Binomial Theorem Class 12 Mathematics Solutions has been updated according to the latest syllabus of 2080. It means the solutions of Binomial Theorem chapter provided in this article contains all the new exercise that has recently been updated. Now you don’t need to go anywhere searching for the notes of binomial theorem because we are here to serve you.
Chapter – 2
Binomial Theorem
A binomial expression, we mean an expression consisting of two terms. For example, x + a, x + y x + b are all binomial expressions. It is not difficult to expand the power of a binomial expression, when the power is a very small positive integer such as 2, 3, 4.
But it is difficult to expand, when the power is big. So, we need a formula for the expansion of binomial expression with any index or power. That formula is called Binomial Theorem. Here we shall be just content with a proof, when the index of the binomial expression is any positive integer. The Binomial theorem was first discovered by Issac Newton.
Class 12 Binomial Theorem chapter has 3 exercises in total and in each exercise we will different proofs as well as theory related to binomial theorem. We have listed the notes of all exercises. You can click on the buttons given below to view exercise-wise notes.
Binomial Theorem for a Postive Integer
This principle can be best understood with some examples. Let us consider two letters A₁ and A₂ and see in how many ways they can be arranged in a row. This consideration leads us to the following principle known as the basic principle of counting.
For any positive integer n, (a+x)ⁿ = C(n,0) * aⁿ + C(n,1) * a⁽ⁿ⁻¹⁾ * x + C(n,2) * a⁽ⁿ⁻²⁾ * x² +……. + C(n,r) * a⁽ⁿ⁻ʳ⁾ * xʳ +…..+C(n,n)x^ n .
Binomial Theorem Class 12 Mathematics Solutions PDF
This PDF will provide the solutions of every question from the 1st exercise of class 12 binomial theorem. If you want the solutions of other exercises then you can select the exercise from the button given below.
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Proof:
We have, (a+x)ⁿ = (a + x)(a + x)(a + x)(a + x) …… to n factors.
In the process of multiplication of n factors in the right hand side, we shall choose either a or x from each factor and get n letters. If we choose a from each factor we get aⁿ. If we choose a from each of n – 1 factors and x from the remaining one, we get c * r ^ (n – 1) * x.
In this way we get a term a’ – x by choosing a from n – r factors and x from the other r factors. Moreover, the number of times a ^ (rr) – xʳ appears in the expression is equal to the number of combination of n taken r at a time which is C(n,r).
Now if we allow r to vary from 0 to n we get the required expansion. Now let us note some the properties of the expression, when n is any positive integer.
- The number of terms in the expansion is n + 1
- In each term, the sum of the exponents is n.
- The expansion starts with the first term aⁿ. and ends with the last term Xⁿ prime prime , When we came across from one term to the next, we find that the exponent of a decreases by one and that of x increases by one.
- The coefficients of the terms equidistant from the beginning and the end are always equal.
General Terms
The (r + 1)th term in the expansion of (a + x)ⁿ is usually called its general term, because any required term may be obtained from it, by a suitable value to r. The (r + 1) th term is denoted by Tᵣ₊₁.
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