Tricky one with surds
Q.1 $$\frac{1}{\sqrt{15+\sqrt{60}-\sqrt{140}-\sqrt{84}}}=?$$ A. \(\frac{\sqrt{7}+\sqrt{5}-\sqrt{3}}{1+2\sqrt{15}}\) B. \(\frac{\sqrt{7}+\sqrt{5}+\sqrt{3}}{1+2\sqrt{15}}\) C. \(\frac{\sqrt{7}-\sqrt{5}+\sqrt{3}}{1+2\sqrt{15}}\) D. \(\frac{\sqrt{7}+\sqrt{5}+\sqrt{3}}{1-2\sqrt{15}}\) \(\sqrt{60}=2 \times \sqrt{3} \times \sqrt{5}\) \(\sqrt{140}=2 \times \sqrt{5} \times \sqrt{7}\) \(\sqrt{84}=2 \times \sqrt{3} \times \sqrt{7}\) The RHS suggests the form: 2ab + 2bc + 2ac In which identity do we have (2ab + 2bc + 2ac)? B. \(\frac{\sqrt{7}+\sqrt{5}+\sqrt{3}}{1+2\sqrt{15}}\)
Escalator
Q. Mohan takes 8 seconds to go up a certain escalator going up, while Sohan takes 12 seconds to go down the same escalator. If Sohan takes 3 steps for every 2 steps that Mohan takes, how long will Sohan take to go up the same escalator? Assume Sohan's speed remains same whether he goes up the escalator or down the escalator. A. 2.4 sec B. 4 sec C. 4.8 sec D. 6 sec Ratio of speeds of Sohan and Mohan is given. Assume speeds of Sohan, Mohan and Escalator in terms of variables and write the two equations for they taking 8 secs and 12 secs. Solve simultaneously to get relation between the speeds of the individuals and that of the escalator. Now knowing escalators speed wrt to Sohan's speed, it is simple proportionality between time and speed. D. 6 sec Since [...]
Maximise Area of Triangle 75, x, 2x
This question here is a pretty tough one and is actually not relevant for CAT. However it is an interesting one, can help you apply a nice maximising approach and can also introduce you to a new formula for area of triangle. Please dont hesitate to have a look at the hint, it only gives a rarely known formula for area of triangle and yet leaves a lot of work for you to do. Q. One side of a triangle has length 75. Of the other two sides, the length of one is double the length of the other. What is the maximum possible area for this triangle? A. 1125 B. \(\frac{1875\sqrt{3}}{2}\) C. 1875 D. 1350 Area of a triangle, A, having sides a, b, c is given by ... $$A^2=\frac{\left(a+b+c\right)\left(a+b–c\right)\left(a–b+c\right)\left(–a+b+c\right)}{16}$$ C. 1875 The sides of the triangle can [...]
Maximising/Minimising scenarios in Weighted Average
Questions are made tough when instead of a precise value, a range of value is given. And instead of a unique answer, either the maximum or minimum value is asked. Try this question as an example. Q. In a group, there are 30 people whose age is 50 years or above. The average age of all the people in the group is 32 years. If the number of people in the group is not more than 75, find the maximum average age of all those who are less than 50 year old. TITA type i.e. Type In The Answer type There are a lot of unknowns here. We would need to know how do the different unknowns affect the average age of those below 50 years. But first lets express the data in math terms ... There are two groups, one [...]
CAT 2022 Slot 3 – VARC
Qs 1-4: Read the following passage and answer the questions that follow. Sociologists working in the Chicago School tradition have focused on how rapid or dramatic social change causes increases in crime. Just as Durkheim, Marx, Toennies, and other European sociologists thought that the rapid changes produced by industrialization and urbanization produced crime and disorder, so too did the Chicago School theorists. The location of the University of Chicago provided an excellent opportunity for Park, Burgess, and McKenzie to study the social ecology of the city. Shaw and McKay found . . . that areas of the city characterized by high levels of social disorganization had higher rates of crime and delinquency. In the 1920s and 1930s Chicago, like many American cities, experienced considerable immigration. Rapid population growth is a disorganizing influence, but growth resulting from in-migration of very different people is particularly disruptive. Chicago's in-migrants were both [...]
