$x\cos3\theta - 8y - 12z = 0$
$x\cos2\theta + 3y + 3z = 0$
$x + y + 3z = 0$
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 A = {-3, -2, -1, 0, 1, 2, 3}. Let R be a relation on A defined by xRy if and only if $0\le x^2+2y\le 4$. Let $l$ be the number of elements in R and $m$ be the minimum number of elements required to be added in R to make it a reflexive relation. Then $l + m$ is equal to
Let $\alpha$ and $\beta$ be the roots of $x^2 + \sqrt{3}x-16=0$, and $\gamma$ and $\delta$ be the roots of $x^2 + 3x - 1 = 0$. If $P_n = \alpha^n + \beta^n$ and $Q_n = \gamma^n + \delta^n$, then $\frac{P_{25} + \sqrt{3}P_{24}}{2P_{23}} + \frac{Q_{25}-Q_{23}}{Q_{24}}$ is equal to
Let the domain of the function $f (x) = \log_2 \log_4 \log_6(3 + 4x -x^2)$ be $(a, b)$.
If $\int_{b-a}^{b+a}[x^2] dx=p-\sqrt{q}-\sqrt{r}$, $p,q,r\in N$, $gcd(p,q,r) = 1$, where $[.]$ is the greatest integer function, then $p + q + r$ is equal to
Let A be a matrix of order $3 \times 3$ and $|A| = 5$. If $|2\text{adj} (3A \text{adj} (2A))| = 2^\alpha \cdot 3^\beta \cdot 5^\gamma$, $\alpha, \beta, \gamma \in N$, then $\alpha + \beta + \gamma$ is equal to
Let $a_1, a_2, a_3,....$ be a G.P. of increasing positive numbers. If $a_3a_5 = 729$ and $a_2 + a_4 = \frac{111}{4}$, then $24 (a_1 + a_2 + a_3)$ is equal to
All five letter words are made using all the letters A, B, C, D, E and arranged as in an English dictionary with serial numbers. Let the word at serial number $n$ be denoted by $W_n$. Let the probability $P(W_n)$ of choosing the word $W_n$ satisfy $P(W_n) = 2P(W_{n-1})$, $n > 1$.
If $P(CDBEA) = \frac{2^\alpha}{2^\beta-1}$, $\alpha, \beta\in N$, then $\alpha + \beta$ is equal to :
- The largest $n \in N$ such that $3^n$ divides $50!$ is :
- The number of sequences of ten terms, whose terms are either 0 or 1 or 2, that contain exactly five 1s and exactly three 2s, is equal to :
- Let A be the set of all functions $f: Z \to Z$ and R be a relation on A such that $R = \{(f, g) : f(0) = g(1) \text{ and } f(1) = g(0)\}$. Then R is :
Let $a \in \mathbf{R}$ and $A$ be a matrix of order $3 \times 3$ such that $\det (A) = -4$ and $A + I = \begin{bmatrix} 1 & a & 1 \\ 2 & 1 & 0 \\ a & 1 & 2 \end{bmatrix}$, where $I$ is the identity matrix of order $3 \times 3$. If $\det ((a+1)\text{adj}((a-1)A))$ is $2^m 3^n$, $m, n \in \{0, 1, 2, \dots, 20\}$, then $m+n$ is equal to :
Important Questions from Algebra
Let A = {-3, -2, -1, 0, 1, 2, 3}. Let R be a relation on A defined by xRy if and only if $0\le x^2+2y\le 4$. Let $l$ be the number of elements in R and $m$ be the minimum number of elements required to be added in R to make it a reflexive relation. Then $l + m$ is equal to
Let $\alpha$ and $\beta$ be the roots of $x^2 + \sqrt{3}x-16=0$, and $\gamma$ and $\delta$ be the roots of $x^2 + 3x - 1 = 0$. If $P_n = \alpha^n + \beta^n$ and $Q_n = \gamma^n + \delta^n$, then $\frac{P_{25} + \sqrt{3}P_{24}}{2P_{23}} + \frac{Q_{25}-Q_{23}}{Q_{24}}$ is equal to
Let the domain of the function $f (x) = \log_2 \log_4 \log_6(3 + 4x -x^2)$ be $(a, b)$.
If $\int_{b-a}^{b+a}[x^2] dx=p-\sqrt{q}-\sqrt{r}$, $p,q,r\in N$, $gcd(p,q,r) = 1$, where $[.]$ is the greatest integer function, then $p + q + r$ is equal to
Let A be a matrix of order $3 \times 3$ and $|A| = 5$. If $|2\text{adj} (3A \text{adj} (2A))| = 2^\alpha \cdot 3^\beta \cdot 5^\gamma$, $\alpha, \beta, \gamma \in N$, then $\alpha + \beta + \gamma$ is equal to
Let $a_1, a_2, a_3,....$ be a G.P. of increasing positive numbers. If $a_3a_5 = 729$ and $a_2 + a_4 = \frac{111}{4}$, then $24 (a_1 + a_2 + a_3)$ is equal to
