The vectors $\vec a (x) = \cos x\hat i + \sin x\hat j$ and $\vec b (x) = x\hat i + \sin x\hat j$ are collinear for
Answer & explanation
Correct answer: option 2
$\vec a (x)$ and $\vec b (x)$ will be collinear if and only if
$\cos x = x$
Let $f(x) = x - \cos x$. Then,
$f'(x)=1+ \sin x > 0$ for all x
$⇒ f(x)$ is an increasing function
$⇒f(x)=0$ for a unique value of x
Clearly, $f (x) > 0$ for $x>\frac{π}{3}$ and $f (x) <0$ for $x <\frac{π}{6}$
Thus, $f (x) = 0$ for a unique value of $x ∈ (π/6, π/3)$.
Hence, $\cos x = x$ for a unique value of $x ∈ (π/6, π/3)$.