CAT 2022 Slot 2 – VARC
Qs 1-4: Read the following passage and answer the questions that follow. Humans today make music. Think beyond all the qualifications that might trail after this bald statement: that only certain humans make music, that extensive training is involved, that many societies distinguish musical specialists from nonmusicians, that in today’s societies most listen to music rather than making it, and so forth. These qualifications, whatever their local merit, are moot in the face of the overarching truth that making music, considered from a cognitive and psychological vantage, is the province of all those who perceive and experience what is made. We are, almost all of us, musicians — everyone who can entrain (not necessarily dance) to a beat, who can recognize a repeated tune (not necessarily sing it), who can distinguish one instrument or one singing voice from another. I will often use an antique word, recently revived, [...]
CAT 2022 Slot 1 – VARC
Qs 1-4: Read the following passage and answer the questions that follow. Stories concerning the Undead have always been with us. From out of the primal darkness of Mankind's earliest years, come whispers of eerie creatures, not quite alive (or alive in a way which we can understand), yet not quite dead either. These may have been ancient and primitive deities who dwelt deep in the surrounding forests and in remote places, or simply those deceased who refused to remain in their tombs and who wandered about the countryside, physically tormenting and frightening those who were still alive. Mostly they were ill-defined-strange sounds in the night beyond the comforting glow of the fire, or a shape, half-glimpsed in the twilight along the edge of an encampment. They were vague and indistinct, but they were always there with the power to terrify and disturb. They had the power to [...]
Partnership
One of the topic which frequently appears in mocks. This is pretty straightforward topic and is dependent basically on two quantities : 1. Amount Invested (P) 2. Time of investment (t) The ratio of product of these two for each is basically the ratio in which profit is divided. So profit ratio is P*t ratio of each. Enough of theory, let’s deal with few Qs to get better understanding. Q. A and B invested Rs. 2000 and Rs. 3000 respectively in some business. At the end of the year they made a profit of Rs. 1500. What is A’s share of profit? Solution . Investment Ratio A: B = 2000:3000 = 2:3 Time ratio A:B = 12:12(Number of months) = 1:1 The ratio in which profit is divided is basically product of the two-investment and time: So profit ratio of A and B is: = 2*1:3*1 = 2:3 So 1500 [...]
Finding the last two digits of a^x form numbers .
You must have encountered questions asking to find the remainder when 45327 (just an example) is divided by 100. What has been your approach till now? Was it by breaking 100 as 25*4 and then applying Chinese remainder theorem? If yes, then go through this good alternative for such questions. Finding the remainder when divided by 100 is nothing but finding the last two digits. Like when 53 is divided by 100, the remainder is 25 and 53 = 125 and 25 is the last two digit. So basically you should approach it like this way, and not by CRT. Let’s start then: We will deal with it in three parts 1. Numbers ending in 2/4/8 (2k form) 2. Numbers ending in 1/3/7/9 3. Numbers ending in 6/5 Numbers ending in 2/4/8 Two things to remember before we start 1) The last two digits of 24even will always be 76. [...]
Finding the right most non zero digit of N!
Finding the right most non zero digit of N!. If we have to find the right most non zero digit of N! we have two ways: 1. Removing all the 10's. 2. Using direct formula. Let’s understand this with an example : Q.Find the right most non zero digit of 16! Approach 1: We will try to remove all 10’s so that we can get non zero digit . 16! = 215 × 36 × 53 × 72 × 11 × 13 How many 10’s will be made ? As 10 = 2 × 5 So limiting factor (factor which will decide ) will be 5 as exponent of 5 is lesser than that of 2 So 53 so three 10’s will be made So 103 will be made So now rewrite 16! as = 103 × {212 × 36 × 72 × 11 × 13} So terms inside { [...]