑆𝑛, 𝑡ℎ then the 𝑛 term 𝑇 = 𝑆𝑛 − 𝑆𝑛−1 𝑛 Create Create Solved CAT Previous Papers PDF 84 Arithmetic Mean: 𝑆𝑢𝑚 𝑜𝑓 𝑎𝑙𝑙 𝑡ℎ𝑒 𝑡𝑒𝑟𝑚𝑠 The arithmetic mean = 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑡𝑒𝑟𝑚𝑠 If two number A and B are in A.P then arithmetic 𝑎+𝑏 mean = 2 Inserting ‘n’ means between two numbers a and b The total terms will become n+2, a is the first term and b is the last term 𝑏−𝑎 Then the common difference d = 𝑛+1 The last term b = a+(n+1)d The final series is a, a+d, a+2d…. Take Free CAT Mock Tests 85 Geometric Mean: If a, b, c,… n terms are in G.P then 𝑛 G.M = 𝑎 × 𝑏 × 𝑐 ×........ × 𝑛 𝑡𝑒𝑟𝑚𝑠 If two numbers a, b are in G.P then their G.M = 𝑎×𝑏 Inserting ‘n’ means between two quantities a and b with common ratio ‘r’ 2 The final series is 𝑎, 𝑎𝑟, 𝑎𝑟 Harmonic Mean: If a, b, c, d,.. are the given numbers in H.P then the Harmonic mean of 𝑁𝑢𝑚𝑏𝑒𝑟𝑠 𝑜𝑓 𝑡𝑒𝑟𝑚𝑠 ‘n’ terms = 1 1 1 𝑎 + 𝑏 + 𝑐 +....... Create Create Solved CAT Previous Papers PDF 86  If two numbers a and b are in H.P then the Harmonic 2𝑎𝑏 Mean = 𝑎+𝑏 Relationship between AM, GM and HM for two numbers a and b, 𝑎+𝑏 A.M= 2 G.M= 𝑎 * 𝑏 2𝑎𝑏 H.M= 𝑎+𝑏 G.M= 𝐴𝑀 * 𝐻𝑀 A.M ≥ G.M ≥ H.M Take Free CAT Mock Tests 87 Success Stories Deveswar Mandava CAT 2023 - 99.92%ile The extensive question bank based on difficulty level helped me practice a lot based on the areas where I was struggling and what level of questions were those. Priyadarshini Das CAT 2023 - 99.91%ile Cracku helped me to maintain regularity with their daily tests. The materials and sectional tests helped to brush up my concepts. The mock tests are also at par with the level of questions in the actual exam and solving and analysing these mock tests helped to improve my score. Rishab Ram CAT 2023 - 99.89%ile Cracku had some of the best test material and the dash cat test series was the most accurate to the actual cat and prepared me in the best way, and not to mention the daily target question quality was really very good and it helped me keep up on a daily basis. Trisha Awari CAT 2023 - 99.33%ile I am very grateful for Sayali ma'am and Maruti sir's course videos and livestreams; they helped me strengthen my concepts and build confidence. Dashcats, daily targets and the extensive study material ensure that I was prepared for whatever the exam threw at me. Read more success stories here +91-6303239042 cracku.in 89  CAT Logarithms, Surds & Indices Formulas “Logarithms, Surds and Indices” is one of the easiest topics in the quantitative section of the CAT exam. Although the number of formulas is high, the basic concepts are very simple to understand and apply. There are no shortcuts to remember and the scope of the questions that can be asked is very limited. The accuracy of answering questions from this section is very high and good students tend to score very well here. Create Create Solved CAT Previous Papers PDF 90  If X,Y > 0 and m,n are rational numbers then 𝑚 𝑛 𝑚+𝑛 ➔ 𝑋 ×𝑋 =𝑋 0 ➔ 𝑋 =1 𝑚 𝑋 𝑚−𝑛 ➔ 𝑛 =𝑋 𝑋 𝑚 𝑛 𝑚𝑛 ➔ ( ) 𝑋 =𝑋 𝑚 𝑚 𝑚 ➔ 𝑋 × 𝑌 = (𝑋 × 𝑌) 𝑋 