Homogeneous System of Equations

Q.

The sum of distinct values of $\lambda$ for which the systems of equations
$(\lambda -1)x+(3\lambda +1)y+2\lambda z=0$
$(\lambda -1)x+(4\lambda -2)y+(\lambda +3)z=0$
$2x+(3\lambda +1)y+3(\lambda -1)z=0$
has non-zero solutions, is

Q. Let $\lambda$ be a real number for which the system of linear equations
$x+y+z=6$
$4x+\lambda y-\lambda z=\lambda -2$
$3x+2y-4z=-5$
has infinitely many solutions. Then $\lambda$ is a root of the quadratic equation :

1. ${\lambda }^2+3\lambda -4=0$
2. ${\lambda }^2-3\lambda -4=0$
3. ${\lambda }^2+\lambda -6=0$
4. ${\lambda }^2-\lambda -6=0$

Q. Let ${\alpha }_1,{\alpha }_2$ and ${\beta }_1,{\beta }_2$ be the roots of $a{x}^2+bx+c=0$ and $p{x}^2+qx+r=0$ respectively. If the system of equations ${\alpha }_{1}y+{\alpha }_{2}z=0$ and ${\beta }_{1}y+{\beta }_{2}z=0$ has a non-trivial solution, then which of the following options is CORRECT ?

1. $abc=pqr$
2. $a^{2}pr=q^{2}bc$
3. $b^{2}pr=q^{2}ac$
4. $a^{2}qr=p^{2}bc$

Q. Consider the planes
$P_1:cy+bz=x$
$P_2:az+cx=y$
$P_3:bx+ay=z.$
$P_1,P_2$ and $P_3$ pass through one line, if

1. $a^2+b^2+c^2=ab+bc+ca$
2. $a^2+b^2+c^2+2abc=1$
3. $a^2+b^2+c^2=1$
4. $a^2+b^2+c^2+2ab+2bc+2ca+2abc=1$

Q. The set of equations
$\lambda x-y+\left(\cos \theta \right)z=0$
$3x+y+2z=0$
$\left(\cos \theta \right)x+y+2z=0,$
where $0\le \theta <2\pi ,$ has non-trivial solutions for

1. all values of $\lambda$ and $\theta$
2. no value of $\lambda$ and $\theta$
3. all values of $\lambda$ and only two values of $\theta$
4. only one value of $\lambda$ and all values of $\theta$

Q.

The sum of distinct values of $\lambda$ for which the systems of equations
$(\lambda -1)x+(3\lambda +1)y+2\lambda z=0$
$(\lambda -1)x+(4\lambda -2)y+(\lambda +3)z=0$
$2x+(3\lambda +1)y+3(\lambda -1)z=0$
has non-zero solutions, is

Q. Consider the planes
$P_1:cy+bz=x$
$P_2:az+cx=y$
$P_3:bx+ay=z.$
$P_1,P_2$ and $P_3$ pass through one line, if

1. $a^2+b^2+c^2=ab+bc+ca$
2. $a^2+b^2+c^2+2abc=1$
3. $a^2+b^2+c^2=1$
4. $a^2+b^2+c^2+2ab+2bc+2ca+2abc=1$

Q. The set of all values of $\lambda$ for which the system of linear equations
$x-2y-2z=\lambda x$
$x+2y+z=\lambda y$
$-x-y=\lambda z$
has a non-trivial solution :

1. is a singleton
2. is an empty set
3. contains exactly two elements
4. contains more than two elements

Q. Let $\lambda$ be a real number for which the system of linear equations
$x+y+z=6$
$4x+\lambda y-\lambda z=\lambda -2$
$3x+2y-4z=-5$
has infinitely many solutions. Then $\lambda$ is a root of the quadratic equation :

1. ${\lambda }^2+3\lambda -4=0$
2. ${\lambda }^2-3\lambda -4=0$
3. ${\lambda }^2+\lambda -6=0$
4. ${\lambda }^2-\lambda -6=0$

Q. The set of equations
$\lambda x-y+\left(\cos \theta \right)z=0$
$3x+y+2z=0$
$\left(\cos \theta \right)x+y+2z=0,$
where $0\le \theta <2\pi ,$ has non-trivial solutions for

1. all values of $\lambda$ and $\theta$
2. no value of $\lambda$ and $\theta$
3. all values of $\lambda$ and only two values of $\theta$
4. only one value of $\lambda$ and all values of $\theta$