Cheat Sheet: Complex Numbers and Quadratic Equation | Mathematics (Maths) for JEE Main & Advanced PDF Download

 Page 1


1 Complex Num b er s
1.1 Key Definitions
• Complex Num b er : z = a + ib , where a (real part), b (imaginary part),
i =
v
-1 .
• Conjugate : z = a-ib .
• Mo dulus : |z| =
v
a
2
+b
2
.
• Argumen t : arg(z) = tan
-1
(
b
a
)
, adjusted for quadran t.
• P olar F orm : z = r(cos? +isin?) , where r =|z| , ? = arg(z) .
1.2 Op erations
• A ddition : (a+ib)+(c+id) = (a+c)+i(b+d) .
• Multiplication : (a+ib)(c+id) = (ac-bd)+i(ad+bc) .
• Division :
a+ib
c+id
=
(a+ib)(c-id)
c
2
+d
2
=
(ac+bd)+i(bc-ad)
c
2
+d
2
.
• Conjugate Prop erties : z
1
+z
2
= z
1
+z
2
, z
1
z
2
= z
1
·z
2
, |z
1
z
2
| =|z
1
||z
2
| .
1.3 P olar F orm and De Moivre’s Theorem
• P olar F orm : z = r(cos? +isin?) , r =|z| , ? = arg(z) .
• Multiplication : z
1
z
2
= r
1
r
2
[cos(?
1
+?
2
)+isin(?
1
+?
2
)] .
• De Moivre’s Theorem : (cos? +isin?)
n
= cos(n?)+isin(n?) .
• n -th Ro ots : z
1/n
= r
1/n
[
cos
(
?+2kp
n
)
+isin
(
?+2kp
n
)]
, k = 0,1,...,n-1 .
1
Complex	Numbers	and	Quadratic	Equations	
Cheat	Sheet
Page 2


1 Complex Num b er s
1.1 Key Definitions
• Complex Num b er : z = a + ib , where a (real part), b (imaginary part),
i =
v
-1 .
• Conjugate : z = a-ib .
• Mo dulus : |z| =
v
a
2
+b
2
.
• Argumen t : arg(z) = tan
-1
(
b
a
)
, adjusted for quadran t.
• P olar F orm : z = r(cos? +isin?) , where r =|z| , ? = arg(z) .
1.2 Op erations
• A ddition : (a+ib)+(c+id) = (a+c)+i(b+d) .
• Multiplication : (a+ib)(c+id) = (ac-bd)+i(ad+bc) .
• Division :
a+ib
c+id
=
(a+ib)(c-id)
c
2
+d
2
=
(ac+bd)+i(bc-ad)
c
2
+d
2
.
• Conjugate Prop erties : z
1
+z
2
= z
1
+z
2
, z
1
z
2
= z
1
·z
2
, |z
1
z
2
| =|z
1
||z
2
| .
1.3 P olar F orm and De Moivre’s Theorem
• P olar F orm : z = r(cos? +isin?) , r =|z| , ? = arg(z) .
• Multiplication : z
1
z
2
= r
1
r
2
[cos(?
1
+?
2
)+isin(?
1
+?
2
)] .
• De Moivre’s Theorem : (cos? +isin?)
n
= cos(n?)+isin(n?) .
• n -th Ro ots : z
1/n
= r
1/n
[
cos
(
?+2kp
n
)
+isin
(
?+2kp
n
)]
, k = 0,1,...,n-1 .
1
Complex	Numbers	and	Quadratic	Equations	
Cheat	Sheet
• Ro ots of Unit y : Solutions of z
n
= 1 , giv en b y z = cos
(
2kp
n
)
+ isin
(
2kp
n
)
,
k = 0,1,...,n-1 .
2 Quadratic Equa tions
2.1 Key Concepts
• Standard F orm : ax
2
+bx+c = 0 , a?= 0 .
• Ro ots : x =
-b±
v
b
2
-4ac
2a
, where D = b
2
-4ac (discriminan t).
• Nature of Ro ots :
– D > 0 : Real and distinct ro ots.
– D = 0 : Real and equal ro ots.
– D < 0 : Complex ro ots (x = p±iq ).
