*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 QUADRATICEQUATIONS | 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
| 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/x _{3}, 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) ^{n} = ^{n}C_{0} a^{n} + ^{n}C_{1} a^{n – 1} b^{1} + ^{n}C_{2} a^{n – 2} b^{2} + … … + ^{n}C_{r} a^{n – r} b^{r} + … + ^{n}C_{n} b^{n} , where ^{n}C_{r} = ⌊n /⌊r ⌊(n-r) for 0 ≤ r ≤ n *

*The general term or (r + 1) ^{th} term in the expansion is given by T_{r + 1} = ^{n}C_{r} a^{n – r} b^{r}*

*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 T_{r + 1} = ^{n}C_{r} a^{n – r} b^{r }*

*Some particular cases If n is a positive integer, then *

*(a + b) ^{n} = ^{n}C_{0} a^{n} b^{0} + ^{n}C_{1} a^{n} b^{1} + ^{n}C_{2} a^{n-2} b^{2}+ … + ^{n}C_{r} a^{n – r} b^{r } + … + ^{n}C_{n} a^{0 }b^{n} … (1) *

*In particular*

*1. Replacing b by – b in (i), we get*

*(a + b) ^{n} = ^{n}C_{0} a^{n} b^{0} – ^{n}C_{1} a^{n} b^{1} + ^{n}C_{2} a^{n-2} b^{2}+ … +(-1)^{r} ^{n}C_{r} a^{n – r} b^{r } + … +(-1)^{n} ^{n}C_{n} a^{0 }b^{n} … (2)*

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

*(a + b) ^{n} + (a – b)^{n} = 2 [^{n}C_{0} a^{n} b^{0} + ^{n}C_{2} a^{n-2} b^{2} + ^{n}C_{4} a^{n-4} b^{4} + … ] *

*= 2 [terms at odd places]*

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

*(a + b) ^{n} + (a – b)^{n} = 2 [^{n}C_{1} a^{n-1} b^{1} + ^{n}C_{3} a^{n-3} b^{3} + … ] = 2 [sum of terms at even places]*

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

*(1 + x) ^{n} =*

^{n}C_{0}*x*+

^{0}

^{n}C_{1}*x*+

^{1}

^{n}C_{2}*x*+ … +

^{2}

^{n}C_{r}*x*+ … +

^{r}

^{n}C_{n-1}*x*+

^{n-1}

^{n}C_{n}*x*

^{n}*i.e. (1 + x) ^{n} = ∑^{n}_{r=0} ^{n}C_{r} x^{r}*

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

*(1 – x) ^{n} =*

^{n}C_{0}*x*–

^{0}

^{n}C_{1}*x*+

^{1}

^{n}C_{2}*x*+ … +

^{2}

^{n}C_{r}*x*+ … +

^{r}

^{n}C_{n-1}*x*+

^{n-1}

^{n}C_{n}*x*

^{n}*i.e. (1 – x) ^{n} = ∑^{n}_{r=0} (-1)^{r n}C_{r} x^{r}*

*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} = ^{n}C_{0} an + ^{n}C_{1} a^{n – 1} b + ^{n}C2 a^{n – 2} b^{2} + … + ^{n}C_{n} b^{n} … *

*(1) The coefficients ^{n}C_{0}, ^{n}C_{1} , ^{n}C2, … , ^{n}C_{n} are known as binomial or combinatorial coefficients. *

*Putting a = b = 1 in (1), we get ^{n}C_{0}* +

^{n}C_{1}+^{n}C2 +*… +*

^{n}C_{n}= 2^{n}*Thus the sum of all the binomial coefficients is equal to 2n . *

*Again, putting a = 1 and b = –1 in (i), we get ^{n}C_{0} + ^{n}C_{2} + ^{n}C_{4} + … = ^{n}C_{1} + ^{n}C_{3} + ^{n}C_{5} + … *

*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 2 ^{n} / 2 = 2 ^{n-1}. *

* ^{n}C_{0} + ^{n}C_{2} + ^{n}C_{4} + … = ^{n}C_{1} + ^{n}C_{3} + ^{n}C_{5} + … = 2 ^{n-1}.*

*for more on ***Karnataka Class 11 Commerce Maths Binomial Theorem** *click 1st PU Maths Book*

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