Binomial Theorem - Algebra - Mathematics Previous Year Questions

The ninth term of the expansion \({\left( {3x - {1 \over {2x}}} \right)^8}\) is

Using mathematical induction, the numbers \({a_n}\)'s are defined by \({a_0} = 1,{a_{n + 1}} = 3{n^2} + n + {a_n},(n \ge 0)\). Then, \({a_n}\) is equal to

The coefficient of \(x^{10}\) in the expansion of \(1+(1+x)+...+(1+x)^{20}\) is

If \(49^n+16n+P\) is divisible by 64 for all \(n\in N\), then the least negative integral value of P is

Number of terms in the binomial expansion of \((x+a)^{53}+(x-a)^{53}\) is

Middle term in the expansion of \({\left( {{x^2} + {1 \over {{x^2}}} + 2} \right)^n}\) is

\({2^{3n}} - 7n - 1\) is divisible by

The total number of terms in the expansion of \({(x + y)^{100}} + {(x - y)^{100}}\) is

If \(U_{n+1}=3 U_n-2 U_{n-1}\) and \(U_0=2, U_1=3\), then \(U_n\) is equal to

The coefficient of \({x^{20}}\) in the expansion of \({(1 + 3x + 3{x^2} + {x^3})^{20}}\) is

If \(4^n+15n+P\) is divisible by 9 for all \(n\in N\), then the least negative integral value of P is

In the expansion of \({(1 - 3x + 3{x^2} - {x^3})^{2n}}\), the middle term is

If the sum of the coefficients of the first three terms in the expansion of \(\left(x-\frac{a}{x^2}\right)^{12}, x \neq 0\) is 559. Find the value of '\(a\)' if '\(a\)' belongs to positive integers

In the expansion \(\left(\frac{1}{x}+x \sin x\right)^{10}, \quad\) the co - efficient of \(6^{\text {th }}\) term is equal to \(7 \frac{7}{8}\), then the principal value of \(x\) is

The coefficient of the third term in the expansion of \(\left(x^2-\frac{1}{4}\right)^n\), when expanded in the descending power of \(x\) is 31, then \(n\) is

If \(49^n+16^n+k\) is divisible by 64 for \(n \in N\), then the least negative integral value of \(k\) is

The coefficient of \(x^{29}\) in the expansion of \(\left(1-3 x+3 x^2-x^3\right)^{15}\) is

In the expansion of \(\left(1+3 x+3 x^2+x^3\right)^{2 n}\), the term which has greatest binomial coefficient, is

$$2^{3 n}-7 n-1 \text { is divisible by }$$