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
Q17
jee_main_2026_21_jan_morning
Cube Roots of Unity
If x^2 + x + 1 = 0$x^{2} + x + 1 = 0$ , then the value of left(mathrmx+frac1mathrmxright)^4+left(mathrmx^2+frac1mathrmx^2right)^4+left(mathrmx^3+frac1mathrmx^3right)^4+ldots+left(mathrmx^25+frac1mathrmx^25right)^4$\left(\mathrm{x}+\frac{1}{\mathrm{x}}\right)^{4}+\left(\mathrm{x}^{2}+\frac{1}{\mathrm{x}^{2}}\right)^{4}+\left(\mathrm{x}^{3}+\frac{1}{\mathrm{x}^{3}}\right)^{4}+\ldots+\left(\mathrm{x}^{25}+\frac{1}{\mathrm{x}^{25}}\right)^{4}$ is :
- A. 128
- B. 162
- C. 175
- D. 145
Solution
### Related Formula
x^2 + x + 1 = 0 Rightarrow x = omega, omega^2$$x^2 + x + 1 = 0 \Rightarrow x = \omega, \omega^2$$
Properties of cube roots of unity:
omega^3 = 1$\omega^3 = 1$ and
1 + omega + omega^2 = 0$1 + \omega + \omega^2 = 0$.
### Core Logic
Let
alpha = omega$\alpha = \omega$. Then
frac1x = frac1omega = omega^2$\frac{1}{x} = \frac{1}{\omega} = \omega^2$.
The series is
sum_k=1^25 (omega^k + omega^2k)^4$\sum_{k=1}^{25} (\omega^k + \omega^{2k})^4$.
Evaluate the term
T_k = (omega^k + omega^2k)^4$T_k = (\omega^k + \omega^{2k})^4$ based on the modulo of
k$k$ with 3.
### Step 1: Cyclic Evaluation
Case 1:
k = 3m$k = 3m$ (multiples of 3)
T_3m = (omega^3m + omega^6m)^4 = (1 + 1)^4 = 2^4 = 16$$T_{3m} = (\omega^{3m} + \omega^{6m})^4 = (1 + 1)^4 = 2^4 = 16$$
There are
8$8$ such multiples up to 25 (
3, 6, 9, dots, 24$3, 6, 9, \dots, 24$).
Case 2:
k neq 3m$k \neq 3m$ (non-multiples of 3)
For
k = 1, 2, 4, 5, dots$k = 1, 2, 4, 5, \dots$
omega^k + omega^2k$\omega^k + \omega^{2k}$ will always be
omega + omega^2$\omega + \omega^2$ or
omega^2 + omega$\omega^2 + \omega$.
Since
1 + omega + omega^2 = 0$1 + \omega + \omega^2 = 0$, we have
omega + omega^2 = -1$\omega + \omega^2 = -1$.
T_k neq 3m = (-1)^4 = 1$$T_{k \neq 3m} = (-1)^4 = 1$$
There are
25 - 8 = 17$25 - 8 = 17$ such non-multiples up to 25.
### Step 2: Total Sum
textSum = 17 times 1 + 8 times 16$$\text{Sum} = 17 \times 1 + 8 \times 16$$
= 17 + 128 = 145$$= 17 + 128 = 145$$
### Pattern Recognition
Powers of
x+1/x$x+1/x$ when
x$x$ solves
x^2 pm x + 1 = 0$x^2 \pm x + 1 = 0$ perfectly orbit around periods of 3 or 6. Isolate the
3m$3m$ resonant beats (which hit pure scalars like
1+1=2$1+1=2$) versus the out-of-phase beats (which collapse to
-1$-1$ or
1$1$ via basic
omega$\omega$ identities).
### Evaluation Rubric / Model Answer
null
### Chapter Mix
Class 11 Maths: Complex Numbers and Quadratic Equations
Q
jee_main_2025_02_april_morning
Geometry of Complex Numbers
Let z$z$ be a complex number such that |z| = 1$|z| = 1$. If frac2 + k^2zk + overlinez = kz, k in mathbbR$\frac{2 + k^2z}{k + \overline{z}} = kz, k \in \mathbb{R}$, then the maximum distance of k + ik^2$k + ik^2$ from the circle |z - (1 + 2i)| = 1$|z - (1 + 2i)| = 1$ is:
- A. sqrt5 + 1$\sqrt{5} + 1$
- B. 2$2$
- C. 3$3$
- D. sqrt3 + 1$\sqrt{3} + 1$
Solution
### Related Formula
For a complex number lying on the unit circle:
|z| = 1 implies zoverlinez = 1 implies overlinez = frac1z$$|z| = 1 \implies z\overline{z} = 1 \implies \overline{z} = \frac{1}{z}$$
Maximum distance from a point P$P$ to a circle with center C$C$ and radius r$r$ is:
d_max = PC + r$$d_{\max} = PC + r$$
### Core Logic
Simplify the algebraic condition using overlinez = 1/z$\overline{z} = 1/z$ to uniquely determine the value of the parameter k$k$, then compute geometric distances.
