NIMCET 2015 — Mathematics PYQ

1. Establish the basic range

For any real value of $\theta$, we know:

• $0\le {\sin }^2\theta \le 1$

• $0\le {\cos }^2\theta \le 1$

Since $P$ is a sum of even powers, $P$ must be non-negative. If $\sin \theta =0$ and $\cos \theta =1$ (or vice versa), then $P=1$. If both $\sin \theta$ and $\cos \theta$ are non-zero, $P$ will be positive. Thus, P > 0.

2. Compare with the fundamental identity

We know the identity:

Now, consider the individual terms of $P$:

• Since $0\le {\sin }^2\theta \le 1$, raising it to a higher power (like $10$) makes it smaller or equal:

  $${\sin }^{20}\theta =({\sin }^2\theta {)}^{10}\le {\sin }^2\theta$$

• Similarly, since $0\le {\cos }^2\theta \le 1$:

  $${\cos }^{48}\theta =({\cos }^2\theta {)}^{24}\le {\cos }^2\theta$$

3. Combine the inequalities

Adding the two inequalities together:

4. Determine the lower bound

The only way $P$ could be $0$ is if $\sin \theta =0$ and $\cos \theta =0$ simultaneously, which is impossible because ${\sin }^2\theta +{\cos }^2\theta =1$. Therefore, $P$ is always strictly greater than $0$.

Combining these findings:

Correct Option: (b)