Practice Problems : Binomial Theorem

Level – I

1. Find the middle term in the expansion of:

(a) $ \displaystyle (\frac{a}{x}+ bx)^{12}$

(b) $ \displaystyle (3x – \frac{x^3}{6})^9 $

2. Find the term independent of x in the expansion of

(a) $ \displaystyle (\frac{1}{2}x^{1/3} + x^{-1/5})^8 $

(b) $ \displaystyle (\sqrt{\frac{x}{3}} + \frac{3}{2x^2})^{10}$

3. Find the coefficient of x7 in

$ \displaystyle (a x^2 + \frac{1}{bx})^{11}$ and x-7 in $ \displaystyle (a x – \frac{1}{bx^2})^{11}$

and find the relation between a and b so that their coefficients are equal.

4. If n is an in integer greater than 1 , prove that

a – C1(a – 1) + C2(a -2) …. + (-1)n Cn (a – n) = 0

5. Show that

$ \displaystyle C_1 -\frac{1}{2}C_2 + \frac{1}{3}C_3 – ….+ (-1)^{n-1}\frac{1}{n} C_n$

$ \displaystyle = 1 + \frac{1}{2} + \frac{1}{3} + ….+ \frac{1}{n} $

6. If the coefficients of the 2nd, 3rd and 4th terms in the expansion of (1 + x)n are in A.P, find the value of n

7. If x4r occurs in the expansion of (x + 1/x2)4n, prove that its coefficient is

$ \displaystyle \frac{(4n)!}{(\frac{4}{3}(n-r))! (\frac{4}{3}(2n+r))!} $

8. If (1 + x)n = C0 + C1x + C2 x2 + Cnxn show that

$ \displaystyle (C_0 + C_1)(C_1 + C_2)(C_2 + C_3) ….(C_{n-1} + C_n)$

$ \displaystyle = \frac{(n+1)^n}{n!}C_1 .C_2 ….C_n $

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