𝑚 𝑚 ➔ 𝑋 𝑌 𝑚 = ( ) 𝑌 −𝑚 1 ➔ 𝑋 = 𝑚 𝑋 Take Free CAT Mock Tests 91  If X and Y are positive real numbers and a,b are rational numbers 𝑋 −𝑎 𝑌 𝑎 ➔ ( ) ( ) 𝑌 = 𝑋 1/𝑎 𝑎 ➔ 𝑋 = 𝑋 𝑎/𝑏 𝑏 𝑎 ➔ 𝑋 = 𝑋 𝑎 𝑎 𝑎 ➔ 𝑋 × 𝑌 = 𝑋𝑌 𝑎 𝑋 𝑎 𝑋 ➔ 𝑎 = 𝑌 𝑌 1 ➔ = 𝑁 + 1+ 𝑁 𝑁+1− 𝑁 Create Create Solved CAT Previous Papers PDF 92  Surds is an irrational number involving a root. 3 5 Ex: 5, 7, 2 Like surds are two surds having the same number under radical sign. Like surds can be added or subtracted. 6 2+ 3 2= 9 2 If 𝑎 + 𝑏 = 𝑐 + 𝑑 , then a = c and b = d. The conjugate of 𝑎 + 𝑏 is 𝑎 − 𝑏 𝑎 𝑎 𝑎..... ∞ = a 1−⎡⎢ 𝑥 ⎤⎥ 1 ⎣2 ⎦ 𝑎 𝑎 𝑎..... 𝑥 𝑡𝑖𝑚𝑒𝑠 = 𝑎 Take Free CAT Mock Tests 93  To find 𝑥 + 𝑦, 𝑥 + 𝑦 should be written in the form of 𝑚 + 𝑛 + 2 𝑚𝑛 where 𝑥 = 𝑚 + 𝑛 and 4𝑚𝑛 = 𝑦 and 𝑥+ 𝑦=± ( 𝑚 + 𝑛) 𝑥 If N =𝑎 then, x is defined as the logarithm of N to base or 𝑥 = 𝑙𝑜𝑔𝑎𝑁 a logarithm of a negative number or zero is not defined 𝑙𝑜𝑔𝑎1 = 0 𝑙𝑜𝑔𝑎𝑥𝑦 = 𝑙𝑜𝑔𝑎𝑥 + 𝑙𝑜𝑔𝑎𝑦 𝑐 𝑙𝑜𝑔𝑎𝑏 = 𝑐 𝑙𝑜𝑔𝑎𝑏 𝑙𝑜𝑔𝑎𝑎 = 1 Create Create Solved CAT Previous Papers PDF 94  𝑙𝑜𝑔𝑏𝑦 𝑙𝑜𝑔𝑏𝑥 𝑋 =𝑌 𝑛 𝑙𝑜𝑔𝑎𝑏 𝑙𝑜𝑔𝑎 𝑏 = 𝑛 1 𝑙𝑜𝑔𝑎𝑥 = 𝑙𝑜𝑔𝑥𝑎 𝑙𝑜𝑔𝑏𝑥 𝑏 = 𝑥 𝑙𝑜𝑔𝑐𝑏 𝑙𝑜𝑔𝑎𝑏 = 𝑙𝑜𝑔𝑐𝑎 𝑙𝑜𝑔𝑎𝑏 * 𝑙𝑜𝑔𝑏𝑎 = 1 𝑙𝑜𝑔𝑎 ( ) = 𝑙𝑜𝑔 𝑋 − 𝑙𝑜𝑔 𝑌 𝑋 𝑌 𝑎 𝑎 If 0 < a < 1, then 𝑙𝑜𝑔 𝑥 < 𝑙𝑜𝑔𝑎𝑦(𝑖𝑓 𝑥 > 𝑦) 𝑎 If a > 1 then 𝑙𝑜𝑔𝑎𝑥 > 𝑙𝑜𝑔𝑎𝑦 (if x>y) Take Free CAT Mock Tests 95  Our Results Students scored 99.9+ Percentile 45 in CAT 2023 Students scored 99.50+ Percentile 280 in CAT 2023 Students scored 99+ Percentile in 510 CAT 2023 Our Faculty Maruti Konduri Sayali Kale 4 Time CAT 100%iler CAT 99.97 %iler IIM Ahmedabad Alumnus IIM Ahmedabad Alumna +91-6303239042 cracku.in 97  CAT Geometry Formulas Geometry is one of the hardest sections to crack without preparation and one of the easiest with preparation. With so many formulas to learn and remember, this section is going to take a lot of time to master. Remember, read a formula, try to visualize the formula and solve as many questions related to the formula as you can. Knowing a formula and knowing when to apply it are two different abilities. The first will come through reading the formulae list and theory but the latter can come only through solving many different problems. So in this document we are going to provide an exhaustive list of formulas and tips for making the geometry section a lot easier. Try to remember all of them and don’t forget to share. Create Create Solved CAT Previous Papers PDF 98  Quadrants Quadrant I X is Positive Y is Positive Quadrant II X is Negative Y is Positive Quadrant III X is Negative Y is Negative Quadrant IV X is Positive Y is Negative Take Free CAT Mock Tests 99  Lines and Angles Collinear points: Three or more points lying on the single straight line. In this diagram the three points A,B and C are collinear Concurrent lines: If three or more lines lying in the same plane intersect at a single point then those lines are called concurrent lines. The three lines X, Y and Z are concurrent lines here. Create Create Solved CAT Previous Papers PDF 100  The distance between two points with coordinates 2 2 (𝑋1,𝑌1), (𝑋2,𝑌2) is given by D = (𝑋2 − 𝑋1) + (𝑌2 − 𝑌1) 𝑦2−𝑦1 Slope, m= 𝑥2−𝑥1 (If 𝑥2=𝑥1then the lines are perpendicular to each other) Mid point between two points A(𝑥 ,𝑦1) and 1 𝑥1+𝑥2 𝑦1+𝑦2 B (𝑥 2 , 𝑦2) is ( 2 , 2 ) When two lines are parallel, their slopes are equal i.e. 