Related Formula
Sum of roots (α + β) = -(b)/(a)$$\text{Sum of roots } (\alpha + \beta) = -\frac{b}{a}$$
Product of roots (α β) = (c)/(a)$$\text{Product of roots } (\alpha \beta) = \frac{c}{a}$$
Core Logic
For the equation x² - 70x + λ = 0$x^2 - 70x + \lambda = 0$:
α + β = 70$\alpha + \beta = 70$
α β = λ$\alpha \beta = \lambda$
Since α, β in N$\alpha, \beta \in \mathbb{N}$, their sum is 70. This gives α(70 - α) = λ$\alpha(70 - \alpha) = \lambda$.
We are given λ / 2 N$\lambda / 2 \notin \mathbb{N}$ and λ / 3 N$\lambda / 3 \notin \mathbb{N}$.
This means λ$\lambda$ is not divisible by 2 or 3. So λ$\lambda$ must be an odd number and not a multiple of 3.
Thus, neither α$\alpha$ nor β$\beta$ can be a multiple of 2 or 3.
Step 1: Finding minimum lambda
We need to find the minimum value of λ$\lambda$. λ = α(70 - α)$\lambda = \alpha(70 - \alpha)$.
This parabola opens downwards, so the minimum product occurs when the integers α$\alpha$ and β$\beta$ are as far apart as possible.
Let's test small values for α$\alpha$:
If α = 1$\alpha = 1$, β = 69 ⇒ λ = 69$\beta = 69 \Rightarrow \lambda = 69$, but 69 = 3 × 23$69 = 3 \times 23$, which is divisible by 3. Rejected.
If α = 2$\alpha = 2$, divisible by 2. Rejected.
If α = 3$\alpha = 3$, divisible by 3. Rejected.
If α = 4$\alpha = 4$, divisible by 2. Rejected.
If α = 5$\alpha = 5$, β = 65 ⇒ λ = 325$\beta = 65 \Rightarrow \lambda = 325$.
Check divisibility: 325 is odd (not div by 2). 3+2+5 = 10$3+2+5 = 10$ (not div by 3).
So minimum λ = 325$\lambda = 325$ with roots α = 5, β = 65$\alpha = 5, \beta = 65$.
Step 2: Evaluating the target expression
We need to compute:
E = (√(α - 1) + √(β - 1))(λ + 35)|α - β|$$E = \frac{(\sqrt{\alpha - 1} + \sqrt{\beta - 1})(\lambda + 35)}{|\alpha - \beta|}$$
Substitute the values α = 5$\alpha = 5$, β = 65$\beta = 65$, λ = 325$\lambda = 325$:
|α - β| = |5 - 65| = 60$$|\alpha - \beta| = |5 - 65| = 60$$
√(α - 1) = √(4) = 2$$\sqrt{\alpha - 1} = \sqrt{4} = 2$$
√(β - 1) = √(64) = 8$$\sqrt{\beta - 1} = \sqrt{64} = 8$$
E = ((2 + 8)(325 + 35))/(60) = ((10)(360))/(60) = 10 × 6 = 60$$E = \frac{(2 + 8)(325 + 35)}{60} = \frac{(10)(360)}{60} = 10 \times 6 = 60$$
The result is exactly 60.
Pattern Recognition
Minimizing the product of two numbers with a fixed sum requires them to be as far apart as possible. Divisibility constraints are quickly verified using prime modulus filters (%2 ≠ 0$%2 \neq 0$, %3 ≠ 0$%3 \neq 0$).
Chapter Mix
Class 11 Maths: Complex Numbers and Quadratic Equations