• Sum and Pro duct : Sum = -
b
a
, Pro duct =
c
a
.
• Quadratic F ormation : If ro ots are a,ß , equation is x
2
-(a+ß)x+aß = 0 .
2.2 Complex Ro ots
• If D < 0 , ro ots are x =
-b±i
v
4ac-b
2
2a
.
• Conjugate P air: If one ro ot is p+iq , other is p-iq (for real c o e?icien ts).
3 Quic k Reference T able
Concept F orm ula/Relation
Mo dulus |z| =
v
a
2
+b
2
Argumen t arg(z) = tan
-1
(
b
a
)
Euler’s F orm z = re
i?
Quadratic Ro ots x =
-b±
v
b
2
-4ac
2a
Sum of Ro ots -
b
a
Pro duct of Ro o ts
c
a
Cub e Ro ots of U nit y 1,? =-
1
2
+i
v
3
2
,?
2
=-
1
2
-i
v
3
2
, where ?
3
= 1 ,
1+? +?
2
= 0
2
Page 3


1 Complex Num b er s
1.1 Key Definitions
• Complex Num b er : z = a + ib , where a (real part), b (imaginary part),
i =
v
-1 .
• Conjugate : z = a-ib .
• Mo dulus : |z| =
v
a
2
+b
2
.
• Argumen t : arg(z) = tan
-1
(
b
a
)
, adjusted for quadran t.
• P olar F orm : z = r(cos? +isin?) , where r =|z| , ? = arg(z) .
1.2 Op erations
• A ddition : (a+ib)+(c+id) = (a+c)+i(b+d) .
• Multiplication : (a+ib)(c+id) = (ac-bd)+i(ad+bc) .
• Division :
a+ib
c+id
=
(a+ib)(c-id)
c
2
+d
2
=
(ac+bd)+i(bc-ad)
c
2
+d
2
.
• Conjugate Prop erties : z
1
+z
2
= z
1
+z
2
, z
1
z
2
= z
1
·z
2
, |z
1
z
2
| =|z
1
||z
2
| .
1.3 P olar F orm and De Moivre’s Theorem
• P olar F orm : z = r(cos? +isin?) , r =|z| , ? = arg(z) .
• Multiplication : z
1
z
2
= r
1
r
2
[cos(?
1
+?
2
)+isin(?
1
+?
2
)] .
• De Moivre’s Theorem : (cos? +isin?)
n
= cos(n?)+isin(n?) .
• n -th Ro ots : z
1/n
= r
1/n
[
cos
(
?+2kp
n
)
+isin
(
?+2kp
n
)]
, k = 0,1,...,n-1 .
1
Complex	Numbers	and	Quadratic	Equations	
Cheat	Sheet
• Ro ots of Unit y : Solutions of z
n
= 1 , giv en b y z = cos
(
2kp
n
)
+ isin
(
2kp
n
)
,
k = 0,1,...,n-1 .
2 Quadratic Equa tions
2.1 Key Concepts
• Standard F orm : ax
2
+bx+c = 0 , a?= 0 .
• Ro ots : x =
-b±
v
b
2
-4ac
2a
, where D = b
2
-4ac (discriminan t).
• Nature of Ro ots :
– D > 0 : Real and distinct ro ots.
– D = 0 : Real and equal ro ots.
– D < 0 : Complex ro ots (x = p±iq ).
• Sum and Pro duct : Sum = -
b
a
, Pro duct =
c
a
.
• Quadratic F ormation : If ro ots are a,ß , equation is x
2
-(a+ß)x+aß = 0 .
2.2 Complex Ro ots
• If D < 0 , ro ots are x =
-b±i
v
4ac-b
2
2a
.
• Conjugate P air: If one ro ot is p+iq , other is p-iq (for real c o e?icien ts).
3 Quic k Reference T able
Concept F orm ula/Relation
Mo dulus |z| =
v
a
2
+b
2
Argumen t arg(z) = tan
-1
(
b
a
)
Euler’s F orm z = re
i?
Quadratic Ro ots x =
-b±
v
b
2
-4ac
2a
Sum of Ro ots -
b
a
Pro duct of Ro o ts
c
a
Cub e Ro ots of U nit y 1,? =-
1
2
+i
v
3
2
,?
