Advertisements
Advertisements
Question
If the term free from x in the expansion of `(sqrt(x) - k/x^2)^10` is 405, find the value of k.
Advertisements
Solution
The given expression is `(sqrt(x) - k/x^2)^10`
General term `"T"_(r + 1) = ""^n"C"_r x^(n - r) y^r`
= `""^10"C"_r (sqrt(x))^(10 - r) ((-k)/x^2)^r`
= `""^10"C"_r (x)^((10 - r)/2) (-k)^r (1/x^(2r))`
= `""^10"C"_r (x)^((10 - r)/2 - 2r) (-k)^r`
= `""^10"C"_r (x)^((10 - r - 4r)/2) (- k)^r`
= `""^10"C"_r (x)^((10 - 5r)/2) (- k)^r`
For getting term free from x
`(10 - 5r)/2` = 0
⇒ r = 2
On putting the value of r in the above expression
We get `""^10"C"_2 (-k)^2`
According to the condition of the question, we have
`""^10"C"_2 k^2` = 405
⇒ `(10*9)/(2*1) k^2` = 405
⇒ 45k2 = 405
⇒ k2 = `405/45` = 9
∴ k = `+- 3`
Hence, the value of k = ±3
APPEARS IN
RELATED QUESTIONS
Write the general term in the expansion of (x2 – yx)12, x ≠ 0
Find the 4th term in the expansion of (x – 2y)12 .
Find the middle terms in the expansions of `(x/3 + 9y)^10`
Prove that the coefficient of xn in the expansion of (1 + x)2n is twice the coefficient of xn in the expansion of (1 + x)2n–1 .
Find n, if the ratio of the fifth term from the beginning to the fifth term from the end in the expansion of `(root4 2 + 1/ root4 3)^n " is " sqrt6 : 1`
Find the middle term in the expansion of:
(iii) \[\left( x^2 - \frac{2}{x} \right)^{10}\]
Find the middle terms in the expansion of:
(ii) \[\left( 2 x^2 - \frac{1}{x} \right)^7\]
Find the middle terms in the expansion of:
(iii) \[\left( 3x - \frac{2}{x^2} \right)^{15}\]
Find the middle terms(s) in the expansion of:
(vi) \[\left( \frac{x}{3} + 9y \right)^{10}\]
Find the middle terms(s) in the expansion of:
(x) \[\left( \frac{x}{a} - \frac{a}{x} \right)^{10}\]
Find the term independent of x in the expansion of the expression:
(i) \[\left( \frac{3}{2} x^2 - \frac{1}{3x} \right)^9\]
If the coefficients of \[\left( 2r + 4 \right)\text{ th and } \left( r - 2 \right)\] th terms in the expansion of \[\left( 1 + x \right)^{18}\] are equal, find r.
Find the coefficient of a4 in the product (1 + 2a)4 (2 − a)5 using binomial theorem.
If in the expansion of (1 + x)n, the coefficients of three consecutive terms are 56, 70 and 56, then find n and the position of the terms of these coefficients.
If a, b, c and d in any binomial expansion be the 6th, 7th, 8th and 9th terms respectively, then prove that \[\frac{b^2 - ac}{c^2 - bd} = \frac{4a}{3c}\].
If the 6th, 7th and 8th terms in the expansion of (x + a)n are respectively 112, 7 and 1/4, find x, a, n.
If the term free from x in the expansion of \[\left( \sqrt{x} - \frac{k}{x^2} \right)^{10}\] is 405, find the value of k.
Write the middle term in the expansion of \[\left( x + \frac{1}{x} \right)^{10}\]
Write the coefficient of the middle term in the expansion of \[\left( 1 + x \right)^{2n}\] .
The middle term in the expansion of \[\left( \frac{2 x^2}{3} + \frac{3}{2 x^2} \right)^{10}\] is
If in the expansion of (1 + y)n, the coefficients of 5th, 6th and 7th terms are in A.P., then nis equal to
In the expansion of \[\left( \frac{1}{2} x^{1/3} + x^{- 1/5} \right)^8\] , the term independent of x is
If the sum of odd numbered terms and the sum of even numbered terms in the expansion of \[\left( x + a \right)^n\] are A and B respectively, then the value of \[\left( x^2 - a^2 \right)^n\] is
The number of terms with integral coefficients in the expansion of \[\left( {17}^{1/3} + {35}^{1/2} x \right)^{600}\] is
Find the middle term (terms) in the expansion of `(p/x + x/p)^9`.
Find the coefficient of `1/x^17` in the expansion of `(x^4 - 1/x^3)^15`
Show that the middle term in the expansion of `(x - 1/x)^(2x)` is `(1 xx 3 xx 5 xx ... (2n - 1))/(n!) xx (-2)^n`
If xp occurs in the expansion of `(x^2 + 1/x)^(2n)`, prove that its coefficient is `(2n!)/(((4n - p)/3)!((2n + p)/3)!)`
Find the term independent of x in the expansion of (1 + x + 2x3) `(3/2 x^2 - 1/(3x))^9`
If the middle term of `(1/x + x sin x)^10` is equal to `7 7/8`, then value of x is ______.
Middle term in the expansion of (a3 + ba)28 is ______.
The last two digits of the numbers 3400 are 01.
The coefficient of x256 in the expansion of (1 – x)101(x2 + x + 1)100 is ______.
If the 4th term in the expansion of `(ax + 1/x)^n` is `5/2` then the values of a and n respectively are ______.
The coefficient of y49 in (y – 1)(y – 3)(y – 5) ...... (y – 99) is ______.
