CBSE Grade 11 Mathematics Ch 5 Complex Numbers And Quadratic Equations Notes

COMPLEX NUMBERS AND QUADRATIC EQUATIONS
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COMPLEX NUMBERS & QUADRATIC EQUATIONS
Some Important Results
1. Solution of x2
+ 1 = 0 with the property i2
= -1 is called the imaginary unit.
2. Square root of a negative real number is called an imaginary number.
3. If a and b are positive real numbers, then √−a  √−b = -√ab
4. If a is a positive real number, then we have √−a = i√a.
5. Powers of i
i = √−1;
i2
= -1;
i3
= -i
i4
= 1
6. If n>4, then i-n
= = where k is the reminder when n is divided by 4.
7. We have io
= 1.
8. A number in the form a + ib, where a and b are real numbers, is said to be a complex number.
9. In complex number z = a + ib, a is the real part, denoted by Re z and b is the imaginary part
denoted by Im z of the complex number z.
10.√−1 =i is called iota, which is a complex number.
11.The modulus of a complex number z = a + ib denoted by |z| is defined to be a non-
negative real number
12.For any non-zero complex number z = a + ib (a ≠ 0, b ≠ 0), there exists a complex
denoted by or z-1
, called the multiplicative inverse of z such that
number
13.Conjugate of a complex number z = a + ib, denoted as z, is the complex number a − ib.

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14.The number z = r(cos θ +isin θ) is the polar form of the complex number z = a + ib.
Here is called the modulus of z and is called the argument or
amplitude of z, which is denoted by arg z.
15.The value of θ such that –π < θ ≤ π called principal argument of z.
16.The Eulerian form of z is z = reiθ
, where e = cosθ + isinθ
17.The plane having a complex number assigned to each of its points is called the Complex
plane orArgand plane.
18.Leta0,a1,a2,...be real numbers and x is a real variable. Then, the real polynomial of a real
variable with real coefficients is given as
f(x) = a0 + a1x + a2x2
+ …. anxn
19.Leta0,a1,a2,...be complex numbers and x is a complex variable. Then, the real polynomial of
a complex variable with complex coefficients is given as
f(x) = a0 + a1x + a2x2
+ …. anxn
20.Apolynomialf(x)=a0 +a1x +a2x2
+….anxn
is apolynomialof degree n.
21.Polynomial of second degree is called a quadratic polynomial.
22.Polynomials of degree 3 and 4 are known as cubic and biquadratic polynomials.
23.If f(x) is a polynomial, then f(x) = 0 is called a polynomial equation.
24.If f(x) is a quadratic polynomial, then f(x) = 0 is called a quadratic equation.

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25.The general form of a quadratic equation is ax2
+ bx + c = 0, a  0.
26.The values of the variable satisfying a given equation are called its roots.
27.A quadratic equation cannot have more than two roots.
28.Fundamental Theorem of Algebra states that ‘A polynomial equation of degree n has n
roots.’
Top Concepts
1. Addition of two complex numbers: If z1 = a + ib and z2 = c + id be any two complex
numbers, then the sum
z1 + z2 = (a + c) + i(b + d).
2.Sum of two complex numbers is also a complex number. This is known as the closure
property.
3. The addition of complex numbers satisfy the following properties:
i. Addition of complex numbers satisfies the commutative law. For any two complex
numbers z1 andz2, z1 + z2 = z2 + z1.
ii. Addition of complex numbers satisfies associative law for any three complex
numbers z1, z2, z3,(z1 + z2) + z3 = z1 + (z2 + z3).
iii. There exists a complex number 0 + i0 or 0, called the additive identity or the zero
complex number,such that for every complex number z,
z + 0 = 0 + z = z.
iv. To every complex number z = a + ib, there exists another complex number –z =–a +
i(-b) calledthe additive inverse of z.
z+(-z)=(-z)+z=0
4. Difference of two complex numbers: Given any two complex numbers If z1 = a +
ib andz2 = c +id the difference z1 – z2 is given by
z1 – z2 = z1 + (-z2) = (a - c) + i(b - d).
5. Multiplication of two complex numbers Let z1 = a + ib and z2 = c + id be any two

