Explanation

If aₙ means the coefficient of xⁿ in the power series/Taylor series of f(x) = x - x³, then:

f(x) = x - x³ = 0 + 1·x + 0·x² + (-1)·x³ + 0·x⁴ + 0·x⁵ + ...

So:

a₀ = 0

a₁ = 1

a₂ = 0

a₃ = -1

a₄ = 0, a₅ = 0, ... all higher terms = 0

0 is also a finite value.

Last verified on 04-06-2026

Explanation

Formula: Ā × B̄ = |Ā||B̄| sinθ 

For non-zero vectors, |Ā||B̄| is never 0.  

So cross product = 0 only when sinθ = 0 → θ = 0° or 180°

That’s exactly same direction or opposite direction = parallel.  

This works for ALL parallel cases.

So correct answer is = Ā × B̄ = 0̄

Last verified on 04-06-2026

Explanation

f(x) = e^x - ax makes a U-shape because e^x is always curving up. 

U-shape has only 1 lowest point at x = ln a. 

So the minimum value = f(ln a) = a - a ln a

Last verified on 04-06-2026

Explanation

System:

x1 + 2x2 + x3 = 9

2x1 + x2 + 3x3 = 7

Matrix form AX = B:

[1 2 1] [x1] [9]

[2 1 3] [x2] = [7]

          [x3]

Augmented matrix [A|B]:

[1 2 1 | 9]

[2 1 3 | 7]

Last verified on 04-06-2026

Explanation

x = A cos(mt - a) is a wave/oscillation.

If you differentiate it twice, you get the same x back but with a minus sign and m² in front.

So the equation becomes: d²x/dt² = -m²x

Clear rule: Cos or Sin → 2nd derivative = -ω² × function

Last verified on 04-06-2026