𝑚1= 𝑚2 When two lines are perpendicular, product of their slopes = -1 i.e, 𝑚 1 ∗𝑚2 = −1 Take Free CAT Mock Tests 101  If two intersecting lines have slopes m1 and m2 then the angle between two lines will be 𝑚1−𝑚2 tan θ = 𝑚1𝑚2 (where θ is the angle between the lines) The length of perpendicular from a point(𝑋 , 𝑌 ) 1 1 𝐴𝑋1+ 𝐵𝑌2+ 𝐶 on the line AX+BY+C = 0 is 𝑃= 2 2 𝐴 +𝐵 The distance between two parallel lines 𝐴𝑥 + 𝐵𝑦 + 𝐶1 = 0 and 𝐴𝑥 + 𝐵𝑦 + 𝐶2 = 0 is | 𝐶1−𝐶2 | D=| | | 𝐴 +𝐵 | 2 2 Coordinates of a point P that divides the line joining A (x1,y1) and B (x2,y2) internally in the ratio Create Create Solved CAT Previous Papers PDF 102  l:m : ( 𝑙𝑥2+ 𝑚𝑥1 𝑙+𝑚 , 𝑙𝑦2+ 𝑚𝑦1 𝑙+𝑚 ) Coordinates of a point P that divides the line joining A (x1,y1) and B (x2,y2) externally in the ratio l:m : ( 𝑙𝑥2− 𝑚𝑥1 𝑙−𝑚 , 𝑙𝑦2− 𝑚𝑦1 𝑙−𝑚 ) For a triangle ABC, A (x1,y1), B (x2,y2), C (x3,y3): (𝑥1+ 𝑥2+ 𝑥3) (𝑦1+ 𝑦2+ 𝑦3) Centroid = ( 3 , 3 ) (𝑎𝑥1+ 𝑏𝑥2+ 𝑐𝑥3) (𝑎𝑦1+ 𝑎𝑦2+ 𝑎𝑦3) Incentre = = ( 3 , 3 ) ; where a, b and c are the lengths of the BC, AC and AB respectively. Take Free CAT Mock Tests 103  Equations of a lines General equation 𝐴𝑥 + 𝐵𝑦 = 𝐶 of a line Slope intercept 𝑦 = 𝑚𝑥 + 𝑐 form (𝑐 𝑖𝑠 𝑦 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡) Point-slope form 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1) 𝑥 𝑦 Intercept form 𝑎 + 𝑏 = 1 𝑦−𝑦1 𝑥−𝑥1 Two point form 𝑦2−𝑦1 = 𝑥2−𝑥1 Create Create Solved CAT Previous Papers PDF 104  General equation of a Circle The general equation of a circle is 2 2 𝑥 + 𝑦 + 2𝑔𝑥 + 2𝑓𝑦 + 𝑐 = 0 Centre of the circle is (-g,-f) 2 2 Radius of the circle = 𝑔 +𝑓 − 𝑐 If the origin is the center of the circle then 2 2 2 equation of the circle is 𝑥 + 𝑦 = 𝑟 When two angles A and B are complementary, sum of A and B is 90° When two angles A and B are supplementary, sum of A and B is 180° When two lines intersect, opposite angles are equal. Adjacent angles are supplementary Take Free CAT Mock Tests 105  When any number of lines intersect at a point, the sum of all the angles formed = 360° Consider parallel lines AB, CD and EF as shown in the figure. XY and MN are known as transversals ∠XPQ = ∠PRS = ∠RTU as corresponding angles are equal Interior angles on the side of the transversal are supplementary. i.e. ∠PQS + ∠QSR = 180° Create Create Solved CAT Previous Papers PDF 106  Exterior angles on the same side of the transversal are supplementary. i.e. ∠MQB + ∠DSU = 180° Two transversals are cut by three parallel lines 𝑃𝑅 𝑄𝑆 in the same ratio i.e. 