$x\cos3\theta - 8y - 12z = 0$
$x\cos2\theta + 3y + 3z = 0$
$x + y + 3z = 0$
has a non-trivial solution, is equal to :
System Equations Non-Trivial Solution Condition
A system of homogeneous linear equations has a non-trivial solution if and only if the determinant of the coefficient matrix is equal to zero.
Coefficient Matrix and Determinant Calculation
The given system of equations is:
- $x\cos3\theta - 8y - 12z = 0$
- $x\cos2\theta + 3y + 3z = 0$
- $x + y + 3z = 0$
The coefficient matrix $A$ is:
$ A = \begin{pmatrix} \cos3\theta & -8 & -12 \\ \cos2\theta & 3 & 3 \\ 1 & 1 & 3 \end{pmatrix} $For a non-trivial solution, $\det(A) = 0$. Calculating the determinant:
$ \det(A) = \cos3\theta(3 \cdot 3 - 3 \cdot 1) - (-8)(\cos2\theta \cdot 3 - 3 \cdot 1) + (-12)(\cos2\theta \cdot 1 - 3 \cdot 1) $ $ = \cos3\theta(9 - 3) + 8(3\cos2\theta - 3) - 12(\cos2\theta - 3) $ $ = 6\cos3\theta + 24\cos2\theta - 24 - 12\cos2\theta + 36 $ $ = 6\cos3\theta + 12\cos2\theta + 12 $Setting the determinant to zero:
$ 6\cos3\theta + 12\cos2\theta + 12 = 0 $Divide by 6:
$ \cos3\theta + 2\cos2\theta + 2 = 0 $Solving Trigonometric Equation for Theta
Use the identities $\cos3\theta = 4\cos^3\theta - 3\cos\theta$ and $\cos2\theta = 2\cos^2\theta - 1$. Let $c = \cos\theta$. Substitute these into the equation:
$ (4c^3 - 3c) + 2(2c^2 - 1) + 2 = 0 $ $ 4c^3 - 3c + 4c^2 - 2 + 2 = 0 $ $ 4c^3 + 4c^2 - 3c = 0 $Factor out $c$:
$ c(4c^2 + 4c - 3) = 0 $This gives two possibilities:
- $c = \cos\theta = 0$
- $4c^2 + 4c - 3 = 0$
Case 1: $\cos\theta = 0$
In the interval $\theta \in [0, 2\pi]$, the values are:
$ \theta = \frac{\pi}{2}, \frac{3\pi}{2} $Case 2: $4c^2 + 4c - 3 = 0$
Solve the quadratic equation for $c$ using the quadratic formula $c = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$:
$ c = \frac{-4 \pm \sqrt{4^2 - 4(4)(-3)}}{2(4)} = \frac{-4 \pm \sqrt{16 + 48}}{8} = \frac{-4 \pm \sqrt{64}}{8} = \frac{-4 \pm 8}{8} $The possible values for $c$ are:
$ c_1 = \frac{-4 + 8}{8} = \frac{4}{8} = \frac{1}{2} $ $ c_2 = \frac{-4 - 8}{8} = \frac{-12}{8} = -\frac{3}{2} $Now, consider $\cos\theta = c$:
- If $\cos\theta = \frac{1}{2}$: In the interval $\theta \in [0, 2\pi]$, the values are $\theta = \frac{\pi}{3}, \frac{5\pi}{3}$.
- If $\cos\theta = -\frac{3}{2}$: This is impossible since $-1 \le \cos\theta \le 1$.
Sum of Possible Theta Values
The possible values for $\theta$ in the interval $[0, 2\pi]$ are $\frac{\pi}{2}, \frac{3\pi}{2}, \frac{\pi}{3}, \frac{5\pi}{3}$.
Sum $= \frac{\pi}{2} + \frac{3\pi}{2} + \frac{\pi}{3} + \frac{5\pi}{3}$
Sum $= \left(\frac{\pi}{2} + \frac{3\pi}{2}\right) + \left(\frac{\pi}{3} + \frac{5\pi}{3}\right)$
Sum $= \frac{4\pi}{2} + \frac{6\pi}{3}$
Sum $= 2\pi + 2\pi = 4\pi$.
Similar Questions
- Let $\alpha = \frac{-1 + i\sqrt{3}}{2}$ and $\beta = \frac{-1 - i\sqrt{3}}{2}$, $i = \sqrt{-1}$. If $(7 - 7\alpha + 9\beta)^{20} + (9 + 7\alpha - 7\beta)^{20} + (-7 + 9\alpha + 7\beta)^{20} + (14 + 7\alpha + 7\beta)^{20} = m^{10}$, then $m$ is _________
Let S be the set of the first 11 natural numbers. Then the number of elements in $A = \{B \subseteq S : n(B) \geq 2$ and the product of all elements of B is even is ________.
- The number of $3 \times 2$ matrices A, which can be formed using the elements of the set $\{-2, -1, 0, 1, 2\}$ such that the sum of all the diagonal elements of $A^T A$ is 5, is ______
- The number of numbers greater than 5000, less than 9000 and divisible by 3, that can be formed using the digits 0, 1, 2, 5, 9, if the repetition of the digits is allowed, is ______
- Let $P = [p_{ij}]$ and $Q = [q_{ij}]$ be two square matrices of order 3 such that $q_{ij} = 2^{(i + j - 1)} p_{ij}$ and $\det(Q) = 2^{10}$. Then the value of $\det(\text{adj}(\text{adj } P))$ is:
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- Let $f$ be a function such that $3f(x) + 2f\left(\frac{m}{19x}\right) = 5x, x \neq 0$, where $m = \sum_{i=1}^{9} (i)^2$. Then $f(5) - f(2)$ is equal to
- The smallest positive integral value of $a$, for which all the roots of $x^4 - ax^2 + 9 = 0$ are real and distinct, is equal to
- $\left(\frac{1}{3} + \frac{4}{7}\right) + \left(\frac{1}{3^2} + \frac{1}{3} \times \frac{4}{7} + \frac{4^2}{7^2}\right) + \left(\frac{1}{3^3} + \frac{1}{3^2} \times \frac{4}{7} + \frac{1}{3} \times \frac{4^2}{7^2} + \frac{4^3}{7^3}\right) + \dots$ upto infinite terms, is equal to
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Important Questions from Algebra
- Let $\alpha = \frac{-1 + i\sqrt{3}}{2}$ and $\beta = \frac{-1 - i\sqrt{3}}{2}$, $i = \sqrt{-1}$. If $(7 - 7\alpha + 9\beta)^{20} + (9 + 7\alpha - 7\beta)^{20} + (-7 + 9\alpha + 7\beta)^{20} + (14 + 7\alpha + 7\beta)^{20} = m^{10}$, then $m$ is _________
Let S be the set of the first 11 natural numbers. Then the number of elements in $A = \{B \subseteq S : n(B) \geq 2$ and the product of all elements of B is even is ________.
- The number of $3 \times 2$ matrices A, which can be formed using the elements of the set $\{-2, -1, 0, 1, 2\}$ such that the sum of all the diagonal elements of $A^T A$ is 5, is ______
- The number of numbers greater than 5000, less than 9000 and divisible by 3, that can be formed using the digits 0, 1, 2, 5, 9, if the repetition of the digits is allowed, is ______
- Let $P = [p_{ij}]$ and $Q = [q_{ij}]$ be two square matrices of order 3 such that $q_{ij} = 2^{(i + j - 1)} p_{ij}$ and $\det(Q) = 2^{10}$. Then the value of $\det(\text{adj}(\text{adj } P))$ is:
