     Question Answers

# If$f(x) = {\cos ^2}x + {\cos ^2}2x + {\cos ^2}3x$, then number of values of $x \in [0,2\pi ]$for which $f(x) = 0$ areA.4B.6C.8D.0  Hint: All the terms in the right hand side are squared. Think about what we can deduce from here.

Given, $f(x) = {\cos ^2}x + {\cos ^2}2x + {\cos ^2}3x$.
Since, every term in the right-hand side of the equation is squared, the value of each individual term could be either zero or greater than zero. We are interested in $x$ where $f(x) = 0$. For$f(x)$to be zero, each individual on the right-hand side has to be zero. That is${\cos ^2}x = 0,{\cos ^2}2x = 0{\text{ and }}{\cos ^2}3x = 0$.Now let’s solve them one by one.
${\cos ^2}x = 0 \Leftrightarrow \cos x = 0 \Leftrightarrow x = \frac{\pi }{2},\frac{{3\pi }}{2}$
${\cos ^2}2x = 0 \Leftrightarrow \cos 2x = 0 \Leftrightarrow 2x = \frac{\pi }{2},\frac{{3\pi }}{2} \Leftrightarrow x = \frac{\pi }{4},\frac{{3\pi }}{4}$
${\cos ^2}3x = 0 \Leftrightarrow \cos 3x = 0 \Leftrightarrow 3x = \frac{\pi }{2},\frac{{3\pi }}{2} \Leftrightarrow x = \frac{\pi }{6},\frac{{3\pi }}{6}$
Observe that, there is no common value of $x$ in all the above terms. The question should come in our mind as to why we are finding the common values. It’s just because we want $x$ where$f(x) = 0$ and $f(x) = 0$ when all the individual terms on the right hand side will be zero. It means for a single value of $x$, all the terms on the right-hand side has to vanish simultaneously. That’s why we are looking at the common value of $x$ where ${\cos ^2}x = 0,{\cos ^2}2x = 0{\text{ and }}{\cos ^2}3x = 0$.But, there is no such common value in the given domain. So, there is no $x$ for which $f(x) = 0$.
Hence the correct option is D

Note: When you are finding the roots of something, keep domain in your mind. Here we have given our domain as $f(x) = 0$. So, we only considered such x where $f(x) = 0$ in the given domain. One should not step out of the domain.
View Notes
Cos 90 Degrees  Sin 2x Cos 2x  Cos 2x Formula  Cos 30 Degrees  Cos 60 degrees  Cos 90 Value  Cos 0  2 Cos A Cos B Formula  Cos 360  Value of Cos 120  