Linear System of Equations

Q. Let $S$ be the set of all column matrices $\left[\begin{array}{c}{b}_1\\ b_2\\ b_3\end{array}\right]$
such that $b_1,b_2,b_3\in R$ and the system of equations (in real variables)
$$\begin{array}{c}-x+2y+5z=b_1\\ 2x-4y+3z=b_2\\ x-2y+2z=b_3\end{array}$$
has at least one solution. Then, which of the following system(s) (in real variables) has (have) at least one solution for each $\left[\begin{array}{c}{b}_1\\ b_2\\ b_3\end{array}\right]\in S$?

1. $x+2y+3z=b_1,4y+5z=b_2\text{and}x+2y+6z=b_3$
2. $x+y+3z=b_1,5x+2y+6z=b_2\text{and}-2x-y-3z=b_3$
3. $-x+2y-5z=b_1,2x-4y+10z=b_2\text{and}x-2y+5z=b_3$
4. $x+2y+5z=b_1,2x+3z=b_2\text{and}x+4y-5z=b_3$

Q. If the system of linear equations
$\begin{array}{c}x-2y+kz=1\\ 2x+y+z=2\\ 3x-y-kz=3\end{array}$
has a solution $(x,y,z),z\ne 0$, then $(x,y)$ lies on the straight line whose equation is :

1. $4x-3y-1=0$
2. $3x-4y-1=0$
3. $3x-4y-4=0$
4. $4x-3y-4=0$

Q. Match the following for system of linear equations
2x -3y + 5z =12
$3x+y+\lambda z=\mu$
x -7y + 8z =17
$\begin{array}{cccc}& \text{Column - I}& & \text{Column - I}\\ \left(P\right)& \text{Unique solution}& \left(1\right)& \lambda =2,\mu =7\\ \left(Q\right)& \text{Infinite solution}& \left(2\right)& \lambda \ne 2,\mu =7\\ \left(R\right)& \text{No solution}& \left(3\right)& \lambda \ne 2,\mu \ne 7\\ \left(S\right)& \text{Consistent system}& \left(4\right)& \lambda \in R,\mu \ne 7\\ & \text{equation}& & \\ & & \left(5\right)& \lambda =2,\mu \ne 7\end{array}$

1. $P\to 2;Q\to 1;R\to 5;S\to 1,2$
2. $P\to 2,3;Q\to 1;R\to 5;S\to 1,2,3$
3. $P\to 2,3;Q\to 1;R\to 4,5;S\to 1,2,3$
4. $P\to 3;Q\to 1;R\to 4,5;S\to 2,3$

Q.

Consider the following system of equations:

$\begin{array}{rcl}x+2y–3z& =& a\\ 2x+6y–11z& =& b\\ x–2y+7z& =& c,\end{array}$

Where $a,b$ and $c$ are real constants. Then the system of equations

1. has a unique solution when $5a=2b+c$

2. has an infinite number of solutions when $5a=2b+c$

3. has no solution for all $a,b$ and $c$

4. has a unique solution for all $a,b$ and $c$

Q. The system of equations,
$\alpha x-y-z=\alpha -1,x-\alpha y-z=\alpha -1,x-y-\alpha z=\alpha -1$
has no solution if $\alpha$ is

1. Either $-2$ or $1$
2. $-2$
3. $1$
4. Not $-2$

Q. An ordered pair $(\alpha ,\beta )$ for which the system of linear equations
$(1+\alpha )x+\beta y+z=2\alpha x+(1+\beta )y+z=3\alpha x+\beta y+2z=2$
has a unique solution, is:

1. $(1,-3)$
2. $(-3,1)$
3. $(-4,2)$
4. $(2,4)$

Q. The value of $\left|\alpha \right|$ for which the system of equation $\alpha x+y+z=\alpha -1,x+\alpha y+z=\alpha -1,x+y+\alpha z=\alpha -1$ has no solution, is ..............

Q. If the system of linear equations
$x+ay+z=3$
$x+2y+2z=6$
$x+5y+3z=b$
has no solution, then :

1. $a\ne -1,b=9$
2. $a=1,b\ne 9$
3. $a=-1,b\ne 91$
4. $a=-1,b\ne 9$

Q. The system of equations $6x+5y+\lambda z=0$ , $3x-y+4z=0$ , $x+2y-3z=0$ has

1. exactly one nontrivial solution for some real $\lambda$
2. Only a trivial solution for $\lambda ϵR$
3. infinite number of nontrivial solutions for one value of $\lambda$
4. only one solution for$\lambda \ne -5$

Q. The system of equations
$kx+(k+1)y+(k-1)z=0$
$(k+1)x+ky+(k+2)z=0$
$(k-1)x+(k+2)y+kz=0$ has a non-trivial solution for

1. Exactly three values of k
2. Exactly two real values of k
3. Exactly one real value of k
4. Infinite real values of k

Q. The following system of linear equations
$7x+6y-2z=0$,
$3x+4y+2z=0$
$x-2y-6z=0$, has

1. infinitely many solutions, $(x,y,z)$ satisfying $y=2z$
2. infinitely many solutions $(x,y,z)$ satisfying $x=2z$
3. no solution
4. only the trivial solution

Q. If the equations
$(b+c)x+(c+a)y+(a+b)=0,cx+ay+b=0$ and $ax+by+c=0$ are consistent, then show that either $a+b+c=0$ or $a=b=c$.

Q. The following system of linear equations
$2x+3y+2z=9$
$3x+2y+2z=9$
$x-y+4z=8$

1. does not have any solution
2. has an unique solution
3. $\text{has a solution}(\alpha ,\beta ,\gamma )\text{satisfying}\alpha +{\beta }^2+{\gamma }^3=12$
   
4. has infinitely many solutions

Q. If the system of linear equations
$x+y+z=5x+2y+2z=6x+3y+\lambda z=\mu ,$
$(\lambda ,\mu \in R),$ has infinitey many solutions, then the value of $\lambda +\mu$ is :

1. $12$
2. $10$
3. $9$
4. $7$