FP2

Question 1

z (sin & cos form )

Answer 1

z = r(cosø+isinø)

where r=√(x^(2)+y^(2)) & ø=argz

Question 2

z (e form)

Answer 2

z = re^(iø)

where r=√(x^(2)+y^(2)) & ø=argz

Question 3

z1*z2=?

For complex nums z1=r1(cosø1+isinø1) & z2=r2(cosø2+isinø2)

Answer 3

z1z2 = r1r2(cos(ø1+ø2) + isin(ø1+ø2))

Question 4

(z1 ⁄ z2)=?

For complex nums z1=r1(cosø1+isinø1) & z2=r2(cosø2+isinø2)

Answer 4

(z1 ⁄ z2) = (r1 ⁄ r2)(cos(ø1–ø2) + isin(ø1–ø2))

Question 5

lz1*z2l=?

For complex nums z1=r1(cosø1+isinø1) & z2=r2(cosø2+isinø2)

Answer 5

lz1z2l = lz1llz2l

Question 6

arg(z1*z2)=?

For complex nums z1=r1(cosø1+isinø1) & z2=r2(cosø2+isinø2)

Answer 6

arg(z1*z2) = arg(z1) + arg(z2)

Question 7

arg((z1 ⁄ z2)=?

For complex nums z1=r1(cosø1+isinø1) & z2=r2(cosø2+isinø2)

Answer 7

arg((z1 ⁄ z2)) = arg(z1) – arg(z2)

Question 8

z^(n)=?

For complex num z=r(cosø+isinø)

Answer 8

z(^n) = r(^n)(cosnø+sinnø)

Question 9

z(^n)–(1 ⁄ z(^n))=?

Answer 9

2isinnø

Question 10

z(^n)+(1 ⁄ z(^n))=?

Answer 10

2cosnø

Question 11

For (dy ⁄ dx)+Py=Q where P and Q are functions of x, what is the integrating factor and what is the general solution

Answer 11

I.F. = e^(∫P dx)
Find gen solution with
(e^(∫P dx))y = ∫(e^(∫P dx))Qdx + C

Question 12

PI f(x)=k

Answer 12

λ

Question 13

PI f(x)=kx

Answer 13

λ+µx

Question 14

PI f(x)=kx^(2)

Answer 14

λ+µx+vx(^2)

Question 15

PI f(x)=ke^(px)

Answer 15

λe^(px)

Question 16

PI f(x)=mcoskx

Answer 16

λcoskx+µsinkx

Question 17

PI f(x)=nsinkx

Answer 17

λcoskx+µsinkx

Question 18

PI f(x)=mcoskx+nsinkx

Answer 18

λcoskx+µsinkx

Question 19

auxiliary equation has TWO REAL DISTINCT ROOTS

Answer 19

y = Ae(^(root1)x) + Be(^(root2)x)

Question 20

auxiliary equation has TWO EQUAL ROOTS

Answer 20

y = (A + Bx)e(^(root)x)

Question 21

auxiliary equation has TWO IMAGINARY ROOTS

Answer 21

y = AcosØx + BsinØx

Question 22

auxiliary equation has TWO COMPLEX ROOTS

Answer 22

y = e^(px)(Acosqx+Bsinqx)

when complex root = p±iq

Question 23

Work out tangents PARALLEL to initial line- polar coordinates

Answer 23

d/dø(rsinø)=0

Question 24

Work out tangents PERPENDICULAR to initial line- polar coordinates

Answer 24

d/dø(rcosø)=0