If [x] denotes the greatest integer leq x,the | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) The given system of equations is:$[\sin 	heta ]x + [-\cos 	heta ]y = 0 \dots (1)$$[\cot 	heta ]x + y = 0 \dots (2)$Case $I$: When $	heta \in \

Vedclass · Accepted Answer

have infinitely many solutions if $	heta \in \left( {\frac{\pi }{2},\frac{{2\pi }}{3}} ight)$ and has a unique solution if $	heta \in \left( {\pi ,\frac{{7\pi }}{6}} ight)$

Vedclass · Answer

(A) The given system of equations is:$[\sin 	heta ]x + [-\cos 	heta ]y = 0 \dots (1)$$[\cot 	heta ]x + y = 0 \dots (2)$Case $I$: When $	heta \in \left( \frac{\pi }{2}, \frac{2\pi }{3} ight)$$\sin 	heta \in \left( \frac{\sqrt{3}}{2}, 1 ight) \implies [\sin 	heta ] = 0$$-\cos 	heta \in \left( 0, \frac{1}{2} ight) \implies [-\cos 	heta ] = 0$$\cot 	heta \in \left( -\frac{1}{\sqrt{3}}, 0 ight) \implies [\cot 	heta ] = -1$Substituting these into the equations:$0x + 0y = 0$$-x + y = 0$The first equation is satisfied for all $(x, y)$,and the second implies $y = x$. Thus,the system has infinitely many solutions.Case $II$: When $	heta \in \left( \pi, \frac{7\pi}{6} ight)$$\sin 	heta \in \left( -\frac{1}{2}, 0 ight) \implies [\sin 	heta ] = -1$$-\cos 	heta \in \left( -\frac{\sqrt{3}}{2}, 1 ight) \implies [-\cos 	heta ] = 0$$\cot 	heta \in (\sqrt{3}, \infty) \implies [\cot 	heta ] = k$,where $k \in \{1, 2, 3, \dots\}$Substituting these into the equations:$-x + 0y = 0 \implies x = 0$$kx + y = 0 \implies k(0) + y = 0 \implies y = 0$Thus,the system has a unique solution $(0, 0)$.

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

Similar Questions

If the system of equations $2x + py + 6z = 8$,$x + 2y + qz = 5$,and $x + y + 3z = 4$ has infinitely many solutions,then $p=$

If $A = \begin{bmatrix} -1 & 2 \\ 2 & -1 \end{bmatrix}$ and $B = \begin{bmatrix} 3 \\ 1 \end{bmatrix}$,$AX = B$,then $X = $

The number of $\theta \in (0, 4\pi)$ for which the system of linear equations $3(\sin 3\theta)x - y + z = 2$,$3(\cos 2\theta)x + 4y + 3z = 3$,and $6x + 7y + 7z = 9$ has no solution is:

If $x=\alpha, y=\beta, z=\gamma$ is the solution for the system of equations:
$\begin{aligned} 2x-y+8z &= 13 \\ 3x+4y+5z &= 18 \\ 5x-2y+7z &= 20 \end{aligned}$
then $\alpha\beta+\beta\gamma+\gamma\alpha=$

Let $\alpha, \beta \in R$ be such that the system of linear equations $x+2y+z=5, 2x+y+\alpha z=5, 8x+4y+\beta z=18$ has no solution. Then $\frac{\beta}{\alpha}$ is equal to:

Vedclass Test Series

Exam Paper Generator

Online Exam Module

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