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# Karnataka Class 11 Commerce Maths Binomial Theorem

## Karnataka Class 11 Commerce Maths Binomial Theorem

### Karnataka Class 11 Commerce Maths Binomial Theorem

Karnataka Class 11 Commerce Maths Binomial Theorem : 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 Pre-University board is a government body which organizes the higher secondary examination in the state.  The board functions under the Department of Primary & Secondary Education. The board has a total of 1202 government pre-university colleges, 165 unaided Pre-University colleges and about 13 Corporation pre-university colleges.

### Karnataka Class 11 Commerce Maths Binomial Theorem

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 Binomial Theorem

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 Binomial Theorem

Binomial Theorem Topics :

• History
• statement and proof of the binomial theorem for positive integral indices.
• Pascal’s triangle, general and middle term in binomial expansion,
• Problems based on expansion, finding any term, term independent of x, middle term, coefficient of xr .

An expression consisting of two terms, connected by + or – sign is called a binomial expression.

For example, x + a, 2x – 3y, 1/x, 1/x3, 7x – 4/5y,  etc., are all binomial expressions.

### Karnataka Class 11 Commerce Maths Binomial Theorem

Binomial theorem

If a and b are real numbers and n is a positive integer, then

(a + b) nnC0 an + nC1 an – 1 b1 + nC2 an – 2 b2 + … … + nCr an – r br + … +  nCn  bn , where  nCr = ⌊n /⌊r ⌊(n-r)  for 0 ≤ r ≤ n

The general term or (r + 1)th term in the expansion is given by Tr + 1 = nCr an – r br

### Karnataka Class 11 Commerce Maths Binomial Theorem

Some important observations

1. The total number of terms in the binomial expansion of (a + b) n is n + 1, i.e. one more than the exponent n.

2. In the expansion, the first term is raised to the power of the binomial and in each subsequent terms the power of a reduces by one with simultaneous increase in the power of b by one, till power of b becomes equal to the power of binomial, i.e., the power of a is n in the first term, (n – 1) in the second term and so on ending with zero in the last term. At the same time power of b is 0 in the first term, 1 in the second term and 2 in the third term and so on, ending with n in the last term.

3. In any term the sum of the indices (exponents) of ‘a’ and ‘b’ is equal to n (i.e., the power of the binomial).

4. The coefficients in the expansion follow a certain pattern known as pascal’s triangle.

Karnataka Class 11 Commerce Maths Binomial Theorem

### Karnataka Class 11 Commerce Maths Binomial Theorem

Each coefficient of any row is obtained by adding two coefficients in the preceding row, one on the immediate left and the other on the immediate right and each row is bounded by 1 on both sides.

The (r + 1)th term or general term is given by Tr + 1 = nCr an – r br

Some particular cases If n is a positive integer, then

(a + b)n = nC0 an b0 + nC1 an b1 + nC2 an-2 b2+ … + nCr an – r b + … + nCn abn                            … (1)

In particular

1. Replacing b by – b in (i), we get

(a + b)n = nC0 an b0 – nC1 an b1 + nC2 an-2 b2+ … +(-1)r nCr an – r b + … +(-1)n nCn abn           … (2)

2. Adding (1) and (2), we get

(a + b)n + (a – b)n = 2 [nC0 an b0 + nC2 an-2 b2 + nC4 an-4 b4 + … ] = 2 [terms at odd places]

3. Subtracting (2) from (1), we get

(a + b)n + (a – b)n = 2 [nC1 an-1 b1 + nC3 an-3 b3 + … ] = 2 [sum of terms at even places]

4. Replacing a by 1 and b by x in (1), we get

(1 + x) n = nC0 x0 +  nC1 x1 +  nC2 x2 + … +  nCr xr + … +  nCn-1 xn-1 +  nCn xn

i.e. (1 + x)n = ∑nr=0 nCr xr

5. Replacing a by 1 and b by –x in … (1), we get

(1 – x) n = nC0 x0 –  nC1 x1 +  nC2 x2 + … +  nCr xr + … +  nCn-1 xn-1 +  nCn xn

i.e. (1 – x)n = ∑nr=0 (-1)r nCr xr

### Karnataka Class 11 Commerce Maths Binomial Theorem

The pth term from the end

The pth term from the end in the expansion of (a + b) n is (n – p + 2)th term from the beginning.

Middle terms

The middle term depends upon the value of n.

(a) If n is even: then the total number of terms in the expansion of (a + b)n is n + 1 (odd). Hence, there is only one middle term,

i.e., [(n/2) + 1]th  term is the middle term.

(b) If n is odd: then the total number of terms in the expansion of (a + b)n is n + 1 (even). So there are two middle terms

i.e., [(n + 1)/2] and [(n + 3)/2]th  are two middle terms.

### Karnataka Class 11 Commerce Maths Binomial Theorem

Binomial coefficient

In the Binomial expression, we have (a + b)n = nC0 an + nC1 an – 1 b + nC2 an – 2 b2 + … + nCn bn

(1) The coefficients nC0, nC1 , nC2, … , nCn  are known as binomial or combinatorial coefficients.

Putting a = b = 1 in (1), we get nC0  +  nC1 + nC2 + … + nCn = 2n

Thus the sum of all the binomial coefficients is equal to 2n .

Again, putting a = 1 and b = –1 in (i), we get  nC0  +  nC2 + nC4 + …  =  nC1  +  nC3 + nC5 + …

Thus, the sum of all the odd binomial coefficients is equal to the sum of all the even binomial coefficients and each is equal to 2n / 2 = 2 n-1.

nC0  +  nC2 + nC4 + …  =  nC1  +  nC3 + nC5 + … = 2 n-1.

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