The number of common terms in the progressions

$4,9,14,19, \ldots \ldots$, up to $25^{\text {th }}$ term and

$3,6,9,12, \ldots \ldots$, up to $37^{\text {th }}$ term is :

Let $\mathrm{a}_1, \mathrm{a}_2, \ldots \mathrm{a}_{10}$ be 10 observations such that $\sum\limits_{\mathrm{k}=1}^{10} \mathrm{a}_{\mathrm{k}}=50$ and $\sum\limits_{\forall \mathrm{k} < \mathrm{j}} \mathrm{a}_{\mathrm{k}} \cdot \mathrm{a}_{\mathrm{j}}=1100$. Then the standard deviation of $\mathrm{a}_1, \mathrm{a}_2, \ldots, \mathrm{a}_{10}$ is equal to :

Let $\overrightarrow{\mathrm{a}}=\hat{i}+2 \hat{j}+\hat{k}, $
$\overrightarrow{\mathrm{b}}=3(\hat{i}-\hat{j}+\hat{k})$.
Let $\overrightarrow{\mathrm{c}}$ be the vector such that $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{b}}$ and $\vec{a} \cdot \vec{c}=3$.
Then $\vec{a} \cdot((\vec{c} \times \vec{b})-\vec{b}-\vec{c})$ is equal to :

If A denotes the sum of all the coefficients in the expansion of $\left(1-3 x+10 x^2\right)^{\mathrm{n}}$ and B denotes the sum of all the coefficients in the expansion of $\left(1+x^2\right)^n$, then :