𝑅𝑇 = 𝑆𝑈 Equations of a lines General equation of a line 𝐴𝑥 + 𝐵𝑦 = 𝐶 𝑦 = 𝑚𝑥 + 𝑐 Slope intercept form (𝑐 𝑖𝑠 𝑦 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡) Point-slope form 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1) 𝑥 𝑦 Intercept form 𝑎 + 𝑏 =1 𝑦−𝑦1 𝑥−𝑥1 Two point form 𝑦2−𝑦1 = 𝑥2−𝑥1 Take Free CAT Mock Tests 107  Triangles ➔ Sum of all angles in a triangle is 180° ➔ An angle less than 90° is called an acute angle. An angle greater than 90° is called an obtuse angle. ➔ A triangle with all sides unequal is called scalene triangle ➔ A triangle with two sides equal is called an isosceles triangle. The two angles of an isosceles triangle that are not contained between the equal sides are equal. ➔ A triangle with all sides equal is called an equilateral triangle. All angles of an equilateral triangle equal 60°. ➔ If in a triangle all of its angles are less than 90° than that triangle is called as acute angled triangle ➔ A triangle with one of its angle equal to 90° than that triangle is called as Right angled triangle Create Create Solved CAT Previous Papers PDF 108 ➔ A triangle with one of its angles greater than 90° than that triangle is called an Obtuse angled triangle. ➔ If one side of a triangle is produced then that exterior angle formed is equal to the sum of opposite remote interior angles ➔ A line joining the mid point of a side with the opposite vertex is called a median. (Here D is the midpoint of the AC side or AD = DC). BD is the median of this triangle ABC. ➔ A perpendicular drawn from a vertex to the opposite side is called the altitude Take Free CAT Mock Tests 109 ➔ A line that bisects and also makes right angle with the same side of the triangle is called perpendicular bisector ➔ A line that divides the angle at one of the vertices into two parts is called angular bisector ➔ All points on an angular bisector are equidistant from both arms of the angle. ➔ All points on a perpendicular bisector of a line are equidistant from both ends of the line. Create Create Solved CAT Previous Papers PDF 110 ➔ In an equilateral triangle, the perpendicular bisector, median, angle bisector and altitude (drawn from a vertex to a side) coincide. ➔ The point of intersection of the three altitudes is the Orthocentre. ➔ The point of intersection of the three medians is the centroid. ➔ The three perpendicular bisectors of a triangle meet at a point called the Circumcentre. A circle drawn from this point with the circumradius would pass through all the vertices of the triangle. ➔ The three angle bisectors of a triangle meet at a point called the incentre of a triangle. The incentre is equidistant from the three sides and a circle drawn from this point with the inradius would touch all the sides of the triangle. Take Free CAT Mock Tests 111 ➔ Sum of any two sides of a triangle is always greater than its third side ➔ Difference of any two sides of a triangle is always lesser than it’s third side Pythagoras theorem: In a right angled triangle ABC where ∠B= 90°, 2 2 2 𝐴𝐶 = 𝐴𝐵 + 𝐵𝐶 Apollonius theorem: In a triangle ABC, if AD is the median to side BC then by Apollonius theorem, 2 2 2 2 2∗(𝐴𝐷 + 𝐵𝐷 ) = 𝐴𝐶 + 𝐴𝐵 Create Create Solved CAT Previous Papers PDF 112 Mid Point Theorem : The line joining the midpoint of any two sides in a triangle is parallel