Numbers-18 (CAT-2003ii)

If a,a+2,a+4 are prime numbers, then the number of possible solutions for a is:

(1)   one                  (2) two            (3) three         (4) more than three

Solution follows here:

Containers – Items (P and C Concepts)

There are 3 containers and 5 items. In how many ways can these items be placed in the containers?

This is a conceptual one and we can see four different cases here:

Type-I:    Similar Containers – Similar Items:

X	X	X
5	0	0
4	1	0
3	1	1
3	2	0
2	2	1

     

5 ways

Type-II:     Different Containers – Similar Items:

Type-II(A) Each container having at least one item:

X	Y	Z
3	1	1	3!/2! = 3 ways
2	2	1	3!/2! = 3 ways

Income Tax - 1 (XAT-2011)

Answer questions 1 and 2 based on the following information:

Income-Tax rates for the financial year 2009-10
Individual & HUF below age of 65 Years	Women below age of 65 Years	Tax Rates
Income upto Rs. 1,60,000/-	Income upto Rs. 1,90,000/-	Nil
Rs. 1,60,001/- to Rs. 3,00,000/-	Rs. 1,90,001/- to Rs. 3,00,000/-	10%
Rs. 3,00,001/- to Rs. 5,00,000/-	Rs. 3,00,001/- to Rs. 5,00,000/-	20%
Above Rs. 5,00,001/-	Above Rs. 5,00,001/-	30%

Triangle Inequality Theorem

“Length of any side of a triangle is less than the sum of the lengths of other two sides”

a < b+c            b < a+c            c < a+b                                    ---(1)

From this we can easily derive the following:

a > b~c            b > a~c            c > a~b                                    ---(2)

From (1) and (2),

b~c < a < b+c             a~c < b < a+c             a~b < c < a+b

This theorem is most important for competitive exams. You can find Geometry-12 problem on this blog, which is based on this theorem.