A ratio of the \(5^{th}\) term from the beginning to the \(5^{th}\) term from the end in the binomial expansion of \(\left( {{2^{1/3}} + \frac{1}{{2{{\left( 3 \right)}^{1/3}}}}} \right)^{10}\) is

The area of the bounded region enclosed by the curve \(y=3-\left|x-\frac{1}{2}\right|-|x+1|\) and the \(x-\)axis is

The number of triplets \((x, y, z)\). where \(x, y, z\) are distinct non negative integers satisfying \(x+y+z=15\), is

The integral \(\int \frac{ e ^{3 \log _{e} 2 x }+5 e ^{2 \log _{ e } 2 x }}{ e ^{4 \log _{e} x }+5 e ^{3 \log _{e} x }-7 e ^{2 \log _{e} x }} dx , x > 0\), is equal to ....... . (where \(c\) is a constant of integration)

The area (in sq. units) of the region \(\mathrm{S}=\{\mathrm{z} \in \mathbb{C} ;|\mathrm{z}-1| \leq 2 ;(\mathrm{z}+\overline{\mathrm{z}})+\mathrm{i}(\mathrm{z}-\overline{\mathrm{z}}) \leq 2, \operatorname{lm}(\mathrm{z}) \geq 0\}\) is

If the function \(f(x)=2 x^3-9 a x^2+12 a^2 x+1, a>0\) has a local maximum at \(\mathrm{x}=\alpha\) and a local minimum \(x=\alpha^2\), then \(\alpha\) and \(\alpha^2\) are the roots of the equation :

Let \(f: R \rightarrow R\) be defined as \(f(x)=\left\{\begin{array}{ll}\frac{x^{3}}{(1-\cos 2 x)^{2}} \log _{e}\left(\frac{1+2 x e^{-2 x}}{\left(1-x e^{-x}\right)^{2}}\right), & x \neq 0 \\ \,\alpha & , x=0\end{array}\right.\) If \(\mathrm{f}\) is continuous at \(\mathrm{x}=0\), then \(\alpha\) is equal to :

Let M denote the set of all real matrices of order \(3 \times 3\) and let \(\mathrm{S}=\{-3,-2,-1,1,2\}\). Let
\(\mathrm{S}_1=\left\{\mathrm{A}=\left[a_{\mathrm{ij}}\right] \in \mathrm{M}: \mathrm{A}=\mathrm{A}^{\mathrm{T}} \text { and } a_{\mathrm{ij}} \in \mathrm{~S}, \forall \mathrm{i}, \mathrm{j}\right\}, \)
\( \mathrm{S}_2=\left\{\mathrm{A}=\left[a_{\mathrm{ij}}\right] \in \mathrm{M}: \mathrm{A}=-\mathrm{A}^{\mathrm{T}} \text { and } a_{\mathrm{ij}} \in \mathrm{~S}, \forall \mathrm{i}, \mathrm{j}\right\}, \)
\( \mathrm{S}_3=\{\mathrm{A}=\left[a_{\mathrm{ij}}\right] \in \mathrm{M}: a_{11}+a_{22}+a_{33}=0\) and \(a_{\mathrm{ij}} \in \mathrm{~S}, \forall \mathrm{i}, \mathrm{j}\}\)
If \(n\left(\mathrm{~S}_1 \cup_2 \mathrm{US}_3\right)=125 \alpha\), then \(\alpha\) equals _______

If \(\left(\sin ^{-1} x\right)^{2}-\left(\cos ^{-1} x\right)^{2}=a ; 0\,<\,x\,<\,1, a \neq 0\), then the value of \(2 \mathrm{x}^{2}-1\) is :

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 \(.............\).

Let sets \(A\) and \(B\) have \(5\) elements each. Let the mean of the elements in sets \(A\) and \(B\) be \(5\) and \(8\) respectively and the variance of the elements in sets \(A\) and \(B\) be \(12\) and \(20\) respectively \(A\) new set \(C\) of \(10\) elements is formed by subtracting \(3\) from each element of \(A\) and adding 2 to each element of B. Then the sum of the mean and variance of the elements of \(C\) is \(.......\).

Let \(S=\left\{x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right): 9^{1-\tan ^2 x}+9^{\tan ^2 x}=10\right\}\) and \(\beta=\sum_{x \in S} \tan ^2\left(\frac{x}{3}\right)\), then \(\frac{1}{6}(\beta-14)^2\) is equal to

Let \(\overrightarrow{ a }=\alpha \hat{ i }+2 \hat{ j }-\hat{ k }\) and \(\overrightarrow{ b }=-2 \hat{ i }+\alpha \hat{ j }+\hat{ k }\), where \(\alpha \in R\). If the area of the parallelogram whose adjacent sides are represented by the vectors \(\vec{a}\) and \(\vec{b}\) is \(\sqrt{15\left(\alpha^{2}+4\right)}\), then the value of \(2|\vec{a}|^{2}+(\vec{a} \cdot \vec{b})|\vec{b}|^{2}\) is equal to