### Step 1: Solve for k
Cross-multiply the given expression:
2 + k^2z = kz(k + overlinez) = k^2z + kzoverlinez$$2 + k^2z = kz(k + \overline{z}) = k^2z + kz\overline{z}$$
Since zoverlinez = |z|^2 = 1$z\overline{z} = |z|^2 = 1$:
2 + k^2z = k^2z + k(1) implies k = 2$$2 + k^2z = k^2z + k(1) \implies k = 2$$
### Step 2: Locate Point and Circle Parameters
Substitute k=2$k=2$ into the target point expression P = k + ik^2$P = k + ik^2$:
P = 2 + 4i equiv (2,4)$$P = 2 + 4i \equiv (2,4)$$
The circle equation is |z - (1 + 2i)| = 1$|z - (1 + 2i)| = 1$, which represents a circle centered at C = (1, 2)$C = (1, 2)$ with radius r = 1$r = 1$.
### Step 3: Compute Geometric Distances
Find the Euclidean distance between P(2,4)$P(2,4)$ and center C(1,2)$C(1,2)$:
PC = sqrt(2-1)^2 + (4-2)^2 = sqrt1 + 4 = sqrt5$$PC = \sqrt{(2-1)^2 + (4-2)^2} = \sqrt{1 + 4} = \sqrt{5}$$
The maximum distance from the point to the circle boundary is:
d_max = PC + r = sqrt5 + 1$$d_{\max} = PC + r = \sqrt{5} + 1$$
### Pattern Recognition
Notice how k^2z$k^2z$ cancels perfectly on both sides during expansion due to the unique property of uni-modular complex numbers (zoverlinez=1$z\overline{z}=1$), rendering the calculation of k$k$ trivial.
### Evaluation Rubric / Model Answer
null
### Chapter Mix
Class 11 Mathematics: Complex Numbers and Quadratic Equations
Q
jee_main_2025_02_april_morning
Theory of Equations
Let P_n = alpha^n + beta^n$P_n = \alpha^n + \beta^n$, n in mathbbN$n \in \mathbb{N}$. If P_10 = 123$P_{10} = 123$, P_9 = 76$P_9 = 76$, P_8 = 47$P_8 = 47$ and P_1 = 1$P_1 = 1$, then the quadratic equation having roots frac1alpha$\frac{1}{\alpha}$ and frac1beta$\frac{1}{\beta}$ is:
- A. x^2 - x + 1 = 0$x^2 - x + 1 = 0$
- B. x^2 + x - 1 = 0$x^2 + x - 1 = 0$
- C. x^2 - x - 1 = 0$x^2 - x - 1 = 0$
- D. x^2 + x + 1 = 0$x^2 + x + 1 = 0$
Solution
### Related Formula
Newton's Sums for the roots of a quadratic equation ax^2 + bx + c = 0$ax^2 + bx + c = 0$:
a P_n + b P_n-1 + c P_n-2 = 0$$a P_n + b P_{n-1} + c P_{n-2} = 0$$
### Core Logic
Observe the recurrence relation from the given numerical values of P_n$P_n$ to construct the base quadratic equation satisfied by alpha$\alpha$ and beta$\beta$, then invert the roots.
### Step 1: Identify the Linear Recurrence Relation
Compare the provided sequence values:
P_8 + P_9 = 47 + 76 = 123 = P_10$$P_8 + P_9 = 47 + 76 = 123 = P_{10}$$
This fits the general sequence relation:
P_n = P_n-1 + P_n-2 implies P_n - P_n-1 - P_n-2 = 0$$P_n = P_{n-1} + P_{n-2} \implies P_n - P_{n-1} - P_{n-2} = 0$$
### Step 2: Construct the Base Quadratic Equation
The characteristic equation corresponding to this recurrence relation is:
x^2 - x - 1 = 0$x^2 - x - 1 = 0$
Thus, alpha$\alpha$ and \(\beta\) are roots of x^2 - x - 1 = 0$x^2 - x - 1 = 0$, giving sum alpha+beta = 1$\alpha+\beta = 1$ and product alphabeta = -1$\alpha\beta = -1$ (which matches P_1 = alpha+beta = 1$P_1 = \alpha+\beta = 1$).
### Step 3: Construct Equation with Reciprocal Roots
To find the equation with roots frac1alpha$\frac{1}{\alpha}$ and frac1beta$\frac{1}{\beta}$, apply the transformations:
textSum of new roots = frac1alpha + frac1beta = fracalpha + betaalphabeta = frac1-1 = -1$$\text{Sum of new roots} = \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\alpha + \beta}{\alpha\beta} = \frac{1}{-1} = -1$$
textProduct of new roots = frac1alphabeta = frac1-1 = -1$$\text{Product of new roots} = \frac{1}{\alpha\beta} = \frac{1}{-1} = -1$$
The new quadratic equation is:
x^2 - (textSum)x + (textProduct) = 0 implies x^2 - (-1)x + (-1) = 0 implies x^2 + x - 1 = 0$$x^2 - (\text{Sum})x + (\text{Product}) = 0 \implies x^2 - (-1)x + (-1) = 0 \implies x^2 + x - 1 = 0$$
### Pattern Recognition
The recurrence pattern P_n = P_n-1 + P_n-2$P_n = P_{n-1} + P_{n-2}$ is the Fibonacci sequence recurrence line. Its roots generate the Golden Ratio layout from x^2-x-1=0$x^2-x-1=0$. Inverting roots swaps the coefficients of x^2$x^2$ and the constant term, yielding x^2+x-1=0$x^2+x-1=0$ instantly.
