Advertisements
Advertisements
प्रश्न
Find the expansion of (3x2 – 2ax + 3a2)3 using binomial theorem.
Advertisements
उत्तर
Using binomial theorem, the given expression (3x* -2ax +3a* ) can be expanded
[3x2 - a (2x - 3a)]3
= 3C0 (3x2 - 2ax)3 + 3C1(3x2 - 2ax)2 (3a2)+ 3C2(3x2 - 2ax) (3a2)2 + 3C3(3a2)3
= (3x2 - 2ax)3 + 3(9x4 - 12ax3 + 4a2x2)(3a2)+3(3x2 - 2ax)(9a4) + 27a6
= (3x2 - 2ax)3 + 81a2x4 - 108a3x3 + 36a4x2 + 81a4x2 - 54a5x + 27a6
= (3x2 - 2ax)3 + 81a2x4 - 108a3x3 + 117a4x2 - 54a5x + 27a6
Again, by using binomial theorem, we obtain
(3x2 - 2ax)3
= 3C0 (3X2)3 - 3C1 (3X2)2 (2ax) + 3C2 (3X2)(2ax)2 - 3C3 (2ax)3
= 27x6 - 3(9x4) (2ax) + 3 (3x2) (4a2x2) -8a3x3
= 27x6 - 54ax5 + 36a2x4 - 8a3x3
From (1) and (2), we obtain
(3x2 - 2ax + 3a2)3
= 27x6 - 54ax5 + 36a2 x4 - 8a3x3 + 81a2x4 - 108a3x3 + 117a4 x2 - 54a5x + 27a6
= 27x6 - 54ax5 + 117a2 x4 - 116a3 x3 + 117a4 x2 - 54a5x + 27a6
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 + 1/x)^6`
Using Binomial Theorem, evaluate the following:
(96)3
Using binomial theorem, evaluate the following:
(99)5
Find (x + 1)6 + (x – 1)6. Hence or otherwise evaluate `(sqrt2 + 1)^6 + (sqrt2 -1)^6`
Show that 9n+1 – 8n – 9 is divisible by 64, whenever n is a positive integer.
Find a, b and n in the expansion of (a + b)n if the first three terms of the expansion are 729, 7290 and 30375, respectively.
Find a if the coefficients of x2 and x3 in the expansion of (3 + ax)9 are equal.
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`
Expand using Binomial Theorem `(1+ x/2 - 2/x)^4, x != 0`
Show that \[2^{4n + 4} - 15n - 16\] , where n ∈ \[\mathbb{N}\] is divisible by 225.
Find the rth term in the expansion of `(x + 1/x)^(2r)`
Evaluate: `(x^2 - sqrt(1 - x^2))^4 + (x^2 + sqrt(1 - x^2))^4`
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)`
If the coefficients of x7 and x8 in `2 + x^n/3` are equal, then n is ______.
The number of terms in the expansion of (a + b + c)n, where n ∈ N is ______.
If z = `sqrt(3)/2 + i^5/2 + sqrt(3)/2 - i^5/2`, then ______.
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.
Find the coefficient of x4 in the expansion of (1 + x + x2 + x3)11.
In the expansion of (x + a)n if the sum of odd terms is denoted by O and the sum of even term by E. Then prove that O2 – E2 = (x2 – a2)n
In the expansion of (x + a)n if the sum of odd terms is denoted by O and the sum of even term by E. Then prove that 4OE = (x + a)2n – (x – a)2n
Let the coefficients of x–1 and x–3 in the expansion of `(2x^(1/5) - 1/x^(1/5))^15`, x > 0, be m and n respectively. If r is a positive integer such that mn2 = 15Cr, 2r, then the value of r is equal to ______.
If the coefficients of (2r + 4)th, (r – 2)th terms in the expansion of (1 + x)18 are equal, then r is ______.
Let `(5 + 2sqrt(6))^n` = p + f where n∈N and p∈N and 0 < f < 1 then the value of f2 – f + pf – p is ______.
The positive integer just greater than (1 + 0.0001)10000 is ______.
