SATHEE: Matrices and Determinants 4 Question 1

Matrices and Determinants 4 Question 1

1. If $[x]$ denotes the greatest integer $\leq x$ , then the system of liner equations $[\sin θ] x + [- \cos θ] y = 0$ , $[\cot θ] x + y = 0$

(2019 Main, 12 April II)

(a) have infinitely many solutions if $θ \in \frac{π}{2}, \frac{2 π}{3}$ and has a unique solution if $θ \in π, \frac{7 π}{6}$ .

(b) has a unique solution if

$θ \in \frac{π}{2}, \frac{2 π}{3} \cup π, \frac{7 π}{6}$

(c) has a unique solution if $θ \in \frac{π}{2}, \frac{2 π}{3}$ and have infinitely many solutions if $θ \in π, \frac{7 π}{6}$

(d) have infinitely many solutions if

$θ \in \frac{π}{2}, \frac{2 π}{3} \cup π, \frac{7 π}{6}$

Show Answer

Answer:

Correct Answer: 1. (a)

Solution:

Given system of linear equations is

$[\sin θ] x + [- \cos θ] y = 0$

and $[\cot θ] x + y = 0$

where, $[x]$ denotes the greatest integer $\leq x$ .

Here, $Δ = | \begin{array}{cc} [\sin θ] & [- \cos θ] \\ [\cot θ] & 1 \end{array} |$

$\Rightarrow Δ = [\sin θ] - [- \cos θ] [\cot θ]$

When $θ \in \frac{π}{2}, \frac{2 π}{3}$

$\sin θ \in \frac{\sqrt{3}}{2}, 1$

$[\sin θ] = 0$

$- \cos θ \in 0, \frac{1}{2}$

$\Rightarrow [- \cos θ] = 0$

and $\cot θ \in - \frac{1}{\sqrt{3}}, 0$

$\Rightarrow [\cot θ] = - 1$

So, $Δ = [\sin θ] - [- \cos θ] [\cot θ]$

$- (0 \times (- 1)) = 0 [from Eqs. (iii), (iv) and (v)]$

Thus, for $θ \in \frac{π}{2}, \frac{2 π}{3}$ , the given system have infinitely many solutions.

When $θ \in π, \frac{7 π}{6}, \sin θ \in - \frac{1}{2}, 0$

$\Rightarrow [\sin θ] = - 1$

$- \cos θ \in \frac{\sqrt{3}}{2}, 1 \Rightarrow [\cos θ] = 0$

and $\cot θ \in (\sqrt{3}, \infty) \Rightarrow [\cot θ] = n, n \in N$ .

So, $Δ = - 1 - (0 \times n) = - 1$

Thus, for $θ \in π, \frac{7 π}{6}$ , the given system has a unique solution.

Matrices and Determinants 4 Question 1

1. If $[x]$ denotes the greatest integer $\leq x$ , then the system of liner equations $[\sin θ] x + [- \cos θ] y = 0$ , $[\cot θ] x + y = 0$

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Matrices and Determinants 4 Question 1

1. If [x] denotes the greatest integer ≤x, then the system of liner equations [sin⁡θ]x+[−cos⁡θ]y=0, [cot⁡θ]x+y=0

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Menu

1. If $[x]$ denotes the greatest integer $\leq x$ , then the system of liner equations $[\sin θ] x + [- \cos θ] y = 0$ , $[\cot θ] x + y = 0$