Let a, b, c ∈ Q, (a, b) /= (0, 0), and consider the line L...

Let a, b, c ∈ Q, (a, b) /= (0, 0), and consider the line L = {(x, y) ∈ R2 ; ax + by = c}. State and prove a criterion in terms of the data a, b, c ∈ Q to check whether L∩Z2 = ∅, i.e., the line passes through no point with integer coordinates in the plane. Give an explicit example of such a line that is neither parallel to the x-axis nor to the y-axis!

Expert Solution

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Colby Messinger answered 7 months ago

There is a line of negative charge (-Q) along the y axis (from 0 to L)....

There is a line of negative charge (-Q) along the y axis (from 0 to L). There is a line of positive charge (+Q) along the x-axis (from 0 to L). A) Find the electric field at the point (L/2, L/2). Sketch the direction of the field at this point on a diagram of the situation. B) Find the potential at the point (L/2, L/2).

7. Let C be the oriented curve consisting of four line segments from (0, 0, 0)...

7. Let C be the oriented curve consisting of four line segments from (0, 0, 0) to (0, 2, 0), from (0, 2, 0) to (0, 0, 1), from (0, 0, 1) to (1, 0, 0), from (1, 0, 0) to (0, 0, 0). Consider the vector field F (x, y, z) = <2z(1 + y) + e^x^10 , 3xz, 4(x + 1)y>. (a) Compute the curl of F . (b) Compute the line integral C F dr . (Hint:...

Let the total cost function be C (q) = 50 + 2q + 0:5q2. For what...

Let the total cost function be C (q) = 50 + 2q + 0:5q2. For what values of q we have economies of scale?

Prove that L = {a + b √ 5i | a, b ∈ Q} is a...

Prove that L = {a + b √ 5i | a, b ∈ Q} is a field containing the roots of x2 + 5. Moreover, prove that if Q ⊆ K ⊆ C is a field containing the roots of x2 + 5, then L ⊆ K.

Consider the following production function q = K a L b. Assume that a+b>1. Assume that...

Consider the following production function q = K a L b. Assume that a+b>1. Assume that the firm takes price of labor w, price of capital r and price of the final product p as given and minimizes costs to produce a given level of output q. Find the share of labor cost in total value of the product wL/pq as a function of q, input prices, a and b (there should not be p in your function). How does...

Let L = {x = a r b s c t | r + s =...

Let L = {x = a r b s c t | r + s = t, r, s, t ≥ 0}. Give the simplest proof you can that L is not regular using the pumping lemma.

1. Set L = 5 mH, C = 8 µF, R = 0 Ω, and Q...

1. Set L = 5 mH, C = 8 µF, R = 0 Ω, and Q = 2E-6 C. Measure the maximum current through the circuit, and the period of the oscillation. Now increase the inductance of the inductor by 50% to 7.5 mH calculate max current = ________ A period T = ________ s. 2. Reset to L = 5 mH, C = 8 µF, keep R = 0 Ω, and Q = 2E-6 C. Keeping the charge on the...

Let A, B and C be events. Express for k = 0, 1, 2, 3, the...

Let A, B and C be events. Express for k = 0, 1, 2, 3, the probabilities that: a) Exactly k of events A, B and C occur. b) At least k of events A, B and C occur.

Consider the following AC circuit. Let R = 1.0kΩ, C = 150µF and L = 10.0...

Consider the following AC circuit. Let R = 1.0kΩ, C = 150µF and L = 10.0 mH. The AC voltage source is of frequency 60 HZ and has a maximum voltage Vmax of 100 V. Find a) the capacitive reactance, b) inductive reactance, c) total impedance, d) RMS current, e) voltage drop across the inductor, f) voltage drop across the capacitor, g) phase angle, h) draw the phasor diagram for all voltage drops and the resulting phasor, and i) is...

If L= {0a1b0c : b ≠ a + c; a, b, c ≥ _0}, Is this...

If L= {0a1b0c : b ≠ a + c; a, b, c ≥ _0}, Is this language context free? If so can you give me PDA, CFG .

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