NIMCET 2010 — Mathematics PYQ

In the interval 0 < x < 1:

• We know that for any $x$ between 0 and 1, a higher power results in a smaller value.

• Therefore, x^2 > x^3.

• Since the base (2) is greater than 1, the function $f(t)=2^t$ is increasing.

• Thus, 2^{x^2} > 2^{x^3}.

Integrating both sides from $0$ to $1$:

Case 2: Comparing $I_3$ and $I_4$ (Interval $x\in [1,2]$)

In the interval x > 1:

• A higher power results in a larger value.

• Therefore, x^3 > x^2.

• Since the base (2) is greater than 1, 2^{x^3} > 2^{x^2}.

Integrating both sides from $1$ to $2$:

Conclusion:

Comparing our findings with the options:

• (a) $I_3=I_4$ is False.

• (b) I_3 > I_4 is False ($I_4$ is actually greater).

• (c) I_2 > I_1 is False ($I_1$ is actually greater).

• (d) I_1 > I_2 is True.

Correct Option:

(d) I_1 > I_2