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} $

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Answer:

Correct Answer: 1. (a)

Solution:

  1. 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.



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