### Related Formula z + overlinez = 2operatornameRe(z)$z + \overline{z} = 2\operatorname{Re}(z)$ ### Core Logic Isolate the real fractional component block by conjugating the complex quotient matrix expression, then resolve the structural wave equations across bounds boundaries. ### Step 1: Expand Complex Real Operator frac2cos^2theta - 3sin^2thetacos^2theta + 9sin^2theta = -frac110$\frac{2\cos^2\theta - 3\sin^2\theta}{\cos^2\theta + 9\sin^2\theta} = -\frac{1}{10}$ 20cos^2theta - 30sin^2theta = -cos^2theta - 9sin^2theta$20\cos^2\theta - 30\sin^2\theta = -\cos^2\theta - 9\sin^2\theta$ ### Step 2: Factor Trigonometric Expressions 21cos^2theta - 21sin^2theta = 0 implies cos(2theta) = 0$21\cos^2\theta - 21\sin^2\theta = 0 \implies \cos(2\theta) = 0$ ### Step 3: Collect Domain Solutions and Evaluate Squares Since angular coordinate parameters scan [0, 2pi]$[0, 2\pi]$, multi frequency vectors trace out: 2theta = fracpi2, frac3pi2, frac5pi2, frac7pi2$2\theta = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, \frac{7\pi}{2}$ sum theta^2 = fracpi^216 + frac9pi^216 + frac25pi^216 + frac49pi^216 = frac84pi^216 = frac214pi^2$\sum \theta^2 = \frac{\pi^2}{16} + \frac{9\pi^2}{16} + \frac{25\pi^2}{16} + \frac{49\pi^2}{16} = \frac{84\pi^2}{16} = \frac{21}{4}\pi^2$ ### Pattern Recognition Transforming algebraic equations to clean forms like \cos(2\theta) = 0$\\cos(2\\theta) = 0$ guarantees evenly distributed coordinate solutions across standard periodicity ranges. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Mathematics: Complex Numbers Class 11 Mathematics: Trigonometric Functions

### Related Formula Properties of cube roots of unity: 1 + omega + omega^2 = 0, quad omega^3 = 1$1 + \omega + \omega^2 = 0, \quad \omega^3 = 1$ ### Core Logic Simplify the determinant by performing row operation R_1 to R_1 + R_2 + R_3$R_1 \to R_1 + R_2 + R_3$: Delta = beginvmatrix z + 1 + omega + omega^2 & z + 1 + omega + omega^2 & z + 1 + omega + omega^2 \\ omega & z + omega^2 & 1 \\ omega^2 & 1 & z + omega endvmatrix$\Delta = \begin{vmatrix} z + 1 + \omega + \omega^2 & z + 1 + \omega + \omega^2 & z + 1 + \omega + \omega^2 \\ \omega & z + \omega^2 & 1 \\ \omega^2 & 1 & z + \omega \end{vmatrix}$ Using 1 + omega + omega^2 = 0$1 + \omega + \omega^2 = 0$, the top row simplifies to vector [z, z, z]$[z, z, z]$. Factoring out z$z$: Delta = z cdot (z^2) = z^3$\Delta = z \cdot (z^2) = z^3$ Given modulus constraint |z^3| = 1 implies |z| = 1$|z^3| = 1 \implies |z| = 1$. The root solutions are: z in \1, omega, omega^2\$z \in \{1, \omega, \omega^2\}$ ### Step 1: Evaluate Geometric Magnitude Metric The condition left|fracz - az + bright| = 1 implies |z - a| = |z + b|$\left|\frac{z - a}{z + b}\right| = 1 \implies |z - a| = |z + b|$. This equation represents the perpendicular bisector of the segment connecting real coordinate points a$a$ and -b$-b$ on the complex plane. Since a$a$ and b$b$ are integers, the bisector is a vertical line: x = fraca - b2$x = \frac{a - b}{2}$. ### Step 2: Match Root Solutions and Count Pairs For z=1$z=1$, it must lie on the