If \(I_{m, n}=\int_{0}^{1} x^{m-1}(1-x)^{n-1} d x,\) for \(m, n \geq 1\) and \(\int_{0}^{1} \frac{x^{m-1}+x^{n-1}}{(1+x)^{m+n}} d x=\alpha I_{m, n}, \alpha \in R,\) then \(\alpha\) equals .... .

Suppose A and B are the coefficients of \(30^{\text {th }}\) and \(12^{\text {th }}\) terms respectively in the binomial expansion of \((1+x)^{2 \mathrm{n}-1}\). If \(2 \mathrm{~A}=5 \mathrm{~B}\), then n is equal to :

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

Let \(q\) be the maximum integral value of \(p\) in \([0,10]\) for which the roots of the equation \(x ^2- px +\frac{5}{4} p =0\) are rational. Then the area of the region \(\{(x, y): 0 \leq y\) \(\left.\leq(x-q)^2, 0 \leq x \leq q\right\}\) is

Let \(\vec{\alpha}=4 \hat{ i }+3 \hat{ j }+5 \hat{ k }\) and \(\vec{\beta}=\hat{ i }+2 \hat{ j }-4 \hat{ k }\). Let \(\vec{\beta}_1\) be parallel to \(\vec{\alpha}\) and \(\vec{\beta}_2\) be perpendicular to \(\vec{\alpha}\). If \(\vec{\beta}=\vec{\beta}_1+\vec{\beta}_2\), then the value of \(5 \vec{\beta}_2 \cdot(\hat{ i }+\hat{ j }+\hat{ k })\) is

Let \(A = \{2, 3, 4, 5, 6\}\). Let R be a relation on the set \(A \times A\) given by \((x, y) R (z, w)\) if and only if \(x\) divides \(z\) and \(y \leq w\). Then the number of elements in R is _______.

Let \(\vec a = \hat i - \hat j,\) \(\vec b = \hat i + \hat j + \hat k\) and \(\vec c\) be a vector such that \(\vec a \times \vec c + \vec b = 0\) and \(\vec a.\vec c = 4\), then \({\left| {\vec c} \right|^2}\) is equal to

If two straight lines whose direction cosines are given by the relations \(l+m-n=0,3l^{2}+m^{2}+c n l =0\) are parallel, then the positive value of \(c\) is

Let \(x\) and \(y\) be real numbers such that \(50\left(\dfrac{2x}{1+3i} - \dfrac{y}{1-2i}\right) = 31 + 17i\), \(i = \sqrt{-1}\). Then the value of \(10(x - 3y)\) is :

All five letter words are made using all the letters \(\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}, \mathrm{E}\) and arranged as in an English dictionary with serial numbers. Let the word at serial number \(n\) be denoted by \(W_n\). Let the probability \(\mathrm{P}\left(\mathrm{W}_{\mathrm{n}}\right)\) of choosing the word \(\mathrm{W}_{\mathrm{n}}\) satisfy \(\mathrm{P}\left(\mathrm{W}_{\mathrm{n}}\right)=2 \mathrm{P}\left(\mathrm{W}_{\mathrm{n}-1}\right), \mathrm{n} \gt 1\).
If \(\mathrm{P}(\mathrm{CDBEA})=\frac{2^\alpha}{2^\beta-1}, \alpha, \beta \in \mathbb{N}\), then \(\alpha+\beta\) is equal to : _______

Number of integral solutions to the equation \(x+y+z=21\), where \(x \geq 1, y \geq 3, z \geq 4\), is equal to \(..........\).

Let \((\lambda, 2,1)\) be a point on the plane which passes through the ponit \((4,-2,2) .\) If the plane is perpendicular to the line joining the points \((-2,-21,29)\) and \((-1,-16,23),\) then \(\left(\frac{\lambda}{11}\right)^{2}-\frac{4 \lambda}{11}-4\) is equal to

Let \(d \in R\), and  \(A = \left[ {\begin{array}{*{20}{c}} { - 2}&{4 + d}&{\left( {\sin \,\theta } \right) - 2}\\ 1&{\left( {\sin \,\theta } \right) + 2}&d\\ 5&{\left( {2\sin \,\theta } \right) - d}&{\left( { - \sin \,\theta } \right) + 2 + 2d} \end{array}} \right]\), \(\theta  \in \left[ {0,2\pi } \right]\). If the minimum value of det \((A)\) is \(8\), then a value of \(d\) is