Matrices And Determinants Ques 87
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: 87.(a)
Solution:
Formula:
System of equations with 3 variables:
- Given system of linear equations is
$ [\sin \theta] x+[-\cos \theta] y=0 \quad ……(i) $
and $\quad[\cot \theta] x+y=0 \quad ……(ii)$
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.
BETA
