NIMCET 2012 — Mathematics PYQ

1. Comparing $I_1$ and $I_2$ (Interval 0 < x < 1):

In the interval $x\in (0,1)$, higher powers of $x$ result in smaller values.

Since the base ($2$) is greater than $1$, the inequality remains the same when used as an exponent:

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

Therefore, I_1 > I_2. (This makes options (a) and (b) incorrect).

2. Comparing $I_3$ and $I_4$ (Interval 1 < x < 2):

In the interval $x\in (1,2)$, higher powers of $x$ result in larger values.

Again, since the base ($2$) is greater than $1$:

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

Therefore, I_4 > I_3.

Conclusion:

By comparing the growth of the functions $2^{x^2}$ and $2^{x^3}$ in the specific intervals, we find that $I_4$ is definitely greater than $I_3$.

Correct Option: (d)