NCERT Exemplar Solutions
Class 11 - Mathematics

Chapter 8: BINOMIAL THEOREM

Find the term independent of \(x\), \(x \neq 0\), in the expansion of \(\left(\dfrac{3x^{2}}{2}-\dfrac{1}{3x}\right)^{15}\).

If the term free from \(x\) in the expansion of \(\left(\sqrt{x}-\dfrac{k}{x^{2}}\right)^{10}\) is \(405\), find the value of \(k\).

Find the coefficient of \(x\) in the expansion of \((1-3x+7x^{2})(1-x)^{16}\).

Find the term independent of \(x\) in the expansion of \(\left(3x-\dfrac{2}{x^{2}}\right)^{15}\).

Find the middle term(s) in the following expansions:

(i) \(\left(\dfrac{x}{a}-\dfrac{a}{x}\right)^{10}\)

(ii) \(\left(3x-\dfrac{x^{3}}{6}\right)^{9}\)

Find the coefficient of \(x^{15}\) in the expansion of \((x-x^{2})^{10}\).

Find the coefficient of \(\dfrac{1}{x^{17}}\) in the expansion of \(\left(x^{4}-\dfrac{1}{x^{3}}\right)^{15}\).

Find the sixth term of the expansion \(\left(\dfrac{1}{y^{2}}+\dfrac{1}{x^{3}}\right)^{n}\), if the binomial coefficient of the third term from the end is \(45\).

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

If the coefficients of the second, third and fourth terms in the expansion of \((1+x)^{2n}\) are in A.P., show that \(2n^{2}-9n+7=0\).

Find the coefficient of \(x^{4}\) in the expansion of \((1+x+x^{2}+x^{3})^{11}\).

If \(p\) is a real number and if the middle term in the expansion of \(\left(\dfrac{p}{2}+2\right)^8\) is 1120, find \(p\).

Show that the middle term in the expansion of \(\left(x-\dfrac{1}{x}\right)^{2n}\) is \(\dfrac{1\times3\times5\times\dots\times(2n-1)}{n!}\times(-2)^n\).

Find \(n\) in the binomial \(\left(\sqrt[3]{2}+\dfrac{1}{\sqrt[3]{3}}\right)^n\) if the ratio of the 7th term from the beginning to the 7th term from the end is \(\dfrac{1}{6}\).

In the expansion of \((x+a)^n\) if the sum of odd terms is denoted by \(O\) and the sum of even terms by \(E\), prove that:

(i) \(O^2-E^2=(x^2-a^2)^n\)

(ii) \(4OE=(x+a)^{2n}-(x-a)^{2n}\)

If \(x^p\) occurs in the expansion of \(\left(x^2+\dfrac{1}{x}\right)^{2n}\), prove that its coefficient is \(\dbinom{2n}{\dfrac{2n+p}{3}}\) (for integral \(\dfrac{2n+p}{3}\)).

Find the term independent of \(x\) in the expansion of \(\left(\dfrac{3}{2}x^2-\dfrac{1}{3x}\right)^9\).

The largest coefficient in the expansion of \((1 + x)^{30}\) is ________.

The number of terms in the expansion of \((x + y + z)^n\) is ________.

In the expansion of \(\left(x^2 - \dfrac{1}{x^2}\right)^{16}\), the value of constant term is ________.

If the seventh terms from the beginning and the end in the expansion of \(\left(\sqrt[3]{2} + \dfrac{1}{\sqrt[3]{3}}\right)^n\) are equal, then \(n\) equals ________.

The coefficient of \(a^{-6} b^{4}\) in the expansion of \(\left(\dfrac{1}{a} - \dfrac{2b}{3}\right)^{10}\) is ________.

Middle term in the expansion of \((a^3 + ba)^{28}\) is ________.

The ratio of the coefficients of \(x^p\) and \(x^q\) in the expansion of \((1 + x)^{p+q}\) is ________.

The position of the term independent of \(x\) in the expansion of \(\left(\dfrac{x}{3} + \dfrac{3}{2x^2}\right)^{10}\) is ________.

If \(25^{15}\) is divided by 13, the remainder is ________.

The sum of the series \(\sum_{r=0}^{10} \binom{20}{r}\) is \(2^{19} + \dfrac{\binom{20}{10}}{2}\).

The expression \(7^9 + 9^7\) is divisible by 64.

The number of terms in the expansion of \([(2x + y^3)^4]^7\) is 8.

The sum of coefficients of the two middle terms in the expansion of \((1 + x)^{2n - 1}\) is equal to \(\binom{2n - 1}{n}\).

The last two digits of the number \(3^{400}\) are 01.

If the expansion of \((x - \dfrac{1}{x^2})^{2n}\) contains a term independent of \(x\), then \(n\) is a multiple of 2.

The number of terms in the expansion of \((a + b)^n\) where \(n \in \mathbb{N}\) is one less than the power \(n\).