Consistency of Linear System of Equations

Q.

If x = cy + bz, y = az + cx, z = bx + ay where x, y, z are not all zeros, then the value of $a^2+b^2+c^2+2abc$ is

1. 0

2. 1

3. None of these

4. -1

Q. Consider the system of equations
$kx+(c-1)y+z=2$
$cx+(k+1)y+kz=4$
$x+cy+z=1$

Then correct statement is/are

1. If $c=1$, then the system will be inconsistent.
2. If $c=1$, then the may have infinite solutions.
3. If $k=2\cos \frac{\pi }{5}$ then the system will be inconsistent.
4. The system can never possess infinite solutions.

Q. If the system of linear equations
$2x+2y+3z=a$
$3x-y+5z=b$
$x-3y+2z=c$
where $a,b,c$ are non-zero real numbers, has more than one solution, then:

1. $b+c-a=0$
2. $b-c+a=0$
3. $b-c-a=0$
4. $a+b+c=0$

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

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

Q. The number of values of $\theta \in (0,\pi )$ for which the system of linear equations $x+3y+7z=0$
$-x+4y+7z=0$
$\left(\sin 3\theta \right)x+\left(\cos 2\theta \right)y+2z=0$ has a non-trivial solution, is :

1. four
2. three
3. two
4. one

Q. Let $a,\lambda ,\mu \in R.$ Consider the system of linear equations
$ax+2y=\lambda 3x-2y=\mu$
Which of the following statement(s) is(are) correct?

1. If $a=-3,$ then the system has infinitely many solutions for all values of $\lambda$ and $\mu$
2. If $a\ne -3,$ then the system has a unique solution for all values of $\lambda$ and $\mu$
3. If $\lambda +\mu =0,$ then the system has infinitely many solutions for $a=-3$
4. If $\lambda +\mu \ne 0,$ then the system has no solution for $a=-3$

Q. Let $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ and $B=\left[\begin{array}{c}p\\ q\end{array}\right]\ne \left[\begin{array}{c}0\\ 0\end{array}\right]$. If $AB=B$ and $a+d=2$, then the value of $ad-bc$ is

Q. The number of all possible value(s) of $k$ for which the system of equations $kx+y+z=k-1,x+ky+z=k-1,x+y+kz=k-1$ has no solution is

Q. If the system of equations $ax+y+z=0;x+by+z=0$ and $x+y+cz=0$ has a non-trivial solution, then the value of $\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}$ is

1. $1$
2. $2$
3. $0$
4. $-1$

Q. For the system of linear equations :
$x-2y=1,x-y+kz=-2,ky+4z=6,k\in R,$
consider the following statements :
$\text{(}A)$ The system has unique solution if $k\ne 2,k\ne -2.$
$\left(B\right)$ The system has unique solution if $k=-2.$
$\left(C\right)$ The system has unique solution if $k=2.$
$\left(D\right)$ The system has no-solution if $k=2.$
$\left(E\right)$ The system has infinite number of solutions if $k\ne -2.$

Which of the following statements are correct ?

1. $\left(B\right)$ and $\left(E\right)$ only
2. $\left(C\right)$ and $\left(D\right)$ only
3. $\left(A\right)$ and $\left(D\right)$ only
4. $\left(A\right)$ and $\left(E\right)$ only