Solution & Explanation
### Related Formula
textFor a quadratic equation A x^2 + B x + C = 0 text to have two positive real roots:$$\text{For a quadratic equation } A x^2 + B x + C = 0 \text{ to have two positive real roots:}$$
text1. Real roots: D = B^2 - 4AC ge 0$$\text{1. Real roots: } D = B^2 - 4AC \ge 0$$
text2. Sum of roots: -fracBA > 0$$\text{2. Sum of roots: } -\frac{B}{A} > 0$$
text3. Product of roots: fracCA > 0$$\text{3. Product of roots: } \frac{C}{A} > 0$$
### Core Logic
We write down three systems of inequalities based on real and positive root conditions, find their intersection, and map the boundaries to solve for the parameters.
### Step 1: Apply the discriminant condition (Real roots)
For real roots, the discriminant D ge 0$D \ge 0$:
D = left[ 2(a - 3) right]^2 - 4(1 - a)(9) ge 0$$D = \left[ 2(a - 3) \right]^2 - 4(1 - a)(9) \ge 0$$
4(a^2 - 6a + 9) - 36(1 - a) ge 0$$4(a^2 - 6a + 9) - 36(1 - a) \ge 0$$
(a^2 - 6a + 9) - 9(1 - a) ge 0$$(a^2 - 6a + 9) - 9(1 - a) \ge 0$$
a^2 - 6a + 9 - 9 + 9 a ge 0$$a^2 - 6a + 9 - 9 + 9 a \ge 0$$
a^2 + 3a ge 0 implies a(a + 3) ge 0$$a^2 + 3a \ge 0 \implies a(a + 3) \ge 0$$
Thus, the interval is:
a in (-infty, -3] cup [0, infty) quad text--- (1)$$a \in (-\infty, -3] \cup [0, \infty) \quad \text{--- (1)}$$
### Step 2: Apply the sum of roots condition (Positive sum)
For positive roots, the sum of roots must be positive:
-fracBA = frac-2(a - 3)1 - a = frac2(a - 3)a - 1 > 0$$-\frac{B}{A} = \frac{-2(a - 3)}{1 - a} = \frac{2(a - 3)}{a - 1} > 0$$
Using the wavy curve method for fraca-3a-1 > 0$\frac{a-3}{a-1} > 0$:
a in (-infty, 1) cup (3, infty) quad text--- (2)$$a \in (-\infty, 1) \cup (3, \infty) \quad \text{--- (2)}$$
### Step 3: Apply the product of roots condition (Positive product)
For positive roots, the product of roots must be positive:
fracCA = frac91 - a > 0 implies 1 - a > 0 implies a < 1$$\frac{C}{A} = \frac{9}{1 - a} > 0 \implies 1 - a > 0 \implies a < 1$$
Thus, the interval is:
a in (-infty, 1) quad text--- (3)$$a \in (-\infty, 1) \quad \text{--- (3)}$$
### Step 4: Find the intersection of all conditions
Intersecting equations (1), (2), and (3):
- First, intersect (2) and (3):
( (-infty, 1) cup (3, infty) ) cap (-infty, 1) = (-infty, 1)$$( (-\infty, 1) \cup (3, \infty) ) \cap (-\infty, 1) = (-\infty, 1)$$
- Next, intersect with (1):
( (-infty, -3] cup [0, infty) ) cap (-infty, 1) = (-infty, -3] cup [0, 1)$$( (-\infty, -3] \cup [0, \infty) ) \cap (-\infty, 1) = (-\infty, -3] \cup [0, 1)$$
Comparing this with (-infty, -alpha] cup [beta, gamma)$(-\infty, -\alpha] \cup [\beta, \gamma)$:
- alpha = 3$\alpha = 3$
- beta = 0$\beta = 0$
- gamma = 1$\gamma = 1$
Now calculate the target sum:
2alpha + beta + gamma = 2(3) + 0 + 1 = 7$$2\alpha + \beta + \gamma = 2(3) + 0 + 1 = 7$$
### Pattern Recognition
Location of roots: When both roots are positive, checking sum and product signs along with D ge 0$D \ge 0$ is the standard and fastest set of inequalities, avoiding complex vertex projections.
### Evaluation Rubric / Model Answer
null
### Chapter Mix
Class 11 Chemistry: Practical Chemistry
More Complex Numbers and Quadratic Equations Previous-Year Questions — Page 2
Q61
jee_main_2025_07_april_morning
Geometry of Complex Numbers
Among the statements
(S1): The set \zin mathbbC - \-i\:|z| = 1text and fracz - iz + itext is purely real\$\{z\in \mathbb{C} - \{-i\}:|z| = 1\text{ and }\frac{z - i}{z + i}\text{ is purely real}\}$ contains exactly two elements, and
(S2) : The set \z in mathbbC - \-1\ : |z| = 1text and fracz - 1z + 1text is purely imaginary\$\{z \in \mathbb{C} - \{-1\} : |z| = 1\text{ and }\frac{z - 1}{z + 1}\text{ is purely imaginary}\}$ contains infinitely many elements.
