If the letters of the word \(MOTHER\) be permuted and all the words so formed (with or without meaning) be listed as in a dictionary, then the position of the word \(MOTHER\) is

Let A and B be the two points of intersection of the line \(y+5=0\) and the mirror image of the parabola \(y^2=4 x\) with respect to the line \(x+y+4=0\). If d denotes the distance between A and B , and a denotes the area of \(\triangle S A B\), where \(S\) is the focus of the parabola \(y^2=4 x\), then the value of \((a+d)\) is \(\qquad\) -

The integral \(\int \frac{\left(x^8-x^2\right) d x}{\left(x^{12}+3 x^6+1\right) \tan ^{-1}\left(x^3+\frac{1}{x^3}\right)}\) is equal to :

Let \(\theta\) be the angle between the planes \(P_1=\vec{r} \cdot(\hat{ i }+\hat{ j }+2 \hat{ k })=9\) and \(P _2=\overrightarrow{ r } \cdot(2 \hat{ i }-\hat{ j }+\hat{ k })=15\).Let \(L\) be the line that meets \(P _2\) at the point \((4,-2,5)\) and makes an angle \(\theta\) with the normal of \(P_{2^*}\) If \(\alpha\) is the angle between \(L\) and \(P_2\) then \(\left(\tan ^2 \theta\right)\left(\cot ^2 \alpha\right)\) is equal to \(...........\).

If the third term in the binomial expansion of \({\left( {1 + {x^{{{\log }_2}\,x}}} \right)^5}\) equals \(2560\), then a possible value of \(x\) is

If the four letter words (need not be meaningful) are to be formed using the letters from the word \("MEDITERRANEAN"\)  such that the first letter is \(R\) and the fourth letter is \(E,\) then the total number of all such words is

A vector \(\vec{a}\) is parallel to the line of intersection of the plane determined by the vectors \(\hat{i}, \hat{i}+\hat{j}\) and the plane determined by the vectors \(\hat{i}-\hat{j}, \hat{i}+\hat{k}\). The obtuse angle between \(\vec{a}\) and the vector \(\vec{b}=\hat{i}-2 \hat{j}+2 \hat{k}\) is

Statement \(-1\) : The system of linear equations \(x + \left( {\sin \,\alpha } \right)y + \left( {\cos \,\alpha } \right)z = 0\) \(x + \left( {\cos \,\alpha } \right)y + \left( {\sin \alpha } \right)z = 0\) \(x - \left( {\sin \,\alpha } \right)y - \left( {\cos \alpha } \right)z = 0\) has a non-trivial solution for only one value of \(\alpha \) lying in the interval \(\left( {0\,,\,\frac{\pi }{2}} \right)\)  Statement \(-2\) : The equation in \(\alpha \) \(\left| {\begin{array}{*{20}{c}}
  {\cos {\mkern 1mu} \alpha }&{\sin {\mkern 1mu} \alpha }&{\cos {\mkern 1mu} \alpha } \\ 
  {\sin {\mkern 1mu} \alpha }&{\cos {\mkern 1mu} \alpha }&{\sin {\mkern 1mu} \alpha } \\ 
  {\cos {\mkern 1mu} \alpha }&{ - \sin {\mkern 1mu} \alpha }&{ - \cos {\mkern 1mu} \alpha } 
\end{array}} \right| = 0\) has only one solution lying in the interval \(\left( {0\,,\,\frac{\pi }{2}} \right)\)

Let the sum of two positive integers be \(24\) . If the probability, that their product is not less than \(\frac{3}{4}\) times their greatest positive product, is \(\frac{\mathrm{m}}{\mathrm{n}}\), where \(\operatorname{gcd}(\mathrm{m}, \mathrm{n})=1\), then \(\mathrm{n}-\mathrm{m}\) equals :

If the matrices \(A=\left[\begin{array}{ccc}{1} & {1} & {2} \\ {1} & {3} & {4} \\ {1} & {-1} & {3}\end{array}\right], B=\operatorname{adj} A\) and \(\mathrm{C}=3 \mathrm{A},\) then \(\frac{|\mathrm{adjB}|}{|\mathrm{C}|}\) is equal to

A hyperbola whose transverse axis is along the major axis of then conic, \(\frac{{{x^2}}}{3} + \frac{{{y^2}}}{4} = 4\) and has vertices at the foci of this conic . If the eccentricity of the hyperbola is \(\frac{3}{2}\) , then which of the following points does \(NOT\) lie on it ?

Let \(P = \{\theta \in [0, 4\pi] : \tan^2\theta \neq 1\}\) and \(S = \{a \in \mathbb{Z} : 2(\cos^8\theta - \sin^8\theta)\sec 2\theta = a^2, \theta \in P\}\). Then \(n(S)\) 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 \(.............\).