SATHEE: Matrices and Determinants 4 Question 10

Matrices and Determinants 4 Question 10

10. The number of values of $θ \in (0, π)$ for which the system of linear equations

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

has a non-trivial solution, is

(2019 Main, 10 Jan II)

(a) two

(b) three

(d) one

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Answer:

Correct Answer: 10. (a)

Solution:

We know that,

the system of linear equations

$\begin{aligned} a_{1} x + b_{1} y + c_{1} z = 0 \\ a_{2} x + b_{2} y + c_{2} z = 0 \\ a_{3} x + b_{3} y + c_{3} z = 0 \end{aligned}$

has a non-trivial solution, if

$| \begin{array}{lll} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{array} | = 0$

Now, if the given system of linear equations

$\begin{aligned} x + 3 y + 7 z & = 0 \\ - x + 4 y + 7 z & = 0 \end{aligned}$

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

has non-trivial solution, then

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

$\Rightarrow 1 (8 - 7 \cos 2 θ) - 3 (- 2 - 7 \sin 3 θ)$ $+ 7 (- \cos 2 θ - 4 \sin 3 θ) = 0$

$\Rightarrow 8 - 7 \cos 2 θ + 6 + 21 \sin 3 θ$ $- 7 \cos 2 θ - 28 \sin 3 θ = 0$

$\Rightarrow - 7 \sin 3 θ - 14 \cos 2 θ + 14 = 0$

$\Rightarrow - 7 (3 \sin θ - 4 \sin^{3} θ) - 14 (1 - 2 \sin^{2} θ) + 14 = 0$

$[∵ \sin 3 A = 3 \sin A - 4 \sin^{3} A and$

$\cos 2 A = 1 - 2 \sin^{2} A]$

$\Rightarrow 28 \sin^{3} θ + 28 \sin^{2} θ - 21 \sin θ - 14 + 14 = 0$

$\Rightarrow 7 \sin θ [4 \sin^{2} θ + 4 \sin θ - 3] = 0$

$\Rightarrow \sin θ [4 \sin^{2} θ + 6 \sin θ - 2 \sin θ - 3] = 0$

$\Rightarrow \sin θ [2 \sin θ (2 \sin θ + 3) - 1 (2 \sin θ + 3)] = 0$

$\Rightarrow (\sin θ) (2 \sin θ - 1) (2 \sin θ + 3) = 0$

Now, either $\sin θ = 0$ or $\frac{1}{2}$

$∵ \sin θ \neq - \frac{3}{2} as - 1 \leq \sin θ \leq 1$

In given interval $(0, π)$ ,

$\begin{aligned} \sin θ & = \frac{1}{2} \\ \Rightarrow θ & = \frac{π}{6}, \frac{5 π}{6} [∵ \sin θ \neq 0, θ \in (0, π)] \end{aligned}$

Hence, 2 solutions in $(0, π)$

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Matrices and Determinants 4 Question 10

10. The number of values of $θ \in (0, π)$ for which the system of linear equations

Answer:

इंजीनियरिंग की तैयारी

चिकित्सा तैयारी

एनसीईआरटी पुस्तकें और समाधान

महत्वपूर्ण संसाधन

अन्य संसाधन

पाठ्यक्रम

परीक्षा मंच

नमूना मॉक टेस्ट

सदस्यता

एनसीईआरटी व्यायाम वीडियो समाधान

Matrices and Determinants 4 Question 10

10. The number of values of θ∈(0,π) for which the system of linear equations

Answer:

इंजीनियरिंग की तैयारी

चिकित्सा तैयारी

एनसीईआरटी पुस्तकें और समाधान

महत्वपूर्ण संसाधन

अन्य संसाधन

पाठ्यक्रम

परीक्षा मंच

नमूना मॉक टेस्ट

सदस्यता

एनसीईआरटी व्यायाम वीडियो समाधान

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10. The number of values of $θ \in (0, π)$ for which the system of linear equations