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 (1– 2x)5
Expand the expression: `(x + 1/x)^6`
Using Binomial Theorem, evaluate the following:
(96)3
Using binomial theorem, evaluate f the following:
(101)4
Find (a + b)4 – (a – b)4. Hence, evaluate `(sqrt3 + sqrt2)^4 - (sqrt3 - sqrt2)^4`
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 the value of `(a^2 + sqrt(a^2 - 1))^4 + (a^2 - sqrt(a^2 -1))^4`
If n is a positive integer, prove that \[3^{3n} - 26n - 1\] is divisible by 676.
Find the value of (1.01)10 + (1 − 0.01)10 correct to 7 places of decimal.
Find the rth term in the expansion of `(x + 1/x)^(2r)`
Find the 4th term from the end in the expansion of `(x^3/2 - 2/x^2)^9`
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?
Find the term independent of x in the expansion of `(sqrt(x)/sqrt(3) + sqrt(3)/(2x^2))^10`.
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 .
The total number of terms in the expansion of (x + a)51 – (x – a)51 after simplification is ______.
If the coefficients of x7 and x8 in `2 + x^n/3` are equal, then n is ______.
If (1 – x + x2)n = a0 + a1 x + a2 x2 + ... + a2n x2n , then a0 + a2 + a4 + ... + a2n equals ______.
The coefficient of xp and xq (p and q are positive integers) in the expansion of (1 + x)p + q are ______.
Find the coefficient of x15 in the expansion of (x – x2)10.
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
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 number of terms in the expansion of (x + y + z)n ______.
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 ______.