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complex numbers. Then, the product z1 z2 is defined as follows:
z1 z2 = (ac – bd) + i(ad + bc)
6. Properties of multiplication of complex numbers: Product of two complex numbers is a
complexnumber; the product z1 z2 is a complex number for all complex numbers z1 and z2.
i. Product of complex numbers is commutative, i.e. for any two complex numbers
z1 and z2,z1 z2 = z2 z1
ii. Product of complex numbers is associative, i.e. for any three complex numbers
z1, z2, z3,(z1 z2) z3 = z1 (z2 z3).
iii. There exists a complex number 1 + i0 (denoted as 1), called the multiplicative
identitysuch that z.1 = z for every complex number z.
iv. For every non-zero complex number, z = a + ib or a + bi (a ≠ 0, b ≠ 0), there
is a complex number called the multiplicative inverse of z such that
v. distributive law: For any three complex numbers z1, z2, z3,
a. z1 (z2 + z3) = z1.z2 + z1.z3
b. (z1 + z2) z3 = z1.z3 + z2.z3
7. Division of two complex numbers: Given any two complex numbers z1 = a + ib
and z2 = c + id, where z2 ≠ 0, the quotient is defined by
8. Identities for complex numbers
i. (z1 + z2)² = z1² + z2² + 2z1.z2, for all complex numbers z1 and z2.
ii. (z1 − z2)² = z1² − 2z1z2 + z2²
iii. (z1 + z2)³ = z1³ + 3z1²z2 + 3z1z2² + z2³
iv. (z1 − z2)³ = z1³ − 3z1²z2 + 3z1z2
2
− z2³
v. z1² − z2² = (z1 + z2) (z1 − z2)
9.Properties of modulus and conjugate of complex numbers
For any two complex numbers z1 and z2,

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10.The order of a relation is not defined in complex numbers. Hence there is no meaning in z1 
z.
11.Two complex numbers z1 and z2 are equal iff Re (z1) = Re (z2) and Im (z1) Im (z2).
12.The sum and product of two complex numbers are real if and only if they are conjugate of
each other.
13.For any integer when a<0 and b<0.
14.The polar form of the complex number z = x + iy is r (cos θ + i sinθ), where r is the modulus
of z and θ is known as the argument of z.
15.For a quadratic equation ax² + bx + c = 0 with real coefficients a, b and c and a ≠ 0. If the
discriminant D = b² − 4ac  0, then the equation has two real roots given by
16. Roots of the quadratic equation ax² + bx + c = 0, where a, b and c  R, a ≠ 0, when
discriminant b² − 4ac < 0, are imaginary given by
17.Complex roots occur in pairs.

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18.If a, b and c are rational numbers and b2
- 4ac is positive and a perfect square, then
is a rational number and hence the roots are rational and unequal.
19.If b2
- 4ac = 0, then the roots of the quadratic equation are real and equal.
20.If b2
- 4ac = 0 but it is not a perfect square, then the roots of the quadratic equation are
irrational and unequal.
21.Irrational roots occur in pairs.
22.A polynomial equation of n degree has n roots. These n roots could be real or complex.
23. Complex numbers are represented in the Argand plane with X-axis being real and Y-axis
being imaginary.
24.Representation of complex number z = x + iy in the Argand plane.
25. Multiplication of a complex number by i results in rotating the vector joining the origin
to the pointrepresenting z through a right angle.
26.Argument  of the complex number z can take any value in the interval [0, 2π). Different
orientations of z are as follows

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Important Questions
Multiple Choice questions-
Question 1. Let z1 and z2 be two roots of the equation z² + az + b = 0, z being
complex. Further assume that the origin, z1 and z2 form an equilateral triangle. Then
(a) a² = b
(b) a² = 2b
(c) a² = 3b
(d) a² = 4b
Question 2. The value of ii
is
(a) 0
(b) e-π
(c) 2e-π/2
(d) e-π/2
Question 3. The value of √(-25) + 3√(-4) + 2√(-9) is
(a) 13 i
(b) -13 i
(c) 17 i
(d) -17 i
So, √(-25) + 3√(-4) + 2√(-9) = 17 i
Question 4. If the cube roots of unity are 1, ω and ω², then the value of (1 + ω / ω²)³
is
(a) 1
(b) -1
(c) ω
(d) ω²
Question 5. If {(1 + i)/(1 – i)}ⁿ = 1 then the least value of n is
(a) 1
(b) 2
(c) 3
(d) 4

