IIT-JEE 2008 · Paper 1 · Mathematics Q13

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MathematicsAssertion and reason+3 / -1Question 13 of 69

Consider the system of equations
$x-2y+3z=-1,\qquad -x+y-2z=k,\qquad x-3y+4z=1.$

STATEMENT-1: The system of equations has no solution for $k\ne3$ .

and

STATEMENT-2: The determinant $\begin{vmatrix}1&3&-1\\-1&-2&k\\1&4&1\end{vmatrix}\ne0$ , for $k\ne3$ .

STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1

STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1

STATEMENT-1 is True, STATEMENT-2 is False

STATEMENT-1 is False, STATEMENT-2 is True

Answer and worked solution

Official answer: A

Coefficient determinant.
$\begin{vmatrix}1&-2&3\\-1&1&-2\\1&-3&4\end{vmatrix}=1(4-6)+2(-4+2)+3(3-1)=0.$
So the system has either no solution or infinitely many.

Statement 2. The given determinant (the coefficient matrix with the $y$ -column replaced by the constants) equals $3-k$ , non-zero for $k\ne3$ . True.

Statement 1. With $D=0$ and $D_y\ne0$ (Cramer), the system is inconsistent for $k\ne3$ . True, and it follows from Statement 2.

(A)

Topics:Systems of linear equations Determinants and properties

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IIT-JEE 2008 · Paper 1, Mathematics question 13