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Binomial Theorem

Multiple Choice Questions

91.

To find the coefficient of x⁴ in the expansion of $\frac{3 x}{(x - 2) (x - 1)}$ , the interval in which the expansion is valid, is

$- 2 < x < \infty$
$- \frac{1}{2} < x < \frac{1}{2}$
$- 1 < x < 1$
$- \infty < x < \infty$

92.

$Let α > 0, β > 0 be such that α^{3} + β^{2} = 4 . If the maximum value of the term independent of x in the binomial expansion of {({αx}^{\frac{1}{9}} + {βx}^{- \frac{1}{6}})}^{10} is 10 k, then k is equal to :$

93.
If for some positive integer n, the coefficients of three consecutive terms in the binomial expansion of
(1 + x)^{n + 5} are in the ratio 5 : 10 : 14, then the largest coefficient in the expansion is :

330

252

792

462

462

$Let three consecutive term are T_{r}, T_{r + 1}, T_{r + 2} Hence \frac{T_{r}}{T_{r + 1}} = \frac{5}{10} and \frac{T_{r + 1}}{T_{r + 2}} = \frac{10}{24} \frac{T_{r + 1}}{T_{r}} = 2 \frac{{}^{n + 5}C_{r}}{{}^{n + 5}C_{r + 1}} = \frac{5}{7} \frac{{}^{n + 5}C_{r}}{{}^{n + 5}C_{r - 1}} = 2 \frac{{}^{n + 5}C_{r + 1}}{{}^{n + 5}C_{r}} = \frac{7}{5} \frac{(n + 5) - r + 1}{r} = 2 \frac{(n + 5) - (r + 1) + 1}{r + 1} = \frac{7}{5} n - r + 6 = 2 r \frac{n - r + 5}{r + 1} = \frac{7}{5} n - 3 r + 6 = 0 . . . (i) 5 n - 5 r + 25 = 7 r + 7 5 n - 12 r + 18 = 0 . . . (ii) Multiply equation (i) by 5 5 n - 15 r + 20 = 0 5 n - 12 r + 18 = 0 - + -____________________- 3 r + 12 = 0 \Rightarrow r = 4, n = 6 hence greatest coefficient will be of middle term =^{n + 5} C_{5} =^{11} C_{5} = 462$

94.

If the constant term in the binomial expansion of ${(\sqrt{x} - \frac{k}{x^{2}})}^{10} is 405, then |k| = ?$

Previous Year Papers

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Multiple Choice Questions

93. If for some positive integer n, the coefficients of three consecutive terms in the binomial expansion of(1 + x)n + 5 are in the ratio 5 : 10 : 14, then the largest coefficient in the expansion is : 330 252 792 462

93.
If for some positive integer n, the coefficients of three consecutive terms in the binomial expansion of
(1 + x)^{n + 5} are in the ratio 5 : 10 : 14, then the largest coefficient in the expansion is :

330

252

792

462