Chapter 1

Expand \((1-2 x)^{5}\) by the binomial theorem.

Expand \(\left(x^{2}+2 y\right)^{5}\) by the binomial theorem.

If the coefficient of 7 th and 13 th term in the expansion of \((1+x)^{n}\) are equal, then \(n=\) (a) 10 (b) 15 (c) 18 (d) 20

Expand \(\left(\frac{2}{x}-\frac{x}{2}\right)^{5}\) by the binomial theorem.

Expand \(\left(2 x-\frac{3}{y}\right)^{5}\) by the binomial theorem.

In the expansion of \((1-x)^{5}\), coefficient of \(x^{5}\) will be (a) 1 (b) \(-1\) (c) 5 (d) \(-5\)

Expand \((2 x-3)^{6}\) by the binomial theorem.

Find the value of \(r\), if the coefficients of \((2 r+4)\) th and \((r-2)\) th terms in the expansion of \((1+x)^{18}\) are equal.

If the ratio of the coefficient of third and fourth term in the expansion of \(\left(x-\frac{1}{2 x}\right)^{n}\) is 1: 2 , then the value of \(n\) will be (a) 18 (b) 16 (c) 12 (d) \(-10\)

Using binomial theorem, evaluate (101) \(^{4}\).

If the coefficients of rth term and \((r+4)\) th term are equal in the expansion of \((1+x)^{20}\), then the value of \(r\) will be (a) 7 (b) 8 (c) 9 (d) 10

Using binomial theorem, evaluate (99)'.

Sum of odd terms is \(A\) and sum of even terms is \(B\) in the expansion \((x+a)^{n}\), then [RPET - 1987, 1992; UPSEAT - 2004; Roorkee - 1986] (a) \(A B=\frac{1}{4}\left[(x-a)^{2 n}-(x+a)^{2 n}\right]\) (b) \(2 A B=(x+a)^{2 n}-(x-a)^{2 n}\)(c) \(4 A B=(x+a)^{2 n}-(x-a)^{2 n}\) (d) none of these

Find \(a\) if the 17 th and 18 th terms of the expansion of \((2+a)^{50}\) are equal.

Find the coeffcient of \(x^{6} y^{3}\) in the expansion of \((x+2 y)^{9}\).

9 th term in the expansion of \(\left(\frac{y}{2}+2 x\right)^{12}\) is (a) \(7920 x^{7} x^{5}\) (b) \(7920 x^{6} y^{6}\) (c) \(7920 x^{8} y^{4}\) (d) \(7816 x^{8} x^{4}\)

The coefficients of three consecutive terms in the expansion of \((1+a)^{n}\) are in the ratio \(1: 7: 42\). Find \(n\).

Find the number of terms in the expansions of the following. \((2 x-3 y)^{9}\)

If \(A\) and \(B\) are the coefficient of \(x^{n}\) in the expansions of \((1+x)^{2 n}\) and \((1+x)^{2 n-1}\) respectively, then (a) \(A=B\) (b) \(A=2 B\) (c) \(2 A=B\) (d) none of these

Expand \((x+y)^{5}\)

Find the 7 th term in the expansion \(f\left(\frac{4 x}{5}-\frac{5}{2 x}\right)^{9}\)

The total number of terms in the expansion of \((x+a)^{100}+(x-a)^{100}\) after simplification will be (a) 202 (b) 51 (c) 50 (d) none of these

Expand the following \(\left(1-x+x^{2}\right)^{4}\).

If the coefficients of \((r-1)\) th, \(r\) th and \((r+1)\) th terms in the expansion of \((x+1)^{n}\) are in the ratio \(1: 3: 5\) find \(n\) and \(r\).

If \(p\) and \(q\) be positive, then the coefficient of \(x^{p}\) and \(x^{7}\) in the expansion of \((1+x)^{p+a}\) will be (a) equal (b) equal in magnitude but opposite in sign (c) reciprocal to each other (d) none of these

Expand the following expressions. (i) \((1-x)^{6}\) (ii) \(\left(x-\frac{1}{y}\right)^{11}, y \neq 0\)

Expand \(\left(x^{2}+2 a\right)^{5}\) by binomial theorem.

If the coefficients of 5 th, 6 th and 7 th terms in the expansion of \((1+x)^{n}\) be in A.P.m then \(n=\) (a) 7 only (b) 14 only (c) 7 or 14 (d) none of these

The value of \((\sqrt{5}+1)^{5}-(\sqrt{5}-1)^{5}\) (a) 252 (b) 352 (c) 452 (d) 532

If the three consecutive coefficient in the expansion of \((1+x)^{n}\) are 28,56 and 70 , then the value of \(n\) is (a) 6 (b) 4 (c) 8 (d) 10

In the expansion of \(\left(x^{2}-2 x\right)^{10}\), the coefficient of \(x^{16}\) is (a) \(-1680\) (b) 1680 (c) 3360 (d) 6720

If \(T_{2} / T_{3}\) in the expansion of \((a+b)^{n}\), and \(T_{3} /\) \(T_{4}\) in the expansion of \((a+b)^{n+3}\) are equal, then \(n=\) (a) 3 (b) 4 (c) 5 (d) 6

If the coefficients of \(x^{7}\) and \(x^{8}\) in \(\left(2+\frac{x}{3}\right)^{n}\) are equal, then \(n\) is

If the coefficient of \((2 r+4)\) th and \((r-2)\) th terms in the expansion of \((1+x)^{18}\) are equal, then \(r=\) (a) 12 (b) 10 (c) 8 (d) 6

If the coefficients of \(T_{r}, T_{r}+1, T_{r}+2\) terms of \((1+x)^{14}\) are in A.P., then \(r=\) (a) 6 (b) 7 (c) 8 (d) 9

The expansion \(\left[x+\left(x^{3}-1\right)^{\frac{1}{2}}\right]^{5}+\left[x+\left(x^{3}-1\right)^{\frac{1}{2}}\right]^{5}\) is \(a\) polynomial of degree (a) 5 (b) 6 (c) 7 (d) 8

In the expansion of \((x+a)^{n}\), the sum of odd terms is \(P\) and Sum of even terms is \(Q\), then the value of \(\left(P^{2}-Q^{2}\right)\) will be: (a) \(\left(x^{2}+a^{2}\right)^{n}\) (b) \(\left(x^{2}-a^{2}\right)^{n}\) (c) \((x-a)^{2 n}\) (d) \((x+a)^{2 n}\)