The ratio of the coefficient of \(x^{15}\) to the term independent of \(x\) in the expansion of \({\left( {{x^2} + \frac{2}{x}} \right)^{15}}\) is

The integral \(\int {\frac{{2{x^3} - 1}}{{{x^4} + x}}} \,dx\) is equal to (Here \(C\) is a constant of integration)

The number of distinct solutions of the equation \(\log _{\frac{1}{2}}|\sin x|=2-\log _{\frac{1}{2}}|\cos x|\) in the interval \([0,2 \pi],\) is

The number of seven-digit numbers, that can be formed by using the digits \(1, 2, 3, 5\) and \(7\) such that each digit is used at least once, is :

The number of ways in which an examiner can assign \(30\) marks to \(8\) questions, giving not less than \(2\) marks to any question, is

If \({ }^{2 n+1} P_{n-1}:{ }^{2 n-1} P_n=11: 21\), then \(n^2+n+15\) is equal to:

Let \(\vec{a}, \vec{b}\) and \(\vec{c}\) be three non-zero vectors such that \(\vec{b}\) and \(\vec{c}\) are non-collinear if \(\vec{a}+5 \vec{b}\) is collinear with \(\overrightarrow{\mathrm{c}}, \overrightarrow{\mathrm{b}}+6 \overrightarrow{\mathrm{c}}\) is collinear with \(\overrightarrow{\mathrm{a}}\) and \(\vec{a}+\alpha \vec{b}+\beta \vec{c}=\overrightarrow{0}\), then \(\alpha+\beta\) is equal to

Line \(L_1\) of slope 2 and line \(L_2\) of slope \(\frac{1}{2}\) intersect at the origin O . In the first quadrant, \(\mathrm{P}_1, \mathrm{P}_2, \ldots . \mathrm{P}_{12}\) are 12 points on line \(L_1\) and \(Q_1, Q_2, \ldots . . Q_9\) are 9 points on line \(L_2\). Then the total number of triangles, that can be formed having vertices at three of the 22 points \(\mathrm{O}, \mathrm{P}_1, \mathrm{P}_2, \ldots \mathrm{P}_{12}\), \(\mathrm{Q}_1, \mathrm{Q}_2, \ldots . \mathrm{Q}_9\), is:

If \((2,3,9),(5,2,1),(1, \lambda, 8)\) and \((\lambda, 2,3)\) are coplanar, then the product of all possible values of \(\lambda\) is.

If \(\operatorname{I}(m, n)=\int_0^1 x^{m-1}(1-x)^{n-1} d x, m, n\gt0\), then \(I(9,14)+I(10,13)\) is

Let \(\vec{a}=-5 \hat{i}+\hat{j}-3 \hat{k}, \vec{b}=\hat{i}+2 \hat{j}-4 \hat{k}\) and \(\vec{c}=(((\vec{a} \times \vec{b}) \times \hat{i}) \times \hat{i}) \times \hat{i}\). Then \(\vec{c} \cdot(-\hat{i}+\hat{j}+\hat{k})\) is equal to

If \(b _{ n }=\int \limits_{0}^{\frac{\pi}{2}} \frac{\cos ^{2} nx }{\sin x } dx , n \in N\), then

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 : _______