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In this won’t A and C both be correct answers as A can be obtained by considering only central part of C?

Also, I tried this by brute force by putting some values; is there any better method for solving these?

yes, Considering it as a rough graph (since scale is not given), both A and C are correct but C gives more information than A.

Given function is an odd function since $f(-x) = -f(x)$ So, graph should be symmetric about origin, so you can eliminate options (B) and (D).

One more way tp eliminate option (D) as:

Since in (D) $f(x)$ is tending to zero when $x \rightarrow -\infty$ but $\lim_{x \rightarrow - \infty} x^2 sin(2x)$ Does not exist because sin oscillates between +1 and -1.

There is no simple trick to plot the graph of product of two functions as you can do for $f(x) \pm a$ or $f(x \pm a)$ etc. But by observing something you can try to plot it like finding whether the function goes when x tends to +infinity or -infinity, finding maxima, minima and here sin oscillates between values -1 and +1, so you can say $f(x)$ oscillates between $\pm x^2$ values. You can plot graph of sin(2x) easily and then observe for what values of x, it is going from -1 to +1 and for what values of x , it goes from +1 to -1 and then you can say, for what values of $x$, $x^2 sin(2x)$ goes from -1 to +1 and +1 to -1.
These are great insights to tackle such problems. Thank you so much

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