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JEE Main 201226 MayOnline
Exam
JEE Main
Year
2012
Date
26 May
Mode
Online
Subject
Mathematics
Chapter
Determinants

JEE Main 2012 · 26 May · Online · Mathematics · Determinants

Statement 1: If the system of equations x+k y+ 3 z=0,3 x+k y-2...

Statement 1: If the system of equations x+ky+x+k y+ 3z=0,3x+ky−2z=0,2x+3y−4z=03 z=0,3 x+k y-2 z=0,2 x+3 y-4 z=0 has a nontrivial solution, then the value of kk is 312\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.

Options

  1. AStatement 1 is false, Statement 2 is true.
  2. BStatement 1 is true, Statement 2 is true, Statement 2 is a correct explanation for Statement 1.
  3. CStatement 1 is true, Statement 2 is true,, Statement 2 is not a correct explanation for Statement 1.
  4. DStatement 1 is true, Statement 2 is false.

Correct answer

A. Statement 1 is false, Statement 2 is true.

Step-by-step solution

Given system of equations is x+ky+3z=03x+ky−2z=02x+3y−4z=0\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 ∴∣1k33k−223−4∣=0⇒1(−4k+6)−k(−12+4)+3(9−2k)=0⇒4k+33−6k=0⇒k=332\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.
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