If alpha+ibeta$\alpha+i\beta$ and gamma+idelta$\gamma+i\delta$ are the roots of x^2-(3-2i)x-(2i-2)=0,$x^{2}-(3-2i)x-(2i-2)=0,$ i=sqrt-1$i=\sqrt{-1}$ then alphagamma+betadelta$\alpha\gamma+\beta\delta$ is equal to :

### Related Formula A complex number w$w$ is purely real if w = barw$w = \bar{w}$. A complex number w$w$ is purely imaginary if w + barw = 0$w + \bar{w} = 0$. ### Core Logic Let's evaluate statement **(S1)**: w = fracz - iz + i$$w = \frac{z - i}{z + i}$$ If w$w$ is purely real, then w = barw$w = \bar{w}$: fracz - iz + i = fracbarz + ibarz - i$$\frac{z - i}{z + i} = \frac{\bar{z} + i}{\bar{z} - i}$$ (z - i)(barz - i) = (z + i)(barz + i)$$(z - i)(\bar{z} - i) = (z + i)(\bar{z} + i)$$ |z|^2 - iz - ibarz - 1 = |z|^2 + iz + ibarz - 1$$|z|^2 - iz - i\bar{z} - 1 = |z|^2 + iz + i\bar{z} - 1$$ -i(z + barz) = i(z + barz) implies 2i(z + barz) = 0 implies z + barz = 0$$-i(z + \bar{z}) = i(z + \bar{z}) \implies 2i(z + \bar{z}) = 0 \implies z + \bar{z} = 0$$ Since z + barz = 2textRe(z) = 0$z + \bar{z} = 2\text{Re}(z) = 0$, z$z$ must lie on the imaginary axis (y-axis). Given the condition |z| = 1$|z| = 1$, the only points are z = i$z = i$ and z = -i$z = -i$. However, the domain excludes z = -i$z = -i$. Let's test z = i$z = i$: For z = i$z = i$, fraci - ii + i = 0$\frac{i - i}{i + i} = 0$, which is purely real. So it contains elements on the unit circle. But the condition z + barz = 0$z + \bar{z} = 0$ alongside |z|=1$|z|=1$ explicitly limits it to z=i$z=i$ only, which is one element, not two. Thus, (S1) is incorrect. ### Step 1: Evaluate Statement S2 Let's evaluate statement **(S2)**: u = fracz - 1z + 1$$u = \frac{z - 1}{z + 1}$$ If u$u$ is purely imaginary, then u + baru = 0$u + \bar{u} = 0$: fracz - 1z + 1 + fracbarz - 1barz + 1 = 0$$\frac{z - 1}{z + 1} + \frac{\bar{z} - 1}{\bar{z} + 1} = 0$$ frac(z - 1)(barz + 1) + (z + 1)(barz - 1)(z + 1)(barz + 1) = 0$$\frac{(z - 1)(\bar{z} + 1) + (z + 1)(\bar{z} - 1)}{(z + 1)(\bar{z} + 1)} = 0$$ (|z|^2 + z - barz - 1) + (|z|^2 - z + barz - 1) = 0$$(|z|^2 + z - \bar{z} - 1) + (|z|^2 - z + \bar{z} - 1) = 0$$ 2|z|^2 - 2 = 0 implies |z|^2 = 1 implies |z| = 1$$2|z|^2 - 2 = 0 \implies |z|^2 = 1 \implies |z| = 1$$ This condition holds true for ALL points on the unit circle |z| = 1$|z| = 1$ except z = -1$z = -1$ (which makes the denominator zero). Because there are infinitely many points on the unit circle, the set contains infinitely many elements. Thus, (S2) is correct. ### Pattern Recognition Geometric shortcut: The transformation w = fracz-1z+1$w = \frac{z-1}{z+1}$ maps the unit circle |z|=1$|z|=1$ directly onto the imaginary axis textRe(w)=0$\text{Re}(w)=0$. Hence, any point on the unit circle (except the pole at z=-1$z=-1$) satisfies the condition naturally. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Mathematics: Complex Numbers and Quadratic Equations

### 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