The number of real values of $\lambda$ for which the system of linear equations $$2 x + 4 y - \lambda z = 0$$ $$4 x + \lambda y + 2 z = 0$$ $$\lambda x + 2 y + 2 z = 0$$ has infinitely many solutions, is:
(1) 0
(2) 1
(3) 2
(4) 3

The number of real values of $\lambda$ for which the system of linear equations, $2 x + 4 y - \lambda z = 0, 4 x + \lambda y + 2 z = 0$ and $\lambda x + 2 y + 2 z = 0$, has infinitely many solutions, is:
(1) 3
(2) 1
(3) 2
(4) 0

The set of all values of $\lambda$ for which the system of linear equations $$\begin{aligned} & x - 2 y - 2 z = \lambda x \\ & x + 2 y + z = \lambda y \\ & - x - y = \lambda z \end{aligned}$$ has a non-trivial solution :
(1) is an empty set
(2) contains more than two elements
(3) is a singleton
(4) contains exactly two elements

Let the system of equations $x + 2y + 3z = 5$, $2x + 3y + z = 9$, $4x + 3y + \lambda z = \mu$ have infinite number of solutions. Then $\lambda + 2\mu$ is equal to:
(1) 28
(2) 17
(3) 22
(4) 15

Consider the system of linear equation $x + y + z = 4 \mu , x + 2 y + 2 \lambda z = 10 \mu , x + 3 y + 4 \lambda ^ { 2 } z = \mu ^ { 2 } + 15$, where $\lambda , \mu \in \mathrm { R }$. Which one of the following statements is NOT correct?
(1) The system has unique solution if $\lambda \neq \frac { 1 } { 2 }$ and $\mu \neq 1$
(2) The system is inconsistent if $\lambda = \frac { 1 } { 2 }$ and $\mu \neq 1, 15$
(3) The system has infinite number of solutions if $\lambda = \frac { 1 } { 2 }$ and $\mu = 15$
(4) The system is consistent if $\lambda \neq \frac { 1 } { 2 }$

If the system of linear equations $$\begin{aligned} & x - 2 y + z = - 4 \\ & 2 x + \alpha y + 3 z = 5 \\ & 3 x - y + \beta z = 3 \end{aligned}$$ has infinitely many solutions, then $12 \alpha + 13 \beta$ is equal to
(1) 60
(2) 64
(3) 54
(4) 58

Q70. $\quad x + ( \sqrt { 2 } \sin \alpha ) y + ( \sqrt { 2 } \cos \alpha ) z = 0$ If the system of equations $x + ( \cos \alpha ) y + ( \sin \alpha ) z = 0 \quad$ has a non-trivial solution, then $\alpha \in \left( 0 , \frac { \pi } { 2 } \right)$ is
$$x + ( \sin \alpha ) y - ( \cos \alpha ) z = 0$$
equal to :
(1) $\frac { 11 \pi } { 24 }$
(2) $\frac { 5 \pi } { 24 }$
(3) $\frac { 7 \pi } { 24 }$
(4) $\frac { 3 \pi } { 4 }$