Linear System of Equations

Q. If the system of equations
$x-2y+3z=9$
$2x+y+z=b$
$x-7y+az=24,$ has infinitely many solutions, then $a-b$ is equal to

Q. The system of equations
$-2x+y+z=a,x-2y+z=b,x+y-2z=c$ has:

1. no solution if $a+b+c\ne 0$
2. unique solution if $a+b+c=0$
3. infinte number of solutions if $a+b+c=0$
4. infinte solution if $a+b+c\ne 0$

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 system of equations,
$ax-y-z=\alpha -1,x-ay-z=\alpha -1x-y-az=\alpha -1$
has no solution if $\alpha$ is

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

Q. The values of $k\in R$ for which the system of equations
$x+ky+3z=0,$
$kx+2y+2z=0$
$2x+3y+4z=0$
admits a non-trivial solution is

1. $2$
   
2. $\frac{5}{2}$
   
3. $3$
   
4. $\frac{5}{4}$

Q. Given $a,b\in \{0,1,2,3,\mathrm{4.....20}\}$. Consider the system of equations
x + y + 2z =5
2x + y + 3z = 6
x + 2y + az =b
Let
'A' denotes number of ordered pairs (a, b) so that the system of equations has unique solution.
'B' denotes number of ordered pairs (a, b) so that the system of equations has no solution.
'C' denotes number of ordered pairs (a, b) so that the sytem of equations has infinite solutions.
then which of the following is/are correct?

1. A = 400
2. B = 20
3. C = 21
4. A+ B = 440

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 system of equations
$x+y+z=6,$
$x+2y+3z=10$ and
$x+2y+\lambda z=\mu$
Statement $1$: If the system has infinite number of solutions, then $\mu =10.$
Statement $2$: The value of $\left|\begin{array}{ccc}1& 1& 6\\ 1& 2& 10\\ 1& 2& \mu \end{array}\right|=0\text{for}\mu =10.$

1. Statement $1$ is false and Statement $2$ is true.
2. Both the statements are true and Statement $2$ is the correct explanation of Statement $1$.
3. Statement $1$ is true and Statement $2$ is false.
4. Both the statements are true but Statement $2$ is not the correct explanation of Statement $1$.

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

Q. If the system of linear equations
$2x+2ay+az=02x+3by+bz=02x+4cy+cz=0,$

where $a,b,c\in R$ are non-zero and distinct; has non-zero solution, then

1. $a+b+c=0$
2. $a,b,c$ are in A.P.
3. $\frac{1}{a},\frac{1}{b},\frac{1}{c}$ are in A.P.
4. $a,b,c$ are in G.P.