Question

85. Let S be the set of all a € R for which the system oflinear
equations
[Sep. 02, 2020 (1)]
2.x - y + 2z = 2
x-2y+z=-4
x+y+z=4
hen
11)]
has no solution. Then the set S
(a) contains more than two elements.
(b) is an empty set.
(c) is a singleton.
(d) contains exactly two elements.​

Answer 1

Answer:

2 will be the answer because xy is not Factor

Answer 2

Answer:

d) contains exactly two elements

Step-by-step explanation:

For no solution, determinant formed by the coefficients of the equations will be 0.

$\left|\begin{array}{ccc} 2 & -1 & 2 \\ 1 & -2 & \lambda \\ 1 & \lambda & 1 \end{array}\right|=0\\\\2\left({-2-{\lambda^2}}\right)-\left({-1}\right)\left({1-\lambda}\right)+2\left({\lambda-\left({-2}\right)}\right)=0\\\\\Rightarrow-4-2{\lambda^2}+1-\lambda+2\lambda+4=0\\\\\Rightarrow-2{\lambda^2}+\lambda+1=0\\\\\Rightarrow2{\lambda ^2} - \lambda- 1 = 0\\\\\Rightarrow2 \lambda^2 -2 \lambda+ \lambda-1=0$

$\\\\\Rightarrow 2 \lambda( \lambda-1)+1(\lambda-1)=0\\\\\Rightarrow(2\lambda+1)(\lambda-1) = 0\\\\\boxed{\lambda \in{\left\{1,\dfrac{-1}{2}}\right\}}$

Hence, S contains exactly two elements.