SATHEE: Matrices and Determinants 4 Question 6

Matrices and Determinants 4 Question 6

6. The greatest value of $c \in R$ for which the system of linear equations $x - c y - c z = 0, c x - y + c z = 0$ ,

$c x + c y - z = 0$ has a non-trivial solution, is

(a) -1

(b) $\frac{1}{2}$

(d) 0

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Answer:

Correct Answer: 6. (b)

Solution:

Key Idea: A homogeneous system of linear equations have non-trivial solutions if $Δ = 0$

Given system of linear equations is

$\begin{aligned} x - c y - c z = 0 \\ c x - y + c z = 0 \end{aligned}$

and $c x + c y - z = 0$

We know that a homogeneous system of linear equations have non-trivial solutions if

$\begin{aligned} Δ = 0 \\ \Rightarrow | \begin{array}{rrr} 1 & - c & - c \\ c & - 1 & c \\ c & c & - 1 \end{array} | = 0 \\ \Rightarrow 1 (1 - c^{2}) + c (- c - c^{2}) - c (c^{2} + c) = 0 \\ \Rightarrow 1 - c^{2} - c^{2} - c^{3} - c^{3} - c^{2} = 0 \\ \Rightarrow - 2 c^{3} - 3 c^{2} + 1 = 0 \\ \Rightarrow 2 c^{3} + 3 c^{2} - 1 = 0 \\ \Rightarrow (c + 1) [2 c^{2} + c - 1] = 0 \\ \Rightarrow (c + 1) [2 c^{2} + 2 c - c - 1] = 0 \\ \Rightarrow (c + 1) (2 c - 1) (c + 1) = 0 \\ \Rightarrow c = - 1 or \frac{1}{2} \end{aligned}$

Clearly, the greatest value of $c$ is $\frac{1}{2}$ .

Matrices and Determinants 4 Question 6

6. The greatest value of $c \in R$ for which the system of linear equations $x - c y - c z = 0, c x - y + c z = 0$ ,

Answer:

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Matrices and Determinants 4 Question 6

6. The greatest value of c∈R for which the system of linear equations x−cy−cz=0,cx−y+cz=0,

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

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NCERT Exercise Video Solution

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6. The greatest value of $c \in R$ for which the system of linear equations $x - c y - c z = 0, c x - y + c z = 0$ ,