Question

A cricketer scores (x-1) runs in each match and scores a total of (x³-1) runs. How many matches did he play?​

Answer 1

$\large\underline{\sf{Given- }}$

↝ A cricketer scores (x-1) runs in each match and scores a total of (x³-1) runs.

$\blue{\large\underline{\sf{To\:Find - }}}$

↝ Number of matches he played.

$\green{\large\underline{\sf{Solution-}}}$

Given that,

• A cricketer scores (x-1) runs in each match.

• A cricketer scores a total of (x³-1) runs.

Let assume that

• Number of matches cricketer played be n.

Thus,

Total run scored in 'n' matches =

Number of matches played × Run scored in each match

So,

$\rm :\longmapsto\: {x}^{3} - 1 = n \: \times \: (x - 1)$

$\rm :\longmapsto\:n = \dfrac{ {x}^{3} - 1}{x - 1}$

can be rewritten as

$\rm :\longmapsto\:n = \dfrac{ {x}^{3} - {1}^{3} }{x - 1}$

We know,

$\boxed{ \tt{ \: {x}^{3} - {y}^{3} = (x - y)( {x}^{2} + xy + {y}^{2}) \: }}$

So, using this identity, we get

$\rm :\longmapsto\:n \: = \: \dfrac{(x - 1)( {x}^{2} + x + 1) }{(x - 1)}$

$\red{\rm \implies\:\boxed{ \tt{ \: n \: = \: {x}^{2} \: + \: x \: + \: 1 \: }}}$

▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬

More Identities to know :-

$\red{\rm :\longmapsto\: {x}^{2} + 2xy + {y}^{2} = {(x + y)}^{2} \: }$

$\red{\rm :\longmapsto\: {x}^{2} - 2xy + {y}^{2} = {(x - y)}^{2} \: }$

$\red{\rm :\longmapsto\: {(x + y)}^{3} = {x}^{3} + 3 {x}^{2}y + 3 {xy}^{2} + {y}^{3} \: }$

$\red{\rm :\longmapsto\: {(x - y)}^{3} = {x}^{3} - 3 {x}^{2}y + 3 {xy}^{2} - {y}^{3} \: }$

$\red{\rm :\longmapsto\: {x}^{3} + {y}^{3} = (x + y)( {x}^{2} - xy + {y}^{2}) \: }$

$\red{\rm :\longmapsto\: {x}^{3} - {y}^{3} = (x - y)( {x}^{2} + xy + {y}^{2}) \: }$

$\red{\rm :\longmapsto\: {x}^{2} - {y}^{2} = (x + y)(x - y) \: }$

Answer 2

The number of matches played by the cricketer are x² + x + 1.

Given:
Runs scored by the cricketer $=(x-1)$

Total runs scored by the cricketer $=(x^{3} -1)$

To find:
Number of matches played by the cricketer.

Solution:

We have been given the runs scored by the cricketer in each match as  $(x-1)$ and the number of matches played by him as $(x^{3} -1)$. Let's assume matches played by the cricketer be $y$.

Hence,

Total runs scored by the cricketer will be the product of number of matches played by him and the runs scored by him per match.

$(x^{3} -1)=y$ × $(x-1)$

Solving further, we get

$y=\frac{x^{3}-1 }{x-1}$

We know, an identity

$a^{3}- b^{3}= (a-b)(a^{2}+ab+ b^{2})$

Hence, $x^{3}-1$ can be written as $(x^{3}- 1^{3} )$

Therefore,

$y=\frac{(x^{3}- 1^{3}) }{(x-1)}$

$y=\frac{(x-1)(x^{2}+x(1)+ 1^{2} )}{(x-1)}$

$y=x^{2} +x+1$

Final answer:

Hence, the number of matches played by the cricketer are x² + x + 1.