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# Karnataka Class 11 Commerce Maths Complex Numbers and Quadratic Equations

## Karnataka Class 11 Commerce Maths Complex Numbers and Quadratic Equations

### Karnataka Class 11 Commerce Maths Complex Numbers and Quadratic Equations

Karnataka Class 11 Commerce Maths Complex Numbers and Quadratic Equations : The Karnataka Secondary Education examination board is a state level examination which is conducted in the month of March-April. Also known as the Pre-University Examination, it offers Science, Arts and commerce streams. The question paper is usually of three hours duration and as per the Science stream, it is divided into practical and written. Written examination is purely of subjective category.

### Karnataka Class 11 Commerce Maths Complex Numbers and Quadratic Equations

Eligibility: All candidates who are presuming to take the Pre-university examination are required to clear the SSLC examination successfully. The results declared are purely based on the merit criteria.

### Karnataka Class 11 Commerce Maths Complex Numbers and Quadratic Equations

Syllabus for Class 11 Commerce Mathematics

• Need for complex numbers, especially √ -1, to be motivated by inability to solve every quadratic equation.
• Brief description of algebraic properties of complex numbers.
• Argand plane and polar representation of complex numbers and problems
• Statement of Fundamental Theorem of Algebra, solution of quadratic equations in the complex number system,
• Square-root of a Complex number given in supplement and problems.

### Karnataka Class 11 Commerce Maths Complex Numbers and Quadratic Equations

Design of the Question Paper

MATHEMATICS CLASS : I PUC

Time: 3 hours 15 minute; Max. Mark:100

The weightage of the distribution of marks over different dimensions of the question paper shall be as follows:

I. Weightage to Objectives:

 Objective Weightage Marks Knowledge 40% 60/150 Understanding 30% 45/150 Application 20% 30/150 Skill 10% 15/150

II. Weightage to level of difficulty:

 Level Weightage Marks Easy 35% 53/150 Average 55% 82/150 Difficult 10% 15/150

III. Weightage to content:

 Chapter No. Chapter No. of teaching Hours Marks 1. SETS 8 8 2. RELATIONS AND FUNCTIONS 10 11 3. TRIGONOMETRIC FUNCTIONS 18 19 4. PRINCIPLE OF MATHEMATICAL INDUCTION 4 5 5. COMPLEX NUMBERS AND QUADRATIC EQUATIONS 8 9 6. LINEAR INEQUALITIES 8 7 7. PERMUTATION AND COMBINATION 9 9 8. BINOMIAL THEOREM 7 8 9. SEQUENCE AND SERIES 9 11 10. STRAIGHT LINES 10 10 11. CONIC SECTIONS 9 9 12. INTRODUCTION TO 3D GEOMETRY 5 7 13. LIMITS AND DERIVATIVES 14 15 14. MATHEMATICAL REASONING 6 6 15. STATISTICS 7 7 16. PROBABILITY 8 9 Total 150 150

IV. Pattern of the question paper:

 PART Type of questions Number of questions to be set Number of questions to be answered Remarks A 1  mark questions 10 10 Compulsory part B 2  mark questions 14 10 —————— C 3  mark questions 14 10 —————— D 5  mark questions 10 6 Questions must be asked from the specific set of topics as mentioned below, under section V E 10  mark questions(Each question with two subdivisions namely)a) 6 mark andb) 4 mark. 2 1

### Karnataka Class 11 Commerce Maths Complex Numbers and Quadratic Equations

GUIDELINES TO THE QUESTION PAPER SETTER

1. The question paper must be prepared based on the individual blue print without changing the weightage of marks fixed for each chapter.

