NIMCET 2009 — Mathematics PYQ

1. Simplify the Exponential Equation

The given equation is:

Using the laws of exponents, we can rewrite the terms:

• $2^{2a}$ can be written as $(2^a{)}^2$.

• $2^{a+2}$ can be written as $2^a\cdot 2^2$, which is $4\cdot 2^a$.

• $2^5$ is equal to $32$.

Substituting these into the equation:

2. Substitute to Form a Quadratic Equation

Let $2^a=t$. The equation becomes:

3. Factorize the Quadratic Equation

We need two numbers that multiply to $32$ and add up to $-12$. Those numbers are $-8$ and $-4$.

So, the possible values for $t$ are:

4. Find the Integral Values of '$a$'

Now, substitute back $t=2^a$:

• Case 1: $2^a=8$

  $$2^a=2^3$$
  $$a=3$$

  (This is an integer)

• Case 2: $2^a=4$

  $$2^a=2^2$$
  $$a=2$$

  (This is also an integer)

5. Conclusion

The distinct integral values of $a$ are $\{2,3\}$. Therefore, there are 2 such values.

Correct Option:

(c) 2