Maths

If the orthocenter and circumcenter of a triangle both lie on the y-axis, the centroid also lies on the y-axis.
The x-coordinate of the centroid is (x1 + x2 + x3) / 3. If the vertices are (cos α, sin α), (cos β, sin β), (cos γ, sin γ), then the x-coordinate of the centroid is (cos α + cos β + cos γ) / 3.
Since the centroid lies on the y-axis, its x-coordinate is 0.
cos α + cos β + cos γ = 0

Using the identity: If a + b + c = 0, then a^3 + b^3 + c^3 = 3abc.
Let a = cos α, b = cos β, c = cos γ.
Then, cos^3 α + cos^3 β + cos^3 γ = 3 * cos α * cos β * cos γ.

We need to evaluate the expression:
(cos 3α + cos 3β + cos 3γ) / (cos α * cos β * cos γ)

Using the identity cos 3θ = 4cos^3 θ - 3cos θ:
cos 3α + cos 3β + cos 3γ = (4cos^3 α - 3cos α) + (4cos^3 β - 3cos β) + (4cos^3 γ - 3cos γ)
= 4(cos^3 α + cos^3 β + cos^3 γ) - 3(cos α + cos β + cos γ)

Substitute the known values:
= 4(3 * cos α * cos β * cos γ) - 3(0)
= 12 * cos α * cos β * cos γ

So, the expression becomes:
(12 * cos α * cos β * cos γ) / (cos α * cos β * cos γ) = 12.
The value of the expression squared is 12^2 = 144.

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.

I = ∫[0 to 10] [sin(2πx)] / e^(x-[x]) dx.
The period of the integrand involves [sin(2πx)] which depends on the sign of sin(2πx) and {x} = x - [x], which has a period of 1.
Let f(x) = [sin(2πx)] / e^{x}.
The integral is ∫[0 to 10] f(x) dx = 10 * ∫[0 to 1] f(x) dx due to the periodicity of {x} and the integer period of sin(2πx).
In the interval (0, 1/2), sin(2πx) is between 0 and 1, so [sin(2πx)] = 0.
In the interval (1/2, 1), sin(2πx) is between -1 and 0, so [sin(2πx)] = -1.

At x=0, 1/2, 1, the value is 0.
So, ∫[0 to 1] f(x) dx = ∫[0 to 1/2] 0 dx + ∫[1/2 to 1] -1/e^x dx
= 0 + [-e^(-x) * (-1)] from 1/2 to 1 = [e^(-x)] from 1/2 to 1 = e?¹ - e?¹/².
The total integral is 10 * (e?¹ - e?¹/²).
The text calculates: 10 * ∫[1/2 to 1] (-1)e^(-x) dx = -10[-e^(-x)] from 1/2 to 1 = 10(e?¹ - e?¹/²).
This seems to lead to α = 10, β = -10, γ = 0. Then α+β+γ = 0. The calculation in the text has a sign error in the final result.

Parabola: y² = 4x - 20 = 4(x - 5). Vertex at (5,0).
Line: The text seems to derive the tangent equation y = x - 4. This is not a tangent to the given parabola. The standard tangent to y²=4aX is Y=mX+a/m. Here X=x-5, a=1. So y = m(x-5)+1/m.
The other curve is an ellipse: x²/a² + y²/b² = 1.
The text says x²/2 + (x-4)²/b² = 1. This assumes a² = 2.
x²/2 + (x²-8x+16)/b² = 1
x²(1/2 + 1/b²) - (8/b²)x + (16/b² - 1) = 0.
For tangency, the discriminant (D) of this quadratic equation must be zero.
D = (8/b²)² - 4(1/2 + 1/b²)(16/b² - 1) = 0.
64/b? - 4(8/b² - 1/2 + 16/b? - 1/b²) = 0.
16/b? - (7/b² - 1/2 + 16/b?) = 0.
-7/b² + 1/2 = 0 ⇒ b² = 14.
The calculation in the text is different: 64/b? - 4(1/2 + 1/b²)(16/b² - 1) = 0 implies b=14. This should be b²=14.