2. The question paper pattern provided should be adhered to.

• Part A : 10 compulsory questions each carrying 1 mark;
• Part B : 10 questions to be answered out of 14 questions each carrying 2 mark ;
• Part C : 10 questions to be answered out of 14 questions each carrying 3 mark;
• Part D : 6 questions to be answered out of 10 questions each carrying 5 mark
• Part E : 1 question to be answered out of 2 questions each carrying 10 mark with subdivisions (a) and (b) of 6 mark and 4 mark respectively. (The questions for PART D and PART E should be taken from the content areas as explained under section V in the design of the question paper)

3. There is nothing like a single blue print for all the question papers to be set. The paper setter should prepare a blue print of his own and set the paper accordingly without changing the weightage of marks given for each chapter.

4. Position of the questions from a particular topic is immaterial.

5. In case of the problems, only the problems based on the concepts and exercises discussed in the text book (prescribed by the Department of Pre-university education) can be asked. Concepts and exercises different from text book given in Exemplar text book should not be taken. Question paper must be within the frame work of prescribed text book and should be adhered to weightage to different topics and guidelines.

6. No question should be asked from the historical notes and appendices given in the text book.

7. Supplementary material given in the text book is also a part of the syllabus.

8. Questions should not be split into subdivisions. No provision for internal choice question in any part of the question paper.

9. Questions should be clear, unambiguous and free from grammatical errors. All unwanted data in the questions should be avoided.

10. Instruction to use the graph sheet for the question on LINEAR PROGRAMMING in PART E should be given in the question paper.

11. Repetition of the same concept, law, fact etc., which generate the same answer in different parts of the question paper should be avoided.

### Karnataka Class 11 Commerce Maths Complex Numbers and Quadratic Equations

Complex numbers:

A number of the form a + ib where a, b ∈ R, the set of real numbers and i = √-1 is called as a complex number. Where ‘a’ is called as real part and ‘b’ is called the imaginary part.
Example: 1 + 6i, 3 – 4i, 2 + √3i etc. are complex numbers.

Algebra of complex numbers:

Addition and subtraction of two complex numbers:

Definition:

If z1 = a + ib and z2 = c + id are two complex numbers, then z1 + z2 = (a + ib) + (c +id) is the complex number = (a + c) + i (b + d);
z1 – z2 = (a + ib) – (c + id) = (a – c) + i (b – d).

Eg. z1 = 4 + 5i , z2 = 1 + 7i.

i) z1 + z2 = 5 + 12i.

ii) z1 – z2 =3 – 2i.

MULTIPLICATION OF TWO COMPLEX NUMBERS.

If z1 = a + ib and z2 = c + id are two complex numbers then z1 z2 = (a + ib) (c + id) = (ac  bd) + i (ad + bc)

Example: (2 + 3i) (4 + 3i) = (8 – 9) + i (18) = -1 + 18i.

CONJUGATE OF A COMPLEX NUMBER

If z = a + ib is a complex number then a – ib is called as its conjugate and is denoted by z .

Example if z = 2 + 3i then z¯ = 2 – 3i

If z = 5 – 4i then z¯ = 5 + 4i

DIVISION OF A COMPLEX NUMBER BY A NON-ZERO COMPLEX NUMBER.

Let z1 = a + ib and z2 =c + id be two complex numbers

⇒ z1/z2 = a + ib/c + id.

⇒  a + ib/c + id = c – id/ c – id.

⇒ [ ac + bd/c2 + d ] + i[bc – ac/c2 + d2].

IDENTITY PROPERTY

z + 0 = 0 + z = z
∀ z ≠ 0 which is a complex number, 1 + 0i is the multiplicative identity.
1 × z = z × 1 = z

INVERSE PROPERTY :

For every complex number z there exists some – z belonging to the set of complex numbers such that z + – z = – z + z.

– z is the additive inverse of z.

For every complex number z except zero there exists some 1/z belonging to the set of complex numbers such that
z × 1/z = 1/z × z = 1 ;

1/z is the multiplicative inverse of z.

