Reference no: EM132655464

Question 1. Find a matrix

Question 2. Determine all values of the constants a and b for which the following system has
(a) No solution
(b) A unique solution
(c) An infinite number of solutions

x + y - 4z = 10
3x + 5y + 6z = 20.
2x + 3y + az = b

Question 3. If v₁ = (3, -1, 2), v₂ = (-9, 5, -2), and v₃ = (-9, 9, 6), show that {v₁, v₂, v₃} is linearly dependent in R³, and determine the linear dependency relationship.

Question 4. Determine whether the given functions are LI or LD on the given interval (show your work):
f₁(x) = sin x, f₂ (x) = cos x, f₃ (x) = tan x, I = (-Π/2, Π / 2).

Question 5. Determine a basis for nullspace of A, NS(A), rowspace of A, RS(A), and columnspace of A, CS(A).

 

Question 6. Solve the given IVP:

(3x² ln x + x² - y)dx - xdy = 0, y(1) = 5.

Question 7. Solve the given ODE:

2xy dy/dx - (x²e^-y2/x2 + 2y² = 0.

Question 8. Solve the given ODE:

dy/dx + 4xy = 4x³y^1/2

Question 9. Solve the given ODE:

y" + 2y^-1(y')² = y'.

Question 10. Use Euler's method with specified step size to determine the solution to the given IVP at the specified point:

y" = 2xy², y(0) = 0.5, h = 0.1, y(0.5).

Question 11. A hot object is placed in a room whose temperature is 72°F. After 1 minute, the temperature of the object is 150°F and its rate of change of temperature is 20°F per minute. Find the initial temperature of the object and the rate at which its temperature is changing after 10 minutes.

Question 12. A container initially contains 10 L of a salt solution. Water flows into the container at a rate of 3 L/min, and the well-stirred mixture flows out at a rate of 2 L/min. After 5 minutes the concentration of salt in the container is 0.2 g/L. Find the amount of the salt in the container initially and the volume of solution in the container when the concentration of salt is 0.1 g/L.