If in the expansion of $(1+x)^{\mathrm{p}}(1-x)^{\mathrm{q}}$, the coefficients of x and $x^{2}$ are 1 and -2 , respectively, then $p^{2}+q^{2}$ is equal to :

The least value of n for which the number of integral terms in the Binomial expansion of ${(\sqrt[3]{7} + \sqrt[{12}]{{11}})^n}$ is 183, is :

If $\alpha = 1 + \sum\limits_{r = 1}^6 {{{( - 3)}^{r - 1}}} {\quad ^{12}}{{\rm{C}}_{2r - 1}}$, then the distance of the point $(12,\sqrt 3)$...

Let $^{\rm{n}}{{\rm{C}}_{{\rm{r}} - 1}} = 28{,^{\rm{n}}}{{\rm{C}}_{\rm{r}}} = 56$ and $^{\rm{n}}{{\rm{C}}_{{\rm{r}} + 1}} = 70$. Let ${\rm{A}}(4{\mathop{\rm cost}\nolimits} ,4\sin {\rm{t}}),{\rm{B}}(2\sin {\rm{t}}, - 2\cos {\rm{t}})$ and $...

For some ${\rm{n}} \ne 10$, let the coefficients of the 5th, 6th and 7th terms in the binomial expansion of ${(1 + x)^{n + 4}}$...

The sum of all rational terms in the expansion of ${\left( {1 + {2^{1/3}} + {3^{1/2}}} \right)^6}$ is equal to ......

If $\sum_{r=1}^{30} \frac{r^2\left({ }^{30} C_r\right)^2}{{ }^{30} C_{r-1}}=\alpha \times 2^{29}$, then $\alpha$ is equal to

If $\sum_{r=0}^5 \frac{{ }^{11} C_{2 r+1}}{2 r+2}=\frac{\mathrm{m}}{\mathrm{n}}, \operatorname{gcd}(\mathrm{m}, \mathrm{n})=1$, then $\mathrm{m}-\mathrm{n}$ is equal to ..........