### Karnataka Class 11 Commerce Maths Complex Numbers and Quadratic Equations

COMMUTATIVE PROPERTY:

For every complex numbers z1 , z2
1) z1 + z2 = z2 + z1
2) z1 × z2 = z2 × z1.

a) LEFT DISTRIBUTIVE PROPERTY

For every complex numbers z1 , z2, z3
z1 (z2 + z3) = z1 z2 + z1 z3

b) RIGHT DISTRIBUTIVE PROPERTY:

(z1 + z2) z3 = z1 z3 + z2 z3

MODULUS OF A COMPLEX NUMBER:

If z = a + ib
|z| = √(a2 + b2 )

EQUALITY OF COMPLEX NUMBERS

DEFINITION:

If a + ib and c + id are two complex numbers then a + ib = c + id if and only if a = c and b = d

Example:

1. a + ib = 4 + 3i
⇒ a = 4, b = 3

2. a + ib = 0 + 0i
⇒ a = 0, b = 0

### Karnataka Class 11 Commerce Maths Complex Numbers and Quadratic Equations

PROPERTIES OF COMPLEX NUMBERS:

Let C be the set of complex numbers then

1. Closure property :

For every complex numbers z1 and z2 ,

z1 + z2 belongs to the set of complex numbers.

Also z1 . z2 belongs to the set of complex numbers

2. Associative property:

For every complex numbers z1 , z2 and z3,

1) z1 + (z2 + z3) = (z1 + z2 ) + z3

2) z1  (z2  z3) = (z1  z2)  z3

3. Identities :

For every complex number z,

0 + 0i is the additive identity.

### Karnataka Class 11 Commerce Maths Complex Numbers and Quadratic Equations

The equation of the form ax2 + bx + c = 0 (a ≠ 0) containing x2 as the highest power of x is called quadratic equation or a second degree equation.

The quadratic equation has two and only two roots. These two roots may be equal or unequal. If α and β are the 2 roots. Then the equation will be of the form: (x – α) (x- β) =0.

Ex. Form the Quadratic – Equation whose roots are 2 & 3
(x – 2) (x – 3) = 0

There are two methods of finding the roots of the quadratic equation.

(1) Factorization method

(2) Formula method.

Factorization Method:

First, the quadratic equation is reduced to the standard form. Factorize the expression on the left side. Equate each factors to zero, solve the corresponding linear equations

Ex. 1) x2 – 2x – 3 = 0
⇒  x2 – 3x + x – 3 = 0
⇒ x (x – 3) + 1 (x – 3) = 0
⇒ (x + 1) (x – 3) = 0 ⇒  x = -1 or x = 3

2) 2x2  – 5x + 2 = 0
⇒  2x2  – 4x – x + 2 = 0
⇒  2x (x – 2) – 1 (x – 2) = 0
⇒  (2x – 1) (x – 2) = 0 ⇒  x = 1/2 or x = 2

### Karnataka Class 11 Commerce Maths Complex Numbers and Quadratic Equations

Formula Method:

The solutions of the quadratic equation ax2  + bx + c = 0 is obtained using the formula.
x = (-b ± √b2 – 4ac) / 2a

If α & β are the two root of the quadratic equation.
Then α = (-b + √b2 – 4ac) / 2a    and    β = (-b – √b2 – 4ac) / 2a.

Nature of the roots of ax2 + bx + c = 0

The quantity b2 – 4ac on which the nature of the roots depend, is called the discriminant of the quadratic equation ax2 + bx + c = 0 and is denoted by Δ or D.
1. If b2 – 4ac = 0. Then the roots are real and equal.
2. If b2 – 4ac is positive. Then the roots are real and unequal.
3. If b2 – 4ac is negative. Then the two roots are unequal and imaginary.

Relation between the roots and the co-efficients of the quadratic equation ax2 + bx + c = 0:

Let α and  β  are the roots of the equation ax2 + bx + c = 0
Dividing throughout by a
⇒ x2 + (b/a)x + c/a ……….(1)

⇒ x2 – (α + β)x + α.β = 0 ……….(2)

⇒ ( x – α )( x -β) = 0

Comparing (1) and (2), we get:

Sum of the roots = α + β = -b/a = -coefficient of  x / coefficient of x2

Product of the roots = α.β = c/a = constant / coefficient of x

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