CAT Exam Model Paper 1 with solutions for free online practice

Question : 67 of 100

Marks: +1, -0

Section III Quantitative Aptitude

Sone lal has 'n' magical eggs whose average weight is 'k’ gm. Each of the 'n' eggs produces 'n' eggs next day such that the average weight of 'n' eggs produced is same as that of the parental (previous generation) egg for each 'n' groups individually i.e., each egg produces 'n' eggs in its next generation and the average weight of all the 'n' eggs of next generation is same as the weight of the mother egg.
This process is continued without any change in pattern. What is the total weight of all the eggs of r

^{th}

generation, where the initial number of eggs with Sone Lal are considered as the eggs of first generation

rnk
n $^\r$ k
nk $^\r$
n $^{\r+1}$ k

Validate

Solution:

(b) The average weight of eggs of first generation is k gm and the no. of eggs is ‘n’.
Let

a _ { 1 } , a _ { 2 } , a _ { 3 } ... , a _ { n }

be the weights of n eggs of the first generation
∴

k = { z _ { 1 } + z _ { 2 } + z _ { 3 } + ... + z_ { n } }/ { n }

∴

n k = z _ { 1 } + z _ { 2 } + z _ { 3 } +.. + z _ { n }\;\;\;\;\;\;

...1
where

z _ { 1 }

is the average weight of its ‘n’ child eggs,

z _ { 2 }

is the average weight of its own ‘n’ child eggs and so on. Child egg is referred to the egg of next generation produced by its mother eggs.
∴

z _ { 1 } = { a _ { 1 } + b _ { 1 } + c _ { 1 } + ... + n _ { 1 } } /{ n }

z _ { 2 } = { a _ { 2 } + b _ { 2 } + c _ { 2 } + ...+ n _ { 2 } } /{ n }

z _ { 3 } = { a _ { 3 } +b _ { 3 } + c _ { 3 } + ... + n _ { 3 } }/ { n }

etc.
Substituting the values of

z _ { 1 } + z _ { 2 } + z _ { 3 }

...in 1

n k = ( { a _ { 1 } + b _ { 1 } + c _ { 1 } + ... } /{ n } ) +

( { a _ { 2 } + b _ { 2 } + c _ { 2 } + ...} /{ n } )

+ ( { a _ { 3 } + b _ { 3 } + c _ { 3 } + ... }/ { n } )

+... ( { a _ { n } + b _ { n } + c _ { n } + ...} /{ n } )

∴

n k = { ( a _ { 1 } + b _ { 1 } + c _ { 1 } + ....) + ( a _ { 2 } + b _ { 2 } + c _ { 2 } + ...) ( { a _ { 3 } + b _ { 3 } + c _ { 3 } + ... })} /{ n }

⇒ Therefore,

n^2k

is the total weight of all the eggs of second generation.
Similarly, each of

a _ { 1 } , b _ { 1 } , c _ { 1 } , a _ { 2 } , b _ { 2 } , c _ { 2 } ....

are the average weights of n eggs of their child eggs. Hence in the third generation total weight will be

n ^ { 3 } k

. Thus the total weight of all the eggs of

r ^ { { th } }

generation is

n ^ { r } k

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