Matrices and Determinants 4 Question 1
1. If $[x]$ denotes the greatest integer $\leq x$, then the system of liner equations $[\sin \theta] x+[-\cos \theta] y=0$, $[\cot \theta] x+y=0$
(2019 Main, 12 April II)
(a) have infinitely many solutions if $\theta \in \frac{\pi}{2}, \frac{2 \pi}{3}$ and has a unique solution if $\theta \in \pi, \frac{7 \pi}{6}$.
(b) has a unique solution if
$ \theta \in \frac{\pi}{2}, \frac{2 \pi}{3} \cup \pi, \frac{7 \pi}{6} $
(c) has a unique solution if $\theta \in \frac{\pi}{2}, \frac{2 \pi}{3}$ and have infinitely many solutions if $\theta \in \pi, \frac{7 \pi}{6}$
(d) have infinitely many solutions if
$ \theta \in \frac{\pi}{2}, \frac{2 \pi}{3} \cup \pi, \frac{7 \pi}{6} $
Show Answer
Answer:
Correct Answer: 1. (a)
Solution:
- Given system of linear equations is
$ [\sin \theta] x+[-\cos \theta] y=0 $
and $\quad[\cot \theta] x+y=0$
where, $[x]$ denotes the greatest integer $\leq x$.
Here, $\quad \Delta=\left|\begin{array}{cc}{[\sin \theta]} & {[-\cos \theta]} \\ {[\cot \theta]} & 1\end{array}\right|$
$\Rightarrow \Delta=[\sin \theta]-[-\cos \theta][\cot \theta]$
When $ \theta \in \frac{\pi}{2}, \frac{2 \pi}{3} $
$ \sin \theta \in \frac{\sqrt{3}}{2}, 1 $
$\quad[\sin \theta]=0$
$-\cos \theta \in 0, \frac{1}{2}$
$\Rightarrow \quad[-\cos \theta]=0$
and $\quad \cot \theta \in-\frac{1}{\sqrt{3}}, 0$
$\Rightarrow \quad[\cot \theta]=-1$
So, $\quad \Delta=[\sin \theta]-[-\cos \theta][\cot \theta]$
$ -(0 \times(-1))=0 \quad \text { [from Eqs. (iii), (iv) and (v)] } $
Thus, for $\theta \in \frac{\pi}{2}, \frac{2 \pi}{3}$, the given system have infinitely many solutions.
When $\theta \in \pi, \frac{7 \pi}{6}, \sin \theta \in-\frac{1}{2}, 0$
$\Rightarrow \quad[\sin \theta]=-1$
$ -\cos \theta \in \frac{\sqrt{3}}{2}, 1 \Rightarrow[\cos \theta]=0 $
and $\quad \cot \theta \in(\sqrt{3}, \infty) \Rightarrow[\cot \theta]=n, n \in N$.
So, $\quad \Delta=-1-(0 \times n)=-1$
Thus, for $\theta \in \pi, \frac{7 \pi}{6}$, the given system has a unique solution.
