JEE MAIN - Mathematics (2021 - 24th February Evening Shift - No. 4)
For the system of linear equations:
$$x - 2y = 1,x - y + kz = - 2,ky + 4z = 6,k \in R$$,
consider the following statements :
(A) The system has unique solution if $$k \ne 2,k \ne - 2$$.
(B) The system has unique solution if k = $$-$$2
(C) The system has unique solution if k = 2
(D) The system has no solution if k = 2
(E) The system has infinite number of solutions if k $$ \ne $$ $$-$$2.
Which of the following statements are correct?
$$x - 2y = 1,x - y + kz = - 2,ky + 4z = 6,k \in R$$,
consider the following statements :
(A) The system has unique solution if $$k \ne 2,k \ne - 2$$.
(B) The system has unique solution if k = $$-$$2
(C) The system has unique solution if k = 2
(D) The system has no solution if k = 2
(E) The system has infinite number of solutions if k $$ \ne $$ $$-$$2.
Which of the following statements are correct?
(B) and (E) only
(C) and (D) only
(A) and (E) only
(A) and (D) only
Explanation
$$x - 2y + 0.z = 1$$
$$x - y + kz = - 2$$
$$0.x + ky + 4z = 6$$
$$\Delta = \left| {\matrix{ 1 & { - 2} & 0 \cr 1 & { - 1} & k \cr 0 & k & 4 \cr } } \right| = 4 - {k^2}$$
For unique solution $$4 - {k^2} \ne 0$$
$$ \Rightarrow $$ k $$ \ne $$ $$ \pm $$ 2
For k = 2 :
$$x - 2y + 0.z = 1$$
$$x - y + 2z = - 2$$
$$0.x + 2y + 4z = 6$$
$$\Delta x = \left| {\matrix{ 1 & { - 2} & 0 \cr 2 & { - 1} & 2 \cr 6 & 2 & 4 \cr } } \right| = ( - 8) + 2[ - 20]$$
$$\Delta x = - 48 \ne 0$$
For k = 2, $$\Delta x \ne 0$$
$$ \therefore $$ For K = 2; The system has no solution.
$$x - y + kz = - 2$$
$$0.x + ky + 4z = 6$$
$$\Delta = \left| {\matrix{ 1 & { - 2} & 0 \cr 1 & { - 1} & k \cr 0 & k & 4 \cr } } \right| = 4 - {k^2}$$
For unique solution $$4 - {k^2} \ne 0$$
$$ \Rightarrow $$ k $$ \ne $$ $$ \pm $$ 2
For k = 2 :
$$x - 2y + 0.z = 1$$
$$x - y + 2z = - 2$$
$$0.x + 2y + 4z = 6$$
$$\Delta x = \left| {\matrix{ 1 & { - 2} & 0 \cr 2 & { - 1} & 2 \cr 6 & 2 & 4 \cr } } \right| = ( - 8) + 2[ - 20]$$
$$\Delta x = - 48 \ne 0$$
For k = 2, $$\Delta x \ne 0$$
$$ \therefore $$ For K = 2; The system has no solution.
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