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Question 6. The value of [i19
+ (1/i)25
]² is
(a) -1
(b) -2
(c) -3
(d) -4
Question 7. If z and w be two complex numbers such that |z| ≤ 1, |w| ≤ 1 and |z +
iw| = |z – iw| = 2, then z equals {w is congugate of w}
(a) 1 or i
(b) i or – i
(c) 1 or – 1
(d) i or – 1
Question 8. The value of {-√(-1)}4n+3
, n ∈ N is
(a) i
(b) -i
(c) 1
(d) -1
Question 9. Find real θ such that (3 + 2i × sin θ)/(1 – 2i × sin θ) is real
(a) π
(b) nπ
(c) nπ/2
(d) 2nπ
Question 10. If i = √(-1) then 4 + 5(-1/2 + i√3/2)334
+ 3(-1/2 + i√3/2)365
is equals to
(a) 1 – i√3
(b) -1 + i√3
(c) i√3
(d) -i√3
Very Short Questions:
Evaluate i-39
1. Solved the quadratic equation 𝑥 + 𝑥
√
= 0
2. If = 1, then find the least positive integral value of m.
3. Evaluate (1+ i)4

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4. Find the modulus of −
5. Express in the form of a + ib. (1+3i)-1
6. Explain the fallacy in -1 = i. i. = √−1. √−1 = −1(−1) = √1 = 1.
7. Find the conjugate of
8. Find the conjugate of – 3i – 5.
9. Let z1 = 2 – i, z2 = -2 + i Find Re
Short Questions:
1. If x + iy = Prove that x2
+ y2
= 1
2. Find real θ such that is purely real.
3. Find the modulus of
( )( )
4. If |a + ib|= 1 then Show that = 𝑏 + 𝑎𝑖
5. If x – iy = Prove that (𝑥 + 𝑦 ) =
Long Questions:
1. If z = x + i y and w = Show that |w| = 1 ⇒ z is purely real.
2. Convert into polar form
√
3. Find two numbers such that their sum is 6 and the product is 14.
4. Convert into polar form 𝑧 =
5. If α and β are different complex number with |β| = 1 Then find
Assertion Reason Questions:
1. In each of the following questions, a statement of Assertion is given followed by
a corresponding statement of Reason just below it. Of the statements, mark the
correct answer as.
Assertion (A): If i = then i4k
= 1, i4k + 1
=i, i4k + 2
= − 1 and i4k +3
= −
i.
Reason (R) : i4k
+ i4k + 1
+ i4k + 2
+ i4k + 3
= 1.
(i) Both assertion and reason are true and reason is the correct explanation of

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assertion.
(ii) Both assertion and reason are true but reason is not the correct explanation
of assertion.
(iii) Assertion is true but reason is false.
(iv) Assertion is false but reason is true.
2. In each of the following questions, a statement of Assertion is given followed by
a corresponding statement of Reason just below it. Of the statements, mark the
correct answer as.
Assertion (A): Simplest form of i −35
is −i.
Reason (R) : Additive inverse of (1 − i) is equal to – 1 + i.
(i) Both assertion and reason are true and reason is the correct explanation of
assertion.
(ii) Both assertion and reason are true but reason is not the correct explanation
of assertion.
(iii) Assertion is true but reason is false.
(iv) Assertion is false but reason is true.
Answer Key:
MCQ
1. (c) a² = 3b
2. (d) e-π/2
3. (c) 17 i
4. (b) -1
5. (d) 4
6. (d) -4
7. (c) 1 or – 1
8. (a) i
9. (b) nπ
10.(c) i√3
Very Short Answer:
1.

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2.
3.
4.

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5.
Let z =
6.
7.
is okay but
is wrong.
8.
Let z =

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9. Let z = 3i – 5
10.z1 z2 = (2 – i)(-2 + i)
Short Answer:
1.
taking conjugate both side

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x2
+ y2
= 1
[i2
= -1]
2.
For purely real
Im (z) = 0
3.
.
4.

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5.
Taking conjugate both side
Long Answer:
1.
w =

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2.
.

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Since Re (z) < o, and Im (z) > o
3.
Let x and y be the no.
x + y = 6
xy = 14
4.

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5.
.

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Assertion Reason Answer:
1. (iii) Assertion is true but reason is false.
2. (iv) Assertion is false but reason is true.

CBSE Grade 11 Mathematics Ch 5 Complex Numbers And Quadratic Equations Notes

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CBSE Grade 11 Mathematics Ch 5 Complex Numbers And Quadratic Equations Notes