The value \(\sum \limits_{ r =0}^{22}{ }^{22} C _{ r }{ }^{23} C _{ r }\) is \(.......\)

If \(1^2 \cdot\left({ }^{15} C_1\right)+2^2 \cdot\left({ }^{15} C_2\right)+3^2 \cdot\left({ }^{15} C_3\right)+\ldots\) \(+15^2 \cdot\left({ }^{15} C_{15}\right)=2^m \cdot 3^n \cdot 5^k\), where \(m, n, k \in \mathbf{N}\), then \(\mathrm{m}+\mathrm{n}+\mathrm{k}\) is equal to :

The value of \(\frac{1}{1 ! 50 !}+\frac{1}{3 ! 48 !}+\frac{1}{5 ! 46 !}+\ldots .+\frac{1}{49 ! 2 !}+\frac{1}{51 ! 1 !}\) is \(.............\).

Let  \( \lim _{n \rightarrow \infty}\left(\frac{n}{\sqrt{n^4+1}}-\frac{2 n}{\left(n^2+1\right) \sqrt{n^4+1}}+\frac{n}{\sqrt{n^4+16}}-\frac{8 n}{\left(n^2+4\right) \sqrt{n^4+16}}\right. \) \( \left.+\ldots \ldots+\frac{n}{\sqrt{n^4+n^4}}-\frac{2 n \cdot n^2}{\left(n^2+n^2\right) \sqrt{n^4+n^4}}\right) \text { be } \frac{\pi}{k},\) using only the principal values of the inverse trigonometric functions. Then \(\mathrm{k}^2\) is equal to ..............

Let \(I(x)=\int \sqrt{\frac{x+7}{x}} d x\) and \(I(9)=12+7 \log _e 7\) If I \((1)=\alpha+7 \log _e(1+2 \sqrt{2})\), then \(\alpha^4\) is equal to \(..........\).

Let \(\vec{a}, \vec{b}, \vec{c}\) be three vectors mutually perpendicular to each other and have same magnitude. If a vector \(\overrightarrow{\mathrm{r}}\) satisfies. \(\overrightarrow{\mathrm{a}} \times\{(\overrightarrow{\mathrm{r}}-\overrightarrow{\mathrm{b}}) \times \overrightarrow{\mathrm{a}}\}+\overrightarrow{\mathrm{b}} \times\{(\overrightarrow{\mathrm{r}}-\overrightarrow{\mathrm{c}}) \times \overrightarrow{\mathrm{b}}\}+\overrightarrow{\mathrm{c}} \times\{(\overrightarrow{\mathrm{r}}-\overrightarrow{\mathrm{a}}) \times \overrightarrow{\mathrm{c}}\}=\overrightarrow{0}\) then \(\overrightarrow{\mathrm{r}}\) is equal to:

Let the vectors \(\overrightarrow{ a }, \overrightarrow{ b }, \overrightarrow{ c }\) be such that \(|\overrightarrow{ a }|=2,|\overrightarrow{ b }|=4\) and \(|\overrightarrow{ c }|=4 .\) If the projection of \(\overrightarrow{ b }\) on \(\overrightarrow{ a }\) is equal to the projection of \(\overrightarrow{ c }\) on \(\overrightarrow{ a }\) and \(\overrightarrow{ b }\) is perpendicular to \(\overrightarrow{ c },\) then the value of \(|\overrightarrow{ a }+\overrightarrow{ b }-\overrightarrow{ c }|\) is

If \(\left(2\alpha+1, \alpha^2-3\alpha, \dfrac{\alpha-1}{2}\right)\) is the image of \((\alpha, 2\alpha, 1)\) in the line \(\dfrac{x-2}{3}=\dfrac{y-1}{2}=\dfrac{z}{1}\), then the possible value(s) of \(\alpha\) is (are)

Let the function \(f(x)=\left\{\begin{array}{cc}\frac{\log _{e}(1+5 x)-\log _{e}(1+\alpha x)}{x} & \text { if } x \neq 0 \\ 10 & \text {; if } x=0\end{array}\right.\) be continuous at \(x=0\).The \(\alpha\) is equal to.

Let \(b _{1} b _{2} b _{3} b _{4}\) be a \(4\)-element permutation with \(b _{i} \in\) \(\{1,2,3, \ldots \ldots . .100\}\) for \(1 \leq i \leq 4\) and \(b_{i} \neq b_{j}\) for \(i \neq j\), such that either \(b _{1}, b _{2}, b _{3}\) are consecutive integers or \(b _{2}, b _{3}, b _{4}\) are consecutive integers.

Let \(\alpha > 0\), be the smallest number such that the expansion of \(\left(x^{\frac{2}{3}}+\frac{2}{x^3}\right)^{30}\) has a term \(\beta x^{-\alpha}, \beta \in N\). Then \(\alpha\) is equal to \(.............\).

The curve satisfying the differential equation, \(ydx-(x + 3y^2 )\, dy = 0\) and passing through the point \((1, 1)\) , also passes through the point

Let \(\overrightarrow{\mathrm{a}}=3 \hat{i}-\hat{j}+2 \hat{k}, \overrightarrow{\mathrm{~b}}=\overrightarrow{\mathrm{a}} \times(\hat{i}-2 \hat{k})\) and \(\overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{b}} \times \hat{k}\). Then the projection of \(\overrightarrow{\mathrm{c}}-2 \hat{j}\) on \(\vec{a}\) is :