Advertisements
Advertisements
प्रश्न
The number of terms with integral coefficients in the expansion of \[\left( {17}^{1/3} + {35}^{1/2} x \right)^{600}\] is
पर्याय
100
50
150
101
Advertisements
उत्तर
101
\[\text{ The general term } T_{r + 1} \text{ in the given expansion is given by } \]
\[ ^{600}{}{C}_r ( {17}^{1/3} )^{600 - r} ( {35}^{1/2} x )^r \]
\[ = ^{600}{}{C}_r {17}^{200 - r/3} \times {35}^{r/2} x^r \]
\[\text{ Now,} T_{r + 1} \text{ is an integer if } \frac{r}{2} \text{ and } \frac{r}{3} \text{ are integers for all } 0 \leq r \leq 600\]
\[\text{ Thus, we have } \]
\[ r = 0, 6, 12, . . . 600 (\text{ Multiples of } 6)\]
\[\text{ Since, It is an A . P } \]
\[\text{ So } , 600 = 0 + \left( n - 1 \right)6\]
\[ \Rightarrow n = 101\]
\[\text{ Hence, there are 101 terms with integral coefficients } .\]
APPEARS IN
संबंधित प्रश्न
Write the general term in the expansion of (x2 – y)6
Find the 4th term in the expansion of (x – 2y)12 .
Find the middle terms in the expansions of `(3 - x^3/6)^7`
Find a positive value of m for which the coefficient of x2 in the expansion
(1 + x)m is 6
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:
(ii) \[\left( \frac{a}{x} + bx \right)^{12}\]
Find the middle terms in the expansion of:
(i) \[\left( 3x - \frac{x^3}{6} \right)^9\]
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:
(iii) \[\left( 1 + 3x + 3 x^2 + x^3 \right)^{2n}\]
Find the middle terms(s) in the expansion of:
(viii) \[\left( 2ax - \frac{b}{x^2} \right)^{12}\]
Find the term independent of x in the expansion of the expression:
(iii) \[\left( 2 x^2 - \frac{3}{x^3} \right)^{25}\]
Find the term independent of x in the expansion of the expression:
(v) \[\left( \frac{\sqrt{x}}{3} + \frac{3}{2 x^2} \right)^{10}\]
Find the term independent of x in the expansion of the expression:
(vi) \[\left( x - \frac{1}{x^2} \right)^{3n}\]
If the coefficients of (2r + 1)th term and (r + 2)th term in the expansion of (1 + x)43 are equal, find r.
Prove that the term independent of x in the expansion of \[\left( x + \frac{1}{x} \right)^{2n}\] is \[\frac{1 \cdot 3 \cdot 5 . . . \left( 2n - 1 \right)}{n!} . 2^n .\]
The coefficients of 5th, 6th and 7th terms in the expansion of (1 + x)n are in A.P., find n.
If the coefficients of 2nd, 3rd and 4th terms in the expansion of (1 + x)2n are in A.P., show that \[2 n^2 - 9n + 7 = 0\]
If the coefficients of 2nd, 3rd and 4th terms in the expansion of (1 + x)n are in A.P., then find the value of n.
If 3rd, 4th 5th and 6th terms in the expansion of (x + a)n be respectively a, b, c and d, prove that `(b^2 - ac)/(c^2 - bd) = (5a)/(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 in the expansion of \[\left( x^4 - \frac{1}{x^3} \right)^{15}\] , \[x^{- 17}\] occurs in rth term, then
If in the expansion of (1 + y)n, the coefficients of 5th, 6th and 7th terms are in A.P., then nis equal to
If rth term is the middle term in the expansion of \[\left( x^2 - \frac{1}{2x} \right)^{20}\] then \[\left( r + 3 \right)^{th}\] term is
Find the term independent of x, x ≠ 0, in the expansion of `((3x^2)/2 - 1/(3x))^15`
If the term free from x in the expansion of `(sqrt(x) - k/x^2)^10` is 405, find the value of k.
Find the middle term (terms) in the expansion of `(x/a - a/x)^10`
If p is a real number and if the middle term in the expansion of `(p/2 + 2)^8` is 1120, find p.
If the middle term of `(1/x + x sin x)^10` is equal to `7 7/8`, then value of x is ______.
In the expansion of `(x^2 - 1/x^2)^16`, the value of constant term is ______.
Middle term in the expansion of (a3 + ba)28 is ______.
The position of the term independent of x in the expansion of `(sqrt(x/3) + 3/(2x^2))^10` is ______.
The number of terms in the expansion of [(2x + y3)4]7 is 8.
If the expansion of `(x - 1/x^2)^(2n)` contains a term independent of x, then n is a multiple of 2.
If n is the number of irrational terms in the expansion of `(3^(1/4) + 5^(1/8))^60`, then (n – 1) is divisible by ______.
The coefficient of y49 in (y – 1)(y – 3)(y – 5) ...... (y – 99) is ______.
If the coefficient of x10 in the binomial expansion of `(sqrt(x)/5^(1/4) + sqrt(5)/x^(1/3))^60` is 5kl, where l, k ∈ N and l is coprime to 5, then k is equal to ______.
