arguments and graphs

Question

Question 1

Question 2

 

 

 

Answer 1

(2).

(c). From the figure 3, | z | =4, | w | = 2, | a | = | d | = | f | = 4, | b | = | c | = | e | = 2.

Calculate | z/w | by using formula : | z/w | = | z | / | w |.

Substitute | z | = 4 and | w | = 2 in the above formula.

| z/w | = 4/2 = 2

| z/w | = | b | = | c | = | e | = 2 and | z/w | < | d | < | a | < | d | < | f | < 4.

So, either b, c, e are equals to z/w.

 

From the figure 3, arg(z) ≅ 70 degrees, arg(w) ≅ 110 degrees, arg(b) ≅ 180 degrees, arg(c) ≅ 315 degrees and arg(e) ≅ 40 degrees.

Calculate arg(z/w) by using formula : arg(z/w) = arg(z) - arg(w).

Substitute arg(z) =70 degrees and arg(w) = 110 degrees in the above formula.

arg(z/w) = 70 - 115

            = - 45 degrees

            = - 45 + 360

            = 315 degrees

So, arg(z/w) = arg(c).

Option c is correct choice.

Answer 2

(1).

(a). From the figure 1, arg(z) = 2, arg(w) = 3.5, | z | = 2 and | w | = 3.

Calculate arg(zw) by using formula : arg(zw) = arg(z) + arg(w).

Substitute arg(z) = 2 and arg(w) = 3.5 in the above formula.

arg(zw) = 2 + 3.5

              = 5.5.

Calculate | zw | by using formula : | zw | = | z | * | w |.

Substitute | z | = 2 and | w | = 3 in the above formula.

| zw | = 2 * 3

          = 6.

Answer 3

(1).

(b). From the figure 2, arg(z) = 2π - 0.6 = 2(3.14159) - 0.6 ≅ 5.7, arg(w) = 1.5, | z | = 3 and | w | = 3.

The argument in the range 0 to 2π ≅ 2(3.14159) ≅ 6.2857 ≅ 6.3.

Calculate arg(zw) by using formula : arg(zw) = arg(z) + arg(w).

Substitute arg(z) = 5.7 and arg(w) = 1.5 in the above formula.

arg(zw) = 5.7 + 1.5

            = 7.2

            = 7.2 - 2π

            = 7.2 - 2(3.14159)

            ≅ 0.9

Calculate | zw | by using formula : | zw | = | z | * | w |.

Substitute | z | = 3 and | w | = 3 in the above formula.

| zw | = 3 * 3

          = 9.

 

Answer 4

(1).

(c). From the figure 3, arg(z) = 3 and arg(w) = 2π - 1.4 = 2(3.14159) - 1.4 ≅ 4.9, | z | = 0.6 and | w | = 2.5.

The argument in the range 0 to 2π ≅ 2(3.14159) ≅ 6.2857 ≅ 6.3.

Calculate arg(zw) by using formula : arg(zw) = arg(z) + arg(w).

Substitute arg(z) = 3 and arg(w) = 4.9 in the above formula.

arg(zw) = 3 + 4.9

            = 7.9

            = 7.9 - 2π

            = 7.9 - 2(3.14159)

            ≅ 1.6

Calculate arg | zw | by using formula : | zw | = | z | * | w |.

Substitute | z | = 0.6 and | w | = 2.5 in the above formula.

| zw | = 0.6 * 2.5

          = 1.5.

Answer 5

(2).

(a). From the figure 1, | z | = 2, | w | = 1, | b | = | f | = | e | = 1, | a | = | c | = | d | = 2.

Calculate | zw | by using formula : | zw | = | z | * | w |.

Substitute | z | = 2 and | w | = 1 in the above formula.

| zw | = 2 * 1 = 2

| zw | = | a | = | c | = | d | = 2 and | zw | > | b | = | f | = | e | = 1.

So, either a, c, d are equals to zw.

 

From the figure 1, arg(z) ≅ 75 degrees, arg(w) ≅ 135 degrees, arg(a) ≅ 210 degrees, arg(c) ≅ 300 degrees and arg(d) ≅ 10 degrees.

Calculate arg(zw) by using formula : arg(zw) = arg(z) + arg(w).

Substitute arg(z) =75 degrees and arg(w) = 135 degrees in the above formula.

arg(zw) = 75 + 135

            = 210 degrees

So, arg(zw) = arg(a).

Option a is correct choice.

 

Answer 6

(2).

(b). From the figure 2, | z | = 1/2, | w | = 2, | c | = | f | = | a | = 1, | b | = 1/2, | d | = 1.5, | e | = 2.

Calculate | zw | by using formula : | zw | = | z | * | w |.

Substitute | z | = 1/2 and | w | = 2 in the above formula.

| zw | = 1/2 * 2 = 1

| zw | = | c | = | f | = | a | = 1, | zw | < | d | = 1.5 < | e | = 2 and | zw | > | b | = 1/2 .

So, either c, f, a are equals to zw.

 

From the figure 2, arg(z) ≅ 110 degrees, arg(w) ≅ 320 degrees, arg(c) ≅ 70 degrees, arg(f) ≅ 160 degrees and arg(a) ≅ 220 degrees.

Calculate arg(zw) by using formula : arg(zw) = arg(z) + arg(w).

Substitute arg(z) =110 degrees and arg(w) = 320 degrees in the above formula.

arg(zw) = 110 + 320

            = 430 degrees

            = 430 - 360

            = 70 degrees

So, arg(zw) = arg(c).

Option c is correct choice.