CAT 2006 Question Paper

In triangle CDO, which is a right anged triangle, we can use pythagoras theorem.

$$6^2$$ + $$(r-2)^2$$ = $$r^2$$

=> 36 + $$r^2 - 4r + 4$$ = $$r^2$$

=> 4r = 40

=> r = 10

=> Area = $$\pi *10*10/2$$ = $$50\pi $$

If the first task is assigned to either person 3 or person 4, the second task can be assigned in only 1 way. If the first task is assigned to either person 5 or person 6, the second task can be assigned in 2 ways. Therefore, the number of ways in which the first two tasks can be assigned is 2*1 + 2*2 = 6.
The other 4 tasks can be assigned to 4 people in 4! ways.
The total number of ways of assigning the 6 tasks is, therefore, 6*4! = 144.

The addition of numerator and denominatpr should give a prime no. Only option E gives that. 

3 is a factor of 189 and 183 => A and D eliminated

17 is a factor of 187 and 221 => B and C eliminated

181 is prime.

$$log_y x = ab$$
$$a*log_z y = ab$$ => $$log_z y = b$$
$$b*log_x z = ab$$ => $$log_x z = a$$
$$log_y x$$ = $$log_z y * log_x z$$ => $$log x/log y$$ = $$log y/log z * log z/log x$$
=> $$\frac{log x}{log y} = \frac{log y}{log x}$$
=> $$(log x)^2 = (log y)^2$$
=> $$log x = log y$$ or $$log x = -log y$$
So, x = y or x = 1/y
So, ab = 1 or -1
Option 5) is not possible

Triangle AOD is isosceles. So, angle DAO = angle DOA = 75. Similarly, angle BOC = 75. So, angle AOB = 150

Since we know the average ,  we can calculate the marks Dipan got in English Paper II (x) as ; 98+95+95.5+95+(96+x)/2 = 96*5 . We get 96+x = 96.5*2 . Thus x = 97 . Hence option C . 

From the given data it is clear that Dipan is the only student who obtained Group Scores of at least 95 in every group and obtained the highest Group Score in Social Science Group. Hence option D. 

Among the top ten students, only Dipan scored at least 95 in at least one paper from each of the groups. Hence option A. 

Lets first consider ayesha's marks , her 7 marks will increase in geography . So increase in her average marks is 7/(2*5)  = 0.7. So new average is 96.9 . </br>
Now for dipan , his 5 marks will increase in mathematics . So increase in his average 5/5 = 1 . So his new average is 96+1 = 97 which is greater than ayesha's . Other will clearly have a lower average than these both. Hence option E. 

Had Joseph, Agni, Pritam and Tirna each obtained Group Score of 100 in the Social Science Group , their final score would increase by 0.9, 0.9, 2.2, 2.1 respectively. Adding these to the final score then their standing in decreasing order of final score would be Pritam, Joseph, Tirna, Agni with scores of 96.1,95.9,95.8,95.2 respectively. Hence option A.