Advertisements
Advertisements
प्रश्न
Evaluate `(sqrt3 + sqrt2)^6 - (sqrt3 - sqrt2)^6`
Advertisements
उत्तर
`(a + b)^6 = ^6C_0 a^6 + ^6C_1 a^5 b + ^6C_2 a^4 b^2 + ^6C_3 a^3 b^3 + ^6C_4 a^2 b^4 + ^6C_5a^1b^5 + ^6C_6 b^6`
= `a^6 + 6a^5b + 15a^4 b^2 + 20a^3 b^3 + 15a^2 b^4 + 6ab^5 + b^6`
`(a - b)^6 = ^6C_0 a^6 - ^6C_1 a^5 b + ^6C_2 a^4 b^2 - ^6C_3 a^3 b^3 + ^6C_4 a^2 b^4 - ^6C_5a^1b^5 + ^6C_6 b^6`
= `a^6 - 6a^5b + 15a^4 b^2 - 20a^3 b^3 + 15a^2 b^4 - 6ab^5 + b^6`
∴ `(a + b)^6 - (a -b)^6 = 2(6a^5b + 20a^3 b^3 + 6ab^5)`
Putting a = `sqrt3` and b = `sqrt2`, we obtain
`(sqrt3 + sqrt2)^6 - (sqrt3 + sqrt2)^6` = `2[6(sqrt3)^5 (sqrt2) + 20 (sqrt3)^3 (sqrt2)^3 + 6 (sqrt3)(sqrt2)^5]`
= `2[54sqrt6 + 120 sqrt6 + 24 sqrt6]`
= `2 xx 198 sqrt6`
= `396 sqrt6`
APPEARS IN
संबंधित प्रश्न
Expand the expression (1– 2x)5
Expand the expression: `(2/x - x/2)^5`
Expand the expression: (2x – 3)6
Expand the expression: `(x/3 + 1/x)^5`
Using Binomial Theorem, evaluate the following:
(96)3
Using Binomial Theorem, evaluate of the following:
(102)5
Find (a + b)4 – (a – b)4. Hence, evaluate `(sqrt3 + sqrt2)^4 - (sqrt3 - sqrt2)^4`
Prove that `sum_(r-0)^n 3^r ""^nC_r = 4^n`
If a and b are distinct integers, prove that a – b is a factor of an – bn, whenever n is a positive integer.
[Hint: write an = (a – b + b)n and expand]
Find the value of `(a^2 + sqrt(a^2 - 1))^4 + (a^2 - sqrt(a^2 -1))^4`
Using binomial theorem determine which number is larger (1.2)4000 or 800?
Find the value of (1.01)10 + (1 − 0.01)10 correct to 7 places of decimal.
Show that \[2^{4n + 4} - 15n - 16\] , where n ∈ \[\mathbb{N}\] is divisible by 225.
Evaluate: `(x^2 - sqrt(1 - x^2))^4 + (x^2 + sqrt(1 - x^2))^4`
Find the coefficient of x11 in the expansion of `(x^3 - 2/x^2)^12`
Determine whether the expansion of `(x^2 - 2/x)^18` will contain a term containing x10?
If n is a positive integer, find the coefficient of x–1 in the expansion of `(1 + x)^2 (1 + 1/x)^n`
Which of the following is larger? 9950 + 10050 or 10150
Find the coefficient of x50 after simplifying and collecting the like terms in the expansion of (1 + x)1000 + x(1 + x)999 + x2(1 + x)998 + ... + x1000 .
If a1, a2, a3 and a4 are the coefficient of any four consecutive terms in the expansion of (1 + x)n, prove that `(a_1)/(a_1 + a_2) + (a_3)/(a_3 + a_4) = (2a_2)/(a_2 + a_3)`
The total number of terms in the expansion of (x + a)51 – (x – a)51 after simplification is ______.
Find the coefficient of x in the expansion of (1 – 3x + 7x2)(1 – x)16.
Find the coefficient of x15 in the expansion of (x – x2)10.
Find the sixth term of the expansion `(y^(1/2) + x^(1/3))^"n"`, if the binomial coefficient of the third term from the end is 45.
If the coefficient of second, third and fourth terms in the expansion of (1 + x)2n are in A.P. Show that 2n2 – 9n + 7 = 0.
The total number of terms in the expansion of (x + a)100 + (x – a)100 after simplification is ______.
Given the integers r > 1, n > 2, and coefficients of (3r)th and (r + 2)nd terms in the binomial expansion of (1 + x)2n are equal, then ______.
The two successive terms in the expansion of (1 + x)24 whose coefficients are in the ratio 1:4 are ______.
The number of terms in the expansion of (x + y + z)n ______.
If the coefficients of (2r + 4)th, (r – 2)th terms in the expansion of (1 + x)18 are equal, then r is ______.
The positive integer just greater than (1 + 0.0001)10000 is ______.
