Q50 / 90 · JEE Main 2012 (26 May Online) · Mathematics

Statement 1: If the system of equations $x+k y+$ $3 z=0,3 x+k y-2 z=0,2 x+3 y-4 z=0$ has a nontrivial solution, then the value of $k$ is $\frac{31}{2}$. Statement 2: A system of three homogeneous equations in three variables has a non trivial solution if the determinant of the coefficient matrix is zero.

  • Correct answer
    A
    Statement 1 is false, Statement 2 is true.
  • B
    Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1.
  • C
    Statement 1 is true, Statement 2 is true,, Statement 2 is not a correct explanation for Statement 1.
  • D
    Statement 1 is true, Statement 2 is false.

Solution

Given system of equations is $$ \begin{aligned} & x+k y+3 z=0 \\ & 3 x+k y-2 z=0 \\ & 2 x+3 y-4 z=0 \end{aligned} $$ Since, system has non-trivial solution $$ \begin{aligned} & \therefore\left|\begin{array}{ccc} 1 & k & 3 \\ 3 & k & -2 \\ 2 & 3 & -4 \end{array}\right|=0 \\ \Rightarrow & 1(-4 k+6)-k(-12+4)+3(9-2 k)=0 \\ \Rightarrow & 4 k+33-6 k=0 \Rightarrow k=\frac{33}{2} \end{aligned} $$ Hence, statement - 1 is false. Statement- 2 is the property. It is a true statement.

Hence, the correct answer is option: 'A'.

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