to the third side and is half the length of the third side. If X is the midpoint of CA and Y is the midpoint of CB. Then XY will be parallel to AB and XY = ½ * AB Take Free CAT Mock Tests 113 Basic proportionality theorem : If a line is drawn parallel to one side of a triangle and it intersects the other two sides at two distinct points then it divides the two sides in the ratio of respective sides. If in a triangle ABC, D and E are the points lying on AC and BC respectively and DE is parallel to AB then AD/DC = EC/BE Create Create Solved CAT Previous Papers PDF 114 Success Stories Deveswar Mandava CAT 2023 - 99.92%ile The extensive question bank based on difficulty level helped me practice a lot based on the areas where I was struggling and what level of questions were those. Priyadarshini Das CAT 2023 - 99.91%ile Cracku helped me to maintain regularity with their daily tests. The materials and sectional tests helped to brush up my concepts. The mock tests are also at par with the level of questions in the actual exam and solving and analysing these mock tests helped to improve my score. Rishab Ram CAT 2023 - 99.89%ile Cracku had some of the best test material and the dash cat test series was the most accurate to the actual cat and prepared me in the best way, and not to mention the daily target question quality was really very good and it helped me keep up on a daily basis. Trisha Awari CAT 2023 - 99.33%ile I am very grateful for Sayali ma'am and Maruti sir's course videos and livestreams; they helped me strengthen my concepts and build confidence. Dashcats, daily targets and the extensive study material ensure that I was prepared for whatever the exam threw at me. Read more success stories here +91-6303239042 cracku.in Interior Angular Bisector theorem : In a triangle the angular bisector of an angle divides the side opposite to the angle, in the ratio of the remaining two sides. In a triangle ABC if AD is the angle bisector of angle A then AD divides the side BC in the same ratio as the other two sides of the triangle. i.e. BD/ CD= AB/AC. Take Free CAT Mock Tests 115 Exterior Angular Bisector theorem : The angular bisector of the exterior, the angle of a triangle divides the opposite side externally in the ratio of the sides containing the angle. In a triangle ABC, if CE is the angular bisector of exterior angle BCD of a triangle, then AE/BE = AC/BC Create Create Solved CAT Previous Papers PDF 116  Cyclic Quadrilateral : If a quadrilateral has all its vertices on the circle and its 0 opposite angles are supplementary (here x+y =180 ) then that quadrilateral is called cyclic quadrilateral. In a cyclic quadrilateral the opposite angles are supplementary. Area of a cyclic quadrilateral is A = (𝑠 − 𝑎)(𝑠 − 𝑏)(𝑠 − 𝑐)(𝑠 − 𝑑) (𝑎+𝑏+𝑐+𝑑) where s = 2 Exterior angle is equal to its remote interior opposite angle. (here ∠CBX = ∠ADC) Take Free CAT Mock Tests 117  If x is the side of an equilateral triangle then the 3 Altitude (h) = 2 𝑥 3 2 Area = 4 𝑥 1 Inradius = *h 3 2 Circumradius = *h 3 𝑎 2 2 Area of an Isosceles triangle = 4𝑐 − 𝑎 4 (where a, b and c are the length of the sides of BC, AC and AB respectively and b = c) Similar triangles : If two triangles are similar then their corresponding angles are equal and the corresponding sides will be in proportion. Create Create Solved CAT Previous Papers PDF 118 For any two similar triangles : Ratio of sides = Ratio of medians = Ratio of heights = Ratio of circumradii = Ratio of Angular bisectors Ratio of areas = Ratio of the square of the sides. Tests of similarity : (AA / SSS / SAS) Congruent triangles: If two triangles are congruent then their corresponding angles and their corresponding sides are equal. Tests of congruence : (SSS / SAS / AAS / ASA) Area of a triangle: (𝑎+𝑏+𝑐) A = 𝑠(𝑠 − 𝑎)(𝑠 − 𝑏)(𝑠 − 𝑐) where s = 2 1 A= 2 * 𝑏𝑎𝑠𝑒 * 𝑎𝑙𝑡𝑖𝑡𝑢𝑑𝑒 Take Free CAT Mock Tests 119  Why Cracku is the Ultimate Preparation Platform for CAT? Our Offerings Daily schedule by 60 MBA Mocks experts (CAT+OMET) 100 Concept 1000+ Videos Notes Weekly Live classes Doubt-solving by By IIMA Alumni experts 18,000+ Solved Access till Questions Jan 10th, 2025 Ready to ace CAT 2024? Enroll Now +91-6303239042 cracku.in  1 A= 2 * 𝑎𝑏 * 𝑠𝑖𝑛𝐶 (C is the angle formed between sides a and b) 𝑎𝑏𝑐 A= where R is the circumradius 4𝑅 A = r * s where r is the inradius and s is the semi perimeter. (where a, b and c are the lengths of the sides BC, AC and AB) Special triangles : 0 0 0 30 , 60 , 90 Create Create Solved CAT Previous Papers PDF 120  0 0 0 45 , 45 , 90 Consider the triangle ABC with incentre I, and the incircle touching the triangle at P,Q,R as shown in the diagram. As tangents drawn from a point are equal, AP=AQ, CP=CR and BQ=BR. Take Free CAT Mock Tests 121  In an equilateral triangle, the centroid divides the median in the ratio 2:1. As the median is also the perpendicular bisector, angle bisector, G is also the circumcentre and incentre. If a is the side of an equilateral triangle, circumradius =a/√3 and inradius = a/(2√3 ) Circles The angle subtended by a diameter of circle on the circle 0 = 90 Angles subtended by an equal chord are equal. Also, angles subtended in the major segment are half the angle formed by the chord at the center Equal chords of a circle or equidistant from the center Create Create Solved CAT Previous Papers PDF 122  The radius from the center to the point where a tangent touches a circle is perpendicular to the tangent Tangents drawn from the same point to a circle are equal in length A perpendicular drawn from the center to any chord, bisects the chord θ 2 Area of sector OAXC = 360 * π𝑟 θ 2 1 2 Area of minor segment AXC = 360 π𝑟 - 𝑟 𝑆𝑖𝑛θ 2 Take Free CAT Mock Tests 123 Inscribed angle Theorem : 2∠ACB = ∠AOB The angle inscribed by the two points lying on the circle, at the center of the circle, is twice the angle inscribed at any point on the circle by the same points. Angles subtended by the same segment on the circle will be equal. So, here angles a and b will be equal. Create