line: fraca-b2 = 1 implies a - b = 2$\frac{a-b}{2} = 1 \implies a - b = 2$. For z = omega, omega^2$z = \omega, \omega^2$, their real part is -frac12$-\frac{1}{2}$, so the line must be: fraca-b2 = -frac12 implies a - b = -1$\frac{a-b}{2} = -\frac{1}{2} \implies a - b = -1$. Counting integer pairs (a,b) in [-3, 3]^2$(a,b) \in [-3, 3]^2$ with a+b neq 0$a+b \neq 0$: From a - b = 2$a - b = 2$: valid pairs are (3,1), (1,-1), (0,-2), (-1,-3)$(3,1), (1,-1), (0,-2), (-1,-3)$. Note: (2,0)$(2,0)$ is valid, but a+b=2 neq 0$a+b=2 \neq 0$. Total = 5 pairs. From a - b = -1$a - b = -1$: valid pairs match another 5 configurations. Combining both groups gives a final count of 10 pairs. ### Pattern Recognition Using matrix summation properties (1+omega+omega^2=0$1+\omega+\omega^2=0$) helps simplify large complex variable equations quickly. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Mathematics: Complex Numbers Class 12 Mathematics: Matrices and Determinants

### Related Formula Complex rotation and scaling vector rule: z_2 = frac|z_2||z_1| z_1 e^itheta$z_2 = \frac{|z_2|}{|z_1|} z_1 e^{i\theta}$ ### Core Logic Given structural rotation conditions: z_2 = frac1sqrt3 z_1 e^ifracpi6$z_2 = \frac{1}{\sqrt{3}} z_1 e^{i\frac{\pi}{6}}$ Evaluating the vectors yields coordinates showing |z_1 - z_2| = |z_2|$|z_1 - z_2| = |z_2|$. ### Step 1: Analyzing Geometry Metrics Since |z_1 - z_2| = |z_2|$|z_1 - z_2| = |z_2|$, Delta ABO$\Delta ABO$ forms an isosceles triangle with internal vertex angles evaluating explicitly to fracpi6, fracpi6$\frac{\pi}{6}, \frac{\pi}{6}$, and frac2pi3$\frac{2\pi}{3}$. ### Step 2: Conclusion Since frac2pi3 > fracpi2$\frac{2\pi}{3} > \frac{\pi}{2}$, the triangle is an obtuse-angled isosceles triangle. ### Pattern Recognition Complex argument shifts represent pure coordinate system rotations on the Argand plane diagram matrix. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Maths: Complex Numbers

### Related Formula For a quadratic system equation az^2+bz+c=0$az^2+bz+c=0$ with roots z_1, z_2$z_1, z_2$: 1. z_1 + z_2 = -b/a$z_1 + z_2 = -b/a$ 2. z_1 z_2 = c/a$z_1 z_2 = c/a$ 3. z_1^2 + z_2^2 = (z_1+z_2)^2 - 2z_1z_2$z_1^2 + z_2^2 = (z_1+z_2)^2 - 2z_1z_2$ ### Core Logic Cross-multiply the denominators to configure a linear equation layout [cite: 1355]: z^2 + 3i = (z - 2 + i)(2 + 3i)$z^2 + 3i = (z - 2 + i)(2 + 3i)$ [cite: 1355] z^2 + 3i = z(2 + 3i) + (-2 + i)(2 + 3i)$z^2 + 3i = z(2 + 3i) + (-2 + i)(2 + 3i)$ [cite: 1355] z^2 + 3i = z(2 + 3i) - 4 - 6i + 2i - 3 = z(2 + 3i) - 7 - 4i$z^2 + 3i = z(2 + 3i) - 4 - 6i + 2i - 3 = z(2 + 3i) - 7 - 4i$ [cite: 1355] Formulate the classic quadratic representation layout [cite: 1356]: z^2 - z(2 + 3i) + 7 + 7i = 0$z^2 - z(2 + 3i) + 7 + 7i = 0$ [cite: 1356] ### Step 1: Summing the squared roots Identify coefficients from the structural template [cite: 1357]: z_1 + z_2 = 2 + 3i$z_1 + z_2 = 2 + 3i$ z_1 z_2 = 7 + 7i$z_1 z_2 = 7 + 7i$ Evaluate sum of possible squared values (z_1^2 + z_2^2$z_1^2 + z_2^2$) [cite: 1357]: z_1^2 + z_2^2 = (z_1 + z_2)^2 - 2z_1 z_2$z_1^2 + z_2^2 = (z_1 + z_2)^2 - 2z_1 z_2$ [cite: 1357] = (2 + 3i)^2 - 2(7 + 7i)$= (2 + 3i)^2 - 2(7 + 7i)$ [cite: 1357] = (4 - 9 + 12i) - (14 + 14i) = -5 + 12i - 14 - 14i$= (4 - 9 + 12i) - (14 + 14i) = -5 + 12i - 14 - 14i$ [cite: 1357] = -19 - 2i$= -19 - 2i$ [cite: 1358] ### Pattern Recognition The question asks for the sum of values of z^2$z^2$, meaning z_1^2 + z_2^2$z_1^2 + z_2^2$. Avoid using complex quadratic formulas to solve for z$z$ explicitly; structural expansions save massive computational effort. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Mathematics: Complex Numbers

### Core Logic Let's expand the terms by grouping real and imaginary parts explicitly: omega_1 = (8sintheta + 7costheta) + i(sintheta + 4costheta)$\omega_1 = (8\sin\theta + 7\cos\theta) + i(\sin\theta + 4\cos\theta)$ omega_2 = (sintheta + 4costheta) + i(8sintheta + 7costheta)$\omega_2 = (\sin\theta + 4\cos\theta) + i(8\sin\theta + 7\cos\theta)$ Notice that if we let u = 8sintheta + 7costheta$u = 8\sin\theta + 7\cos\theta$ and v = sintheta + 4costheta$v = \sin\theta + 4\cos\theta$, then: omega_1 = u + iv quad textand quad omega_2 = v + iu$\omega_1 = u + iv \quad \text{and} \quad \omega_2 = v + iu$ ### Step 1: Calculating the Product Multiplying omega_1$\omega_1$ and \omega_2: omega_1omega_2 = (u + iv)(v + iu) = uv + iu^2 + iv^2 - uv = i(u^2 + v^2)$\omega_1\omega_2 = (u + iv)(v + iu) = uv + iu^2 + iv^2 - uv = i(u^2 + v^2)$ Since the product is given as alpha + ibeta$\alpha + i\beta$: alpha = 0$\alpha = 0$ beta = u^2 + v^2 = (8sintheta + 7costheta)^2 + (sintheta + 4costheta)^2$\beta = u^2 + v^2 = (8\sin\theta + 7\cos\theta)^2 + (\sin\theta + 4\cos\theta)^2$ ### Step 2: Simplifying the expression for alpha + beta Expanding the terms for beta$\beta$: beta = (64sin^2theta + 49cos^2theta + 112sinthetacostheta) + (sin^2theta + 16cos^2theta + 8sinthetacostheta)$\beta = (64\sin^2\theta + 49\cos^2\theta + 112\sin\theta\cos\theta) + (\sin^2\theta + 16\cos^2\theta + 8\sin\theta\cos\theta)$ alpha + beta = 0 + beta = 65sin^2theta + 65cos^2theta + 120sinthetacostheta$\alpha + \beta = 0 + \beta = 65\sin^2\theta + 65\cos^2\theta + 120\sin\theta\cos\theta$ Using the identity sin^2theta + cos^2theta = 1$\sin^2\theta + \cos^2\theta = 1$ and 2sinthetacostheta = sin 2theta$2\sin\theta\cos\theta = \sin 2\theta$: alpha + beta = 65 + 60sin 2theta$\alpha + \beta = 65 + 60\sin 2\theta$ ### Step 3: Max and Min Extrema Analysis Since -1 le sin 2theta le 1$-1 \le \sin 2\theta \le 1$: textMaximum value p = 65 + 60(1) = 125$\text{Maximum value } p = 65 + 60(1) = 125$ textMinimum value q = 65 + 60(-1) = 5$\text{Minimum value } q = 65 + 60(-1) = 5$ Sum of maximum and minimum bounds equals: p + q = 125 + 5 = 130$p + q = 125 + 5 = 130$ ### Pattern Recognition Observe the symmetric structure in complex variables: (u+iv)$(u+iv)$ and (v+iu)$(v+iu)$. Their product structurally completely cancels out the real component, saving you from a highly messy component expansion. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Mathematics: Complex Numbers Class 11 Mathematics: Trigonometric Functions