### Evaluation Rubric / Model Answer
null
### Chapter Mix
Class 11 Mathematics: Complex Numbers and Quadratic Equations
Class 11 Mathematics: Sequences and Series
Q65
jee_main_2025_03_april_evening
Geometry of Complex Numbers
If z_1, z_2, z_3 in mathbbC$z_1, z_2, z_3 \in \mathbb{C}$ are the vertices of an equilateral triangle, whose centroid is z_0$z_0$, then sum_k=1^3 (z_k - z_0)^2$\sum_{k=1}^{3} (z_k - z_0)^2$ is equal to
- A. 0$0$
- B. 1$1$
- C. i$i$
- D. -i$-i$
Solution
### Related Formula
For any equilateral triangle with vertices z_1, z_2, z_3$z_1, z_2, z_3$:
z_1^2 + z_2^2 + z_3^2 = z_1 z_2 + z_2 z_3 + z_3 z_1 quad text--- (1)$$z_1^2 + z_2^2 + z_3^2 = z_1 z_2 + z_2 z_3 + z_3 z_1 \quad \text{--- (1)}$$
Centroid equation:
z_0 = fracz_1 + z_2 + z_33 implies z_1 + z_2 + z_3 = 3z_0$$z_0 = \frac{z_1 + z_2 + z_3}{3} \implies z_1 + z_2 + z_3 = 3z_0$$
### Core Logic
Let's expand the target sum:
sum_k=1^3 (z_k - z_0)^2 = (z_1 - z_0)^2 + (z_2 - z_0)^2 + (z_3 - z_0)^2$$\sum_{k=1}^{3} (z_k - z_0)^2 = (z_1 - z_0)^2 + (z_2 - z_0)^2 + (z_3 - z_0)^2$$
### Step 1: Expansion and Algebraic Grouping
Expanding each quadratic term:
= (z_1^2 + z_2^2 + z_3^2) - 2z_0(z_1 + z_2 + z_3) + 3z_0^2$$= (z_1^2 + z_2^2 + z_3^2) - 2z_0(z_1 + z_2 + z_3) + 3z_0^2$$
Substitute z_1 + z_2 + z_3 = 3z_0$z_1 + z_2 + z_3 = 3z_0$:
= (z_1^2 + z_2^2 + z_3^2) - 2z_0(3z_0) + 3z_0^2$$= (z_1^2 + z_2^2 + z_3^2) - 2z_0(3z_0) + 3z_0^2$$
= z_1^2 + z_2^2 + z_3^2 - 3z_0^2$$= z_1^2 + z_2^2 + z_3^2 - 3z_0^2$$
### Step 2: Resolving using Equilateral Condition
Substitute 3z_0^2 = 3 left(fracz_1+z_2+z_33right)^2 = frac(z_1+z_2+z_3)^23$3z_0^2 = 3 \left(\frac{z_1+z_2+z_3}{3}\right)^2 = \frac{(z_1+z_2+z_3)^2}{3}$:
= (z_1^2 + z_2^2 + z_3^2) - fracz_1^2 + z_2^2 + z_3^2 + 2(z_1z_2 + z_2z_3 + z_3z_1)3$$= (z_1^2 + z_2^2 + z_3^2) - \frac{z_1^2 + z_2^2 + z_3^2 + 2(z_1z_2 + z_2z_3 + z_3z_1)}{3}$$
= frac2(z_1^2 + z_2^2 + z_3^2) - 2(z_1z_2 + z_2z_3 + z_3z_1)3$$= \frac{2(z_1^2 + z_2^2 + z_3^2) - 2(z_1z_2 + z_2z_3 + z_3z_1)}{3}$$
= frac23(z_1^2 + z_2^2 + z_3^2 - (z_1 z_2 + z_2 z_3 + z_3 z_1))$$= \frac{2}{3}(z_1^2 + z_2^2 + z_3^2 - (z_1 z_2 + z_2 z_3 + z_3 z_1))$$
Using condition (1) for equilateral triangles, the terms inside the parentheses equal 0$0$. Thus:
= 0$= 0$
### Pattern Recognition
This is a standard invariant of equilateral triangles. Any translation to center of mass coordinates leaves the shape invariant, making the sum of squares of coordinate vectors relative to the centroid equal to zero.
### Evaluation Rubric / Model Answer
null
### Chapter Mix
Class 11 Mathematics: Complex Numbers
