Advertisements
Advertisements
प्रश्न
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.
Advertisements
उत्तर
The given expression is `(y^(1/2) + x^(1/3))^"n"`
Since the binomial coefficient of third term from the end = Binomial coefficient of third term from the beginning = nC2
∴ nC2 = 45
⇒ `("n"("n" - 1))/2` = 45
⇒ n2 – n = 90
⇒ n2 – n – 90 = 0
⇒ n2 – 10n + 9n – 90 = 0
⇒ n(n – 10) + 9(n – 10) = 0
⇒ (n – 10)(n + 9) = 0
⇒ n = 10, n = –9
⇒ n = 10, n ≠ – 9
So, the given expression becomes `(y^(1/2) + x^(1/3))^10`
Sixth term is this expression T6 = T5+1
= `""^10"C"_5 (y^(1/2))^(10 - 5) (x^(1/3))^5`
= `""^10"C"_5 y^(5/2) * x^(5/3)`
= `252 y^(5/2) x^(5/3)`
Hence, the required term = `252 y^(5/2) * x^(5/3)`
APPEARS IN
संबंधित प्रश्न
Expand the expression: (1– 2x)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 f the following:
(101)4
Find (x + 1)6 + (x – 1)6. Hence or otherwise evaluate `(sqrt2 + 1)^6 + (sqrt2 -1)^6`
Find a if the coefficients of x2 and x3 in the expansion of (3 + ax)9 are equal.
Find the coefficient of x5 in the product (1 + 2x)6 (1 – x)7 using binomial theorem.
Evaluate `(sqrt3 + sqrt2)^6 - (sqrt3 - sqrt2)^6`
Find the value of `(a^2 + sqrt(a^2 - 1))^4 + (a^2 - sqrt(a^2 -1))^4`
Find an approximation of (0.99)5 using the first three terms of its expansion.
If n is a positive integer, prove that \[3^{3n} - 26n - 1\] is divisible by 676.
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?
Find the term independent of x in the expansion of `(sqrt(x)/sqrt(3) + sqrt(3)/(2x^2))^10`.
If n is a positive integer, find the coefficient of x–1 in the expansion of `(1 + x)^2 (1 + 1/x)^n`
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 ______.
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 coefficient of x15 in the expansion of (x – x2)10.
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.
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
Number of terms in the expansion of (a + b)n where n ∈ N is one less than the power n.
The sum of the last eight coefficients in the expansion of (1 + x)16 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 ______.