2
=-
1
2
-i
v
3
2
, where ?
3
= 1 ,
1+? +?
2
= 0
2
4 Solv ed Example s
4.1 Example 1: Complex Num b er Op erations
Problem : F or z
1
= 3+4i , z
2
= 1-2i , find z
1
+z
2
and z
1
z
2
.
Solution :
• A ddition: z
1
+z
2
= (3+4i)+(1-2i) = (3+1)+(4-2)i = 4+2i .
• Multiplication: z
1
z
2
= (3+4i)(1-2i) = 3·1+3·(-2i)+4i·1+4i·(-2i) =
3-6i+4i-8i
2
= 3-2i-8(-1) = 11-2i .
Answ er : 4+2i , 11-2i .
4.2 Example 2: Mo dulus and Argumen t
Problem : Find the mo dulus and argumen t of z =-1+i .
Solution :
• Mo dulus: |z| =
v
(-1)
2
+1
2
=
v
2 .
• Argumen t: ? = tan
-1
(
1
-1
)
= tan
-1
(-1) = -
p
4
. Since z is in 2nd quadran t,
arg(z) = p-
p
4
=
3p
4
.
Answ er : |z| =
v
2 , arg(z) =
3p
4
.
4.3 Example 3: De Moivre’s Theorem
Problem : Find (1+i)
4
using De Moivre’s theorem.
Solution :
• P olar form: 1+i =
v
2
(
cos
p
4
+isin
p
4
)
.
• (1+i)
4
= (
v
2)
4
(
cos
p
4
+isin
p
4
)
4
= 4(cosp +isinp) = 4(-1+0i) =-4 .
Answ er : -4 .
4.4 Example 4: Quadratic Equation Ro ots
Problem : Solv e 2x
2
-4x+3 = 0 . Find nature and ro ots.
Solution :
• Discriminan t: D = (-4)
2
-4·2·3 = 16-24 =-8 .
• Nature: D < 0 , complex ro o ts.
• Ro ots: x =
4±
v
-8
4
=
4±2i
v
2
4
= 1±i
v
2
2
.
Answ er : Complex ro ots, 1+i
v
2
2
, 1-i
v
2
2
.
3
Page 4


1 Complex Num b er s
1.1 Key Definitions
• Complex Num b er : z = a + ib , where a (real part), b (imaginary part),
i =
v
-1 .
• Conjugate : z = a-ib .
• Mo dulus : |z| =
v
a
2
+b
2
.
• Argumen t : arg(z) = tan
-1
(
b
a
)
, adjusted for quadran t.
• P olar F orm : z = r(cos? +isin?) , where r =|z| , ? = arg(z) .
1.2 Op erations
• A ddition : (a+ib)+(c+id) = (a+c)+i(b+d) .
• Multiplication : (a+ib)(c+id) = (ac-bd)+i(ad+bc) .
• Division :
a+ib
c+id
=
(a+ib)(c-id)
c
2
+d
2
=
(ac+bd)+i(bc-ad)
c
2
+d
2
.
• Conjugate Prop erties : z
1
+z
2
= z
1
+z
2
, z
1
z
2
= z
1
·z
2
, |z
1
z
2
| =|z
1
||z
2
| .
1.3 P olar F orm and De Moivre’s Theorem
• P olar F orm : z = r(cos? +isin?) , r =|z| , ? = arg(z) .
• Multiplication : z
1
z
2
= r
1
r
2
[cos(?
1
+?
2
)+isin(?
1
+?
2
)] .
• De Moivre’s Theorem : (cos? +isin?)
n
= cos(n?)+isin(n?) .
• n -th Ro ots : z
1/n
= r
1/n
[
cos
(
?+2kp
n
)
+isin
(
?+2kp
n
)]
, k = 0,1,...,n-1 .
1
Complex	Numbers	and	Quadratic	Equations	
Cheat	Sheet
• Ro ots of Unit y : Solutions of z
n
= 1 , giv en b y z = cos
(
2kp
n
)
+ isin
(
2kp
n
)
,
k = 0,1,...,n-1 .
