Class 12th

The problem provides an equation involving the coordinates (α, β, γ) of a point P:
((α + β + γ) / √3)^2 + ((α - nγ) / √(l^2 + n^2))^2 + ((α - 2β + γ) / √6)^2 = 9

The locus of P(α, β, γ) is given by replacing (α, β, γ) with (x, y, z):
((x + y + z) / √3)^2 + ((lx - nz) / √(l^2 + n^2))^2 + ((x - 2y + z) / √6)^2 = 9

This represents the equation of an ellipsoid. The text proceeds by comparing coefficients. By expanding the equation, the coefficients of x^2, y^2, z^2, and cross-product terms are collected. From the given conditions:
Coefficient of x^2: 1/3 + l^2 / (l^2 + n^2) + 1/6 = 1
Coefficient of y^2: 1/3 + 0 + 4/6 = 1
Coefficient of z^2: 1/3 + n^2 / (l^2 + n^2) + 1/6 = 1
Coefficient of xz: -2ln / (l^2 + n^2) + 2/6 = 0
(The document seems to simplify this differently).

The text provides a shortcut:
1/3 + (l^2 / (l^2 + n^2)) + 1/2 = 1 (Combining 1/3 and 1/6 to 1/2)
And also, (ln / (l^2 + n^2)) = 0. This implies ln = 0.

If ln = 0, then from l^2/(l^2+n^2) = 1/2, this means 2l^2 = l^2+n^2, so l^2 = n^2, which implies l = ±n.
The text states l^2 + n^2 - 2ln = 0, which simplifies to (l - n)^2 = 0, therefore l - n = 0 or l = n.

The numbers 1, log10(4^x - 2), and log10(4^x + 18/5) are in an Arithmetic Progression (A.P.).
This means that the corresponding numbers 10^1, 10^(log10(4^x - 2)), and 10^(log10(4^x + 18/5)) are in a Geometric Progression (G.P.).
So, 10, 4^x - 2, and 4^x + 18/5 are in G.P.

For a G.P., the square of the middle term is equal to the product of the other two terms:
(4^x - 2)^2 = 10 * (4^x + 18/5)
Let y = 4^x.
(y - 2)^2 = 10y + 36
y^2 - 4y + 4 = 10y + 36
y^2 - 14y - 32 = 0
(y - 16)(y + 2) = 0
So, y = 16 or y = -2.

Since y = 4^x, y must be positive. Thus, 4^x = 16, which gives x = 2.

The determinant calculation that follows appears to be unrelated to the A.P./G.P. problem.
| [3, 1, 4], [1, 0, 2], [2, 1, 0] | = 3(0 - 2) - 1(0 - 4) + 4(1 - 0) = -6 + 4 + 4 = 2.

Given the integral In = ∫(log|x|)^n / x^19 dx.
Let t = log|x|, which implies x = e^t and dx = e^t dt.

The integral becomes:
In = ∫ e^(-20t) * t^n dt

Using integration by parts, where u = t^n and dv = e^(-20t) dt:
In = [t^n * e^(-20t) / -20] - ∫ n*t^(n-1) * e^(-20t) / -20 dt
In = e^(-20) / -20 - (n / -20) * In-1
20 * In = -e^(-20) + n * In-1

For n = 10: 20 * I10 = e^20 - 10 * I9 (Note: There seems to be a sign inconsistency in the original document's derivation vs. standard integration by parts, the document states e^20 instead of -e^(-20) and proceeds with e^20).
For n = 9: 20 * I9 = e^20 - 9 * I8

From these two equations, we can express e^20 from the second equation and substitute into the first:
e^20 = 20 * I9 + 9 * I8
20 * I10 = (20 * I9 + 9 * I8) - 10 * I9
20 * I10 = 10 * I9 + 9 * I8

Comparing this to the form α * I9 + β * I8, we get α = 10 and β = 9.
Therefore, α - β = 10 - 9 = 1.