Math Answers

Given the linear transformation below:

T (x₁, x₂, x₃) → (x₁-x₂+2x₃, 2x₁-2x₃, -x₁-x₂+4x₃, 3x₁-x₂)

T = R³ → R⁴

1. Determine the transformation matrix of the linear transformation above

2. Determine Ker(T) and Range(T)

Known vectors:

a = (α, 0, 1), b = (1, β, 1), c = (1, 1, γ)

Determine the values of α, β,γ when the three vectors are orthogonal to each other.

Known vectors:

⃗a = (1, 0, 1) , ⃗b = (0, 1, -1) , ⃗c = (0, 0, 1)

Find the angle between:

1. a and b

2. a and c

3. b and c

Solve

x ^ 2 * (d ^ 2 * y)/(d * x ^ 2) - 2x * (dy)/(dx) - 4y = x ^ 2 + 2 * log x

Solve

x ^ 2 * (d ^ 2 * y)/(dx) + x * (dy)/(dx) - 9y = 48x ^ 5

A labor union spokesman claims that 75% of union members will support a strike if their basic demands are not met. A company spokesman believes the true percentage is higher and runs a hypothesis testing at 10% significance level. (a) Compute the test statistic if 100 of 125 union members say they will strike.

Find the area of the surface cut from the bottom of the paraboloid x²+y²-z=0 by the plane z = 4

An inductor of 2 henries, resistor of 16 ohms and capacitor of 0.02 farads are connected in series with a battery of

e.m.f E = 100sin33t. At t=0, the charge on the capacitor and current in the circuit are zero. Find the charge and

current at time t.

Find the isogonal trajectories of the family of curves x²+y²=c if θ=45°.

It is known that the vectors in the vector space R³:

⃗v₁ = (1, 1, 1), v₂ = (2, -1, 1), v₃ = (0, 2, 1)

and

⃗w₁ = (2, -1, 3), w₂ = (3, -1, 7), w₃ = (-1, 1, 1)

The vectors v₁, v₂, v₃ are basis in R³. Transformation T : R³ → R³ is a linear transformation defined by:

 T( ⃗v_i) = ⃗w_i

Define:

1. Matrix transformation of T

2. Basis of Ker(T) and Range(T)