Create Solved CAT Previous Papers PDF 124 The angle made by a chord with a tangent to one of the ends of the chord is equal to the angle subtended by the chord in the other segment. As shown in the figure, ∠ACB = ∠BAT. Consider a circle as shown in the image. Here, 2 𝐴𝑃 * 𝐴𝑄 = 𝐴𝑆 * 𝐴𝑈 = 𝐴𝑇 Take Free CAT Mock Tests 125 Two tangents drawn to a circle from an external common point will be equal in length. So here AZ = AT Direct common tangent : In this figure PQ and RS are the direct common tangents and let AB (Distance between the two centers) = D 2 2 2 2 𝑃𝑄 =𝑅𝑆 = 𝐷 -(𝑟 − 𝑟 ) 1 2 Create Create Solved CAT Previous Papers PDF 126 Transverse common tangent : In this figure PQ and RS are the transverse common tangents and let AB (Distance between the two centers) = D 2 2 2 2 𝑃𝑄 = 𝑅𝑆 = 𝐷 − (𝑟1 + 𝑟2) Take Free CAT Mock Tests 127  Polygons and Quadrilaterals If all sides and all angles are equal, then the polygon is a regular polygon 𝑛(𝑛−3) A regular polygon of n sides has 2 diagonals In a regular polygon of n sides, each exterior 360 angle is degrees. 𝑛 Sum of measure of all the interior angles of a regular polygon is 180 (n-2) degrees (where n is the number of sides of the polygon) Sum of measure of all the exterior angles of regular polygon is 360 degrees Create Create Solved CAT Previous Papers PDF 128 ABCDEF is a regular hexagon with each side equal to ‘x’ then 0 Each interior angle = 120 0 Each exterior angle = 60 0 Sum of all the exterior angles = 360 0 Sum of all the interior angles = 720 3 3 2 Area = 2 𝑎 Take Free CAT Mock Tests 129 Areas of different geometrical figures: 1 Triangles 2 * 𝑏𝑎𝑠𝑒 * ℎ𝑒𝑖𝑔ℎ𝑡 Rectangle 𝑙𝑒𝑛𝑔𝑡ℎ * 𝑤𝑖𝑑𝑡ℎ 1 Trapezoid 2 * 𝑠𝑢𝑚 𝑜𝑓 𝑏𝑎𝑠𝑒𝑠 * ℎ𝑒𝑖𝑔ℎ𝑡 Parallelogram 𝑏𝑎𝑠𝑒 * ℎ𝑒𝑖𝑔ℎ𝑡 Circle π * 𝑟𝑎𝑑𝑖𝑢𝑠 2 1 Rhombus 2 * 𝑝𝑟𝑜𝑑𝑢𝑐𝑡 𝑜𝑓 𝑑𝑖𝑎𝑔𝑜𝑛𝑎𝑙𝑠 2 1 2 Square 𝑠𝑖𝑑𝑒 𝑜𝑟 𝑑𝑖𝑎𝑔𝑜𝑛𝑎𝑙𝑠 2 1 Kite 2 * 𝑝𝑟𝑜𝑑𝑢𝑐𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑑𝑖𝑎𝑔𝑜𝑛𝑎𝑙𝑠 Create Create Solved CAT Previous Papers PDF 130  Solids Volume of different solids: Cube 𝑙𝑒𝑛𝑔𝑡ℎ 3 Cuboid 𝑙𝑒𝑛𝑔𝑡ℎ * 𝑏𝑎𝑠𝑒 * ℎ𝑒𝑖𝑔ℎ𝑡 Prism 𝐴𝑟𝑒𝑎 𝑜𝑓 𝑏𝑎𝑠𝑒 * ℎ𝑒𝑖𝑔ℎ𝑡 Cylinder π𝑟 ℎ 2 Pyramid 1 * 𝐴𝑟𝑒𝑎 𝑜𝑓 𝑏𝑎𝑠𝑒 * ℎ𝑒𝑖𝑔ℎ𝑡 3 Cone 1 2 * π𝑟 * ℎ 3 Cone Frustum (If R is the base radius, r is the upper 1 2 2 surface radius and h is the 3 * πℎ(𝑅 + 𝑅𝑟 + 𝑟 ) height of the frustum) 4 3 Sphere 3 *π * 𝑟 2 3 Hemi-sphere 3 π𝑟 Take Free CAT Mock Tests 131 Total Surface area of different solids: 2 * 𝑏𝑎𝑠𝑒 𝑎𝑟𝑒𝑎 * Prism 𝑏𝑎𝑠𝑒 𝑝𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟 * ℎ𝑒𝑖𝑔ℎ𝑡 Cube 6 * 𝑙𝑒𝑛𝑔𝑡ℎ 2 Cuboid 2(𝑙ℎ + 𝑏ℎ + 𝑙𝑏) Cylinder 2π𝑟ℎ + 2π𝑟 2 1 * 𝑃𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟 𝑜𝑓 𝑏𝑎𝑠𝑒 * Pyramid 2 𝑠𝑙𝑎𝑛𝑡 ℎ𝑒𝑖𝑔ℎ𝑡 + 𝐴𝑟𝑒𝑎 𝑜𝑓 𝑏𝑎𝑠𝑒 Cone (l is the slant π𝑟(𝑙 + 𝑟) height) Cone Frustum (where R & r are the radii 2 2 of the base faces and l is π(𝑅 + 𝑟 + 𝑅𝑙 + 𝑟𝑙) the slant height) Sphere 4π𝑟 2 Hemi-sphere 3π𝑟 2 Create Create Solved CAT Previous Papers PDF 132 Lateral/Curved surface area: Prism 𝑏𝑎𝑠𝑒 𝑝𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟 * ℎ𝑒𝑖𝑔ℎ𝑡 Cube 4 * 𝑙𝑒𝑛𝑔𝑡ℎ 2 2 𝑙𝑒𝑛𝑔𝑡ℎ * ℎ𝑒𝑖𝑔ℎ𝑡 + Cuboid