XAT 2012 : Quant Question 66 : Mensuration Surface Area

Question

A spherical metal of radius 10 cm is molten and made into 1000 smaller spheres of equal sizes. In this process the surface area of the metal is increased by:

A.    1000 times
B.    100 times
C.    10 times
D.    No Change
E.    None of the above

Correct Answer : 10 times. Choice C

Explanatory Answer
Let the radius of the small spheres be ‘r’ cm.
The amount of metal remains the same in both cases.

Hence, volume of the spherical metal of radius 10 cm should be equal to the volume of the 1000 smaller spheres of radius ‘r’ cm put together.

Therefore, r = 1 cm.
Surface area of a spherical metal of radius 10 cm is

 


Surface area of 1000 smaller spheres put together is

Thus the surface area increased by 10 times.

XAT 2012 Q59 : Mean, Median, Mode

This is a 1 - Mark question that appeared in XAT 2012 from Statistics. This is an interesting question that tests your basic understanding of how Mean, Median and Mode can be interpreted.

Question

Ramesh analysed the monthly salary figures of five vice presidents of his company. All the salary figures are in integer lakhs. The mean and the median salary figures are Rs. 5 lakhs, and the only mode is Rs. 8 lakhs. Which of the options below is the sum (in Rs. lakhs) of the highest and the lowest salaries?
A.    9
B.    10
C.    11
D.    12
E.    None of the above

Correct Answer – Choice A. Rs. 9 lakhs

Explanatory Answer

The mean salary of the five vice presidents is Rs.5 lakhs.
So, the sum of their salaries = 5 * 5 = 25 lakhs.

Let their salaries in ascending order be a, b, c, d and e.
So, a + b + c + d + e = 25.
The median salary is Rs.5 lakhs. So, C’s salary is Rs.5 lakhs.

The only  mode is Rs.8 lakhs.
So,  Rs.8 lakhs salary is drawn by the maximum number of VPs.
C’s salary is Rs.5 lakhs. So, d and e have to draw Rs. 8 lakhs each.

Therefore, a + b + 5 + 8 + 8 = 21
or a + b = 4

Their salaries are in integer lakhs.
Therefore, a can draw Rs.1 lakh and b can draw Rs.3 lakhs or a and b can both draw Rs. 2 lakhs each.

However, there is only one mode. So, a and b cannot draw Rs.2 lakhs each.
So, a draws Rs.1 lakh (the least salary) and e draws Rs.8 lakhs (the highest salary).

So, a + e = Rs.9 lakhs

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XAT 2012 Quant - Number Properties: LCM

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XAT 2012 Question 58  : LCM Word Problem

 Three Vice Presidents (VP) regularly visit the plant on different days. Due to labor unrest, VP (HR) regularly visits the plant after a gap of 2 days. VP (Operations) regularly visits the plant after a gap of 3 days. VP (sales) regularly visits the plant after a gap of 5 days. The VPs do not deviate from their individual schedules. CEO of the company meets the VPs when all the three VPs come to the plant together. CEO is on leave from January 5th to January  28th, 2012. Last time CEO met the VPs on January 3, 2012. When is the next time CEO will meet all the VPs?
A.    February 6, 2012
B.    February 7, 2012
C.    February 8, 2012
D.    February 9, 2012
E.    None of the above

Correct Answer February 8, 2012.

Explanatory Answer

VP (HR) visits the plant every 3rd day (the question states the VP visits after a gap of 2 days). 

The most important point in this question is this point. As the VP visits the plant after a gap of 2 days - he visits the plant every third day - not every second day.


If VP (HR) visited the plant on day 0, he will visit again on day 3, day 6 and so on.
i.e., on days that are multiples of 3.

VP (Operations) visits the plant every 4th day (as he visits after a gap of 3 days).
If VP (Operations) visited the plant on day 0, he will visit again on day 4, day 8 and so on.
i.e., on days that are multiples of 4.

VP (Sales) visits the plant every 6th day (as the VP Sales visits after a gap of 5 days).
If VP (Sales) visited the plant on day 0, he will visit again on day 6, day 12 and so on.
i.e., on days that are multiples of 6.

So, all 3 VPs will visit the plant together again on a day that is a multiple of 3, 4 and 6 or on a day that is a common multiple of 3, 4 and 6.

The first such common multiple is the LCM of 3, 4 and 6 = 12.

The last date on which all three VPs met the CEO was on Jan 3, 2012.
The next dates on which their visits will coincide will be after 12 days and multiples therefore. 
i.e., on Jan 15, 2012, Jan 27, 2012, Feb 8, 2012.

The CEO is on leave from Jan 5, 2012 to Jan 28, 2012. 
So, the CEO will meet the VPs on February 8, 2012.