GATE2013 AE: GA-8

Question

If $\mid -2X+9\mid =3$ then the possible value of $\mid -X\mid -X^2$ would be:

• A. $30$
• B. $-30$
• C. $-42$
• D. $42$

Answer 1

$\text{Given }\left | -2X +9 \right | = 3,$

$\Rightarrow -2X +9 = 3 \text{ or } -\left (-2X +9 \right ) = 3,$

$\Rightarrow X = 3 \text{ or } X = 6,$

$\Rightarrow \left | -X \right |-X^{2} = \left | -3 \right |-3^{2} = -6 \text{ or } \left | -X \right |-X^{2} = \left | -6 \right |-6^{2} = -30,$

$\text{ Thus B is the correct option. }$

Comments

a similar type of problem

https://gateoverflow.in/40275/gate2013-ce-7

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@Lakshman Patel RJIT , @ankitgupta.1729 Sir, 

$\Rightarrow -2X +9 = 3 \text{ or } -\left (-2X +9 \right ) = 3,$

$\Rightarrow X = 3 \text{ or } X = 6,$

$\Rightarrow \left | -X \right |-X^{2} = \left | -3 \right |-3^{2} = -6 \text{ or } \left | -X \right |-X^{2} = \left | -6 \right |-6^{2} = -30,$
 

 why $\Rightarrow \left | -X \right |-X^{2}$

this is not opened up for both positive and negative 

like for x>0  ,  $\Rightarrow -X -X^{2}$

and for x<0 ,   $\Rightarrow X -X^{2}$

Then both B and C are correct.

pls help!

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@Pranavpurkar

There are two ways to do it.

First way like you are trying to do i.e. :

$Y= |-X| \ –  \ X^2$

Case 1: when $-X \geq 0$ or $X \leq 0 $ then $Y = -X \  – \ X^2$

Case 2: when $-X \leq 0$ or $X \geq 0 $ then $Y = X \  – \ X^2$ 

Since, $X$ is positive here, so you have to apply case 2 here.

 

Another way is to use one of the other definitions of $|X|$ in which you don’t have to consider cases i.e.

$$|X| = \sqrt{X^2}$$

So, Here, $Y= |-X| \ –  \ X^2 = \sqrt{(-X)^2} \ – \ X^2 = X \  – \ X^2 $

It will handle both positive and negative values of $X.$

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@ankitgupta.1729 Thankyou sir!

i got my mistake :)

Answer 2

Answer : -30

Given :   |-2X + 9|=3

there will be 2 cases :

1. if (-2X+9) >= 0 ,                   2.   if (-2X + 9) < 0

   -2X+9 = 3                               -(-2X+9) = 3 

  -2X = -6                                      2X-9=3

    X = 3                                        X=12/2 = 6

Now, put these values in |-$X$| - $X^{2}$

1.    |-$3$|-$3^{2}$                                   2.    |-$6$|-$6^{2}$

       -(-3)-9                                              -(-6)-36

          3-9                                                  6-36

          -6                                                     -30

so, -30 matches here. 

Answer 3

Given  is |−2X+9|=3

removing mod operator with both sign
case 1 : if |−2X+9|>=0 mod will open with positive sign  i.e. (-2x+9)>=0 

                   hence x<4.5

              ⇒ (−2X+9)=3

              ⇒  x=3   (accepted)

|-X|-X^2 = |3|-9= -6

case 2 : if |−2X+9|<0 mod will open with negative sign  i.e. (-2x+9)<0

              hence x>4.5

              ⇒     (−2X+9)=-3   

              ⇒     x=6 (accepted)

now put value x=6 in 

|-X|-X^2 = |-6|-36= -30

option that matches is B

 Thus B is the correct option. 

Comments

Isn’t 3 < 4.5?