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: `(x/3 + 1/x)^5`
Expand the expression: `(x + 1/x)^6`
Using binomial theorem, evaluate f the following:
(101)4
Using binomial theorem, evaluate the following:
(99)5
Using Binomial Theorem, indicate which number is larger (1.1)10000 or 1000.
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 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]
Evaluate `(sqrt3 + sqrt2)^6 - (sqrt3 - sqrt2)^6`
Find the expansion of (3x2 – 2ax + 3a2)3 using binomial theorem.
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 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`
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
If the coefficients of x7 and x8 in `2 + x^n/3` are equal, then n 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.
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
The two successive terms in the expansion of (1 + x)24 whose coefficients are in the ratio 1:4 are ______.
Number of terms in the expansion of (a + b)n where n ∈ N is one less than the power 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 ______.
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 ______.