2 Quadratic Equa tions
2.1 Key Concepts
• Standard F orm : ax
2
+bx+c = 0 , a?= 0 .
• Ro ots : x =
-b±
v
b
2
-4ac
2a
, where D = b
2
-4ac (discriminan t).
• Nature of Ro ots :
– D > 0 : Real and distinct ro ots.
– D = 0 : Real and equal ro ots.
– D < 0 : Complex ro ots (x = p±iq ).
• Sum and Pro duct : Sum = -
b
a
, Pro duct =
c
a
.
• Quadratic F ormation : If ro ots are a,ß , equation is x
2
-(a+ß)x+aß = 0 .
2.2 Complex Ro ots
• If D < 0 , ro ots are x =
-b±i
v
4ac-b
2
2a
.
• Conjugate P air: If one ro ot is p+iq , other is p-iq (for real c o e?icien ts).
3 Quic k Reference T able
Concept F orm ula/Relation
Mo dulus |z| =
v
a
2
+b
2
Argumen t arg(z) = tan
-1
(
b
a
)
Euler’s F orm z = re
i?
Quadratic Ro ots x =
-b±
v
b
2
-4ac
2a
Sum of Ro ots -
b
a
Pro duct of Ro o ts
c
a
Cub e Ro ots of U nit y 1,? =-
1
2
+i
v
3
2
,?
2
=-
1
2
-i
v
3
2
, where ?
3
= 1 ,
1+? +?
2
= 0
2
4 Solv ed Example s
4.1 Example 1: Complex Num b er Op erations
Problem : F or z
1
= 3+4i , z
2
= 1-2i , find z
1
+z
2
and z
1
z
2
.
Solution :
• A ddition: z
1
+z
2
= (3+4i)+(1-2i) = (3+1)+(4-2)i = 4+2i .
• Multiplication: z
1
z
2
= (3+4i)(1-2i) = 3·1+3·(-2i)+4i·1+4i·(-2i) =
3-6i+4i-8i
2
= 3-2i-8(-1) = 11-2i .
Answ er : 4+2i , 11-2i .
4.2 Example 2: Mo dulus and Argumen t
Problem : Find the mo dulus and argumen t of z =-1+i .
Solution :
• Mo dulus: |z| =
v
(-1)
2
+1
2
=
v
2 .
• Argumen t: ? = tan
-1
(
1
-1
)
= tan
-1
(-1) = -
p
4
. Since z is in 2nd quadran t,
arg(z) = p-
p
4
=
3p
4
.
Answ er : |z| =
v
2 , arg(z) =
3p
4
.
4.3 Example 3: De Moivre’s Theorem
Problem : Find (1+i)
4
using De Moivre’s theorem.
Solution :
• P olar form: 1+i =
v
2
(
cos
p
4
+isin
p
4
)
.
• (1+i)
4
= (
v
2)
4
(
cos
p
4
+isin
p
4
)
4
= 4(cosp +isinp) = 4(-1+0i) =-4 .
Answ er : -4 .
4.4 Example 4: Quadratic Equation Ro ots
Problem : Solv e 2x
2
-4x+3 = 0 . Find nature and ro ots.
Solution :
• Discriminan t: D = (-4)
2
-4·2·3 = 16-24 =-8 .
• Nature: D < 0 , complex ro o ts.
• Ro ots: x =
4±
v
-8
4
=
4±2i
v
2
4
= 1±i
v
2
2
.
Answ er : Complex ro ots, 1+i
v
2
2
, 1-i
v
2
2
.
3
4.5 Example 5: Cub e Ro ots of Unit y
Problem : Find the cub e ro ots of unit y and v erify 1+? +?
2
= 0 .
Solution :
• Cub e ro ots of 1: Solv e z
3
= 1 . Ro ots: z = 1,? = e
i
2p
3
= -
1
2
+ i
v
3
2
, ?
2
=
e
i
4p
3
=-
1
2
-i
v
3
2
.
• Sum: 1+? +?
2
= 1+
(
-
1
2
+i
v
3
2
)
+
(
-
1
2
-i
v
3
2
)
= 1-
1
2
-
1
2
+0i = 0 .
Answ er : Ro ots: 1,-
1
2
±i
v
3
2
, sum = 0.
4