If the system of linear equations,
x + y + z = 6
x + 2y + 3z = 10
3x + 2y + $\lambda$z = $\mu$
has more than two solutions, then $\mu$ - $\lambda$2
is equal to ______.
Answer (integer)
13
Solution
Given system of equation more than
2 solutions.
Hence system of equation has infinite many
solution.
<br><br>$\therefore$ $\Delta$ = $\Delta$<sub>1</sub> = $\Delta$<sub>2</sub> = $\Delta$<sub>3</sub> = 0
<br><br>$\Delta$ = $$\left| {\matrix{
1 & 1 & 1 \cr
1 & 2 & 3 \cr
3 & 2 & \lambda \cr
} } \right|$$ = 0
<br><br>$\Rightarrow$ 1(2λ – 6) – 1(λ – 9) + 1(– 4) = 0
<br><br>$\Rightarrow$ 2λ – 6 – λ + 9 – 4 = 0
<br><br>$\Rightarrow$ λ = 1
<br><br>$\Delta$<sub>1</sub> = $$\left| {\matrix{
6 & 1 & 1 \cr
{10} & 2 & 3 \cr
\mu & 2 & \lambda \cr
} } \right|$$ = 0
<br> 6(2λ – 6) – 1(10λ – 3μ) + 1(20 – 2μ) = 0
<br><br>$\Rightarrow$ 12λ – 36 – 10λ + 3μ + 20 – 2μ = 0
<br><br>$\Rightarrow$ 2λ + μ = 16
<br><br>$\Rightarrow$ 2 + μ = 16
<br><br>$\Rightarrow$ $\mu$ = 14
<br><br> $\therefore$ $\mu$ - $\lambda$<sup>2</sup> = 14 - 1 = 13
About this question
Subject: Mathematics · Chapter: Matrices and Determinants · Topic: Types of Matrices and Operations
This question is part of PrepWiser's free JEE Main question bank. 274 more solved questions on Matrices and Determinants are available — start with the harder ones if your accuracy is >70%.