Firstly, the expression (a+ b)6 - (a - b)6 is simplified by using Binomial Theorem.
This can be done as
\((a+b)^6 = C_0^6a^6 + C_1^61a^5b + C_2^6a^4b^2 + C_3^6a^3b^3 + C_4^6a^2b^4 + C_5^6a^1b^5 + C_6^6b^6\)
\(= a^6 + 6a^5b +15a^4b^2 + 20a^3b^3 + 15a^2b^4 + 6ab^5 + b^6\)
\((a-b)^6 = C_0^6a^6 - C_1^6a^5b + C_2^6a^4b^2 - C_3^6a^3b^3 + C_4^6a^2b^4 - C_5^6a^1b^5 + C_6^6b^6\)
\(= a^6 - 6a^5b+15a^4b^2 - 20a^3b^3 +15a^2b^4- 6ab^5 + b^6\)
\(∴ (a + b)^6 - (a - b)^6 = 2[ 6a^5b+20a^3b^3 +6ab^5]\)
Putting a = √3 and b = √2, we obtain
\((√3+ √2)^6 - (√3 −√2)^6 = 2[6(√3)^5(√2) +20 (√3)^3 (√2)3 + 6(√3)(√2)^5]\)
\(=2[54√6+120√6+24√6]\)
\(=2×198√6\)
\(=396√6\)
If a and b are distinct integers, prove that a - b is a factor of \(a^n - b^n\) , whenever n is a positive integer.
[Hint: write\( a ^n = (a - b + b)^n\) and expand]
Give reasons for the following.
(i) King Tut’s body has been subjected to repeated scrutiny.
(ii) Howard Carter’s investigation was resented.
(iii) Carter had to chisel away the solidified resins to raise the king’s remains.
(iv) Tut’s body was buried along with gilded treasures.
(v) The boy king changed his name from Tutankhaten to Tutankhamun.
Find the mean deviation about the median for the data
xi | 15 | 21 | 27 | 30 | 35 |
fi | 3 | 5 | 6 | 7 | 8 |
The binomial theorem formula is used in the expansion of any power of a binomial in the form of a series. The binomial theorem formula is