- A. textboth are incorrect$\text{both are incorrect}$
- B. textonly (S1) is correct$\text{only (S1) is correct}$
- C. textonly (S2) is correct$\text{only (S2) is correct}$
- D. textboth are correct$\text{both are correct}$
Solution
### 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
Q64
jee_main_2025_08_april_evening
Purely Real/Imaginary Conditions
Let mathrmA = left\theta in [0,2pi ]:1 + 10operatorname Releft(frac2costheta + mathrmisinthetacostheta - 3mathrmisinthetaright) = 0right\$\mathrm{A} = \left\{\theta \in [0,2\pi ]:1 + 10\operatorname {Re}\left(\frac{2\cos\theta + \mathrm{i}\sin\theta}{\cos\theta - 3\mathrm{i}\sin\theta}\right) = 0\right\}$. Then sum_theta in mathrmAtheta^2$\sum_{\theta \in \mathrm{A}}\theta^2$ is equal to
- A. frac214pi^2$\frac{21}{4}\pi^2$
- B. 8pi^2$8\pi^{2}$
- C. frac274pi^2$\frac{27}{4}\pi^2$
- D. 6pi^2$6\pi^{2}$
Solution
### 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
Q74
jee_main_2025_29_jan_evening
Determinants and Roots of Unity
Let integers a, b in [-3, 3]$a, b \in [-3, 3]$ be such that a + b neq 0$a + b \neq 0$. Then the number of all possible ordered pairs (a, b)$(a, b)$, for which left| fracz - az + b right| = 1$\left| \frac{z - a}{z + b} \right| = 1$ and
left| beginarraycccz + 1 & omega & omega^2\\ omega & z + omega^2 & 1\\ omega^2 & 1 & z + omega endarray right| = 1, z in mathbbC$\left| \begin{array}{ccc}z + 1 & \omega & \omega^2\\ \omega & z + \omega^2 & 1\\ \omega^2 & 1 & z + \omega \end{array} \right| = 1, z \in \mathbb{C}$, where omega$\omega$ and omega^2$\omega^2$ are the roots of x^2 + x + 1 = 0$x^{2} + x + 1 = 0$, is equal to
Numerical Answer. Answer: 10 to 10
Solution
### 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
Q68
jee_main_2025_28_jan_morning
Geometry of Complex Numbers
Let O be the origin, the point A be z_1 = sqrt3 + 2sqrt2i$z_1 = \sqrt{3} + 2\sqrt{2}i$, the point B(z_2)$B(z_2)$ be such that sqrt3left|z_2right| = left|z_1right|$\sqrt{3}\left|z_2\right| = \left|z_1\right|$ and arg (z_2) = arg (z_1) + fracpi6$\arg (z_2) = \arg (z_1) + \frac{\pi}{6}$. Then
(1) area of triangle ABO is frac11sqrt3$\frac{11}{\sqrt{3}}$
(2) ABO is a scalene triangle
(3) area of triangle ABO is frac114$\frac{11}{4}$
(4) ABO is an obtuse angled isosceles triangle
- A. area of triangle ABO is frac11sqrt3$\frac{11}{\sqrt{3}}$
- B. ABO is a scalene triangle
- C. area of triangle ABO is frac114$\frac{11}{4}$
- D. ABO is an obtuse angled isosceles triangle
Solution
### 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}}$$
Evaluating the vectors yields coordinates showing |z_1 - z_2| = |z_2|$|z_1 - z_2| = |z_2|$.
### Step 1: Analyzing Geometry Metrics
Since |z_1 - z_2| = |z_2|$|z_1 - z_2| = |z_2|$, Delta ABO$\Delta ABO$ forms an isosceles triangle with internal vertex angles evaluating explicitly to fracpi6, fracpi6$\frac{\pi}{6}, \frac{\pi}{6}$, and frac2pi3$\frac{2\pi}{3}$.
### Step 2: Conclusion
Since frac2pi3 > fracpi2$\frac{2\pi}{3} > \frac{\pi}{2}$, the triangle is an obtuse-angled isosceles triangle.
### Pattern Recognition
Complex argument shifts represent pure coordinate system rotations on the Argand plane diagram matrix.
### Evaluation Rubric / Model Answer
null
### Chapter Mix
Class 11 Maths: Complex Numbers
Q64
jee_main_2025_03_april_morning
Roots of Quadratic Equations in Complex Fields
Let zin mathbbC$z\in \mathbb{C}$ be such that fracz^2 + 3iz - 2 + i = 2 + 3i$\frac{z^2 + 3i}{z - 2 + i} = 2 + 3i$[cite: 627, 629]. Then the sum of all possible values of z^2$z^2$ is[cite: 630]:
- A. 19 - 2i
- B. -19 - 2i
- C. 19 + 2i
- D. -19 + 2i
Solution
### Related Formula
For a quadratic system equation az^2+bz+c=0$az^2+bz+c=0$ with roots z_1, z_2$z_1, z_2$:
1. z_1 + z_2 = -b/a$z_1 + z_2 = -b/a$
2. z_1 z_2 = c/a$z_1 z_2 = c/a$
3. z_1^2 + z_2^2 = (z_1+z_2)^2 - 2z_1z_2$z_1^2 + z_2^2 = (z_1+z_2)^2 - 2z_1z_2$
### Core Logic
Cross-multiply the denominators to configure a linear equation layout [cite: 1355]:
z^2 + 3i = (z - 2 + i)(2 + 3i)$$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
