Consistency of Linear System of Equations

Q.

If $S$ is the set of distinct values of $b$ for which the following system of linear equations
$$\begin{gathered} x+y+z=1 \\ x+ay+z=1 \\ ax+by+z=0 \end{gathered}$$
has no solution, then $\mathrm{S}$ is

1. an infinite set

2. a finite set containing two or more elements

3. a singleton set

4. an empty set

Q.

$x=4\left(1+\cos \theta \right)$ and $y=3\left(1+\sin \theta \right)$ are the parametric equations of

1. $\left[\frac{{\left(x-3\right)}^2}{9}\right]+\left[\frac{{\left(y-4\right)}^2}{16}\right]=1$

2. $\left[\frac{{\left(x+4\right)}^2}{16}\right]+\left[\frac{{\left(y+3\right)}^2}{9}\right]=1$

3. $\left[\frac{{\left(x-4\right)}^2}{16}\right]-\left[\frac{{\left(y-3\right)}^2}{9}\right]=1$

4. $\left[\frac{{\left(x-4\right)}^2}{16}\right]+\left[\frac{{\left(y-3\right)}^2}{9}\right]=1$

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. -1

4. None of these

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. 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 non-trivial solutions of the system $x-y+z=0,x+2y-z=0,2x+y+3z=0$ is

1. $2$
2. $0$
3. $1$
4. infinite

Q. The system of equation $ax+y+z=0,x+by+z=0;x+y+cz=0$ has a non-trivial solution then $\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}=$

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

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

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

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. 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 $\omega (\ne 1)$ is cube root of unity satisfying $\frac{1}{a+\omega }+\frac{1}{b+\omega }+\frac{1}{c+\omega }=2{\omega }^2$ and $\frac{1}{a+{\omega }^2}+\frac{1}{b+{\omega }^2}+\frac{1}{c+{\omega }^2}=2\omega$, then the value of $\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}$ is :

1. $2$
2. $\omega$
3. $-1$
4. $-2$

Q. If $\omega (\ne 1)$ is cube root of unity satisfying $\frac{1}{a+\omega }+\frac{1}{b+\omega }+\frac{1}{c+\omega }=2{\omega }^2$ and $\frac{1}{a+{\omega }^2}+\frac{1}{b+{\omega }^2}+\frac{1}{c+{\omega }^2}=2\omega$, then the value of $\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}$ is :

1. $2$
2. $\omega$
3. $-1$
4. $-2$

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.