SATHEE: Matrices and Determinants 4 Question 28

Matrices and Determinants 4 Question 28

29. Consider the system of linear equations in $x, y, z$

$(\sin 3 θ) x - y + z = 0, (\cos 2 θ) x + 4 y + 3 z = 0$ and $2 x + 7 y + 7 z = 0$

Find the values of $θ$ for which this system has non-trivial solution.

(1986, 5M)

Show Answer

Answer:

Correct Answer: 29. $m < - \frac{15}{2}$ or $m > 30$

Solution:

The system of equations has non-trivial solution, if $Δ = 0$ .

$\Rightarrow | \begin{array}{ccc} \sin 3 θ & - 1 & 1 \\ \cos 2 θ & 4 & 3 \\ 2 & 7 & 7 \end{array} | = 0$

Expanding along $C_{1}$ , we get

$\begin{array}{lr} \sin 3 θ \cdot (28 - 21) & - \cos 2 θ (- 7 - 7) + 2 (- 3 - 4) = 0 \\ \Rightarrow & 7 \sin 3 θ + 14 \cos 2 θ - 14 = 0 \\ \Rightarrow & \sin 3 θ + 2 \cos 2 θ - 2 = 0 \\ \Rightarrow & 3 \sin θ - 4 \sin^{3} θ + 2 (1 - 2 \sin^{2} θ) - 2 = 0 \\ \Rightarrow & \sin θ (4 \sin^{2} θ + 4 \sin θ - 3) = 0 \\ \Rightarrow & \sin θ (2 \sin θ - 1) (2 \sin θ + 3) = 0 \\ \Rightarrow & \sin θ = 0, \sin θ = \frac{1}{2} \\ \Rightarrow & [neglecting \sin θ = - 3 / 2] \\ θ = n π, n π + (- 1)^{n} \frac{π}{6}, n \in Z \end{array}$

Matrices and Determinants 4 Question 28

29. Consider the system of linear equations in $x, y, z$

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Matrices and Determinants 4 Question 28

29. Consider the system of linear equations in x,y,z

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Menu

29. Consider the system of linear equations in $x, y, z$