SATHEE: Trigonometrical Equations 1 Question 27

Trigonometrical Equations 1 Question 27

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

$\begin{aligned} (\sin 3 θ) x - y + z & = 0, \\ (\cos 2 θ) x + 4 y + 3 z & = 0, \\ 2 x + 7 y + 7 z & = 0 \end{aligned}$

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

(1986, 4M)

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Answer:

Correct Answer: 27. $θ = n π$ or $n π + (- 1)^{n} \frac{π}{6}$

Solution:

Since, the given system has non-trivial solution.

$\begin{array}{lc} ∴ & | \begin{array}{ccc} \sin 3 θ & - 1 & 1 \\ \cos 2 θ & 4 & 3 \\ 2 & 7 & 7 \end{array} | = 0 \\ \Rightarrow & \sin 3 θ (28 - 21) + 1 (7 \cos 2 θ - 6) \\ \Rightarrow & + 1 (7 \cos 2 θ - 8) = 0 \\ \Rightarrow & 7 \sin 3 θ + 14 \cos 2 θ - 14 = 0 \\ \Rightarrow & 3 \sin θ - 4 \sin^{3} θ + 2 (1 - 2 \sin^{2} θ) - 2 = 0 \end{array}$

$\begin{array}{lc} \Rightarrow & \sin θ (4 \sin^{2} θ + 4 \sin θ - 3) = 0 \\ \Rightarrow & \sin θ = 0 \\ \Rightarrow & θ = n π \\ or & 4 \sin^{2} θ + 4 \sin θ - 3 = 0 \\ \Rightarrow & (2 \sin θ - 1) (2 \sin θ + 3) = 0 \\ \Rightarrow & \sin θ = \frac{1}{2} [∵ \sin θ = - \frac{3}{2} is not possible] \\ ∴ & θ = n π + (- 1)^{n} \frac{π}{6} \end{array}$

$∴$ From Eqs. (i) and (ii), we get

$θ = n π or n π + (- 1)^{n} \frac{π}{6}$

Trigonometrical Equations 1 Question 27

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

Answer:

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Trigonometrical Equations 1 Question 27

27. 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

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