NIMCET 2009 — Mathematics PYQ

Given the condition x < -1:

• For $|x+1|$: Since x < -1, then x + 1 < 0. Therefore, $|x+1|=-(x+1)$.

• For $|2^x-1|$: Since x < -1, then 2^x < 2^{-1} (which means 2^x < 0.5). Since 0.5 < 1, the term $2^x-1$ is negative. Therefore, $|2^x-1|=-(2^x-1)=1-2^x$.

2. Substitute back into the Equation

The original equation is:

Substituting the absolute value expressions:

3. Test the Options

Since this is a multiple-choice question and we have a specific constraint (x < -1), we can test the options that satisfy the constraint. Only option (a) $x=-2$ satisfies x < -1.

Let's check $x=-2$:

• Left Hand Side (LHS):

  $$2^{-(-2+1)}-2(-2)$$
  $$2^{-(-1)}+4$$
  $$2^1+4=6$$

• Right Hand Side (RHS):

  $$2-2^{-2}$$
  $$2-\frac{1}{4}=1.75$$

Wait, let's re-examine the equation from the image image_741606.png carefully. The equation in the image is $2^{|x+1|}-2^x=|2^x-1|+1$.

Corrected Substitutions for $2^{|x+1|}-2^x=|2^x-1|+1$:

Since the bases are the same, we equate the exponents:

Conclusion:

The value of $x$ is $-2$, which satisfies the initial condition x < -1.

Final Answer:

The correct option is (a) $-2$.