Question

The greatest positive integral value of x for which $200 - x\left( {10 + x} \right)$ is positive is
A. $11$
B. $10$
C. $9$
D. None

Answer 1

Hint: In this question we have to put all the options in place of $x$ one by one and then we have to find with which option we are getting a positive value if we are selecting the greatest value of $x$ from the given options.

Complete step by step answer:
In the above option, we have $200 - x\left( {10 + x} \right)$.Now, we will put all the options in place of x one by one and then we have to find with which option we are getting a positive value if we are selecting the greatest value of x from the given options. In option (A) we have
$x = 11$
On putting the above value in given equation, we get
$ \Rightarrow 200 - 11\left( {10 + 11} \right)$
$ \Rightarrow 200 - 11\left( {21} \right)$
$ \Rightarrow 200 - 231$
$ \Rightarrow - 31$
Here we are getting a negative value but we need a positive value.Therefore, option (A) is incorrect.

In option (B)
We have
$x = 10$
On putting the above value in given equation, we get
$ \Rightarrow 200 - 10\left( {10 + 10} \right)$
$ \Rightarrow 200 - 10\left( {20} \right)$
$ \Rightarrow 200 - 200$
$ \Rightarrow 0$
Here we are getting zero as a value but we need a positive value.Therefore, option (B) is incorrect.

In option (C)
We have
$x = 9$
On putting the above value in given equation, we get
$ \Rightarrow 200 - 9\left( {10 + 9} \right)$
$ \Rightarrow 200 - 9\left( {19} \right)$
$ \Rightarrow 200 - 171$
$ \Rightarrow 29$
Here we are getting a positive value and it is the greatest positive value.

Therefore, option (C) is correct.

Note:We can also do this question by converting the given equation in a quadratic equation and then we will factorize the quadratic equation by middle term splitting and then we will put all the options one by one to get to the correct option. For example: we can write is as $200 - 10x - {x^2}$.
Now, we can split it as $200 - 20x + 10x - {x^2}$.
Now, if we take out common as
$20\left( {10 - x} \right) + x\left( {10 - x} \right)$
$ \Rightarrow \left( {20 + x} \right)\left( {10 - x} \right)$
So, we can easily see that we will get positive value only when the value of x would be less than $10$ and therefore from the options we can easily see that $9$ is the only value that satisfies the given condition.